ANNUAL REPORTSON THEPROGRESS OF CHEMISTRYGENERAL AND PHYSICAL CHEMISTRY1. INTRODUCTIONTHIS year we have selected a small number of topics in Physical Chemistryand reported on them in some detail. The policy is to attempt to cover thevarious aspects of the subject over a period of years, the choice of subjectsdepending on the rate a t which contributions to the subject are growing.Theoretical calculations on the properties of small molecules have notbeen reported for some time, so we include an account of this work and itsmost recent developments in a way which we hope will be of value to physicalchemists.Great advances in the study of molecular energy transfer processes arebeing made a t the present time, and the report on this subject covers rathercompletely the most recent work.The special shock-tube method of study-ing chemical reactions is reported, and recent advances described.I n recent years a great deal of spectroscopic work has been devoted tothe study of intermolecular interactions, so we include a report on studiesof solvent effects in infrared spectra.The relatively new subject of nuclear magnetic resonance has grown ina quite remarkable way and its applications to chemical problems havebeen extremely varied. The availability of commercial spectrometers hasresulted in a tremendous increase in the literature of this subject, so we haveattempted to summarise the present position as thoroughly as possible.R. E. R.2. WAVE-MECHANICAL CALCULATIONS ON ATOMS ANDSMALL MOLECULESALTHOUGH it is written for chemists without any specialized experience ofthe application of rigorous wave-mechanical techniques to the study ofatomic and molecular systems, this report deals exclusively with whatCoulson 1 has catalogued as “ Group I ” quantum chemistry.It describescalculations which do not have recourse to experimental data other thaninteratomic distances and general “ constants,’’ which are limited at presentto the accurate estimation of total electronic energies rather than excitation,ionization, or dissociation energies, and which, at their best, give wave1C. A. Coulson, Rev. Mod. Phys., 1960, 32, 1708 GENERAL -4ND PHYSICAL CHEMISTRYfunctions good enough to permit the reliable evaluation of physical proper-ties dependent on electronic-charge distribution.*An astonishing amount of work of this kind was carried out in the &stdecade of wave mechanics, with a degree of success that now seems almosthumiliating. The subject was then very largely abandoned during a, longperiod of preoccupation with the seemingly simple properties of n-electronsystems. In the past few years, however, the availability of automaticcomputers and increasing dissatisfaction with the often uninterpretableresults of ‘‘ Group I1 ” calculations have combined to restore “ Group I ”quantum chemistry to favour. Whether recent developments are to beregarded as progress in quantum chemistry or progress in numerical analysisis a matter of taste.This Report covers work published between 1959 and 1961, but by nomeans completely, and we have endeavoured to survey recent trends ratherthan recent publications.In general, we have omitted papers concernedsolely with computational techniques, papers describing manifestly unsuc-cessful calculations, and papers describing calculations without an obviousobjective. As the topic was reviewed last year,5 we have not consideredpublications dealing specifically with the electrical and magnetic propertiesof molecules.Because the Report is not intended for the specialist theoretician, thefirst three sections are purely expository. They are concerned mainly withexplaining the term ‘‘ self-consistent field,’’ which is used so indiscriminatelynowadays in molecular quantum chemistry as to be in danger of losing allsignificance. Although these sections are illustrated by reference to theground state of the beryllium atom, the analysis is applicable, mutatismutandis, to other atoms and to molecules.A considerable proportion of the remainder of the Report is devoted towork on atoms, especially small atoms; this is a fair reflexion of currentdevelopments.Apart from their intrinsic chemical interest, calculations onatomic systems are invaluable as models for calculations on molecularsystems.Determinantal Wave Functions.-Except in a small number of highlyspecialized calculations, the wave function, Y, for a single configuration ofan N-electron atomic or molecular system is taken to be a product of N one-electron wave functions (spin-orbitals ?),suitably antisymmetrized in accordance with the general form of Pauli’s*P.-0.Lowdin, Adv. Chem. Phys., 1959, 2, 207.sP.-O. Lowdin, Ann. Rev. Phys. Chem., 1960, 11, 107.4T. Fueno, Ann. Rev. Php. Chern., 1961, 12, 303.‘A. D. Buckingham, Ann. Reports, 1960, 57, 53.* Much more sophisticated accounts of recent developments in ‘‘ Group I ” quantumchemistry have been given by Lowdin; as 3 and “ Group I1 ” quantum chemistry(semi-empirical and semi-quantitative work on large molecules) has been reviewed byFuen0.4 t A spin-orbital is a function of the spatial co-ordinates of a single electron multipliedby one of the two possible spin factors, a or IsSTEWART : WAVE-MECHANICAL CALCULATIONS 9principle.s If N is even and the spin-orbitals comprise N / 2 spatially dis-tinct orbitals having the spin factor a (m, = 6) plus the same N / 2 orbitalswith the spin factor B (ms = -+), the antisymmetrized wave function issimply the Slater 7 determinant *' YlW Yl(2) * Yl")' YN(1) YN(2) - - Y"N>y = ~ Y2W Y2(2) ' - yl2(N) .. . . . . . . . . . .= det{ Yl(l)Y2(2) ' Y N ( W (2)A single Slater determinant suflices also if all the spin-orbitals which do notoccur in spatially identical pairs in the configuration considered have thesame spin factor (all m, = Q or all m, = -8) ; but in general %I linear com-bination of Slater determinants is required. Thus for the ground state andthe lowest excited states of beryllium we write, adopting the abbreviatednotation (2), and using bars to distinguish spin-orbitals with ms = -4 fromthose with ms = Q,W1s22s2 l8) = det(Yls(l)yls(2>Y2,(3)~2,(4) I ; (3)Y/(1S22S2P lp) = det {Yls(l)yls(~>Y2s(3)y2,(4) 1 - det {Yls(l)yls(2>~2,(3)Y2,(4) I ;WS22S2P 3p) = det {YlS(l>YlS(2>Y2,(3)~2~(4) I + det {Yls(l)y1,(2>~2s(3)Y2,(4) 1or det{Y,,s(1)~,(2)ly2,(3)Y2,(4) 1or det {Yls(l>~ls(2)y2s(3)y2~(4) 1.(4)If any of the N spin-orbitals selected for the product wave function ( 1 ) aredegenerate with others not included, the arbitrary selection will not nor-mally provide a satisfactory wave function: all possibilities must be con-sidered; function ( 1 ) must often be replaced by a linear combination of pro-ducts; and the determinantal form of the wave function must be extendedcorrespondingly.This situation occurs notably in atoms with incompleteshells, and in the excited states of symmetrical molecules. For example,y(ls22P2 '8) = det ~Y1,(1)yls(2)Y,(3)y)2(4) 1 + det(yls(l)~l,(z)ly,(3)y,(4) 1- det(Yi,(l)yli,(2)Y,(3)y,(4) 1, (5)where x = 2px, y = 2py, x = 2pz.Superposition of collfigurations (configuration ' I interaction ") causesa further increase in complexity. Even where degeneracy has not to betaken into account, it is seldom sufficient in work of high accuracy to buildup an N-electron wave function from N spin-orbitals. A larger set mustbe considered, and a linear combination formed from products of functionsrepresenting various ways of choosing N members of the set :Y = C&+ C2Y2 + C3Y/3 + . . . (6)C. A. Coulson, " Valence," Oxford University Press,,,2nd Ed., 1961.J.C. Slater, " Quantum Theory of Atomic Structure, McGraw-Hill, New York,* Except where their inclusion serves a useful purpose, normalizing factors areetc., 1960, (a) Vol. I; ( b ) Vol. 11.omit,ted from all wave functions quoted in this Report10 GENERAL AND PHYSICAL CHEMISTRYThe coefficients of the linear combination (6) are determined by the applica-tion of the variation principle; those of linear combinations of the types (4)and (5) are obtained by the use of angular momentum operators.7bSelf-consistent-field Wave Functions.-If the optimum forms of all theorbitals yl,y2, . . . , yN, comprising a single configuration are determined(to within arbitrarily specified limits) by a completely Jexible application ofthe variation principle, the orbitals are known as Hartree-Fock orbitals or(for reasons which are now mainly historical) as self-consistent-field orbitals.The total energy obtained in this way is usually found to be 98-5-99*5%of the observed energy.The term correlation energy is applied to the difference between theHartree-Fock energy of an atomic or molecular system and the energy cor-responding to an exact solution of the Schrodinger equation for the samesystem.[Because of quantum-electrodynamic and relativistic effects, 8which increase rapidly with increasing atomic number, the exact eigenvaluemay not be precisely the same as the observed energy.]There are two ways of determining the best possible forms of the orbitalsyl, y2, .. . , yN. We illustrate them by considering the ground state of theberyllium atom, ls22s2 18. This is a particularly simple system, for there areno incomplete electron shells, and the orbitals are spherically symmetrical;but, where nothing to the contrary is indicated, the results derived for theparticular case of beryllium can safely be generalized.The energy corresponding to the four-electron wave function (3) isE = J YHYdr/J Y2dT [dr = d ~ , d ~ ~ d ~ , d ~ , ] (7)and this is broken down by standard techniques (Slater 7a) into integralsinvolving one-electron wave functions :Jdet (y& )qww2s(3)q2s(4) )H def (~ls(l)qls(2)Y2s(3)W2s(4) Id?Jdet (YlS( 1$ls(2)w2s(3)q2s(4) 1 det (Yls(l)~1,(2)w2s(3)q2s(4) )drs CYlS(1 )~lS(2)W2s(3)q2s(4)1 det ( Y l S ( 1 )q1s(2)Y2s(3)V28(4) IdT= f [Yls(l)qls(2)W2s(3)q2s(4)IH def (Yls(l)qls(2)V2s(3)Y2s(4) w= s ~ ~ Y l S ~ ~ ~ ~ l S ~ ~ ~ Y 2 S ~ ~ ~ - [YlS(l )W2,(2>Wzs(3>qls(4)1+ CY 2s ( 1 )V2s (2)Y IS( 3 )VlS(4) 1 )dTE =- s [Yls(~)~1s(2)w2s(3)~2s(4)lH det (YlS(1 )qls(2)Y2s(3)~2s(4 w-(8)In Hartree atomic units(9)4 1 1 1 1 1 1H = 5 (-a.: - <) + g + < + < + + + G'i=lEqn.(9) is substituted in (8), and the variation principle is then appliedto determine the optimum forms of yls and y2s.If, for computational convenience, we segregate from E all the integrals8A. Froman, Rev. Mod. Phys., 1960, 32, 317.OH. Shull an4 G. G. Hall, Nature, 1959, 184, 1559STEWART : WAVE-MECHANICAL CALCULATIONS 11in which yls appears, calling their sum El,, and do likewise for the otherthree spin-orbitals, we find, if yIs and y2s are orthonormal,yls( l ) y ~ 2 ~ ) ~ l s ( l)dz1dt29-12+ 1y2,( 1)y2“2)~2~)y2s( l)dz1dz2r12+ 2Jyas( l)y1s(2)y1s(2)y2s( 1)dz1dz2r12In (10) and (1 1) we have reduced the set of numerals identifying the electronco-ordinates from four to two, and we have omitted the spin factors, whichgive unity on “integration.”The minimum value of El, (or E2s) clearly represents the kinetic energyof one of the Is (or 2s) electrons plus its potential energy in the field of thenucleus and the other three electrons.To find the optimum form of, say, yl, it is immaterial whether we mini-mize E or El, with respect to variations in yls.There is, however, animportant difference between the minimum energies obtained : whereas Ecorresponds to a measurable quantity (the sum of the four ionization ener-gies), the “ one-electron ” energy El, is a purely artificial quantity.More-over,for each of the electron-repulsion integrals included in E is counted twice onthe left-hand side of the expression (12).Of the two methods for determining the optimum forms of yls and y2sone is of recent developrnent,l0 while the other dates from the early yearsof quantum chemistry.1lRoothn’s method. 12-14 that for atoms of lowatomic number (where comparison with spectroscopic results is possible)remarkably accurate estimates of the total electronic energy (to within 1%)can be obtained from wave functions built up from orbitals of exceedinglyloC. C.J. Roothaan, Rev. Mod. Php., (a) 1951, 23, 69; ( b ) 1960, 32, 179.l1 D. R. Hartree, “ The Calculation of Atomic Structures,” Wiley, New York, 1957.la W. E. Duncanson and C. A. Coulson, Nature, 1949, 164, 1003.C. C. J. Roothaan, Technical Report, Laboratory of Molecular Structure and14R. G. Breene, Phys. Rev., 1958, 111, 1111; 1959, 113, 809; 1960, 119, 1615El, + 4 s + E2, + E 2 , > E, (12)It is well knownSpectra, University of Chicago, 1955, p. 2412 GENERAL AND PHYSICAL CHEMISTRYsimple form. In the case of beryllium ls22s2 W, if the two orbitals aretaken to bey1s = exp(--r),y2s = r exp( -br) + A exp( -cr), (13)where a, b, and c are variational parameters, and A is adjusted to ensureorthogonality between yls and y2s, the calculated 12 energy is -14-552 H.*The observed energy, corrected l5 for experimental error, is -14,66745 H.When functions as simple as (13) give accurate results, it is reasonableto expect that the best possible forms of yls and yyZs might be obtained byvariation of the linear parameters C in a finite expansion(14)with n fairly small and the " basis functions " x of the same form as the right-hand terms in function (13).This expectation has been realized not onlyfor light atoms but (see below) for atoms with electronic configurations upto 417. For beryllium itself Roothaan, Sachs, and Weiss l6 have used thebasis functionsY = ClX1+ (72x2 + * ' + cnxn,x1 = exp(-6*50r), x3 = exp(-3*40r), x5 = exp(-O-90r),x2 = r exp(-6-50r), xa = r exp(-3.40r), x6 = r exp(-O40r). (15)They obtained an energy of -14.57298 H, and showed this to be the bestenergy attainable (within the limits of precision implied) from a wavefunction of the form (14).It is customary in self-consistent-field calculations of this type to use thesame basis functions for all orbitals of a given symmetry (s, p , d, .. .). Thesuccessive quantum levels (e.g., Is, 2438, . . .) corresponding to different setsof coefficients in function (14) are then obtained by the solution of a secularequation (much as when molecular orbitals are expanded as linear combina-t,ions of atomic orbitals 6 ) . Roothaan lo has developed general matrixmethods for evaluating the coefficients (i.e., the orbitals) and the energies.It can be shown 79 11 that,provided the variations are completely flexible and not limited to a fewarbitrarily chosen parameters, the orbitals which minimize the energies (10)and (11) are the same as those which satisfy the Hartree-Fock equationsHartree-Pock method (numerical integration).- _ - - ~~ ____ ___ ___ _____ ___ - -I5R.E. Watson, Phys. Rev., 1960, 119, 170.l6 C. C. J. Roothaan, L. M. Sachs, and A. W. Weiss, Rev. Mod. Phys., 1960,32,186.recommendations in part, we use the symbolsH (= Hartree) and B (= Bohr) in this report for the " reduced " atomic units of energyand length.*Following Shull and Hall'sFor infinite nuclear mass 1 H = 27.210 ev, and 1 B = 0.52097 ASTEWART : WAVE-MECHANICAL CALCULATIONS 13If yls and yps are normalized, equation (10) can be obtained from (16) bypremultiplying the latter by vlS( 1) throughout and integrating over theco-ordinates of electron 1; likewise (11) can be obtained from (17).[Dummynumerals, 1 and 2, are used in (16) and (17) as in (10) and (ll).]The solutions to simultaneous integro-differential equations of the type(16) and (17) are obtained by numerical integration, and are usually pre-sented in the form of a table of values of Pls(r) and P2,(r) as functions of r ,whereP(r) = rR(r),R(r) = 2 d y (for s orbitals). (18)The symbol R (superfluous for s orbitals) is used to represent the radialfactor in orbitals ( p , d, f, . . .) having angular dependence. For s orbitalsR differs from y only because R is normalized in the sense(19)P(r) is the square root of the function 4nr2y2 which measures the prob-ability of finding an electron on the surface of a sphere of radius r (Coulson 6).In principle, solution of the Hartree-Fock equations yields an infinitenumber of orbitals (Is, 2s,3s, .. . in the present case), whereas the numberof orbitals available from a h i t e set of basis functions (as in Roothaan’smethod) is determined by the size of the set. In practice this is not amatter of consequence, except that it is not easy to choose a severely trun-cated basis set which adequately represents wave functions of widelydiffering principal quantum number.For nearly thirty years the numerical integration of the Hartree-Fockequations was the only practicable method of determining the optimumforms of atomic orbitals.Because of lack of precision, it was never anentirely satisfactory method, and it had the disadvantage of yielding atomicwave functions in a form in which they could not be used directly in non-spherical systems (Le., in molecular calculations). In the past two yearsit has been displaced almost completely by developments of Roothaan’smore straightforward procedure. A very recent report, however, suggeststhat the problems of precise numerical integration may have been over-come.17The simple energy expressions given in equations (10) and (11) areobtained only if the 1s and 2s orbitals are orthonormal. That they areorthogonal is readily demonstrated if function (16) is pre-multiplied byy2,(l) throughout, and (17) by ylS(l), integration then carried out over theco-ordinates of electron 1, and the second equation subtracted from thefirst. This gives(20)El, and E,, are not equal; therefore J’ yls(l)y2s(l)drl = 0.Orthogonality isalways ensured automaticdly in systems consisting only of “ doubly occu-pied ” orbitals; in other cases additional terms involving all the orbitals ofthe same symmetry appear on the right-hand sides of the Hartree-Fockequations.J R2r2dr = S P2dr = 1.0 = (El3 - E2S) s Yls(l)Y2s(l)d~l-l7 (a) D. F. Mayers, ( b ) D. A. Goodings, unpublished work quoted in ref. 18.leR. E. Watson and A. J. Freeman, Phys. Rev., 1961, 124, 111714 GENERAL AND PHYSICAL CHEMISTRYIn self-consistent-field calculations involving orbitals with angulardependence ( p , a, f, .. .) the assumption is always made, explicitly orimplicitly, that y = R(r)@(O)@(+), where R(r) is a function of R only, andthe angular factors (spherical harmonics) are the same as for the hydrogenatom (this is indeed true for S states). The problem of determining theoptimum forms of the orbitals y then reduces to that of determining theoptimum forms of functions of r only; the Hartree-Pock equations areusually written in terms of the variable P = rR.Ionimtion energies. Though one-electron energies (e.g., El, and E,, inthe case of beryllium) are artificial quantities defined for computational con-venience, they are puzzlingly closely related to the corresponding observedionization energies. It is easy to show (Koopmans’s theorem 19) that themerence between the self-consistent-field energy for beryllium 1s22s2and that for beryllium 1822s 2 5 is given by expression (lo), provided that ylSand y2s are taken to be the same for the cation as for the neutral atom.Butthis they are not.Ionization energies are obtained in two ways in Hartree-Fock calcula-tions : either the self-consistent-field energies are calculated separately forthe neutral atom and the ion, or otherwise the ionization energy of theneutral atom is taken to be simply the absolute value of the one-electronenergy concerned. Oddly enough, whereas the strict calculation usuallygives poor results, the rough calculation based on Koopmans’s theorem oftengives astonishingly accurate results.It is easy to account for the failure ofthe strict calculation (an ionization energy being calculated as the differ-ence between two very large quantities, each slightly in error), but not forthe success of the rough one.Correlation Energy.-However satisfactorily the forms of the indi-vidual orbitals are determined, a self-consistent-field wave function cannever be an exact solution of the Schrodinger equation for a many-electronsystem. The difference between the Hartree-Fock energy and the energycorresponding to an exact solution is known as the correlation energy, andis usually accounted for as follows. Although the density distribution ofany electron in a system described by a Hartree-Fock wave function neces-sarily depends on the density distributions of all the other electrons, theHartreeFock wave function cannot provide for the “ instantaneous ”repulsive effect of the various electrons on each others’ positions, and itthus tends to overestimate the electron-repulsion energy.The discrepancy caused by the lack of proper correlation of the individualelectronic motions is a relatively small one (usually 0.5-1*5% of the totalelectronic energy), but it is much larger than many energy differences ofchemical interest (e.g., transition, ionization, and dissociation energies), and,moreover, there are many purposes for which a Hartree-Fock wave func-tion is not usually suiliciently accurate (e.g., the calculation of oscillatorstrengths and various quantities involving coupling between the electronsand the nucleus).Now that the self-consistent-field wave functions canbe obtained quite straightforwardly, a great deal of attention is being devotedto the production of really accurate wave functions. Even in systems ofIsT. A. Koopmans, Physica, 1933, 1, 104STEWART WAVE-MECHANICAL CALCULATIONS 15only a few electrons this is a very diflicult problem, despite the constantextension of computing resources. Four procedures for obtaining improvedwave functions are in regular use for molecular as well as atomic systems.The attempt to base a many-electron wave function on an antisymmetrized product of one-electronwave functions can be abandoned altogether, and the many-electron wavefunction expanded instead as a power series in the co-ordinates and relativeco-ordinates of the various electrons.Completely successful applicationsof this technique to the helium atom 2 * ~ 21 and the hydrogen molecule 22 aredescribed below, but the method is not of general application.Because single determinants lead to moretractable numerical analysis than linear combinations of determinants, ithas been customary in self-consistent-field calculations to use the minimumnumber of radial functions consistent with Pauli’s exclusion principle, onefor each shell (Is 2s, 2p, . . .) in the case of atomic systems. If this restric-tion is relaxed, and “ closed ” shells are replaced by “ open ” shells (e.g.,if four orbitals instead of two are used for beryllium ls22s2 W), the calculatedenergy is improved considerably, at least in the case of Is electrons.The only general method of obtainingespecially accurate wave functions which at present offers the promise ofextension to large atoms and molecules is that of “superposition of con-figurations,” also known as < < configuration interaction.” If Yl representsone way of selecting N orbitals from a larger set to make up an N-electronwave function, and Y2, Y3, .. . , YN represent other ways, then the varia-tion principle ensures that the linear combination (6) is a better wave func-tion than any of Yl, Y2, . . . , YN taken separately. Fully automatic com-puting programmes are available for calculations of this type on atomic andmolecular systems (Boys and Cook 23); but the problem of selecting sets ofone-electron functions which give accuracy without unwieldiness has notyet been solved.The most obvious way of improving a“ non-correlated ” wave function is to multiply it by a correlation function, x, which ensures a reduction in amplitude as any of the interelectronic dis-tances decreases.For two-electron systems 249 25 the simplest correlationfunctions which have been used successfully arewhere a, #?, y, 6 are variational parameters.The four methods of obtaining wave functions more satisfactory thanthose resulting from Hartree-Pock calculations are discussed below in rela-tion to specific atoms and molecules. It is interesting to note that thesemethods were all introduced by Hylleraas more than thirty years ago.26(a) Wave functions without an orbital basis.(b) Open-shell wave functions.(c) Superposition of conjiguratiom.(d) Correlated wave functions.x = exp(a9i2), x = 1 + pr12, x = 1 - y exp(-67i,), (21)zoT. Kinoshita, Phys.Rev., 1957, 105, 1490; 1959, 115, 366.21C. L. Pekeris, Phys. Rev., 1958, 112, 1649; 1959, 115, 1216.22 W. Kolos and C. C. J. Roothaan, Rev. Mod. Phys., 1960, 32, 219.23S. F. Boys and G. B. Cook, Rev. Mod. Phys., 1960, 32, 285.24 C. C. J. Roothaan and A. W. Weiss, Rev. Mod. Phys., 1960, 32, 194.45 W. Kolos and C. C. J. Roothaan, Rev. Mod. Phys., 1960, 32, 205.zsE. A. Hylleraas, 2. Physik, 1928, 48, 469; 1929, 54, 347; E. A. Hylleraas andB. Undheim, ibid., 1930, 65, 75916 GENERAL AND PHYSICAL CHEMISTRYA comprehensive review of the correlation problem in quantum chemistryhas been given by L o ~ d i n , ~ and Slater 7b has provided a valuable intro-duction to the subject.Helium Atom.-As in the earliest years of quantum chemistry, thehelium atom and its isoelectronic ions (1 < 2 < 10) have recently beenstudied far more intensively than any other atomic or molecular system.As a guide to the effectiveness of the miscellany of calculations which comewithin the scope of this report, we list in Table 1 the energies correspondingto wave functions of five clearly defined types.TABLE 1.Wace functions for the ground state of heliumWave functionexp(-ccr, - cr2)exp(-ccr, - c’rr,) + exp(-c’r, - w2)[C = 2.1832; C’ = 1.18851$(l)$(2) [SCF: optimum $1Exact[c = 1.68751+(1)$’(2) + $’(1)$(2)[optimum $ and $’I* J.N. Silvennan, 0. Platas, and F. A. t Ref. 16.$ C. C. J. Roothaan and A. W. Weiss,TRef. 20 and 21.% % of.Energy (H) of exact correlationenergy energyincluded- 2.84766 98.069-2.87566 * 99.034 33.25-2.86168 t 98.552 0.00-2.87798 $ 99-1 14 38.77Mat,sen, J. Chem. Phys., 1960, 32, 1402.-2.90372 7 100.000 100~00Rev. Mod. Phys., 1960, 32, 194.The exact value quoted in Table 1 was obtained from calculations ofastonishing precision carried out independently by ICinoshita 20 andPekeris,Z1 both abandoning all attempt a t orbital formulations of the wavefunction. Kinoshita adopted an 80-term expansion,in Hylleraas’s 26 co-ordinatess = rl + r2, t = rl - r2, u = r12, (23)using the variation principle to evaluate the coefficients Clms.The exponentsI , m, 12 were positive integers * (n even) not greater than 10, used in 80 dis-tinct combinations.Aiming at even higher precision than Kinoshita, Pekeris used a 1078-term wave function in James and Coolidge’s z8 “ perimetric ” co-ordinates,determining the coefficients by solving the wave equation. He calculatedthe electronic energy to be -2.903724375 H.The purpose of these extraordinarily elaborate computations was in partt,o match the accuracy of Herzberg’s 29 determination of the ionizationenergy of helium: in this they were completely successful (Table 2).Kinoshita obtained an energy of -2.9037237 H.27 H. M. Schwartz, Phys. Rev., 1960, 120, 483.28 H. M. James and A. S.Coolidge, Phys. Rev., 1937, 51, 857.29 G. Herzberg, Proc. Roy. SOC., 1968, A, 248, 328.* Schwartz 27 showed subsequently that the convergence of Kinoshita’s wave func-tion might be improved by the use of half-integral exponents for 9 and uSTEWART : WAVE-MECHANICAL CALCULATIONS 17TABLE 2. Calculated ionizution energy of helium (cm.-l)Kinoshitrt PekerisNon-relativistic ionization energy 198317.45 & 0-11 198317.3747Mass-polarization correction -4.786 f 0.006 -4.7854Relativistic correction -0.557 f 0.09 -0.5636Electrodynamic correction (Lamb shift) -1.336 f 0.2 -1.339 & 0.2Corrected ionization energy 198310.17 198310.687Observed ionization energy 198310.82 & 0.15Notably successful calculations on the lowest triplet states of heliumhave been carried out by Pekeris 21 and by Traub and F01ey;~O they havegiven values for the s-electron density at the nucleus in excellent agreementwith those obtained from the hyperfine splitting.The Hartree-Fock energy quoted in Table 1 is one of a comprehensivecollection determined by Roothaan's l6 method for the 182, 1s?%, andls22s2 configurations of atoms and ions with 2 ranging from 2 to 10.Com-parison with the neighbouring results in the Table serves as a reminder thatthe label " self-consistent-field " implies a satisfactory and predictablestandard of accuracy rather than a very high one.Open-shell wave functions. A straightforward way of providing for somemeasure of electron correlation in ls2 configurations is to assign differentorbitals (y, y') to the two 1s electrons, writing the " open-shell" wavefunction asy = Y(W'(2) + Y'(UY(2) (24)Even with the simplest orbital forms 31 [ y = exp( -cr), y' = exp(-c'r)],this gives an energy superior to the Hartree-Fock energy (Table 1).In amore sophisticated treatment the optimum forms of the two orbitals can berepresented as different linear combinations of a small set of basis func-t i o n ~ . ~ ~ , 32 The majority of calculations aiming at high accuracy nowadaysuse open-shell wave functions.For excited states of helium such as lsns (n > 1) any orbital-productwave function is necessarily an open-shell wave function, and provision forelectron correlation is not a formidable problem. Ritter and Pauncz 33have obtained excellent results for n = 2 , 3 , 4 in the series from He toC4+ by using the simple wave function= 'v)lS(1)y'7ZS(2) + $%S(1)~1S(2)J (25)with ylS = exp( -w), and yns a four-term expansion in Epstein 34 functions(see below).As would be expected, the error in Ritter and Pauncz's resultsdecreased with increasing separation of the two electrons (0.0029 H forz = 2, n = 2; 0.0004~ for '2 = 2, n = 4).30 J. Traub and H. M. Foley, Phys. Rev., 1959, 116, 914.31 J. N. Silverman, 0. Platas, and F. A. Matsen, J . Chem. Phys., 1960, 32, 1402;C. W. Scherr and J. N. Silverman, ibid., p. 1407; M. Machacek and C. W. Scherr,ibid., 1960, 33, 242.32C. Franconi and J. A. Petruska, Bicerca, 1960, 30, 2152.33Z. Ritter and R. Pauncz, J . Chem. Phys., 1960, 32, 1820.34 P.S. Epstein, Phys. Rev., 1926, 28, 695; L. Pauling and E. B. Wilson, " Int,ro-duction to Quantum Mechanics," McGraw-Hill, New York, 1935, Q 27a18 GENERAL AND PHYSICAL CHEMISTRYSuperposition of configurations. A single-configuration wave functionof the formYI = Yl(l)Yl(2) Or YI = Yl(l)ly1’(2) + Yl’(l)Y1(2) (26)can be regarded as the first member of an infinite serieswhich will constitute an exact solution to the Schrodinger equation pro-vided the basis functions y form a ‘‘ complete ” set, i.e., a set such that anarbitrary function can be expanded as a linear combination of members ofthe set to within any specified limits of precision. By commencing with asingle configuration Yl and gradually superposing others it is possible toapproach nearer and nearer to an exact solution. But unless the basisfunctions y are carefully chosen, the convergence may be unworkably slow.Hydrogen-like basis functions (Is, 2s,2p, .. .) have been used extensivelyin the past in atomic and molecular calculations, but it is clear that they arenot satisfactory.2J 3 5 ~ 36 This is partly because they are incomplete unlessthe continuum is included, and partly because they increase so rapidly in“ size ” as the principal quantum number increases that the higher memberscan contribute but little to the representation of a function closely resemblinga lower member. The rapid increase in the size of hydrogen-like wave func-tions is due to the radial exponential factor, and is avoided by the simplereplacement of exp( - Zr/n) by exp( - Zr) ; this gives a complete set withouta continuum.The members of the set can conveniently be made ortho-gonal by increasing the order of the associated Laguerre polynomial factorsfrom (21 + 1) to (21 + 2 ) , in which case they are now often known asEpstein 3* functions. These functions were introduced into contemporaryquantum chemistry by Shull and Lowdin;37 they have been used successfullyeven in severely truncated sets.Shull and Lowdin 35 have used the lower members of a set of Epsteinfunctions for a thorough investigation of the effects of superposition of con-figurations in the ground state of helium. Including all 21 codgurationsbuilt up from the first six Epstein s functions, they obtained an energy of-2.87897 H, and improved this by about 0-023 H by the addition of p, d,and f functions.The behaviour of the scale parameter c in Shull and Lowdin’s 35 calcula-tions provides a useful illustration of the danger of endowing mere mathe-matical symbols with physical significance.The scale factor is well knownto have the value 1.6875 in the simplest wave function for helium,exp(-cr, - cr2); and the fact that this is leis than the value appropriateto the ion He+ (c = 2 = 2) is often attributed to the “ screening ” of oneelectron by the other. There is no wave-mechanical foundation for a beliefin screening, and it is noteworthy that Shull and Lowdin found c to increaseS5H. Shull and P.-0. Lowdin, J. Chem. Phys., 1959, 30, 617.36H. 0. Pritchard and F.H. Sumner, J . Phys. Chem., 1961, 65, 641.37 H. Shull and P.-0. Lowdin, J . Chem. Phys., 1955, 23, 1362; P.-0. Lowdin, ref.2 and 3STEWART : WAVE-MECHANICAL CALCULATIONS 19from 1.6875 to 2-37 as the number of configurations in their wave functionincreased from 1 to 21.Weiss 39 has applied an extended open-shell version of Shull and Lowdin’scomputations to the ls2 lX and the ls2s 3 5 states of helium and the iso-electronic cations (2 < 2 < 8). He reduced the calculated ground-stateenergy of helium to --2.90320~, and estimated the energy of the tripletstate (in which the correlation problem is not serious) with an error of only0.00002 H.Effective configuration-interaction calculations having simplicity ratherthan high precision as their object have been reported by Silverman, Platas,and Mat~en.~l They obtained an energy of -2.89523 H, using a two-termwave function of type (27), the first term being the same as the secondfunction in Table 1, the second term being built up from 2p orbitals.(See also Howell and Shu1L3*)Correlated wawe functions.Using helium wave functions of the formwithi = O i = O j = ORoothaan and Weiss 24 have made a detailed study of the effect of “ expan-sion length ” on the accuracy of calculated energies. With rn = n = 4they obtained an energy of -2.90319 H (y + y‘), and they concluded thatno improvement would result from the use of higher values of rn and n.[With rn = n = 0 Roothaan and Weiss’s function is, of course, the same asthe first function in Table 1.3Lowdin and Rhdei 40 have combined superposition of codgurations andthe correlation-factor method in one and the same wave fmction,where Yi is the same as in function (27).Building up their superposed con-figurations from Is, 28, and 3s orbitals only, they calculated the ground-stateenergies of the series H- to Be++ with errors of only 0.001--0.002 H.Wave functions of an altogether different type involving rI2 co-ordinateshave been used by Walsh and Borowitz 41 and by Hameka 42 in an exam-ination of Pluvinage’s 43 perturbation method (in which the Hamiltonianoperator is rearranged so that the perturbation term is not l/r12). In rela-tion to the computation required, the results are not especially encouragingfor either the ground state or the lower excited states.A number of papers on helium have appeared recentlywhich do not fall into any of the categories we have chosen for discussion.Amongst them are papers by Dalgarno 44 (expansion of correlation energy in38K.M. Howell and H. Shull, J . Chem. Phys., 1959, 30, 627.39 A. W. Weiss, Phys. Rev., 1961, 122, 1826.40 P.-0. Lowdin and L. RGdei, Phys. Rev., 1959, 114, 752.r l P . Walsh and S. Borowite, Phys. Rev., 1959, 115, 1206; 1960, 119, 1274.42 H. F. Hameka, J. Chem. Phys., 1961, 34, 884.43P. Pluvinage, Ann. Physique, 1950, 5, 145.44 A. Dalgarno, Proc. Phys. SOC., 1960, 75, 439; M. Cohen and A. Dalgarno, ibid.,Miscelkaneous.1961, 77, 16520 GENERAL AND PHYSICAL CHEMISTRYpowers of l/Z), Dalgarno and Stewart 45 (relativistic and radiative correc-tions), Hall 46 (scale factor varying with r), Scherr 47 and Gray and White-head 48 (second-order perturbation theory), Henderson and Scherr 49 (wavefunctions in momentum space).Interelectronic distance.As the short comings of single- configurationclosed-shell wave functions without correlation factors have long been associ-ated with their failure to provide for adequate separation of the two elec-trons, it is surprising how little work has been done on the relation betweenthe calculated interelectronic distance and the character of the wave func-tion. This need has now been met by Coulson and Neil~on,~* who have notonly determined the mean values of rI2 and l/rl2 for various wave functions,but also evaluated the probability distribution function for r12.Theirresults support current views in a most interestkg way, and they are quotedin Table 3 along witch results obtained by Pekeris.21TABLE 3. Electron correlation in the ground state of heliumWave functionexp( -crl - cr2) - 2.84766 1.296 1-055 1.0Self-consistent -field -2.86168 1.311 1-026 1.0exp(-cr, - c'r2).+ exp(-c'r, - cr2) -2.87566 1.394 0.993 1.0Six-term expansion m s, t, u - 2.9032 1.420 0.946 1.1Exact (Pekeris 21) -2.90372 1.422 0.946Criteria for msessing accuracy. Although almost all the energies quotedin this section have been obtained by the application of the variation prin-ciple, a most welcome feature of current work is the interest shown in othercriteria for assessing the qualities of wave functions (Lowdin 2, 3, 61).It isnow customary in calculations of high precision to study the effect of varia-tions in parameters and in expansion length, not only on the calculatedenergy, but also on the mean values of a variety of simple functions of theco-ordinates. This provides a check on internal consistency, and permitscomprehensive comparisons between wave functions of different degrees ofaccuracy. The availability of exact wave functions 2% 2 1 has proved mostvaluable in this respect.Various attempts have been made in recent years to extend the scope ofthe variation principle by using it to calculate lower energy bounds as wellas the usual upper energy bounds; they have not been very successful.52Lithium and Beryllium Atoms.-The most striking feature of recentwork on small atoms is the marked loss of accuracy as the number of elec-trons increases first from 2 to 3, and then from 3 to 4 (Table 4; cf.Table 1).Whereas self-consistent-field calculations almost always give, in theirreliable way, the same relative error irrespective of the number of electrons,45A. Dalgarno and A. L. Stewart, Proc. Phys. SOC., 1960, 75, 441.46 G. G. Hall, Proc. Phys. SOC., 1960, 75, 575.48B. F. Gray and R. Whitehead, J . Chem. Phys., 1961, 34, 1243.40M. G. Henderson and C. W. Scherr, Phys. Rev., 1960, 120, 150.6o C. A. Coulson and A. H. Neilson, Proc. Phys. SOC., 1961, 78, 831.62 G. L. Caldow and C . A. Coulson, Proc. Cambridae Phil. SOC., 1961, 57, 341.W. Scherr, J .Chem. Phys., 1960, 33, 317."P.-O. Lowdin, J. MoZ. Spectroscopy, 1959, 3, 46STEWART : WAVE-MECHANICAL CALCULATIONS 21more sophisticated procedures encounter considerable difficulty with increasein size.TABLE 4. Calculated ground-state energies (H) of lithium and berglliumWave function Lithium BerylliumSimplest analytical [equation (13)] -7.41792 * -14.552 pSelf-consistent -field - 7.43273 - 14.5730217-Term correlated function -7.47608 fSuperposition of configurations 39 - 7.477 10 - 14.66090Observed - 7.47807 - 14.66741Pluvinage 41 - 7-395*Ref. 13.7 Ref. 41.$ H. M. James and A. S. Coolidge, Phys. Rev., 1936, 49, 688.The only method of improving on Hartree-Fock wave functions whichadmits of general extension is the method of superposition of configura-tions.With the use of basis functions of s, p , d , f, and g symmetry, this hasbeen applied by Weiss 39 to lithium (45 configurations) and beryllium (55configurations) and by Watson 53 to beryllium (37 configurations). Theenergies which Weiss calculated-the best so far obtained for these twoatoms-are given in Table 4 for comparison with the results of less formidablecalculations. The value which James and Coolidge obtained for lithium25 years ago is included as a reminder that remarkable successes wereachieved in quantum chemistry before the advent of the electronic com-puter.Walsh and Borowitz’s 41 extension of Pluvinage’s 43 perturbationmethod to three-electron systems provides interesting mathematical analysis,but gives a poorer result than the simplest variational calculation.Using fairly simple open-shell wave functions analogous to those theydevised for two-electron systems,33 Ritter, Pauncz, and Appe1S4 have studiedthe ls2ns 2X states (n = 2,3,4,5) of the series Li to P6+.They obtainedground-state energies in error by about 0.01 H for each unit of 2; but theerrors in their transition energies and ionization energies (in the sense ofKoopmans’s theorem 19) were only about one-tenth as large.Linderberg and Shull 55 have made an interesting analysis of correlationin three- and four-electron atoms and ions, finding K-shell correlation tobe largely radial (as in two-electron systems), but L-shell correlation essen-tially angular; i.e., they found configurations in which the 2 9 electrons arekept on opposite sides of the nucleus much more effective in improving thecalculated energy than configurations in which the 2s2 electrons are kept atdifferent distances from the nucleus.Atoms with 2 > 4.-This section is concerned solely with self-consistent-field calculations, for which automatic computer programmes have beenavailable for some years.Although the possibilities of accurate numericalintegration have not been neglected,17, 56 almost all recent work has been53 R. E. Watson, Phys. Rev., 1960, 119, 170.64Z. W. Ritter, R. Pauncz, and K. Appel, J. Chem. Phys., 1961, 35, 571.65 J. Linderberg and H. Shull, J. Mol. Spectroscopy, 1960, 5, 1.a6B. H. Worsley, Proc. Roy. SOC., 1958, A , 247, 39022 GENERAL AND PHYSICAL CHEMISTRYdeveloped in terms of analytical Hartree-Fock wave functions, in which theorbitals are expressed as linear combinations of a small number (usually5-12) of pre-determined basis functions. Precise self-consistent-field wavefunctions are now available for a great variety of atoms and ions with 2as high as 36, and these have superseded the relatively inexact functionspreviously determined by numerical integration. Calculations have beencarried out not only for free atoms but for atoms in perturbing field~.~7, 58Watson and Freeman 18, 59 have given references to recent publications,and have listed the many uses to which the results have been put (notablythe calculation of scattering factors 60-63).has written a mostinformative introduction to this work.The following results 64 for P and F- are typical of the accuracy attain-able :SlaterTotal electronic energy (H) F F-Calc.- 99.40792 - 99.45891Obs. - 99.8096 - 99.9426The error is about twice as large as in a thorough-going calculation based onsuperposition of configurations. G5For atoms of high atomic number, where no spectroscopic determina-tions of total electronic energy are available, self-consistent-field wave func-tions can be checked by evaluation of ionization energies, transition energies,and multiplet separations. These are severe tests, for they involve energydifferences far smaller than the error in the total electronic energy, but theresults give little cause for lack of confidence. Unaccountably, ionizationenergies (and electron affinities (j4) calculated by means of Koopmans'theorem,lg unlike those calculated as the differences of state energies, areusually in excellent agreement with experiment.An atomic Hartree-Fock calculation is described as " restricted " if itis based on the minimum number of radial functions (one for each of Is, 29,2p, .. .), and " unrestricted '' if this condition is relaxed in the interests offlexibility or orthogonality. Technical problems arise in the solution of anunrestricted Hartree-Fock equation, for the wave function cannot then beexpressed strictly as a single determinant. The problems and means ofevading them have been discussed by various authors.l0* 66Hydrogen Molecule-ion.-Because of its unique simplicity the diatomichydrogen cation, H2+, has always been a popular system on which to testquantum-chemical techniques.Exact solutions 67 to the Schrodingera7L. C. Allen, Phys. Rev., 1960, 118, 167.58 J. H. Wood, Plzys. Rev., 1960, 117, 714.5 9 R . E. Watson and A. J. Freeman, Phys. Rev., 1961, 123, 521.soA. J. Freeman, Acta Cryst., 1959, 12, 274, 929; 1960, 18, 190, 618.R. E. Watson and A. J. Freeman, Acta Cryst., 1961, 14, 27; A. J. Freeman andR. E. Watson, ibid., p. 231; A. J. Freeman and J. H. Wood, ibid., 1959, 12, 271.62B. Dawson, Acta Cryst., 1960, 13, 403; 1961, 14, 1117, 1120, 1271.63 C. M. Womack, J. N. Silverman, and F. A. Matsen, Acta Cryst., 1961, 14, 744.64L. C. Allen, J . Chem. Phys., 1961, 34, 1156.6sM.J. M. Bernal and 8. F. Boys, Phil. Trans., 1952, A, 245, 139.66R. K. Nesbet and R. E. Watson, Ann. Physique, 1960, 9, 260; L. M. Sachs,Phys. Rev., 1960, 117, 1504; W. Marshall, Proc. Phys. SOC., 1961, 78, 113; A. T. Amosand G. G. Hall, Proc. Roy. Soc., 1961, A, 263, 483.67 D. R. Bates, K. Ledsham, and A. L. Stewart, Phil. Trans., 1953, A, 246, 215STEWART : WAVE-MECHANICAL CALCULATIONS 23equation are available for a wide range of internuclear distances, and thesehave been of considerable value in supplementing the minimum-energycriterion for appraising the merits of approximate wave functions.Apart from the lack of spherical symmetry, the feature which makesmolecular calculations so much more difficult than atomic calculations isthe need for evaluating the many-centre integrals which arise when themolecular wave function is built up from atomic wave functions with originsof co-ordinates at the various nuclei. This difficulty can sometimes beovercome by writing the molecular wave function as a linear combinationof basis orbitals having a common origin of co-ordinates (usually the centreof the molecule).In the case of H2+ this consists in treating the molecularwave functions as perturbed “ united-atom ” wave functions for He+.Howell and Shull 38 investigated the expansion of the wave functionfor H,+ as a linear combination of single-centre Epstein 34 functions, butfound the convergence discouragingly slow. Using the &st six basis func-tions of s symmetry they obtained a ground-state energy of -1.01842 H,and this they reduced to -1.09563 H by the addition of the first five dfunctions and the first g function.Spheri-cally symmetrical basis functions are not, of course, well suited to a moleculewith axial symmetry, at least in the ground state. In the excited states,where the effective size of the charge distribution is high in relation to theinternuclear distance, convergence is relatively rapid, and Howell and Shullobtained more promising results.Cohen, Coulson, and Fox 68 have solved the convergence problem bytaking the one-centre basis functions to be products of Legendre angularfactors and completely flexible radial factors (determined by numericalintegration). Using only four basis functions they obtained very goodvalues for the ground-state energy (-1-09994 H) and for the oscillatorstrength of the transition to the first excited CT state.[A discrepancybetween the results of Howell and Shull 38 and Cohen, Coulson, and Fox 68and an earlier result obtained by Chen 69 has been resolved by G&~p&r.~0]In a conventional two-centre molecular-orbital calcul&tion on H, +Pritchard and Sumner 36 have demonstrated convincingly the superiorityof Epstein 34 basis functions (which form a complete set *) over the usua.1hydrogen-like functions.In a straightforward molecular-orbital study of the lowest nu state ofHZ+, Sovers and Kauzmann 71 have shown that the simplest possible 2pnwave function is greatly improved by the superposition of a d configuration.A calculation remarkable for both its simplicity and its success has beencarried out by Scrocco and Tomasi.72 They obtained a ground-state energyThe exact value is - 1.10262 H.68 M.Cohen, C. A. Coulson, and L. Fox, Proc. Cambridge PhiE. Soc., 1961, 57,asT. C. Chen, J . Chem. Phys., 1958, 29, 347.? O R . G&sp&r, Acta Phys. Acad. Sci. Hung., 1960, 11, 295.72E. Scrocco and 6. Tomasi, MoZ. Phys., 1961, 4, 193.96.Sovers and W. Kauzmann, J . Chem. Phys., 1961, 35, 652.*Two complete sets in fact in Pritchard and Sumner’s calculations. “Over-completeness ” is a potential source of d%culty, but not one which Pritchard andSumner experienced24 GENERAL AND PHYSICAL CHEMISTRYofbyin- 1.1005 a merely by replacing the customary 1s atomic orbital exp( -cr).exp(-cr - br cos 0).Jepsen and Hirschfelder 73 have estimated the coupling terms neglectedthe Born-Oppenheimer approximation.Robinson 74 has studied the hydrogen molecule-ion as a problem inperturbation theory.Hydrogen Molecule.-The hydrogen molecule continues to be studiedfar more intensively, and with far greater success, than any other molecularsystem. The results of some recent calculations are given in Table 5, alongwith two early results (recalculated 75) for comparison.TABLE 5. Calculated ground-state energy of the hydrogen molecule( R = Re = 1.4008 B)Wave function Energy (a)Simplest molecular-orbital 75Self-consistent-field : closed shell 22* 25Self-consistent-field : open shell 25Molecular-orbital with 1s configuration interaction 75Molecular-orbital with Is, 28, 2p configuration interaction 76Wave function with correlation factor: closed shell asWave function with correlation factor: open shell a5Exact 28Observed- 1.1282 - 1.13357- 1.14182- 1.14796-1.1672- 1.17257- 1.17296- 1.17445- 1.17444 f 0.00003The well-known wave function which James and Coolidge 77 used in1933 was an expansion in elliptical co-ordinates of the formwhereandR being the internuclear distance, and r, and rl, distances measured fromthe two nuclei.The terms in the expansion (31) arise from various choicesof integral exponents, p , q, r , s, p. By considerably extending the expan-sion lengths which James and Coolidge were able to use, Kolos and Root-haan 22 have obtained wave functions which can be regarded as exact overa wide range of R (subject to the limitations of the Born-Oppenheimerapproximation). The value quoted in Table 5 was obtained from st 50-termexpansion, but a 14-term expansion gives an energy only 0.0002 H in error.The self-consistent-field energies in Table 5 were calculated by omittingthe r,, terms in expression (32), and, in the case of the conventional closed-shell calculation, putting p = ry q = s.It is interesting to note thatCoulson 78 obtained -1.13318 H in the first molecular self-consistent-fieldcalculation ever carried out (by what is now known as Roothaan's lo method).t = (r, + rd/R, q = (r, - rb)/RY73 D. W. Jepsen and J. 0. Hirschfelder, J . Chem. Phys., 1960, 32, 1323.74P.D. Robinson, Proc. Phys. Xoc., 1961, 78, 537.75H. Shull, J . Chem. Phys., 1959, 30, 1405.76A. D. McLean, A. Weiss, and M. Yoshimhe, Rev. Mod. Phys., 1960, 32, 211.7 7 H . M. James and A. S. Coolidge, J . Chem. Phys., 1933, 1, 825.78 C. A. Coulson, Proc. Cambridge Phil. SOC., 1938, 34, 204STEWART : WAVE - M E CH AN I C AL C AL C U L AT1 0 N S 25The possibility of using numerical integration for solving the Hartree-Fockequation for hydrogen has been examined by Berthier and May~t.~QKolos and Roothaan 22 have compared the mean values of r:, 3x; - Y:,lly 1/r12, and rlz given by the self-consistent-field wave function with theexact values, and have found reasonably good agreement in the first threecases. This suggests that self-consistent-field wave functions may give satis-factory values for the diamagnetic susceptibilities (Larmor terms) andmolecular quadrupole moments of larger molecules for which more accuratewave functions are unattainable.The wave functions with correlation factors listed in Table 5 wereobtained 2 5 (by analogy with the corresponding functions for helium z4) bytruncating the expansion (31).Recent accurate work on the ground state of hydrogen not representedin Table 5 includes the use of Gaussian orbitals (Longstaff and Singersingle-centre wave functions (Hagstrom and Shull 81), and " product atomicorbitals " (Ellison and Companion 82).Shull 75 has made a detailed com-parison of various wave functions in terms of " natural spin-orbitals 2,'y andhas discussed the use of the terms " covalent " and " ionic " in the ele-mentary analysis of hydrogen wave functi0ns.~3 Ladik 84 has evaluatedrelativistic corrections.McLean, Weiss, and Yoshimine 76 have compiled a valuable bibliographyof work on the ground state of the hydrogen molecule.Calculations on the lower excited states of hydrogen have been carriedout by Kolos and Roothaan 22 (3&!, lXC,+, Z;), by Davidson 85 (Z;) andby Tschudi and Cohan s6 ('EL).The last two calculations exemplify thedifficulty of describing excited molecular states in terms of atomic groundstates.Using the first allowed transition of hydrogen as an example, Ehrensonand Phillipson 87 have demonstrated the impossibility of calculating oscil-lator strengths from simple wave functions.Diatomic &leculc%.-Allen and Karo s8 have provided a comprehensivesurvey and bibliography of non-empirical calculations on small molecules,radicals, and ions published by mid-1959.Systems studied more recentlyinclude H,- (Fischer-Hjalmars s9), He, + (Csavinszky 9, LiH (Kar0;~1Robinson, Stuart, and Matsen;92 Rfoccia 93), BeH (Moccia;93 Aburto etJ. V. L. Longstaff and K. Singer, Proc. Roy. Soc., 1960, A, 258, 421.S . Hagstrom and H. Shull, J. Chem. Phys., 1959, 30, 1314.79G. Berthier and M. Mayot, J . Chim. phys., 1959, 56, 504.saF. 0. Ellison and A. L. Companion, J . Chem. Phys., 1959, 31, 285.83H. Shull, J . Amer. Chem. SOC., 1960, 82, 1287.84 J. Ladik, Acta Phys. Acad. Sci. Hung., 1961, 13, 123.85E.R. Davidson, J. Chem. Phys., 1961, 35, 1189.86C. S. Tschudi and N. V. Cohan, J . Chenz. Phys., 1961, 34, 401.87 S. Ehrenson and P. E. Phillipson, J . Chem. Phys., 1961, 34, 1224.L. C. Allen and A. M. Karo, Rev. Mod. Phys., 1960, 32, 275.8g I. Fischer-Hjalmars, Arkiv. Fys., 1959, 16, 33.slA. M. Karo, J . Chem. Phys., 1960, 32, 907.s2 J. M.'Robinson, J. D. Stuart, and F. A. Matsen, J. Chenz. Phys., 1960, 32, 988.93R. Moccia, Gazzetta, 1960, 90, 955, 968.s4 S. Aburto, R. Gallardo, R. Muiioz, R. Daudel, and R. Lefebvre, J . Chim. phys.,Csavinszky, J. Chem. Phys., 1959, 31, 178.1959, 56, 56326 GENERAL AND PHYSICAL CHEMISTRYCH (Masse 95), C, (Clementi 96), CO (Brion, Moser, and Ne~bet;~' Hurley 98),N, (Clementi 9g), NO (Brion, Moser, and Yamazaki loo), OH (Freeman 101),OH and OH- (GAsphr and Tamhssy-Lentei lo,), HF (Nesbet lo3), F, andF2+ (Hijikata lo*).The interaction of two helium atoms has been examinedby Moore,lo5 by Brigman, Brient, and Matsen,106 and by Ransil,lo7 andother long-range interactions by Dalgarno and his co-workers, lo8 by Hirsch-felder and L0wdin,lo9 by Eliason and Hirschfelder,llo and by Bingel, Preuss,and Schmidtke .The simplest single-configuration molecular-orbital wave function for theground state of the hydrogen molecule (Table 5, first function),Y = @( l)@(2)= [exP(-cral) 3- exp(--cr,,)lCexp(--cr,,) + exp(--cr,,)l, (33)has served as a model for a considerable number of calculations on the groundstates of homonuclear and heteronuclear diatomic molecules, radicals, andions formed from atoms with 1 < 2 < 10.In the majority of these cal-culations the atomic orbitals from which the molecular orbitals are con-structed are taken to be single ls, 28, and 2p functions of the Slater type(hybridized where necessary), and superposition of configurations is notconsidered. Work of this kind is typified by Ransil's 112 systematic studyof twelve diatomic molecules with closed-shell ground states. Total elec-tronic energies are usually calculated with an error of only 0.5--1.5% (muchas in the corresponding atomic systems); but dissociation energies are badlyunderestimated and excitation energies seldom reported.* It is significantthat the use of improved (e.g., Hartree-Fock) atomic orbitals seems to im-prove total molecular energies but not dissociation energies.88 Calculationsof molecular energy as a function of internuclear distance have been made,and used with some success in the estimation of spectroscopic constants.114J.-L. Masse, J . Chim. phys., 1961, 58, 372.gsE. Clementi, Gazzetta, 1961, 91, 717.97 H. Brion and C. Moser, J . Chem. Phys., 1960, 32, 1194; H. Lefebvre-Brion, C.Moser, and R. K. Nesbet, ibid., 1960, 33, 931; 1961, 34, 1950.98 A. C. Hurley, Rev. Mod. Phys., 1960, 32, 400.g9E. Clementi, Gazzetta, 1961, 91, 722.100 H. Brion, C. Moser, and M. Yamazaki, J . Chem. Phys., 1959, 30, 673; 1960, 33,1871; M. Yamazaki, M. Sakamoto, K. Hijikata, and C. C. Lin, ibid., 1961, 34, 1926.1olA. J. Freeman, Rev. Mod. Phys., 1960, 32, 273.loe R.GBsp&r and I. Tamhssy-Lentei, Acta Phys. Acad. Sci. Hung., 1959, 10, 149.103 R. K. Nesbet, Rev. Mod. Phys., 1960, 32, 272.104K. Hijikata, J . Chem. Phys., 1961, 34, 221, 231.l o s N . Moore, J . Chem. Phys., 1960, 33, 471.lo6 G. H. Brigman, S. J. Brient, and F. A. Matsen, J . Chem. Phys., 1961, 34, 958.107B. J. Ransil, J . Chem. Phys., 1961, 34, 2109.l o 8 A. Dalgarno and A. L. Stewart, Proc. Roy. SOC., 1960, A , 254, 570; A. Dalgarno109 J. 0. Hirschfelder and P.-0. Lowdin, Mol. Phys., 1959, 2, 229.lroM. A. Eliason and J. 0. Hirschfelder, J . Chem. Phys., 1959, 30, 1397.ll1 W. A. Bingel, H. Preuss, and H.-H. Schmidtke, 2. Naturforsch., 1961, Ma, 435.llaB. J. Ransil, Rev. Mod. Phys., 1960, 32, 239, 245; S. Fraga and B. J. Ransil,lrsE.T. Stewart, Proc. Phys. Soc., 1960, 75, 402.ll4S. Fraga and B. J. Ransil, J . Chem. Phys., 1961, 35, 669.* It is, of course, impossible to calculate a wide spectrum of molecular excitationenergies with wave functions built up solely from ground-state atomic orbitals (cf.hydrogen 113).and A. E. Kingston, Proc. Phys. SOC., 1961, 78, 607.J . Chem. Phys., 1961, 34, 727STEWART : WAVE-MECHANICAL CALCULATIONS 27The terms ‘‘ Hartree-Fock ” and “ self-consistent-field ” serve a veryuseful purpose in characterizing atomic and molecular orbitals of optimumform in single-configuration wave functions, and it is thus most unfortunatethat they have come to be applied to wave functions of the type describedin the previous paragraph; these are no more like HartreeFock wavefunctions than is function (33) or the first function in Table 1.Thenomenclature of quantum chemistry is cumbrous enough without the intro-duction of tautological absurdities such as “ accurate self-consistent-fieldorbitals.”In one of the few genuine self-consistent-field calculations on molecularsystems to be reported, Nesbet lo3 has found the error in the computedelectronic energy of hydrogen fluoride to be reduced considerably by theuse of orbitals built up from an adequate set of basis functions.Surprisingly few quantum- chemical calculations include an examinationof electron-density distribution. R o u x , ~ ~ ~ however, has calculated thechange in charge distribution which occurs when the molecules H,, O,, N2,F,, NO, and CO are formed from their constituent atoms.The results offerlittle support for elementary theories of chemical bonding.Polyatomic Molecules.-The majority of calculations on polyatomicmolecules, radicals, and ions, as on the corresponding diatomic systems,embody the smallest possible number of atomic orbitals (Is, 28, Zp), and aremodelled on the simplest wave function for the ground state of hydrogen(sometimes with limited superposition of configurations). There is muchthe same agreement between theory and experiment as for diatomic mole-cules, provided three- and four-centre integrals are evaluated accurately.lfsRecent papers not listed in Allen and Karo’s 88 bibliography report workon CH, and CH, (Padgett and Krauss 117); C,H, (McLean;llG Burnelle ll*);C3, N3-, NO,+ (Clementi l19); HF2- (Clementi;l19 Bessis and Bratoi lZ0);H,O (GAspAr and TamAssy-Lentei ; l o 2 McWeeny and Ohno 121) ; NH, +(Lorquet 122); and HCHO (Goodfriend, Birss, and Duncan lZ3).Because of its nearly spherical symmetry, the methane molecule has beenthe subject of more refined calculations l Z 4 than most other polyatomicmolecules. The latest of these (Albasiny and Cooper 125) gives a totalelectronic energy differing by 1.5% ( 0 .6 ~ ) from the observed value.Boys 23 has described the problems involved in the completely automatic115 M. ROUX, J . Chim. phys., 1960,57, 53; M. ROW, M. Cornille, and G. Bessis, &bid.,1961, 58, 389; S. BratoB, R. Daudel, M. ROUX, and M. Allavena, Rev. Mod. Phys.,1960, 32, 412.116 A.D. McLean, J. Chem. Phys., 1960, 32, 1595; A. D. McLean, B. J. Ransil, andR. S. Mulliken, ibid., p. 1873.l17A. Padgett and M. Krauss, J. Chem. Phys., 1960, 32, 189.llSL. Burnelle, J . Chem. Phys., 1960, 32, 1872; 1961, 35, 311.119 E. Clementi, J . Chem. Phys., 1961, 34, 1468.lZo G. Bessis and S. Bratoi, J . Chim. phys., 1960, 57, 769.lZ1R. McWeeny and K. A. Ohno, Proc. Roy. SOC., 1960, A , 255, 367.lZ2 J.-C. Lorquet, Rev. Mod. Phys., 1960, 32, 312; J.-C. Lorquet and H. Lefebvre-Brion, J . Chim. phys., 1960, 5’7, 85.lZ3P. L. Goodfriend, F. W. Birss, and A. B. F. Duncan, Rev. Mod. Phys., 1960,32, 307.1241. M. Mills, Mol. Phys., 1958, 1, 99, 107; 1961, 4, 57; R. K. Nesbet, J. Chsm.Phys., 1960, 32, 1114; A. F. Saturn0 and R. G.Pam, ibid., 1960, 33, 22.lZ5E. L. Albasiny and J. R. A. Cooper, Mol. Phys., 1961, 4, 35328 GENERAL AND PHYSICAL CHEMISTRYcomputation of molecular properties by extensive superposition of configura-tions, and has presented the results of specimen calculations on the methyl-ene radical 126 and the formaldehyde m01ecule.l~~ He has also exploredthe possibility of constructing localized orbitals approximately invariantfrom one molecule to another.128Lowdin 129 has discussed the properties of wave functions which gobeyond the Hartree-Fock formulation.E. T. S.3. IOLECULAR-ENERGY TRANSFER IN GASESTHERMAL energy may be stored by molecules in translational, rotational,and vibrational motion. Transfer of energy from internal modes of motion,particularly vibration, to translation is not necessarily an efficient processcollision-wise. This means that the thermodynamic properties of gases maybe time-dependent, an important phenomenon from the point of view ofgas dynamics.Further, since molecules must become highly excited vibra-tionally in order to decompose, the connexion between energy transfer andchemical kinetics is a close one. The subject is therefore of interest to bothphysicists and chemists. It was last treated in these Reports for 1958,lbut the present Review is more limited in scope than its predecessor. Energytransfer in liquids and solutions was also treated in 1958. The presentReporter believes, however, that it is not necessarily appropriate to treatthe liquid and the gas phase together in this connexion, and so he dealshere exclusively with gases, for the period from late 1958 to October1961.Since 1958 two books have dealt with this subject, one mainly with ultra-sonic methods but including a detailed treatment of the theory of vibrationalenergy transfer,2 the other covering nearly the same ground as the presentR e p ~ r t .~Experimental Methods.-The classical ultrasonic methods of measuringrelaxation times for vibrational-translational energy transfer continue tobe used. Edmonds and Lamb4 have described a new method of measuringsound absorption in gases and applied it to a number of polyatomic mole-cules. The use of the shock tube for energy-transfer studies at high tem-peratures has increased rapidly, and Gaydon and his co-workers 5-7 havedeveloped a method of measuring vibrational temperatures in shocks byJ.M. Foster and S. F. Boys, Rev. Mod. Phys., 1960, 32, 305.wJ. M. Foster and S. F. Boys, Rev. Mod. Phys., 1960, 32, 300, 303.1Z8S. F. Boys, Rev. Mod. Phys., 1960, 32, 296.12@ P.-0. Lowdin, Rev. Mod. Phys., 1960, 32, 328.1 B. Stevens, Ann. Reports, 1958, 55,‘80.8 K. F. Herzfeld and T. A. Litovitz, Absorption and Dispersion of UltrasonicT. L. Cottrell and J. C. McCoubrey, “ Molecular Energy Transfer in Gases,”4 P. D. Edmonds and J. Lamb, Proc. Phgs. SOC., 1958, 72, 940.5 J. G. Clouston, A. G. Gaydon, and I. I. Glass, Proc. Roy.Soc., 1958, A , 248, 429.6 A. G. Gaydon and J. Hurle, Eighth Combustion Symp., Pasadena, in the press.7 A.G. Gaydon and I. R. Hurle, Proc. Roy. SOC., 1961, A , 262, 38.Waves,” Academic Press, New York, 1959.London, Butterworths Scientific Publns., 1961COTTRELL : MOLECULAR-ENERGY TRANSFER I N GASES 29spectrum-line reversal. It appears that excitation temperatures of metalsare in equilibrium with the vibrational rather than the translational tem-peratures. Direct spectroscopic measurement of the population of lowexcited vibrational levels has been achieved by several workers.8, Theoptic-acoustic effect,lO which has long promised to give information aboutrelaxation times, has not yet done so, but a paper on the theory hasappeared in Canada.11 The same results were obtained independently inLondon.Vibra-tional-temperature measurement in a shock tube has yielded values of0-4 psec. at 2280" H, and 0.8 psec.at 2600" K for the vibrational-relaxationtime, z. These times seem surprisingly short. The most recent results l3on rotational relaxation confirm earlier work.Vibrational-relaxation times have been measured by usingthe spectrum-line-reversal method in shocked gas,5, 6 with results in agree-ment witn those of previous work. l4 Water-nitrogen collisions were be-tween 10 and 100 times more effective than nitrogen-nitrogen collisions indeactivating vibrationally excited nitrogen. Rotational-relaxation times inagreement with earlier work have been determined.15Oxygen. Direct measurement of the vibrational-relaxation time bymeans of the propagation of low-frequency sound l8 has confirmed an earlierconclusion by the indirect method 1' that z = 3.2 x loA3 sec.On theother hand, a further determination18 by the indirect method (that is,by extrapolation to zero impurity) gave the considerably longer result,z = 1.8 x 10-2 sec. Mixtures were also studied, hydrogen-oxygen colli-sions being 3 x lo3, deuterium-oxygen 3 x lo2, and helium-oxygen3-5 x 102 times more effective than oxygen-oxygen collisions. Spectro-scopic measurements l9 on shocked oxygen-argon mixtures from 1200" to7000" K gave z for oxygen-oxygen collisions close to those found earlier l4above 2000" K, but a little longer at temperatures around 1200" K. Oxygen-oxygen collisions were about 5 times more effective than oxygen-argoncollisions, the ratio being independent of temperature.Rotational relaxa-tion has also been studied.15Vibrational relaxation in the B2Cf excited state of cyanidehas been detected spectroscopically in a shock tube. z varies from 42 psec.a t 6 3 0 0 " ~ to 14 psec. a t 9 5 5 0 " ~ . ~ ~Curbon monoxide. Shocked carbon monoxide has been studied byExperimental Results €or Low-lying Energy States.-Hydrogen.Nitrogen.Cyanide.N. Basco, A. B. Callear, and R. G. W. Norrish, Proc. Roy. Soc., 1961, A, 260, 459.9F. Robben, J . Chem. Phys., 1959, 31, 420.lo M. E. Delany, Science Progr., 1959, 47, 459.l1 R. Kaiser, Cannd. J . Phys., 1959, 37, 1499.12M. E. Delany, Thesis, London, 1959.13H. D. Parbrook and W. Tempest, J . Acoust. SOC. Amer., 1958, 30, 985.l* S. J. Lukasik and J. E.Young, J . Chem. Phys., 1957, 27, 1149; V. Bhckman,16M. Greenspan, J . Acoust. SOC. Amer., 1959, 31, 155.l6 F. A. Smith and W. Tempest, J . Acoust. SOC. Amer., 1961, 33, 1626.I7H. Knotzel and L. Knotzel, Ann. Phys., 1948, 2, 393.18J. G. Parker, J . Chem. Phys., 1961, 34, 1763.19M. Camac, J . Chem.. Phys., 1961, 34, 448.2OW. Roth, J. Chem. Phys., 1959, 31, 720.J . Fluid Mechanics, 1956, 1, 6130 GENERAL AND PHYSICAL CHEMISTRYseveral workers.6, ' 9 21 The longest reported z is about 6 x 10-4 sec. at1500" K , ~ O decreasing to about 2 psec. at 4900" K. Water produces a markedreduction in z, and carbon dioxide produces a slight reduction; hydrogen,nitrogen and oxygen are stated to have no effect.22Nitric oxide. Vibrational relaxation in the ground ( 2Li!) electronic statehas been studied by sound absorption at room temperature,23 by kineticspectroscopy of flashed gas at room temperature,8 and by spectroscopicshock-tube experiments from 450 to 1300" K.' All these investigations showrelaxation times many powers of ten shorter than predicted theoretically orfound for nitrogen, oxygen, or carbon monoxide, and this anomaly has beendiscussed.24 The values obtained are in the microsecond region at roomtemperature.The egect of added gas has also been studied:8 water isabout 20 times as efficient as nitric oxide, carbon dioxide about half asefficient, carbon monoxide about a tenth as efficient, and nitrogen andhydrogen are much less efficient. The situation here is complicated byspin-orbit relaxation.Vibrational relaxation in the excited (A2X +) state of nitric oxide has beenstudied spectroscopically for the shocked gas.25 The relaxation times forthe 2-1 and 1-0 vibrational transitions at 6950" K are 6.7 and 12.5 psec.,respectively. For the 1-0 transition, the transition probabilities percollision are 3 x a t 10,000"~.Chlorine. Vibrational-relaxation times in chlorine have been remeasuredby the sound absorption method.26 z a t low temperature (e.g., 4-9 psec.a t 2 9 8 " ~ ) is in agreement with earlier work, but a t higher temperatures(400-500" K) is rather longer than was thought earlier (e.g., 2.3 psec. at4-40" E, compared with an earlier value of 1-6 psec. a t 415" K 27). A newmeasurement 28 of velocity dispersion over a good range gives a slightlyshorter value for z at room temperature than was obtained by others.at 5 0 0 0 " ~ and 3 xHydrogen chloride.Rotational relaxation only has been studied. 29Bromine and iodine. Vibrational relaxation has been studied by soundabsorption 26 and dispersion.28 For bromine, dispersion gives z twice asgreat as is given by absorption, and for iodine about seven times as great.The only new results since 1958 are some observationsa t ordinary density 30 in general agreement with previous work, and somemeasurements at high density 31 which show that if intermolecular attractiveforces are taken into account good agreement is obtained between the col-lision efficiencies in the gas and the liquid phase.The relaxation time of the 2223 cm.-l band has beenCarbon dioxide.Nitrous oxide.21M.Windsor, N. Davidson, and R. Taylor, 7th Sjmp. on Combustion, Butter-22D. L. Mathews, J . Chem. Phys., 1961, 34, 639.23 H. J. Bauer, H. 0. Kneser, and E. Sittig, J . Chem. Phys., 1959, 30, 1119.24F. Robben, P. R. Monson, and J. J. Allport, J . Chem. Phys., 1960, 33, 630.Z5W. Roth, J . Chem. Phys., 1961, 34, 999, 2204.26 F. D. Shields, J . Acoust. SOC. Amer., 1960, 32, 180.27A. Eucken and R. Becker, 2. phys. Chem., 1934, B, 27, 235.2sE. G. Richardson, J . Acoust. SOC. Amer., 1959, 31, 152.=OM. A. Breazeale and H. 0. Kneser, J . Acoust. SOC. Amer., 1960, 32, 885.%OF. D. Shields, J . Acoust. SOC. Amer., 1959, 31, 248.91 W. M. Madigosky and T. A. Litovitz, J . Chem. Phys., 1961, 34, 489.worths Scientific Publns., London, 1958, p.80COTTRELL : MOLECULAR-ENERGY TRANSFER IN GASES 31found by an infrared method 32 to be sec., much longer than the overallz found by ultrasonic methods. The established ultrasonic results havebeen further ~onfirmed.~3Carbon disulphide. z has been redetermined over the temperaturerange 233433'9, by the absorption technique.34 The results a t highertemperatures are new, those at lower temperatures agree with previouswork.Sulphur dioxide. Some doubt has been cast 35 on earlier 36 gas data forsulphur dioxide. Until evidence for these doubts has been fully published,the earlier data must stand, particularly as they have recently been inde-pendently confirmed. 37 There are two relaxation times for this molecule,and the more recent values for these quantities are z1 = 6 x sec.,z, = 1.2 x 10-6 sec.The previous 36 result for z, was 0.6 x sec.,but this relaxation time is the more subject to experimental error. n-Hex-ane is a particularly effective energy-transfer catalyst for sulphur dioxide.37Ammonia. The most recent investigation3* suggests that z for thismolecule is very short.Boron trijuoride. Sound-absorption measurements* lead to z = 0.09psec. a t 298" K.Silane. Ultrasonic velocity measurements 39 at 298" K give z = 0.19psec. for tetradeuteriosilane, and 0.11 psec. for silane. These results showthe same tendency as those for the methane analogues, discussed below.SuZphur hexajluoride. Recent sound-velocity measurements 40 a t 301 " Klead to z = 0.78 psec., in fair agreement with previous work.Hethane.Ultrasonic velocity measurements 41 at 2 9 8 " ~ give z as3.9 psec. for tetradeuteriomethane and 2.1 psec. for methane. The latterfigure is slightly greater than the previous highest result., The CD, : CH,ratio does not agree with expectation based on simple vibrational-transla-t,ional energy transfer, but might be due to rotational effects.Carbon tetrajuoride. Sound-absorption measurements have confirmedearlier sound-velocity results.Methyl chloride. Earlier work from Lambert's school has now beensuperseded.3, *2 Sound-absorption measurements have confirmed othersound-velocity results.43Chlorodijuoromethane, dichlorodi$uoromethane, chlorotrijuoromethune, andbromotrijuormethane.Ultrasonic velocity measurements 43 give relaxationtimes over a temperature range.Methyl bromide. Recent ultrasonic measurements 43 give results in32G. Gauthier and J. Marcoux, Cunad. J . Phys., 1961, 39, 1130.a3R. Holmes, H. D. Parbrook, and W. Tempest, Acustica, 1960, 10, 155.84 J. C. Gravitt, J . Acoust. SOC. Amer., 1960, 32, 560.35M. C. Henderson, quoted in ref. 31.s6 J. D. Lambert and R. Salter, Proc. Roy. SOC., 1957, A, 243, 78.37 J. C. McCoubrey, R. C. Milward, and A. R. Ubbelohde, Proc. Roy. SOC., 1961,38 J. D. Lambert and R. Salter, Proc. Roy. SOC., 1959, A, 253, 277.40 J. C. McCoubrey, unpublished work.41 T. L. Cottrell and A. J. Matheson, Proc. Chem. SOC., 1962, 17.r2A. J. Edwards, B.Sc. Thesis, Oxford, 1959.43R.Amme and S. Legvold, J . Chem. Phys., 1959, 30, 163.A, 264, 299.L. Cottrell and A. J. Matheson, unpublished work32 GENERAL AND PHYSICAL CHEMISTRYagreement with earlier values. z for bromotrideuteriomethane is nearlydouble the value for bromomethane.38Methylene bromide. A set of three separate relaxation times has beenpostulated to explain the ultrasonic-velocity results for this compomd,44and it has been shown that this scheme leads to complete disagreementbetween theory and experiment.4s As the theory is fairly well established,further experimental work seems called for.Ethane. A double-dispersion region in this compound is well estab-l i ~ h e d , ~ ~ , 46 the more reliable results at 2 9 6 " ~ being z1 = 1.4 x 10-8 sec.,tz = 0.12 x sec.Ethyl fluoride, 1,l-difluoroethaneY hexaJEuoroethane, ethyl chloride, and1 , l - and 1,2-dichloroethane.All these compounds have been shown by ultra-sonic velocity measurement 38 to have very short relaxation times.1,2- DichZoro-1,1,2,2-tetra$uoroethane. z has been measured by soundabsorption. *Propane, n-butane, isobutam, and mopentune. These compounds haveall been shown to have very short relaxation times,38 only that of propanea t 298" K (z = 0.4 x sec.) being detectable up to 50 Mc. sec.-1 atm.-1.Ethylene. Two more contributions have been made to the mass of dataon this c o m p o ~ n d , ~ ~ ~ 48 and it seems that z a t 298" K is fairly well estab-lished as 0.26 & 0.02 psec. The relaxation time for tetradeuterioethyleneis about half that value, although simple theory 48 suggests it should benot more than a quarter.Tetra- and di-deuterioethylene, water, and di-deuterium oxide are all energy-transfer catalysts for ethylene.48Acetylene. Two recently determined values of z differ by a factor ofnearly 2, velocity dispersion 38 giving z = 7 x sec., and sound absorp-tionButadiene. The relaxation time has been measured by the sound-absorption m e t h ~ d . ~Dimethyl ether. The relaxation time is very short and it is uncertainwhat modes are concerned.38Cyclopropane. Sound-absorption measurements have confirmed earlierresults.Theory of Vibrational-energy !Crans€er.-No fundamentally new theo-retical discussion of the problem has appeared in the period under review,for the good reason that the basic theory put forward by Zener in 1931 isessentially correct.49 The amplification of it described in detail by Herzfeldand his co-workers puts it in a form which is convenient to use. Themodifications which have been discussed are concerned with refinement intwo main directions: fist, in order to discuss low-energy interactions betweenpolyatomic molecules more accurately and, secondly, in order to extendits usefulness to higher-energy states and eventually t o reaction.giving 12 x 10-8 sec.44N.J . Meyer, J . Chenz. Phys., 1960, 33, 487.45P. G. Dickens and D. Schofield, J. Chem. Phys., 1961, 35, 374.4 6 L. M. Valley and S. Legvold, J . Chcm. Phys., 1960, 33, 627.4 7 J. C. Gravitt, J . Acoust. SOC. Amcr., 1960, 32, 1455.48 G.H. Hudson, J. C. McCoubrey, and A. R. Ubbelohde, Proc. Roy. Soo., 1961,p°C. Zener, Phys. Rev., 1931, 37, 556; 38, 277.A , 284, 289COTTRELL : MOLECULAR-ENERGY TRANSFER IN GASES 33The theory in its simplest form neglects the attractive part of the inter-molecular potential and the (‘ symmetrisation ” of approach and recessionvelocities. It is necessary to take both these factors into account in orderto secure agreement with experiment.50 If this is done by the amplificationof Tanczos’s method 51 in applications to polyatoniic molecules, fair agree-ment between calculated and observed results is generally obtained.52 Useof a Morse potential for the intermolecular interaction improves agreementwith experiment for methyl chloride.53 Use of a Lennard-Jones type 28 : 7potential improves agreement with experiment for halogenomethanes ingeneral.54 Vibrational relaxation in carbon dioxide has been treated withparticular reference to the form of the normal modes, with the predictionof two relaxation times.55The classical treatment of one-dimensional inelastic collision with im-pulsive interactions 56 has been followed by a quantum-mechanical one 57in which it is shown that under these conditions there may be vibrationaljumps of more than one quantum.The study of higher-energy interactionshas been carried further by the use of a Morse function for the vibrator.58When transitions to the continuum are taken into account it appears thatthe activation energy for the thermal dissociation of diatomic moleculeswould be expected to be below the bond energy because of breakdown ofthe Boltzman distribution.This problem has also been tackled by manyother workers.Energy-transfer processes in polyatomic molecules are so complicatedthat there is room for empirical correlations of the experimental results,and a particularly interesting study in this field is due to Lambert andSalter.38 They showed that, for molecules in which the fundamental vibra-tion frequencies are so distributed that there is only a small gap betweenthe lowest and the remaining frequencies, vibrational-energy transfer in-volves a single relaxation process. The chief factors affecting the prob-ability of energy transfer are the frequency of the lowest mode and thepresence or absence of a hydrogen atom in the molecule.Molecules con-taining two or more hydrogen atoms s a e r energy transfer much morereadily than do other molecules. This generalisation is supported by animpressive correlation diagram. No firm explanation of this effect was putforward by its discoverers, but it seems possible that it may be linked withthe greater ease of rotational-vibrational energy transfer in molecules withlow moments of inertia.39It has been common in all this work to quote ‘( the number of collisionsrequired” for energy transfer, and to calculate this quantity from theexperimental observations requires a knowledge of the collision number.6o J. C. McCoubrey, R. C. Milward, and A. R. Ubbelohde, Trans. Furday SOC.,1961, 57, 1472.61F.I. Tanczos, J . Chem. Phys., 1956, 25, 439.62 P. G. Dickens and A. Ripamonti, Trans. Faraday SOC., 1961, 57, 735.63 A. R. Blythe, T. L. Cottrell, and A. W. Read, Trans. Furaday SOC., 1961,57,935.64 R. C. A v e and S. Legvold, J . Chem. Phys., 1960, 33, 91.66 W. J. Wittaman, J . Chsm. Phys., 1961, 35, 1.66B. Widom, J . Chem. Phys., 1959, 30, 238.s7K. E. Shuler and R. Zwanzig, J . Chem. Phys., 1960, 33, 1778.68H. 0. Pritchard, J . Phys. Chem., 1961, 65, 504.34 GENERAL AND PHYSICAL CHEMISTRYThe method of calculating collision numbers in the two recent books 2, 3has been criticised as being too heavily weighted at low temperatures byattractive deflection^.^^Some examples, such asthat for nitric oxide, of energy transfer from higher, identified, states havebeen discussed above.Vibrational-energy transfer from excited statesmay be studied by the examination of fluorescence stabilisation in poly-atomic molecules. The methods involved in such studies were reviewed in1957,60 and furfher experimental results for aromatic systems have becomeavailable.61 The method has also been used for other types of compound,as, for example, in a study of the photochemistry of cyclopentanone.62Added gases did not show great differences in energy-transfer efficiency.Apparently abnormal rotational distributions of hydroxyl in detonationhave been observed 63 and attributed to rotational disequilibrium. Theobservations are now 64 attributed to turbulence. Indeed, it has beenshown 65 that the cross-section for rotational relaxation of hydroxyl inhydrogen-oxygen flames is even larger than the gas-kinetic collisional cross-section.Further related work on higher-energy states has been concerned withthe deactivation of excited radicals (see, for example, ref.66). This typeof investigation is, however, so closely related to chemical kinetics thatit is not considered further here.Energy transfer involving higher-energy states.T. L. C.4. INFRARED SPECTRA AND MOLECULAR INTERACTIONSCONSIDERABLE interest has been shown during the last few years in theefTect of molecular interactions on infrared spectra. This report reviewscertain aspects of the subject, vix., solvent egects, matrix spectra, and thespectra of adsorbed films.Most of the references are from publicationsduring the last four years but in certain cases earlier work is included.Since the last Annual Report, the I.U.P.A.C. Commission on spectro-scopy has published comprehensive sets of tables for the calibration ofgrating and of prism spectrometers. The continued publication of variouscard-index systems, such as that of D.M.S., has reduced the labour of locat-ing spectra, but the appearance of a Beilstein-type index of infrared spectra 2covering the years up to 1957 is an additional help.Intermolecular Forces and Solvent Eff ects.-Frequency shyk Consider-69 J. s. Rowlinson, MoZ. Phys., 1961, 4, 317.G O B . Stevens, Chem. Rev., 1957, 57, 439.61 B. Stevens, Discuss. Faraday SOC., 1959, 27, 34; Mol.Phys., 1960, 3, 589.s2R. Srinivasan, J . Amer. Chem. SOC., 1961, 83, 4348.63G. B. Kistiakowsky and F. D. Tabbutt, J . Chem. Phys., 1959, 30, 577.64 G. B. Kistiakowsky and R. K. Lyon, J . Chem. Phys., 1961, 35, 995.65T. Carrington, J . Chem. Phys., 1959, 31, 1418.66 R. E. Harrington, B. S. Rabinovitch, and M. R. Hoare, J . Chem. Phys., 1960,1 “ Tables of Wavenumbers for the Calibration of Infrared Spectrometers,”“ An Index of Published Infrared Spectra,” Ministry of Aviation, H.M.S.O.,33, 744.I.U.P.A.C., Butterworths Scientific Publns., London, 1961.London, 1960WILLIAMS : INFRARED SPECTRA 35able changes of both frequency andintensity occur in the vibrational spectrumof a molecule on going from the gaseous to the solution phase.One of thesimplest and earliest attempts to explain these effects was that of Kirk-wood 3 and of Bauer and Magat.4 This was based on the model of an oscil-lating point dipole in a spherical cavity in a continuous dielectric medium.The frequency of the oscillator was modified as a result of the field actingon it, owing to polarization induced in neighbouring molecules by its owncharge distribution. The treatment led to the well-known Kirkwood-Bauer-Magat (KBM) relation(1)where Av is the change in frequency on going from vapour to solution phase,Y is the vapour-phase frequency, E is the dielectric constant of the solvent,and C is a constant. The applicability and limitations of this model havesubsequently been examined in considerable detail by Pullin who appliedessentially the same treatment as Bauer and Magat and derived an equa-tion for a diatomic vibratorAV/V = C(E - 1)/(2& + I),AV/V = K - l ( l - A)-1[3f ‘ob,u0M‘/K + f ‘yB~OM’ + f ‘,,Mr2/2 + 3(f ’ 0 + f ’2v)PoJf”lY (2)where K and b are constants of the potential energy function, M’ and M“are the derivatives of the dipole moment ,uo-distance function, A and Bare functions of bond polarizability, and the quantities f‘ are constantsrelated to the reaction field of the dipole.The first and the third term inthis equation are equivalent to those given by Bauer and Magat and thesecond and fourth to those of West and Edwards.3 The equation wasextended to polyatomic molecules and approximate numerical calculationsshowed that the frequency shift, e.g., in vco, of acetone is about one-halfof that observed.Subsequently, P ~ l l i n 5 ~ showed that for acetone, fre-quencies varied linearly with the function(3)where n and E are the refractive index and the dielectric constant respectivelyof the solvent, B is a constant for each particular solute mode, and R is asolvent-dependent radius.BuckinghamJGQ using a model similar to Kirkwood’s, expanded thesolvent-solute interaction energy, U, as a power series in the normal co-ordinates of the solute and then treated the U and the anharmonic termsin the potential-energy function as small perturbations to the harmonicoscillator Hamiltonian. He found that for a diatomic molecule Av oc(U” - 3 U‘g/m,,) where U’ and U” are derivatives of the interaction energywith respect to the vibrational displacement and g/me is an anharmonicconstant.The theory indicated that Av/v should be independent of isotopic[(n2 - 1)/(2n2 + 1)RI + [B(& - + 1)W,J. G. Kirkwood, J . Chem. Phys., 1934, 2, 351; quoted by W. West and R. T.E. Bauer and M. Magat, J . Phys. Radium, 1938, 9, 319.SA. D. E. Pullin, Spectrochim. Acta, ( a ) 1958, 13, 125; ( b ) 1960, 16, 12; (c) Proc.A. D. Buckingham, Proc. Roy. SOC., (a) 1958, A , 248, 169; ( b ) 1960, A , 255, 32;Edwards, ibid., 1937, 5, 14.Roy. SOC., 1960, A, 255, 39.(c) Trans. Paraday SOC., 1960, 56, 75336 GENERAL AND PHYSICAL CHEMISTRYsubstitution and should be the same for both the fundamental and its over-tones. For non-polar solvents it was possible to write an equation(4)(5)A V l V = Cl + w 2 + C3)(& - 1)/W + I),AV/Y = C1 + C ~ ( E - 1)/(2& + 1) + C3(n2 - 1)/(2%2 + I),while for solvents with a dipolewhere Cl, C2, and C3 are constants depending on both solute and solvent.I n later papers@* extension to polyatomic molecules was considered, inaddition to the application of the method to intensities, band widths, andStokes and anti-Stokes Raman lines.A considerable number of publications on empirical approaches to theproblem of solvent shifts have appeared.Allen and Warhurst 7 showedthat AV/Y for the v1 symmetric mode of mercuric chloride, a Raman activefrequency, was a linear function of log E and that the slope of the plot fora number of solutes was a function of the bond ionic character.Grangeet aL8 have shown similarly that the relative frequency shifts (Av/v) for thehydrogen halides in carbon tetrachloride solution were a linear function ofthe Pauling electronegativity of the halogen atom.More recently, Le Fbvre 9 has put forward an empirical method of cal-culating solution frequencies based on equations which relate (i) bondpolarizability to the refractive index of the solvent and (ii) the frequencyin solution to the bond polarizability. Fair agreement is obtained betweenobserved and calculated values for acetone.The validity of the KBM equation has been tested by many workers.Bayliss, Cole, and Little 10 measured the frequencies? intensities, and half-band widths for C=O, C-H, and C-C bands in a range of solvents and foundthe equation inapplicable.Cannon and Stace l1 were unable to relate thehydroxyl shifts of ethanol and trifluoroethanol to the KBM equation.Other studies include those of the -CN,l2 >NH,13 -OH,14 and >CO l5 groups.Arbitrary classification of solvents into three groups has been made,based on their effect on either carbonyl groups l6 or acetylenic CH.l7Acetylenes have also been studied by Gastilovitch and his co-workers lSaand by Shuvalova . sb7 G. Allen and E. Warhurst, Trans. Paraday SOC., 1958, 54, 1786.P. Grange, J. Lascombe, and M. L. Josien, Spectrochim. Acta, 1960, 16, 981.9R. J. W. Le FBvre, Austral. J . Chem., 1961, 14, 312.ION. S. Bayliss, A. R. H. Cole, and L. H. Little, Austral. J . Chem., 1955, 8, 26.11 C.G. Cannon and B. C. Stace, Spectrochim. Acta, 1958, 13, 253.12 J. P. Jesson and H. W. Thompson, Spectrochim. Acta, 1958, 13, 217.l a R. E. Dodd and G. W. Stephenson, " Hydrogen Bonding," ed. D. Hadii andH. W. Thompson, Pergamon Press, London, 1959, p. 177; I. Suzuki, M. Tsuboi, andT. Shimanouchi, J . Chem. Phys., 1960, 32, 1263; K. B. Whetsel, W. E. Roberson,and M. W. Krell, Analyt. Chem., 1960, 32, 1281.14 Y. Sato, J . Chem. SOC. Japan, 1958, '79, 358; P. A. D. De Maine, L. H. Daly, andM. M. De Maine, Canad. J . Chem., 1960, 38, 1921.15L. B. Archibald and A. D. E. Pullin, Spectrochim. Acta, 1958, 12, 34.16 T. V. Yakovleva, A. G. Mmlennikova, and A. A. Petrov, Optics and Spectro-scopy, 1961, 10, 131.17B. Wojtkowiak and R.Romanet, Compt. rend., 1960, 250, 3980.18 (a) E. A. Gastilovich, D. N. Shigorin, E. P. Gracheva, I. A. Chekula, and M. F.Shostakovski, Optics and Spectroscopy, 1961, 10, 312; (b) E. V . Shuvalova, ibid., 1959,6, 452WILLIAMS : INFRARED SPECTRA 37Josien and her co-workers in particular have examined the KBMrelation, which was found to be satisfactory for XH vibrators only with afew non-polar solvents, e.g., carbon tetrachloride or n-hexane. Deviationfrom the equation was attributed to complex formation and this was con-firmed by the presence of two XH stretching bands in mixed solvents.Subsequently, equilibrium constants were determined.20 For example, theassociation constant for the equilibriumPyrrole + Solvent + Pyrrole, Solventlies in the range 0.23 l./mole for chlorobenzene to 2.7 l./mole for pyridine.A different approach has been employed by Bellamy and his co-workers,21who plotted Av/v for given XH vibrators in a series of solvents againstAv/v for a standard substance, e.g., v(NH) of pyrrole in the same series.This method eliminates the effect of the solvent, and straight lines wereobtained whose slopes were related approximately to the pK, value of theX-H bond.It was proposed that the frequency shifts were due to localassociation between the X-H dipole and a negative charge centre in aneighbouring solvent molecule. The lack of discontinuity in the plotsshowed that there was little difference in kind between association witha polar solvent, e.g., ether, and a non-polar solvent, e.g., carbon tetrachloride.This has however been disputed by Grange et aZ.8 I n a second paper 22ait was shown that >cO groups could be treated in the same fashion but sincethe dipole was >C=O the order of solvents along the plot was not the sameas for X-H solutes.The results were later extended 2Zb to groups of thetype -XO, where X = N, P, or S. In all these cases it was found thath / v for a series of solvents gave linear plots against AY/Y for a standard>CO compound, indicating that the mechanism of association with thesolvent was the same. Further, it was found23 that in a series of theacetyl derivatives of some N-heterocyclic compounds, the >CO solvent-sensitivity increased steadily in the order tetrazole, 1,2,4-triazole, imidazole,pyrrole, and pyrrolidine, which is also that of increasing CO dipole.Whenthe vibrating group is already associated 24 as in, e.g., carboxylic acid dimersor o-nitrophenol, local association with the solvent is nearly negligible andthe frequency is virtually solvent-insensitive. Mixed solvents studies, 24e.g., with pyrrole in carbon tetrachloride, benzene, and mesitylene, gave threebands a t frequencies corresponding to single-component solutions. Theill. L. Josien and N. Fuson, J . Chm. Phye., 1954, 22, 1169, 1264; M. L. Josienand J. Lascombe, Compt. rend., 1954, 238, 2414; M. L. Josien and G. Sourisseau, ibid.,1954, 238, 2525; M. L. Josien and J. Lascombe, J . Chim. phya., 1955, 52, 162; 1957,54, 761; M. L. Josien and P. Saumagne, Bull.SOC. chim. France, 1956, 937; 1958, 813;M. L. Josien, J. P. Leicknam, and N. Fuson, &id., 1958, 188.*ON. Fuson, P. Pineau, and M. L. Josien, J . Chim. phya., 1958, 55, 454, 464;“ Hydrogen Bonding,” ed. D. Hadii and H. W. Thompson, Pergamon Press, London,21 L. J. Bellamy, Spectrochim. Acta, 1959, 14, 192; L. J. Bellamy, H. E. Hallam,and R. L. Williams, Trana. Paraday SOC., 1958, 54, 1120.22 (a) L. J. Bellamy and R. L. Williams, Trans. Faraday SOC., 1959, 55, 14; ( b ) L. J.Bellamy, C. P. Conduit, R. J. Pace, and R. L. Williams, ibid., p. 1677.2a L. J. 13ellamy and R. L. Williams, Proc. Roy. Soc., 1960, A , 255, 22.24 L. J. Bellamy and H. E. Hallam, Tram?. Faraday SOC., 1959, 55, 220.8 - 8+6+ 6-6- 6 l1959, p. 16938 GENERAL AND PHYSICAL CHEMISTRYrelative areas of the three bands depended on the proportions of solvents,even though the overall dielectric constant remained constant.The effectof the shape of the solute molecule 25a and the solvent molecule 25b onsolvent-sensitivity has been studied with substituted phenols and ethers,respectively. Similar solvent plots to those mentioned above have beenreported for -CN,26ur >NH, and -OH groups.26c An attempt hasalso been made to use them to measure the basicity of aromatic compounds,correlations being observed between Av/v for a standard XH compoundand such variables as the heat of mixing of chloroform with the solvent,the lowering of vapour pressure, and ionization p0tential.27~ A similarcorrelation between heat of mixing with chloroform and the shift of theOD frequency in CH,*OD in cyclic sulphoxides and ketones27b has beenreported .Coulson 28 has considered theoretically the reasons for the breakdownof the dielectric-constant approach and for the applicability of a local-interaction model.Attempts to predict frequency shifts on the basis ofsuch a model have been made in a number of cases. La Lau 29 calculatedthe change in the CH bending force constant of benzene, assuming electro-static interaction between a positive charge on the hydrogen and a negativecharge on a solvent such as acetone or acetonitrile, comparatively goodagreement being obtained between observed and calculated Av/v values.Drickamer and his co-workers 3O examined the shifts in frequency resultingfrom the application of high pressures to solutions.They interpreted theirresults in terms of the interaction of the vibrator with each bond of thesolvent, the relative contribution from dispersion, orientation, induction,and repulsion forces being estimated from the formuheidisp. = -~ITiaai/[(l + Ii)RF]; etepuls- = Ci/Ri12;where e is the energy of interaction, I ionization potential, a bond polariz-ability, p bond dipole, R the solute-solvent distance, f an orientationfactor, and C a constant. On this basis they found that for >C=O incarbon disulphide the inductive interaction constituted 65% of that observed.Some cases, for which the shift of band position with pressure is claimed toobey the KBM relation, are probably of this type.31A calculation similar to that above has been made by Linevsky 32 for25 (a) L.J. Bellamy and R. L. Williams, Proc. Roy. Soc., 1960, A, 254, 119; (b) L. J.Bellamy, G. Eglington, and J. F. Morman, J., 1961, 4762.(a) H. W. ThompsonandD. J. Jewell, Spectrochim. Ackc, 1958, 13, 254; (b) N. S.Bayliss, A. R. H. Cole, and L. H. Little, ibid., 1959, 15, 12; (c) W. C. Price, W. F. Sher-man, and G. Wilkinson, Proc. Roy SOC., 1960, A , 255, 5.27 (a) M. L. Josien and G. Sowisseau, " Hydrogen Bonding," ed. D. Hadii andH. W. Thompson, Pergamon Press, London, 1959, p. 129; ( 6 ) M. Tamres and S. Seaxles,J . Amer. Chem. SOC., 1959, 81, 2100.2sC. A. Coulson, PTOC. Roy. SOC., 1960, A , 255, 69.29 C. La Lau, Spectrochim. Acta, 1959, 14, 181.3O E.Fishman and H. G. Drickamer, J. Chem. Phys., 1956,24, 548; A. Benson andH. G. Drickamer, ibid., 1957, 2'4, 1164; R. R. Wiederkehr and H. G. Drickamer, ibid.,1958, 28, 311.31 M. L. Josien and P. Dizabo, Compt. rend., 1956, 243, 44; M. L. Josien, J. Las-combe, and N. Fuson, J. Chem. Phys., 1956, 25, 1291.32M. J. Linevsky, J. Chem. Phys., 1961, 34, 587.} (6) eielectrostat. = ,,upi fi2/Ri3; eind. = p2ai fi3/Ri6WILLIAMS : INFRARED SPECTRA 39lithium fluoride in matrices of solid argon, krypton, xenon, or nitrogen.For the first three matrices, the component of the interaction energy due toinduction was calculated from the formula (7):(7) einduct. - - [ap2(3 cos20 + 1)/2r6],from which the frequency shift was found to be -45 cm.-l, leaving a furthershift of 20-30 cm.-1 due to dispersion forces. In the case of nitrogen, therewas also a dipole-quadrupole interaction of -55 cm.-l, which in con-junction with an induction shift of 40 cm.-l left a component of -30 crn.-lto dispersion.Calculations on the basis of an intermolecular interaction with a 6-9potential function have been made for MX,-type molecules by H e i ~ k l e n , ~ ~to determine the frequency changes which occur on going from a gaseousto a condensed phase. A similar method has been applied satisfactorily tothe vibrational structure of the electronic bands of the NH radical in argon,krypton, or xenon matrices, observed in the ultraviolet region.34 Galatryand Schuller 35a have considered a KBM-type model but with the oscillatingdipole inducing a dipole in each of the surrounding molecules, while Schuller,Galatry, and Vodar 35b have calculated the effect of dispersion on hydrogenchloride dissolved in carbon tetrachloride or benzene, by using a Heitler-London wave function with an ionic term.They derived an equationAv/v = - [e2/2nv2m][n~(4g + 7f)]/Bo3,where e is the electronic charge, m is the reduced mass of the oscillator, a isthe polarizability of the solvent, y is the mechanical anharmonicity, R, theminimum distance of approach of the solute and solvent, n = iV/V is thenumerical density of the solvent, and g andf are quantities calculable fromthe wave equation. Insertion of numerical quantities in expression (6)showed that dispersion could account for 30--50% of the observed shift,while induction amounted to only 2-5--5%.Another model for the calculation of frequency shifts has been suggestedby Person and his co-w0rkers,~6 who studied the complexes between varioushalogen compounds, e.g., iodine chloride, iodine cyanide, chlorine, andbromine, and donor solvents.Intensities, frequencies, and half-bandwidths were measured and the changes were interpreted on the basis ofMulliken’s charge-transfer theory. It is possible to write the structure of thecomplex in terms of resonance between D-n-X-Y (no bond) and D +--(X-Y) -.Person estimated the charge transferred in this process, E,, from the inten-sity of the infrared bands of the various halogen compounds in solution andshowed that it gave linear plots against Ak/k, the relative change in forceconstant for the vibration in question.The magnitude of Ak/E (w2Av/v,33 J. Heicklen, Spectrochim. Acta, 1961, 17, 82.34 M. McCarty and G. W. Robinson, J . Amer. Chem. SOC., 1959, 81, 4472.35 (a) L. Galatry and F. Schuller, Compt. rend., 1957, 244, 1749; 1957, 245, 901;( b ) F. Schuller, L. Galatry, and B. Vodar, Spectrochim. Acta, 1960, 16, 789.s6 W. B. Person, R. E. Humphrey, W. A. Deakin, and A. I. Popov, J . Amer.Chem. SOC., 1958, 80, 2049; W. B. Person, R. E. Humphrey, and A. I. Popov, ibid.,1959, 81, 273; W. B. Person, R. E. Erickson, and R. E. Buckles, ibid., 1960, 82, 29;also D. C. Cook, J . Amer. Chern. SOC., 1958, 80, 4940 GENERAL AND PHYSICAL CHEMISTRYfor small changes) depends on the donor strength of D and on the acceptorstrength of the bond X-Y.For a given pair of solutes the difference inacceptor strength is constant so that the plot of Av/v for one acceptor in aseries of solvents against Av/v for a second acceptor should give a straightline, as observed by Bellamy et aL21Successful attempts have been made by Caldow and Thompson 37 tomodify the Buckingham equations (4) and ( 5 ) . They observed that theCH frequency of hydrogen cyanide or phenylacetylene dissolved in solventsof the type RCN, RCl, and R*CO*CH, was a linear function of the Taftinductive factor o* for the residue R. A term C 4 ~ * was therefore added toequations (4) and (5). A set of constants C,, C2, C3, C4 was chosen to givethe best fit between calculated and observed frequency displacements fora given solute in eight selected solvents and was then used to predict thedisplacements for other solvents.Excellent agreement was obtained.Moreover, it was observed that for the CH shifts of hydrogen cyanide theC40* term was dominant, as would be expected for the local associationmodel of Bellamy et aL21 On the other hand, for a number of carbonyl-containing solutes, the dielectric constant terms, involving C, and C3, weresubstantial. A similar result has been obtained for C-C1 bonds by Hallamand Ray.38 It is noteworthy how these conclusions agree with the magni-tude of the dielectric contribution obtained by Pullin 5a and by Drickamer’sschool 3O for the carbonyl group and by Schuller’s school 35 for hydrogenchloride.Caldow and Thompson 37 also examined the effect of isotopicsubstitution on the solvent shift of the CH band of hydrogen cyanide.They found thatHuong et ~ 1 . 3 9 ~ have, however, shown more recently that this is fortuitousand that the relation (10) holds:(AV/Y2)H-CN = (AY/Y2)D-CN. (9)(Av/y)Hc + (A~/Y)Hc=N = (AV/Y)DC + (AY/Y)DC=N. (10)I n a number of cases 3ga3 constancy of Av/v with isotopic substitution wasfound, as predicted by Buckingham.6Though solvent shifts of infrared bands are not often related to thoseof ultraviolet bands, Ito et ~ 1 . ~ 0 ~ have shown that, for ketones, ( A Y / Y ) ~ ~ forthe infrared bands gives linear plots against a similar function for then-n* transition. This is attributed to solute-solvent association in theground state only, since on n-n* excitation the dipole of the carbonyl groupbecomes very small.Becker 40b has similarly interpreted the behaviour ofbemophenone in various solvents.Application of solvent eflects. Solvent effects have been used extensivelyfor diagnostic purposes, Thus the Av/v values for a band should give linear37 G. L. Caldow and H. W. Thompson, Proc. Roy. SOC., 1960, A, 254, 1.3*H. E. Hallam and T. C. Ray, Nature, 1961, 189, 915.39 ( a ) P. V. Huong, J. Lascombe, and M. L. Josien, J . C h h phys., 1961,58, 694;( b ) J. P. Leicknam, J. Lascornbe, N. Fuson, and M. L. Josien, BUZZ. SOC. chim. France,1959, 1516; ( c ) M. L. Josien, J. Lascombe, and J. P. Leicknam, Compt. rend., 1958,246,401418.(a) M.Ito, K. Inuzuka, and S. Imanishi, J . Amer. Chem. Soc., 1960, 82, 1317;( b ) R. S. Becker, J . Mol. Spectroscopy, 1959, 3, 1WILLIAMS : INFRARED SPECTRA 41plots against those of a standard band, e.g., v(C0) of acetophenone, only ifthe band arises from a dipole similar in structure to the standard. Thishas been used to show that one component of CO doublets in ethylenecarbonate 27a and in cyclopentanone 41 results from Fermi resonance of acombination band with the fundamental, while in a series of esters of thetype R*N(N02)*C0,Et 42 it is the result of rotational isomerism.Carbonyl assignments based on solvent shifts have been made in salicyl-aldehydes, 43a arylazonaphthols, 43b pyridones, 43c* and pyrones, 43a andsimilar methods have been used for the thiocarbonyl group >C=S in a rangeof compounds 44a* and for the C-0 band of secondary alkylcyclohexanols.45Conversely, lack of solvent-sensitivity of the hydroxyl band in certain ortho-substituted phenols has been taken to indicate intermolecular b~nding.~GChangeof solvent usually alters the ratio of isomers and it is possible to makedeductions on the conformation of the isomer from the nature of the solvent.Substances examined include a-chloro-ketone~,~7~ a-chloro- and a-bromo-cyclohe~anones,~~~~ a-bromocyclo-octanone,47d 2,3-dihalogenopropene~,~7~dimetkyl phosphonate,22b and alkyl nitrites.22b Other studies involvingsolvent effects include those on acetylenes,48 nitroanilines,49 a~iline,~o car-bony1 cornpounds,51~ 52 hydrogen sulphide and hydrogen deuterium sul-~ h i d e , ~ 3 the deuterium compound being used to olarify the assignment ofthe v1 and v3 stretching frequencies.Inorganic investigations by thismethod include the nature of the metal-carbonyl link in metal ~arbonyls,~~and the metal-hydrogen stretching frequency in some complex platinumand iridium hydrides .Intensity changes. The effects of solvents on band intensities have notbeen examined so extensively as those on frequencies. Earlier referencesmay be found in a review by Browq56 in Annual Reports by Mills,57 andRotational isomerism has been studied in a number of cases.41 G. Allen, P. S. Ellington, and G. D. Meakins, J., 1960, 1909.4zE. H. White and D. W. Grisley, J. Amer. Chem. SOC., 1961, 83, 1191.43 (a) C.J. W. Brooks and J. F. Morman, J., 1961, 3372; (b) K. J. Morgan, J., 1961,2151; ( c ) L. J. Bellamy and P. E. Rogasch, Spectrochim. Acta, 1960, 16, 30; (d) A. R.Katritzky and R. A. Jones, J., 1960, 2947; ( e ) A. R. Katritzky and R. A. Jones, Spectro-c h i m Acta, 1961, 17, 64.(a) L. J. Bellamy and P. E. Rogasch, J., 1960, 2218; (b) L. H. Little, G. W. Poling,and J. Leja, Canad. J. Chern., 1961, 39, 1783.45 G. Chiurdoglu and W. Masschelein, Bull. SOC. chim. belges, 1961, 70, 307.46 J. H. Richards and S. Walker, Trans. Paraday Soc., 1961, 57, 399, 412.47 (a) L. J. Bellamy and R. L. Williams, J., 1957, 4294; (b) J. Allinger and N. L.Allinger, Tetrahedron, 1958, 2, 64; N. L. Allinger and J. Allinger, J. Amer. Chem. SOC.,1958, 80, 5476; (c) K.Kozima and Y. Yamanouchi, ibid., 1959, 81, 4159; ( d ) J. Allingerand N. L. Allmger, ;bid., p. 6736; ( e ) E. B. Whipple, J . Chem. Phyls., 1961, 35, 1039.* 8 J. C. D. Brand, G. Eglington, and J. F. Morman, J., 1960, 2526; R. West andC. S. Kraihanzel, J. Amer. Chem. SOC., 1961, 83, 765.4gL. K. Dyall, Spectrochim. Acta, 1961, 17, 291.50A. G. Moritz, Spectrochim. Acta, 1961, 17, 365.51 K. Inuzuka, M. Ito, and S. Imanishi, Bull. Chem. SOC. Japan, 1961, 34, 467.5z T. V. Yakovleva, A. G. Maslennikova, and A. A. Petrov, Optics and Spectroscopy,53P. Saumagne, J. Lascornbe, and J. Devaure, Conapt. rend., 1961, 253, 632.55D. M. Adams, Proc. Chem. SOC., 1961, 431.66T. L. Brown, Chem. Rev., 1958, 58, 681.67 I. M. Mills, Ann. Reports, 1958, 55, 65.1961, 10, 64.C.Barraclough, J. Lewis, and R. S. Kyholm, J., 1961, 258242 GENERAL AND PHYSICAL CHEMISTRYin a paper by Buckingham.6a I n the last, Buckingham showed that, onthe basis of the dielectric model, band intensities were proportional to afactor involving the refractive index, n, of the solvent and the rate of changeof dipole with normal co-ordinate of the solute in association with its nearestneighbours. For the transition 0 --+ n, (A,)o+Jv(n+l), where A, is theband intensity of the solution, should be constant. In the later papers,6b~cexpression (1 1) was derived :and a number of isotopic substitution relations were also proposed, e.g.,( A , / Y ~ ) is constant; At2/v2 is constant where A, is the half-band width.Caldow et al.58 have shown that equation (11) holds quite well for aceto-nitrile or trichloroacetonitrile, but that it is not satisfactory for hydrogencyanide or deuterium cyanide.Similarly, isotopic substitution constanciesfor (A,/vn) and A$2/v2 were found to be obeyed. In many cases it waspossible to plot &/A, for one solute in a series of solvents against ASIA, fora second solute in the same solvents. A, could also be plotted against theTaft o* factor for the solvent in some instances.Forel, Leicknam, and Josien 59 have shown that, for the CH band ofchloroform, A,/$ is constant. Studies have also been made on the anti-symmetric and symmetric modes of methylene chloride 6o and on the C-Imode of methyl iodide 61 in different solvents.Studies of a number of substances dissolved inrelatively inert solvents such as carbon tetrachloride or stannic chloridehave shown that a considerable degree of rotational freedom is retained.The evidence for rotation is based on band contour.With diatomic mole-cules, e.g., hydrogen halides, carbon monoxide, nitric oxide, and also withammonia and methane, the band is of the same width as that in the gas butwith a strong central maximum and sub-maxima corresponding to P and Rbranch maxima of the gas spectrum.62 The occurrence of the central Qbranch has been explained theoretically by Galatry. 63 For symmetric-topmolecules, e.g., methyl halides, Jones and Sheppard 64@ found that the A,-class bands were sharp and could be fitted by a Lorentz function whereasthe E-class bands varied considerably in width and could not be fitted bythis function.Moreover, the band widths of the latter gave linear plotsagainst the corresponding &-branch separations in the vapour state, indi-cating clearly that there was rotation about the %fold symmetry axis insolution. This work has been extended to the internal rotation of methylRotation in solution.68 G. L. Caldow, D. Cunliffe-Jones, and H. W. Thompson, Proc. Roy. SOC., 1960,59 M. T. Forel, J. 9. Leicknam, and M. L. Josien, J. Chim. phys., 1960, 57, 1103.6oM. L. Josien, N. Fuson, and A. Lafaix, Compt. rend., 1959, 249, 256.6a J. Lascombe, P. V. Huong, and M. L. Josien, Bull. Sac. chim. France, 1959,1175; M. 0. Bulanin and N. D. Orlova, Optics and Spectroscopy, 1958, 4, 569; C.G.Cannon, Spectrochim. Acta, 1958, 10, 425.63 L. Galatry, Spectrochim. Acta, 1959, 15, 849.64 ( a ) W. J. Jones and N. Sheppard, Trans. Faraday SOC., 1960, 56, 625; (b) Proc.Chm. SOC., 1961, 420.A, 254, 17.L. Angell, Spectrochim. Acta, 1961, 17, 1118WILLIAMS : INFRARED SPECTRA 43groups.6@ Partial resolution of fine structure of thiocyanic acid in solutionhas also been observed.65Matrix Studies.-Many applications of the matrix isolation technique ofPimentel and his co-workers have been made. In this method a substanceis mixed in the vapour phase with a large excess of an inert diluent, e.g.,argon, and then condensed on to a transparent plate held a t low tem-peratures.66Studies have included hydrogen bonding in water 67a and in methanol,67bfree rotation of ammonia 6Ba and water in the matrix, the non-forma-tion of carbonic acid from water and carbon and carbon mon-0xide.70~~ In the last,'Oa the invariance of Av/v for isotopic substitution,predicted by Buckingham,6 was checked.The considerable sharpening of bands which occurs on matrix formationhas been used to resolve nearly coincident vibrations in carbonyl chloride,71Qdisiloxane,71b and monomeric 71' and dimeric 71d formic acid.The most important application of the method, however, is in the studyof species which are unstable under normal conditions. Fateley and hisco-~orkers,~z for example, examined the oxides of nitrogen a t low tempera-ture and found that nitric oxide existed not only as a monomer but also ascis- and trans-dimers; that nitrogen dioxide occurred as monomer and twodimers, ONO-NO, and possibly 02N-N02 ; and dinitrogen trioxide occurredas ON-NO, and as ONONO.Photolysis of hydrazoic acid in a matrix hasbeen shown to give intermediates which are believed to be NH 73a or NH,,7*while photolysis of hydrazoic acid-oxygen mixtures has yielded cis- andtrans-isomers of nitrous acid. 73c* Other photochemical reactions whichhave been studied, include: CH3*NO2 or CK3*ON0 --+ HN0;72 CH,N2 -+CH, ; 7 5 HI- or HBr-CO -+ HCO ; 76 CH,*N3 --+ CH,:NH ; 77 decompositionof CfiT3;78 and proton abstraction by CH, generated from CH,I.79 The66N. Legge and A. D. E. Pullin, Proc. Chem. SOC., 1961, 342.66G. C. Pimentsl, D. A. DOWS, and E. Whittle, J .Chem. Phys., 1954, 22, 1943;E. D. Becker and G. C. Phentel, ibid., 1956, 25, 224; G. C. Pimentel, Spectrochim.Acta, 1958, 12, 94.6 7 (a) M. Van Thiel, E. D. Beckor, and G. C. Pimentel, J . Chem. Phys., 1957, 27,486; ( b ) M. Van Thiel, E. D. Becker, and G. C. Pimentel, ibid., p. 95.g8 (a) D. E. Milligan, R. M. Hexter, and K. Dressler, J. Chem. Phys., 1961, 34,1009; ( b ) E. Catalano and D. E. Milligan, ibid., 1959, 30, 45; (c) J. A. Glasel, ibid.,1960, 33, 252.'j9M. E. Jacox and D. E. Milligan, Spectrochim. Acta, 1961, 1'4, 1108.7 0 (a) G. E. Ewing and G. C. Pimentel, J. Ckm. Phys., 1961, 35, 925; ( b ) A. G.Maki, ibid., p. 931.71 ( a ) E. Catalano and K. S. Pitzer, J . Amer. Chem. SOC., 1958, 80, 1054; ( b ) R. F.Curl and K. S. Pitzer, ibid., p.2371; (c) R. C. Millikan and K. S. Pitzer, ibid., p. 3515;( d ) T. Miyazawa and K. S. Pitzer, J . Chem. Phys., 1959, 30, 1076.W. G. Fateley, H. A. Bent, and B. L. Crawford, J . Chem. Phys., 1959, 31, 204.7s (a) E. D. Becker, G. C. Pimentel, and M. Van Thiel, J. Chem. Phys., 1967, 26,145; (b) M. Van Thiel and G. C. Pimentel, ibid., 1960, 32, 133; (c) J. D. Baldeschwielerand G. C. Pimentel, ibid., p. 1008; ( d ) Q. C. Pimentel, J . Amer. Chem. SOC., 1958,80, 62.74H. W. Brown and G. C. Pimentel, J. Chem. Phys., 1958, 29, 883.76 D. E. Milligan and G. C. Pimentel, J . Chem. Phys., 1958, 29, 1405; T. D. Gold-farb and G. C. Pimentel, ibid., 1960, 33, 105; J . Amer. Chem. SOC., 1960, 82, 1865.76 G. E. Ewing, W. E. Thompson, and G. C. Pimentel, J .Chem. Phys., 1960, 32,927.77D. E. Milligan, J . Chem. Phys., 1961, 3!5, 1491.7BC. D. Bass and G. C. Pimentel, J . Amer. Chern. SOC., 1961, 83, 3764.E. Milligan, J . Chem. Phys., 1961, 35, 37244 GEENEBAL AND PHYSICAL CHEMISTRYmathematical theory of free-radical stabilization in matrices has been con-sidered by Golden.80Studies have also been made on polyatomic anions in matrices preparedby compressing powdered mixtures of a salt containing such an anion withthe corresponding alkali halide. Considerable frequency changes take place,depending on the nature of the alkali halide. Maki and Decius 81 examinedthe CNO- ion in this way, and found that the observed frequency changeswere very much greater than those calculated on the basis of dipole-induceddipole interactions.Repulsion forces made a much more important con-tribution. Ketelaar et aL8, showed that for NO,- in alkali halides, the plotof the antisymmetric Y(NO) frequency against the reciprocal of the latticedimension gave a series of parallel lines, corresponding to the different alkali-metal halides. Subsequently, 83 by applying a KBM-type correction, theywere able to reduce the plots to a single line. More recently, however,Strasheim and Buijs84 showed that the frequencies of all vibrations ofNCO-, NO,-, BH,-, and HF,- gave linear plots against the sum of therepulsive, dipole-dipole, and dipole-quadrupole interactions of the alkali-halide lattice, as determined by the Born-Mayer equation. In addition,they stated that the major part of this lattice contribution arises from short-range repulsive forces.A very comprehensive study of ions in alkali-halide lattices has beenreported by Price, Sherman, and Wilkins~n,~~* 26c covering a range of tem-peratures ; high resolution was used and variation of frequencies, intensities,and contour were investigated.A number of Buckingham’s predictions 6were verified and an extension of the equation Av,, JvTn, = constant toexpression (12) was shown to hold.(12) A(~,Y, + ~ 2 ~ 2 + ~ 3 ~ 3 ) = v,Avl + ~ J Y , + v3Av3.Adsorbed Molecules.-The infrared spectra of molecules physically orchemically adsorbed on surfaces such as porous glass, silica, or alumina oron metallic films supported by these substances have provided valuableinformation both on the mechanism of adsorption and on catalysis.Thesubject up to 1958 has been comprehensively reviewed by Eischens andPliskin.86 More recent work has been discussed by Sheppard *’ and byYates.88The rotational freedom of methyl bromide adsorbed on porous glass hasbeen studied by Sheppard et ~ 1 . 8 ~ who used the band contours in a similarway to that used for rotation in solution.64 The surface hydroxyl groupsof porous silicates 90a and y-alumina 90b have been examined, particularly8 0 s . Golden, J . Chem. Phys., 1958, 29, 61.81A. Maki and J. C. Decius, J . Chem. Phys., 1958, 28, 1003; 1959, 31, 772.82 J. A. A. Kehlaar. C. J. H. Schutte, and B. L. Schram, Spectrochim. Acta, 1959,The latter author, however, reviews only physical adsorption.- -13, 336.88 J.A. A. Ketelaar and J. van der Elsken, J . Chem. Phys., 1959, 30. 336. - -84A. Strasheim and K. Buijs, J . Chem. Phys., 1961, 34,-691.85 W. C. Price, W. F. Sherman, and G. R. Wilkinson, Xpectrochim. Ada, 1960,8sR. P. Eischens and W. A. Pliskin, Adv. Catalysis, 1958, 10, 1.87N. Sheppard, Spectrochim. Acta, 1959, 14, 249.88D. J. C. Yates, Adv. Catalysis, 1960, 12, 265.89 N. Sheppard, M. V. Mathieu, and D. J. C. Yates, 2. Elektrochem., 1960, 64, 734.90“() A. Terenin and V. Filimonov, “Hydrogen Bonding,” ed. D. Hrtdii and16, 663WILLIAMS : I N F R A R E D SPECTRA 45with reference to hydroxyl-frequency shifts which occur when other sub-stances are adsorbed on the surface. Such frequency shifts have beenshown by Bellamy and Williams 23 to give linear plots against (AY/Y) for theNH frequency of pyrrole dissolved in these substances, indicating the closesimilarity in mechanism between surface adsorption and local interactionin solution.The physical adsorption of water on synthetic zeolites 91a andon silica gel 91b has also been examined, and Little 92 has used the break-down of selection rules on physical adsorption to show that the infraredinactive v5 of ethylene occurs at 3070 cm.-l.The chemical adsorption of acetylene and ethylene on silica-supportedpalladium or copper has been investigated by Little, Sheppard, and Yates 93who showed from the CH stretching frequencies that, initially, acetylenegave olefhic CH bands while ethylene gave similar bands but with somesaturated CH absorptions in addition.On hydrogenation, Me[ CH,],*CH,type spectra appeared where ?z > 3. Yates and Lucchesi 94 have shown thatacetylene and methylacetylene are strongly adsorbed on alumina, with themolecules perpendicular to the surface, while dimethylacetylene is heldparallel to the surface. Isomeric butenes, on adsorption on porous glass,give the same species.95Several investigations of carbon monoxide chemisorption have beenreported, including that on 1lletals,~6 and the effect of poisoning on theadsorption on nickeL97 Yates and O'Neill have examined the effect of thesupport; on the spectrum of adsorbed carbon monoxide on both nickelQs"and nickel oxide.9* In the latter study, nickel oxide on alumina was foundto hold carbon monoxide as a carbonate complex, but nickel oxide on titaniabound it as 40,-.Silica-supported oxide did not react, but C0,- groupswere formed when oxygen was subsequently admitted. Similar resultshave been found for both carbon monoxide and carbon dioxide adsorbed onanatase and rutile 99 and on zinc oxide.loOOther spectroscopic studies of adsorption include that of formic acidon various surfaces;lol nitric oxide on transition metals, salts, and oxides;102H. W. Thompson, Pergamon Press, London, 1959, p. 545; ( b ) J. B. Peri and R. B.Hannan, J . Phys. Chew., 1960, 64, 1526.91 (a) H. A. Szymanski, D. N. Stamires, and G. R. Lynch, J . Opt. SOC. Avr$er.,1960, 50, 1323; ( b ) A. V. Kiselev and V. I. Lygin, Kolloid.Zhur., 1960, 22, 403.9nL. H. Little, J . Chem. Phys., 1961, 34, 342.98 L. H. Little, N. Sheppard, and D. J. C. Yates, Proc. Roy. SOC., 1960, A , 259,04D. J. C. Yates and P. J. Lucchesi, J . Chem. Phys., 1961, 35, 243.06L. H. Little, H. E. Klauser, and C. H. Amberg, Canad. J . Chem., 1961, 39, 42.e6R. A. Gardner and R. H. Petrucci, J . Amer. Chem. SOC., 1960, 82, 5051.B7 J. T. Yatss and C. W. Garland, J . Phys. Chem., 1961, 65, 617; C. W. Garland,ibid., 1959, 63, 1423.C. E. O'Neill and D. J. C. Yates, (a) J . Phys. Cltem., 1961,65, 901 ; (b) Spectrochiin.Acta, 1961, 17, 953.loo S. Matsushita and T. Nakata, J . Chem. Phys., 1960, 32, 983; J, H. Taylor andC. H. Amberg, Canad. J . Chem., 1961, 39, 535.lol J. Fahrenfort and H. F. Hazebroek, 2.phys. Chem. (Frankjwt), 1969, 20, 105;I(. Hirots, K. Kuwata, and S. Asai, J . Chem. SOC. Japan, 1959, 80, 701; K. Shindo,Y. Nakai, K. Fueki, K. Hirota, and T. Otaki, {bid., p. 1215; J. K. A. Clarke and A. D. E.Pullin, Trans. Paraday SOC., 1960, 56, 534.242.O9D. J. C . Yaks, J . Phys. Chem., 1961, 85, 746.'1oZA. N. Terenin and L. M. Row, Spectrochim. Acta, 1959, 15, 94646 GENERAL AND PHYSICAL CHEMISTRYnitrobenzene on silica;lo3 hydrogen and deuterium on platinum ;lo4 am-monia,106a and the reaction of ammonia lo5* with carbon disulphide onalumina; alcohols on chromic oxide;lo6 methanol and phenol on micro-porous glass;l07 and the spectra of monola,yers on metal mirrors.108R. L. W.5. CHEMICAL REACTIONS STUDIED BY USING SHOCK TUBESA REVIEW on the use of shock tubes has been given recently by Pritchard.1The present Report is concerned with papers published in 1960 and 1961,but it does not consider the use of shock tubes to study gas flows or high-temperature plasmas.Developments in Shock Techniques.-Some reactions can be studied byusing a shock tube to heat reactants for a certain time which is followed bycooling and analysis of the products.A simple shock tube has been usedby Greene et aZ.,2 while the chemical shock tube has been developed byGlick et aZ.3 Alternatively, the reaction can be followed directly in theshock-heated gas. Using the chemical shock tube Boudart and his co-workers have made allowances for the fact that all parts of the gas samplewhich is analysed are not heated for the same time.4 However, most ofthe recent developments have been connected with the second method.Thus the improvement in fast infrared detectors has enabled vibrationalrelaxation times to be compared by using infrared as well as ultravioletmethods.5 The use of absorption spectroscopy has been extended into thevacuum ultraviolet region, which gives an especially sensitive method ofmeasuring the concentration of oxygen.6 For studying reactions at lowdensities DUE has used an electron-beam densitometer.Gaydon andHurle 8 and Bauer et aL8 have extended the method of following a reactionby measuring the change in temperature by a reversal technique by usingchromium lines.8 It cannot be used successfully with monatomic gases, oroxygen, but otherwise is generally applicable.103M.Okuda, J . Chem. SOC. Japan, 1961, 82, 1121.l’J4W. A. Pliskin and R. P. Eischens, 2. phys. Chem. (Fmnkfurt), 1960, 24, 11.lo5 ( a ) D. E. Nicholson, Nature, 1960, 186, 630; ( b ) E. H. Parry and H. Rubalcava,106 L. M. Roev and A. N. Terenin, Doklady Akad. Nauk S.S.S.R., 1959,124, 373.10’ A. M. Bogomolny and Yu. A. Lyubimov, Optics and Spectroscopy, 1960, 8, 131.1 0 8 s . A. Francis and A. H. Ellison, J . Opt. SOC. Amer., 1959, 49, 131.1 H. 0. Pritchard, Quart. Rev., 1960, 14, 46.2E. F. Greene, R. L. Taylor, and W. L. Patterson, jun., J . Phys. Chem., 1958,8 H. S. Glick, W. Squire, and A. Hertzberg, “ Fifth Symposium (International)H. S. Glick, J. J. Klein, and W.4V. Kevorkian, C. E. Heath, and M.Boudart, J . Phys. Chem., 1960, 64, 964.SF. Robben, P. R. Manson, and J. J. Allport, J . Chem. Phys., 1960, 33, 630.6M. Camac, J . Chem. Phys., 1961, 34, 448; S. A. Losev, DokWy Akad. NaukS.S.S.R., 1958, 120, 1291. R. W. Patch, United Aircraft Co., Res. Lab. Res. Rept.No. R 1558-1 (Sept. 1961); 0. L. Anderson, United Aircraft Co. Res. Lab. Res. Rept.7R. E. Duff, Phys. Fluids, 1959, 2, 207.8A. G. Gaydon and I. R. Hurle, Proc. Roy. SOC., 1961, A , 262, 38; S. H. Bauer,J . Phys. Chem., 1960, 64, 955.62, 238.on Combustion,” Reinhold Publ. Corp., New York.Squire, J . Chem. Phys., 1957, 27, 850.NO. R 1828-1 (Aug. 1961).J. H. Kiefer, and B. E. Loader, Canad. J . Phys., 1961, 39, 1113SIMPSON: REACTIONS STUDIED USING SHOCK TUBES 47A general technique for following reactions has been developed byBradley and Kistiakowsky ; this uses a time-of-flight mass spectrometer tofollow the concentrations of species produced by a shock wave.9 Thespectrometer has a time resolution of 50 psec., which is not as good as thatavailable with an optical absorption method when that can be used, but itis not a specific method and it can follow several species simultaneously.These are considerable advantages as the concentration of species such asoxygen atoms cannot be followed readily by other means and it is diEcultto follow several species simultaneously in the short time available in shock-tube experiments, However, there are difEculties such as the possibility ofsome of the sample gas coming from the boundary layer, which is a regionof cool gas spreading from the tube walls into the heated gas after the shockwave has passed.Furthermore, in the present arrangement there is someuncertainty as to the temperature of the reactants owing to the considerableattenuation of the incident shock wave, and in general temperatures behindreflected shock waves are less well known than those behind incident shockwaves though this effect is minimized by using a mixture diluted with aninert gas.In studying kinetics by a shock-tube method, small errors in the shockspeed can lead to errors in the calculated temperatures which have an impor-tant effect in determining rate constants. This means that shock speedshave to be measured very accurately and estimates made as to when shockattenuation and the growth of boundary layers will contribute errors tothe measurements.1° There is more uncertainty about the precise value ofthe temperature behind a reflected shock than behind the incident shock,but it does not differ so much from its value calculated from the incidentshock conditions as was fist considered by Strehlow and Cohen.ll How-ever, Schlieren photographs do show that there are two cases to be considered.In the first, for polyatomic gases, strong shock interactions occur on reflec-tion of the shock wave.Diatomic gases may have an acceleration andbifuration of the reflected shock wave. They are nearer ideal conditions forshocks of moderate strength, but for stronger shocks in nitrogen, theSchlieren photographs by Holder et aZ.12 show considerable interactions be-tween the reflected shock and the boundary layer. The second case is thatof the inert gases where no bifuration of the reflected shock wave occurs, butits velocity is not exactly that calculated.Evidence that reflected shockwave properties in inert gases are more nearly those expected comes fromthe pressure measurements by Skinner and by Brabbs et aL13 and the densitymeasurements by Gardiner and Kistiakowsky and by Strehlow and Case l4which are commented on by Rudinger.15 These observations do show smallJ. N. Bradley and G. B. Kistiakowsky, J . Chem. Phys., 1961, 35, 256.loM. Camac and A. Vaughan, J . Chem. Phys., 1961, 34, 460.IIR. A. Strehlow and A. Cohen, J .Chem. Phys., 1959, 30, 257.12D. W. Holder, C. M. Stuart, and R. J. North, Aeronautical Research Councill3 C. B. Skinner, J . Chem. Phys., 1959, 31, 268; T. A. Brabbs, S. A. Zlatovich, andl4 W. C. Gardiner and G. B. Kistiakowsky, J . Chem. Phys., 1961, 34, 1080; R. A.15G. Rudinger, J . Chem. Phys., 1961, 35, 1507.Report No. 22,891 (1961).F. E. Belles, ibid., 1960, 33, 307.Strehlow and C. T. Case, ibid., 1961, 35, 150648 GENERAL AND PHYSICAL CHEMISTRYchanges in pressure and density with time behind reflected shocks, but theeffect on the temperature is not quite clear. This question is important asreactions are often studied behind reflected shock waves ; l6 reflected shocksare always used in the chemical shock tube and in the method using a massspectrometer.A method of studying atom recombination rates has been developed byBurns and Hornig l7 who have combined a flash photolysis and shock-tubeexperiment.This has the clear theoretical attraction of permitting a studyof a high-temperature recombination rate directly instead of inferring itfrom the dissociation rate and equilibrium constant, but there are manyconsiderable experimental diiliculties.There have been two important developments in producing shocks a thigh temperatures both at high densities and low densities. I n the firstcase, Stalker l8 has developed a free-piston'compression tube to give a drivergas with a high pressure and temperature. In the second, Lin and E'yfe l9have made a shock tube to work at low densities using a 24-inch diametertube.This size is essential because the tube must be large compared withthe boundary layer, as is shown by Duff's results 7 obtained in a l-in. dia-meter tube. The apparatus can be used to study chemical reactions at hightemperatures and low densities. The low densities give long distances overwhich reactions take place, but there are difficulties because of the curvatureand tilt of the shock front.There have been developments in the analysis of results as well as inmethods of obtaining them. Johannesen and Blythe 2o have given an exactmethod for analysing vibrational relaxation regions as an alternative tomethods which use some average temperature during the relaxation pro-cess.21 Computers have been used to calculate the resultant changes to beexpected in some property-for example, density-when certain rate con-stants are assumed for the reactions occurring, and comparison of measuredand calculated results enables values of several rate constants to be found(see, for example, ref.22).There is,however, a considerable number of papers concerning the ionization ofxenon (e.g., ref. 23), and the evidence obtained by Gloersen 2* that luminousshocks can produce photoionization on the walls of the shock tube ahead ofthe shock wave is of great importance in high-temperature studies wherethere is light emission. Examples of the use of a shock tube to study equi-Systems Studied.-This report does not consider ionized gas.l6 F. Freedman and J. W. Daiber, J. Chem.Phys., 1961, 34, 1271; C. E. Treanorl7 G. Burns and D. F. Hornig, Canad. J . Chem., 1960, 38, 1702.18 R. J. Stalker, Aeronautical Research Council Report No. 23,280 (1961).19 S. C. Lin and W. I. Fyfe, Phys. 'Fluids, 1961, 4, 238.2O N. H. Johannesen, Aeronautical Research Council Report No. 22,171 (1960);21D. L. Matthewo, J. Chem. Phys., 1961, 34, 639.ssR. E. Duff and N. Davidson, J. Chem. Phys., 1959, 31, 1018.23 A. P. Dronov, A. G. Sveridov and N. N. Sobolev, Optika i Spectroskop.1:ya7 1961,103, 12; H. J. Johnston and W. Kornegay, Trans. Faraday SOC., 1961, 57, 1563; P.Gloersen, Phys. Fluids, 1961, 4, 790.a* P. Gloersen, General Electric Co., Space Sciences Lab. Tech. Information ReportNo. R60 Sd 364 (April 1960).and W. H. Wurster, ibid., 1960, 32, 758.P. A.Blythe, Aeronautical Research Council Report No. 22,170 (1960)SIMPSON: REACTIONS STUDIED USING SHOCK TUBES 49librium conditions are the papers by Knight and Rink,25 who measured thedissociation energy of cyanogen and related quantities, and Treanor andWurster, l6 who measured transition probabilities in the Schumann Rungesystem of oxygen. This was done by taking absorption spectra of oxygenwhen the upper vibrational and rotational levels were populated in a mannerwhich could be calculated from the equilibrium temperature behind thereflected shock wave. However, most papers are concerned with measuringrate constants rather than equilibrium properties. These can be dividedvery roughly between cases involving an overall process, or complex reaction,and those which consider energy transfer, the dissociation of a diatomicmolecule, or a detailed study of an intermediate in a complex reaction.Overall Reactions.--Shock tubes have been used to extend the tempera-ture range in which decomposition and oxidation reactions can be studied.The pyrolysis of acetylene has been studied by Aten and Greene, using,asimple shock tube, and by Skinner, using a chemical shock tube.26 Thecracking of paraffins has also been studied by Poltorak.27 Shock ignitionof hydrocarbons has been investigated by Yamozaki and Kato.2s The de-composition of nitric acid vapours has been studied 29 and Wellman 3O hasused a chemical shock tube to study the oxidation of hydrocarbons.Chemi-cal shock tubes have been used by Boudart and his co-workers * and Skinneret aL31 to study the decomposition of methane and the pyrolysis of ethaneand ethylene.Boudart obtained a good correlation between the high- andlow-temperature studies, but found that, in contrast to the low-temperaturereactions, the homogeneous high-temperature reaction is not inhibited byhydrogen. Skinner compared his results for ethane with those computedby using a mechanism based on nine free-radical reactions.Bradley and Kistiakowsky have used a mass-spectrometer to studyreactions in a reflected shock and have investigated the polymerization andoxidation of acetylene, and the decomposition of nitrous oxide and nitro-methane.g, 32 This method has the advantage of avoiding the possibilityof reactions occurring during cooling,26 or even in the very rapid coolingeffected in the chemica.1 shock tube.Rather different from these studies is that of the hydrogen bromine reac-tion by Britton and Cole? They followed the reaction by the decrease inbromine concentration with time and were able to separate the effects dueto bromine dissociation and consumption by the formation of hydrogenbromide.Detailed Studies.-A very interesting example of a detailed study ofa reaction is the measurement of the hydroxyl concentration in a mixture of25H. T.Knight and J. P. Rink, J . Chem. Phys., 1961, 35, 199.28 C. F. Aten and E. F. Greene, Combustion and Flume, 1961, 5, 55; G. B. Skinner27 B. A. Poltorak, Zhur. $2. Khim., 1961, 35, 284.2a K.Yamozaki and Y . Kato, J . Chem. Soc. Japan, I n d . Chem. Sect., 1960, 63, 2141.2sH. Harrison, Diss. Abs., 1960, 21, 773.30 W. E. Wellman, Diss. Abs., 1960, 21, 767.s1 G. B. Skinner and W. E. Ball, J . Phys. Chem., 1960, 64, 1025; G. B. Skinnerand E. M. Sokoloski, ibid., p. 1028.3a J. N. Bradley and G. B. Kistiakowsky, J. Chern. Phys., 1961, 35, 264; J. N.Bradley, Trans. Fn~cEay Soc., 1961, 57, 1750; D. Britton and R. M. Cole, J . Phys.Chem., lQcl, 60, 1302.and E. M. Sokoloski, J . Phys. Chem., 1960, 64, 195250 GENERAL AND PHYSICAL CHEMISTRYhydrogen, oxygen, and argon heated in a shock tube.33 The hydroxylconcentration was followed by using an OH-line light source. It was foundto build to a maximum above its final high-temperature equilibrium con-centration in the way to be expected on the basis of Sugden's hypothesis offast bimolecular reactions in a flame reaching a partial equilibrium followedby slower termolecular recombination reactions.34 The results could becorrelated quite well with calculated concentrations based on assumed rateconstants for the five basic bimolecular reactions.An analogous comparison is given by Lin and Pyfe for oxygen concentra-tions in the shock heating of air in strong shocks at low densities.lS Thisis of interest in showing that at these very high temperatures, calculationswhich assume that vibrational equilibrium is complete before chemicalreactions begin do not agree with the results as well as those treating vibra-tional excitation simultaneously with chemical reaction.Vibrational relaxation in carbon dioxide has been studied by Witteman 35who quotes relaxation times and gives calculations to show whether thereshould be separate vibrational relaxation times for the doubly degeneratebending mode and the symmetric stretching mode.A similar question ismultiple vibrational relaxation times for dibromomethane. This has beentreated theoretically and experimentally for low temperatures by Dickensand Schofield36 and Meyer.36 The problem here is whether translationalenergy is transferred to the different vibrational modes by independent pathseach with its own relaxation time, or whether the slow step is the transferof energy to one vibrational mode which is rapidly equilibriated with theothers giving only one overall relaxation time.The vibrational relaxation of carbon monoxide has been studied bylSilatthews21 using a shock tube and interferometer and by Gaydon andHurle 37 from temperature measurements using line reversal.It has beenmeasured also by using a spe~trophone.~~ These shock studies agree fairlywell, but give shorter times than those measured by Windsor, Davidson, andTayl~r.~S The correction suggested by Decius 40 to the results of Windsoret al. makes this difference greater rather than smaller; it may be due toimpurities reducing the relaxation time in the first case, and the results maybe criticized as the relaxation was measured over a condition of changingtemperature.Parker 41 has measured vibrational relaxation times for oxygen acoustic-ally and concludes that the high-temperature shock-tube results obtained byBlackman and by Byron 42 give relaxation times that are too short; they33G.L. Schott, J . Chem. Phys., 1960, 32, 710.34 E. M. Bulewicz, C. G. James, and T. M. Sugden, Proc. Roy. SOC., 1956, A , 235, 89.35 W. J. Witteman, J . Chem. Phys., 1961, 35, 1.36P. G. Dickens and D. Schofield, J . Chem. Phys., 1961, 35, 374; N. J. Meyer,37 A. G. Gaydon and I. B. Hurle, Aeronautical Research Council Report No. 22,55438W. E. Woodmansee, Dim. Abs., 1961, 22, 105.30M. Windsor, N. Davidson, and R. Taylor, J . Chenz. Phys., 1957, 37, 315.40 J. C. Decius, J . Chem. Phys., 1960, 32, 1262.41 J. G. Parker, J . Chem. Phvs.. 1961, 34. 1763.ibid., 1960, 33, 487.(1961).42N.Blackman, J . Fluid Meih.,. 1956,. 1, 61; S. Byron, J . Chem. Phys., 1959, 30,1380SIMPSON : REACTIONS STUDIED USING SHOCK TUBES 51had been consistent with earlier measurements of velocity of sound at roomtemperature.Vibrational relaxation times for nitric oxide computed by using an interac-tion potential from viscosity measurements and the theory of Schwartz ,Slawsky, and Herzfeld are longer than those observed. Experimental valueshave been obtained by using a shock tube and ultraviolet absorption and in-frared emission. The ultraviolet absorption method uses the intensity changeof the yol band in absorption to measure increase in population of the firstvibrational level of the ground state. The infrared emission from the 1-0and 2-0 transitions gives a direct measure of the changes in population ofthese vibrational levels.These results agree with each other, and with ameasurement of the velocity of sound 5 and the results of a flash-photolysisinvestigation. 43 It is interesting that in a shock-tube investigation ofrelaxation in the A2E + excited state of nitric oxide the ratio of the relaxationtimes for the 2-1 and 1-0 transitions have been measured and found toagree with a theoretical calculation. 44The very great interest in the dissociation and recombination reactionsof simple molecules is clear from papers concerned with calculations pub-lished during 1960 and 1961,45 while practical studies have been made withbromine, hydrogen, nitric oxide, and oxygen.The decomposition rate of nitric oxide was followed by ultraviolet absorp-tion using a reflected shock wave.l6 Burns and Hornig l7 have measureda high-temperature recombination rate for bromine.They give one pointfor the rate of bromine atom recombination at 950" c . It would seem tobe worthwhile to try to extend the direct flash photolysis to higher tempera-tures so as to compare these two methods. Britton 46 has studied the rateof dissociation in the presence of argon, helium, nitrogen, carbon monoxide,oxygen, and carbon dioxide. With the diatomic diluents the situation wascomplicated by the simultaneous occurrence of vibrational relaxation anddissociation.Hydrogen dissociation has been studied by Gardiner and Kistiakowsky 47from density changes in hydrogen-xenon mixtures followed by X-rays; byGaydon and Hurle 37 using temperature measurements; and by Patch 6by following the concentration of hydrogen molecules from their vacuum-ultraviolet absorption.Gaydon and Hurle used hydrogen-argon mixturescontaining 50% or 30% of hydrogen and expressed their results as relaxa-tion times, it being assumed that dissociation was not caused by the argon.The argon results agreed with those from earlier studies.43 N. Basco, A. B. Callear, and R. G. W. Norrish, Proc. Roy.Soc., 1961, A, 260,459.44 W. Roth, J . Chem. Phys., 1961, 34, 999, 2204.45 ( a ) E. Bauer and M. Salkoff, J . Chem. Phys., 1960, 33, 1202; ( b ) D. L. Bunker,ibid., 1960, 32, 1001; (c) J. C. Keck, ibid., p. 1035; ( d ) E.E. Nikitin, Doklady Akad.Nauk S.S.S.R., 1960, 132, 395; (e) E. E. Nikitin and N . D. Sokolov, ibid., 1959, 124,366; (f) E. E. Nikitin, 2hur.Ji.z. Khim., 1959,33,1893; (9) E. E. Nikitin and N. Sokolov,J . Chem. Phys., 1959, 31, 1371; (h) A. I. Osipov, Zhur. $2. Khim., 1961, 35, 1524;(i) A. I. Osipov, Doklady Akad. Nauk S.S.S.R., 1961, 13'7, 833; ( j ) G. Porter and J. A.Smith, Proc. Roy. SOC., 1961, A , 261, 28; ( k ) H. 0, Pritchard, J . Phys. Chem., 1961, 65,504; ( I ) H. 0. Pritchard, Cunad. J . Chem., 1960, 38, 319; ( m ) K. E. Shuler and R.Zwanzig, J . Chem. Phys., 1960,33, 1778; (n) K. E . Shuler, ibid., 1959,31, 1375; (0) E. V.Stupochenko and A. I. Osipov, Zhur. Jiz. Khim., 1959, 33, 1526.46D. Bitton, J . Phys. Chem., 1960, 64, 742.4 7 W.C. Gardiner and G. B. Kistiakowsky, J . Chem. Phys., 1961, 35, 176552 GENERAL AND PHYSICAL CHEMISTRYThe temperature dependence of the rate constants for bimolecular dissocia-tions given by Gardiner and Kistiakowsky for the dissociation in xenonsuggest that a t high temperatures argon may not be much less efficient thanhydrogen. They used either 17% or 48% of hydrogen and found that thelarge change in temperature during a dissociation gave some uncertainty inthe temperature dependence of the rate constants. Patch was able to useas little as 1% of hydrogen in argon. He compares recombination rateconstants, calculated from dissociation rates and equilibrium constants, withthe results of the shock-tube studies made by Sutton and Rink who usedabout 3% and 10% of hydrogen in argon. Patch obtains Kr,Ar 2.1 x 1014,with K,,,, 2-1 x 1015 and Kr,= 1.4 x l0l6 cm.6 mole-2 sec.-l at 3500"~.Sutton and Rink's results do not differ by a factor of more than three or fivefrom these values, while Gardiner and Kistiakowsky obtained 1.3 x lo1*for Rr,=*.Patch compared his high-temperature value of Kr,, with recom-bination constants from dischwge- tube experiments at room temperature ;the results are quite similar.The rate of dissociation of oxygen has been determined by using X-ray ab-sorption by Rink, Knight, and Duff 48 and by Cheswick and Kistiak~wsky;~Sby vacuum-ultraviolet absorption by Camac and Vaughan,lo Anderson,Gand Losev;SO and by use of an interferometer by Byron 42 and Matthews.51The results have been compared by Duff as recombination rate constantsa t 3500" K for the third bodies molecular oxygen, xenon, argon, and atomicoxygen.For argon as a third body theresults by Anderson, Camac, and Byron are within a factor of two. Thecomparison should be thought of as one between dissociation rates, as ineach case the recombination constant is derived from the equilibrium con-stant a t 3500"~ and dissociation rates, but there is some question its towhether this is the true recombination rate.Camac and Vaughan's results are of great interest as in experimentsabove 7500" K these authors found coupling between the rates of vibrationalequilibrium and dissociation. This is consistent with Lin and Fyfe's resultson the reactions of high-temperature air.lg Camac and Vaughan concludedthat at these high temperatures lack of vibrational equilibrium reduces therate of dissociations by more than a factor of two.However, their resultsfor dissociation of oxygen in argon can be well expressed by using the dissocia-tion energy D in the classical kinetic theory curveThe results agree quite well.KAr = 6 x 1013(D/RT)exp(-D/RT)for the temperature range 3500-7000" K. This uses measurements whichinvolve the assumption of vibrational equilibration being fast compared todissociation. They have used these data and calculated equilibrium con-stants to calculate recombination rate constants which they have comparedwith those obtained by use of Keck's theory.An outstanding problem in considering recombination rates is the de-crease in rate which occurs as the temperature increases.This is found from48 J. P. Rink, H. T. Knight, and R. E. Duff, J . Chem. Php., 1961, 34, 1942.4* J. P. Cheswick and G. B. Kistiakowsky, J . Chem. Phys., 1958, 28, 956.6 0 S . A. Losev, Doklady Akad. Nauk S.S.S.R., 1958, 120, 1291.6lD. L. Matthews, Ph.ys. Pluids, 1959, 2, 170SIMPSON: REACTIONS STUDIED USING SHOCK TUBES 53flash-photolysis experiments in the temperature range 300-500 O K, fromshock-tube results at high temperatures and by comparisons of these resultswith rates obtained at room temperature by discharge experiments.Bunker 4Bb considers the mechanics of the reaction X + X + M + X, + Mto proceed either by the path X + X --+ X2x ; X,x + M + X, + M,or X + M -+ XM; XM + X -+ X, + M with van der Waals’ interactionbetween X and M.Such mechanisms give negative-temperature coefficients.Porter and Smith 45i also consider recombination between iodine atomswith different third bodies and conclude that charge-transfer complexesare important.Keck has derived results for the maximum rates of atom recombination.He represents the reaction by the motion of a point in phase space acrossa surface dividing the initial and the final state and calculates the rate ofcrossing this in one directi0n.45~ To obtain results he has to specify thepotentials between the particles and, unfortunately, there are few data onthree-body potentials. He compares his calculated maximum recombinationrate constants with experimental results for X + X + Ar --+ X, + Arwhere X, is oxygen, iodine, bromine, and nitrogen.The results of low-temperature flash photolysis fall quite close to the theoretical curves-thecase of bromine being improved by the separation of the third-body effectsof bromine and argon in Givens and Willard’s recent results.52 The shock-tube results, including recent ones on oxygen, fall below the calculatedcurve. The low-temperature results for oxygen considered by Keck andby Bauer 45 are recombination rate constants from discharge experiments.The system has been investigated by Harteck and his co-workers and byKretschner by following the concentrations of oxygen atoms by the 0 + NOreaction;53 references to other papers are given by Linnett and his co-workers.54 The low-temperature results are complicated by the competingreactions 0 + 0 + M-+ 0, + M and 0 + 0, + M-+ O3 + &I, but theozone reaction is not important at the high temperatures of the shock-tubee~periments.~8 This difficulty is avoided by Morgan, Elias, and S ~ h i i T , ~ ~who produced oxygen atoms by the reaction N + NO + Nz + 0 andmeasured the recombination rate with nitrogen as the third body. SinceBunker and Davidson’s calculations 56 agree with experimental results foriodine-atom recombination in assigning nearly equal third-body effects tonitrogen and argon it is reasonable to consider that the rate constant for0 + 0 + Ar+ 0, + Ar will be close to the value found for nitrogen andto use this result obtained by ScM.Keck considers two reasons for thedifference between the calculated and experimental results at high tempera-tures. First, instead of being a simple one-way process, recrossing of thebarrier between initial and final states may become more important a t hightemperatures and, secondly, the differences might be due to the choice ofthe three-body potential. It is possible also that the experimental results63 W. G. Givens and J. E. Willard, J . Amer. Chenz. SOC., 1959, 81, 4773.6s R. R. Reeves, G. Mannella, and P. Harteck, J . Chem. Phys., 1960, 32, 632.64P. G. Dickens, R. D. Gould, J. W. Linnett, and A. Richmond, Nature, 1960,66 J. E. Morgan, L. Elias, and H. I. Schiff, J . Chem. Phys., 1960, 33, 930.6 6 D. L. Bunker and N.Davidson, J . Amer. Chem. SOC., 1958, 80, 6090.187, 68654 GENERAL AND PHYSICAL CHEMISTRYmay be low, for if the rate of dissociation a t high temperatures is limited bythe rate of population of the vibrational levels then Kd will be less than theequilibrium value corresponding to the product of the equilibrium constantand the recombination rate constant, and so K,, calculated as Kd/K, Willbe small.Oxygen-atom recombination rates have been treated by Bauer 45a alsousing quantum mechanics. He considered the more complicated reactionusing an interaction potential derived from experimental results, of vibra-tional relaxation in oxygen. He found that recombination takes place intohighly vibrationally excited states and obtained rate constants which agreewithin a factor of three with the experimental results obtained by Harteckand his co-workers 53 and Matthews’s 51 shock-tube results.This is encourag-ing but it must be remembered that a greatly simplified theoretical modelwas used and there are these uncertainties in the experimental results.The question whether the relation Kd/Kr = K can be used at conditionswhich are not those of equilibrium depends on whether .& equals the equi-librium value, Kd,eq., for examples by Nikitin and Sokolov 45e show thatK, would be close to Kr,eq.. The system usually considered is that of A,dissociating in a large heat bath of molecules M at a constant temperature T.Dissociation probably occurs from vibrational levels near the dissociationlimit and the question is to what extent these levels are depleted comparedwith the Boltzmann value for the temperature T.The question is consideredby Shuler, Nikitin, Osipov, and others,45 and results of calculated Kd arecompared with shock-tube results. Nikitin and Sokolov and Osipov andhis co-workers both consider that the Boltzmann distribution is disturbed athigh temperatures, and derive equations for &. Shuler also calculated thedisturbance of the Boltzmann distribution, using the assumption commonlymade that, except near the dissociation limit, the vibrational energy onlychanges by one quantum steps, Av I = 1. However, he stated that thesecalculations should not be considered too seriously in view of his work withMontroll that showed that a one-step-at-a-time “ ladder climbing ” mech-anism gave dissociation rates that are much too slow compared with thoseobserved.57 The later paper of Shuler and Zwanzig shows that a harmonicoscillator undergoing impulsive hard-sphere collisions has a considerableprobability of acquiring vibrational energy in steps with I AY I > 1.Thisshould be contrasted with the recent paper by Pritchard who based his cal-culations on a formula for transition probabilities given by Jackson and Mottand concluded that excitation occurs by single steps and that the rate ofdissociation is affected by a lack of vibrational equilibration which becomesmore serious as the temperature is raised.45kCamac and Lin’s experimental results do show that at sufficiently hightemperatures a slow rate of vibrational equilibration does affect the rate ofdissociation.As the temperature is lowered, measured vibrational relaxa-tion in oxygen becomes faster relative to dissociation. However, these67E. W. Montroll and K. E. Shuler, “Advances in Chemical Physics,” ed. L.Prigogine, Interscience Publ. Inc., New York, 1958, Vol. 1, p. 361BIS'HOP : NUCLEAR MAGNETIC RESONANCE 55measurements are concerned with the population of low-lying vibrationallevels and it is not clear how far the populations of the upper levels are fromtheir extremely small equilibrium values. Thus, it can be concluded thatdespite considerable progress neither shock-tube results nor theoreticalcalculations yet give a complete picture of the mechanism of vibrationalexcitation and dissociation.It may be true that Ed = K,K only appliesa t equilibrium, but it is not certain under what conditions it is a goodapproximation. However, recent shock-tube studies have given informationon a wide variety of subjects.C. J. S. M. S.6. NUCLEAR MAGNETIC RESONANCETHE present Report is concerned mainly with recent developments in thetheory and applications of the spin coupling and chemical shift parameters,and the ways in which these may be deduced from the high-resolution nuclearmagnetic resonance spectra of liquids and gases. Some chemical aspects ofrelaxation times and of the broad-line spectra of solids are also discussedbriefly. The literature surveyed is mainly that of the parst two years, sincethe previous Report on this subject by Pop1e.lInterpretation of Spectra.-The greater part of molecular informationfrom the nuclear magnetic resonance spectra of liquids and gases is obtainedfrom the chemical shift and spin coupling pararneters.l, For a weaklycoupled system, where the coupling constant Jij between each pair of mag-netically non-equivalent nuclei i, j is small compared with their difference inchemical shift, the parameters (but magnitude only, not sign, of Jij) areevident by inspection of the '' first-order " spectrum of symmetrical multi-plets.In more strongly coupled cases, spin-coupling causes significant mix-ing of the first-order eigenstates with distortion of both line positions andintensities from the symmetrical multiplet patterns, and appearance ofnew (" combination ',) lines which were forbidden in first order.Quantum-mechanical analysis is then necessary, and can often be simplified by useof symmetry properties of the spatial distribution of the nuclei and theircoupling interactions, and also by neglect of small mixing terms. If signi-ficant mixing can thereby be restricted to two of the first-order eigenstatesa t a time, then all transition energies and intensities can be expressed insimple algebraic form and t'he parameters can be found directly. Thesemethods have been reviewed by Roberts and C ~ r i o , ~ and the latter illus-trates the appearance of many spectral systems with variation of couplingstrength. In the nomenclature of Pople et aL19 2 for nuclei of spin 9, suchalgebraic treatment has been extended to the systems ABRX5 and AB,X,6J.A. Pople, Ann. Reports, 1959, 56, 78, and references therein.J. A. Pople, W. G. Schneider, and H. J. Bernstein, " High Resolution N.M.R.,"J. D. R o p t s , " An Introduction to Spin-Spin Splitting in High ResolutionB. D. W. Rao and P. Venkateswarlu, Proc. Indian Acad. Sci., 1960, 52A, 109.(a) R. J. Abraham, E. 0. Bishop, and R. E. Richards, Mol. Phys., 1960, 3, 485;McGraw-Hill, 1959.N.M.R. Spectra, W. A. Benjamin Inc., N.Y., 1961.4P. L. Corio, Chem. Rev., 1960, 60, 363.(b) P. Diehl and I. Granacher, J. Chem. Phys., 1961, 34, 184656 GENERAL AND PHYSICAL CHEMISTRYwhilst Diehl, Pople, and Schaefer 7 have shown that all systems of typeA,B, . . . R, . .. Xu . . . (my n, p, q are any integers) may be included inthis scheme provided that each nucleus of a particular magnetic environmentis equivalently coupled to those of any other environment. Under theseconditions, the spectrum is independent of mutual coupling within each ofthe groups A,, B,, R, . . ., and may be considered as a superposition ofspectra for the various combinations of these groups, considered as “ com-posite particles,” with fixed total spin.8 This is exemplified by analysisof the systems ABX,, ABR3X, and AB,X,. Hoffman and Gronowitz havealso analysed the system ABT, where T = deuteron of spin unity, as aspecial case of ABX, in which the composite particle X, cannot have itsconstituent spins of Q opp~sed.~ The general three-spin system ABC isoften encountered, and although mixing of states in sets of three is involved,exact analytical treatment is discussed in three papers.10, 11, 1 2Slight departures from a more readily soluble case can be convenientlytreated by including the effect of small additional mixing terms as a pertur-bation upon the simpler system.Reilly and Swalen have considered thesystems ABK l3 (intermediate between ABX and ABC) and ABKY 1 4 bysecond-order perturbation, and applied them to an analysis of the protonspectra of some epoxides. Cavanaugh and Dailey15 have analysed theproton spectra of the propyl group (A,B,C,) in a number of compounds, byperturbation to third order. For strongly coupled spectra the analyticalsolution is usually impracticable, however, and is replaced by iterative orfrial-and-error nuinerical solution by computer to match a set of eigenvaluesdeduced from the spectral line positions. Elements for setting up thenuclear-spin Hamiltonian matrix are tabulated for the systems ABCX,16A2B6,17 and AB,C2X l8 and applied to spectral analyses of vinyl fluoride,propane, and fluorobenzene, respectively ; pent-2-yne has been treatedexactly as an A3B,C3 system.lg Analyses of ethyl compounds are discussedin a later section.A novel extension of the deuterium-substitution method of simplifyingcomplex proton spectra has been given recently by Garnett et aL20 Cata-lytic “ massive deuteration ” in favourable cases replaces H largely by Dat random sites, so that the remaining protons are likely to couple only todeuterium.The 1H spectrum then collapses to fairly narrow bands at eachibid., 3, p. 557.7 J. A. Pople and T. Schaefer, MoZ. Phys., 1960, 3, 547; P. Dish1 and J. A. Pople,*R. A. Hoffman, Arkiv Kemi, 1961, 1’4, 1.*R. A. Hoffman and S. Gronowitz, Arkiv Kenti, 1961, 16, 501.10 W. Brugel, T. Ankel, and F. Kruckeberg, 2. Elektrochem., 1960, 64, 1121.11s. Castellano and J. S. Waugh, J . Chem. Phys., 1961, 34, 295.12 S. S. Jha, PTOC. Indian Acad. Sci., 1961, 54A, 13.13C. A. Reilly and J. D. Swalen, J. Chem. Phys., 1960, 32, 1378.14C. A. Reilly and J. D. Swalen, J. Chem. Phys., 1961, 34, 980.15 J. R. Cavanaugh and B. P. Dailey, J. Chem. Phys., 1961, 34, 1094.16 C. W. Banwell and N. Sheppard, Proc. Roy. Xoc., 1961, A , 263, 136.1 7 D.R. Whitman, L. Onsager, M. Saunders, and H. E. Dubb, J . Chem. Phys.,185. Fujiwara and H. Shimizu, J. Chem. Phys., 1960, 32, 1636.19 B. Braillon, J . Chim. phys., 1961, 58, 495.Zo J. L. Garnett, L. J. Henderson, W. A. Sollich, and G. V. D. Tiers, Tetruhedron1960, 32, 67.Letters, 1961, 516BISHOP : NUCLEAR MAGNETIC RESONANCE 57chemical-shift position, and the latter have been found for a series of mono-substituted benzenes.It has been demonstrated recently that, even in quite simple systems,a spectrum containing close and unresolved lines may sometimes bematched, to within the limits of experimental accuracy, by a wide range ofrelative values of coupling constants which on casual inspection appearto be equal. This aspect of systems ABX, A,X2, and ABXY has beenstudied by Abraham and Bernstein,,l and of A2B2 by Grant and Gutowsky,22and consequent errors in previous analyses are noted.Anet 23 has notedrecently that the practice of equating the proton doublet splitting of asecondary methyl group to the coupling constant may give a low estimate ifthe spectrum is not first-order. As the chemical shift between CH and CH,in the grouping CHCH, is diminished, new lines appear mainly within thefirst-order doublet due to CH, and may not be resolved from it.In view of theoretical interest in the sign of coupling constants, deter-mination of their relative signs within a given spin system has becomeincreasingly important. This can be effected by conventional analysis onlyunder conditions for which the analysis itself is fairly =cult-thus a systemof at least three non-equivalent and strongly coupled nuclei is then necessaryto determine all the relative signs.Significantly, a direct method fordetermining relative signs by spin decoupling has been reported recentlyby Maher and Evans 24 and developed by Freeman and Atleast three non-equivalent nuclei are again necessary, but these are notrequired to be strongly coupled. The method is most effective underconditions of weak coupling, and in this sense is complementary to conven-tional analysis. Turner 26 has utilized beat patterns to determine couplingconstants too small to cause resolvable line splitting, whilst Brown andThompson 27 have suggested the use of ‘‘ free precession ” signals a t verylow field for direct determination of coupling constants.Electron-coupled Nuclear Spin-spin Interactions.-The general expressionfor the energy of interaction (J = coupling constant, usually given in c/sec.)has been given by Ramsey.l, Inter-proton coupling JHH has been mostwidely investigated experimentally, and is also the most amenable to theo-retical interpretation since it depends mainly on the Fer& contact term.This is the only term which does not involve use of angular-dependentorbitals, whilst the electron distribution about a hydrogen atom is wellrepresented by the 1s atomic orbital.The contact mechanism for spin-coupling depends on weak admixture of excited triplet electronic states withthe ground singlet state, and in the absence of knowledge of most excitationenergies and excited-state wave functions, it has been usual practice to usean approximation requiring only the mean triplet excitation energy and thetotal ground-state wave function.Valence-bond treatment, mainly by2lR. J. Abraham and H. J. Bernstein, Canad. J . Chem., 1961, 39, 216.22D. M. Grant and H. S. Gutowsky, J . Chem. Phys., 1961, 34, 699.23 F. A. L. Anet, Canad. J . Chem., 1961, 39, 2262.24 J. P. Maher and D. F. Evans, Proc. Chem. SOC., 1961, 208.R. Freeman and D. H. Whiffen, MoZ. Phys., 1961, 4, 321; R. Freemm, {bid.,28 J. J. Turner, Mol. Phys., 1960, 3, 417.27R. J. S. Brown and D. D. Thompson, J . Chem. Phys., 1961, 34, 1580; 35, 1894.p. 38558 GENERAL AND PHYSICAL CHEMISTRYKarplus, achieved notable successes in predicting the magnitudes andmarked steric dependence of JHH upon interbond angle in the system HCH,upon the dihedral angle in substituted ethanes, and upon cis-trans-isomerismin coupling across an olefinic bond.1 According to this treatment, finitecoupling depends on inclusion of non-perfect pairing structures, and onlya-type bonds were used.The results are in good agreement with experi-ment in many cases.1 This is confirmed by more recent studies describedbelow, although other effects are sometimes significant. The validity of theaverage energy approximation has been challenged by McLachlan 28 and byAlexander, 29 who concludes, however, from an alternative valence-bondapproach, that Karplus's procedure is largely justified in the particularsystems studied.This view is maintained by Karplus in recent papers.30The case of proton coupling across a system of single bonds being con-sidered first, the commonest interaction (JVic) observed between protonson adjacent carbon atoms occurs when the re-orientation rate about theC-C bond is considerably greater than Jvic, so that the single observed JHHis the mean value for all conformations. The theoretical coupling predictedby Karplus by averaging over all values of the HCCH dihedral angle (4)is 4.2 c/sec. (positive), which can be compared with the narrow experimentalrange of 6-8 c/sec., including values recently found in propane l7 andpropyl derivatives.15 A more critical test of Karplus's theory is providedby cases of known geometry, having a rigid ring system or at least a highlyfavoured conformation.Here, theory predicts maxima for 4 = 180"(Jtrans) and 0" (Jcis), with Jtrans > Jcis, and approximately zero couplingfor 4 = 90". Earlier data are considered by Jackman (ref. 31, p. 84),and it is noted that some contrary evidence from the inferred equality ofJtrans and Jcis in A,B, systems is invalid 22-e.g., they may be considerablydifferent in trans-dibromocyclopropane, although they are probably similarin ,8-propriolactone. Anet 32 has reported values of JHH for $ = 0", 44",79", and 120" in a camphane bridged-ring system, and finds good agreementwith Karplus's predicted values in each case. A value for r) = 60" reportedby Brownstein 33 in a heavily substituted ethane also agrees well with theory.Musher 34 finds coupling for + = 60", 180" in a cyclohexene system some2-3 c/sec.larger than predicted-cf. the " free rotation " case discussedabove. Extensive study of epoxides 13, 14, 35, 36 has revealed an interestingsituation in that Jcis (3$ to 5 c/sec.) is always larger than Jtrans (2 to 34c/sec.), the opposite of usual behaviour in saturated and unsaturated mole-cules. This is qualitatively consistent with Karplus's theory, since an" eclipsed ethane " structure would have qb considerably less than 180" forthe '' trans " interaction.z*A. D. McLachlan, J . Chem. Phys., 1960, 32, 1263.zOS. Alexander, J . Chem. Phys., 1961, 34, 106.3oM. Karplus, J .Chem. Phys., 1960, 33, 941; 1842.31 L. M. Jackman, " Applications of N.M.R. Spectroscopy in Organic Chemistry,"3aF. A. L. Anet, Canad. J . Chem., 1961, 39, 789.38S. Brownstein, Canad. J . Chem., 1961, 39, 1677.34 J. I. Musher, J . Chem. Phys., 1961, 34, 594.*6C. A. Reilly and J. D. Swalen, J . Chem. Phys., 1961, 35, 1322.36 J. I. Musher, Mol. Phys., 1961, 4, 311.Pergamon, London, 1959BISHOP : NUCLEAR MAGNETIC RESONANCE 59JHH between protons attached to the same carbon atom is predicted byKarplus to decrease steeply with increasing bond angle, from +32-3 c/sec.at loo", through 12.5 c/sec. a t the tetrahedral angle, and becoming negativea t 125". Data presented by Banwell and Sheppard 37 for compounds ofknown HCH angle show a good correlation with the latter as predicted,although theoretical values for olefins are somewhat higher than observed.Experimental values where carbon is in sp3 hybridization are scarce-arecent value of 12.36 c/sec.is in good agreement with theory.34 The theo-retical predictions must, however, be treated with caution in view of tworecent reports of opposite signs of Jge, and Jvic in saturated molecules.Fraser et al. find this in substituted di0xalans,3~ and Kaplan and Robertsin diethyl sulphite.39 The usual range of JHHgem values in the vinyl groupCH,:CHX is from -3 to +2 c/sec. The analogous coupling of 3.92 c/sec.in diketen is reasonably ascribed to a decrease in HCH bond angle below120" due to bond strain in the adjacent ring carbon. Unusually large valuesin organometallic compounds, especially vinyl linked to aluminium (6.3c/sec.)QO and lithium (7.1 c/sec.),*l are almost certainly due to factors otherthan change in bond angle, since they are paralleled by large values for othercoupling constants in the systems-see below.A large value reportedin acrylic acid 42 does not agree with other analyses for this com-pound.10, l1 Hiroike 43 predicts, from an alternative valence-bond treat-ment of saturated systems, that JHH will decrease as the electronegativityof the atom chemically bonded to either proton increases: 12 c/sec. is cal-culated for methane.Excepting the low values of Jgem, JHH through a a-bond system isobserved to attenuate very rapidly with increasing number of bonds betweenthe protons-by a factor of 1/10-1/20 for each additional bond.Couplingacross four a-bonds is observed only in geometrically favourable systems,when it is -1 c/sec.;44 also an exceptionally large value of -7 c/sec. isreported in a bicyclo[2,l,l]hexane ~ystern.~5 Coupling of &0.4 46 or 0.54 47c/sec. between the methyl groups in acetone has been detected from the13CH satellite spectrum, and has been ascribed to a n-electron mechanismarising from the carbonyl group.47 This should be nega,tive, and a similarn-contribution of -1.5 c/sec. to JHHgBm in vinyl compounds is postulated-cf. discussion above.The experi-mental ratio and order of magnitudes of Jcis, Jtrans in the system H*C:C*Hare quite well reproduced in Karplus's treatment of coupling via the37 C.N. Banwell and N. Sheppard, Mo2. Phys., 1960, 3, 351.38 R. R. Fraser, R. V. Lemieux, and J. D. Stephens, J . Amer. Chem. SOC., 1961,39 F. Kaplan and J. D. Roberts, J . Amer. Chem. SOC., 1961, 83, 4666.40D. W. Moore and J. A. Happe, J . Phys. Chem., 1961, 65, 224,JHH across a multiple bond follows a different pattern.83, 3901.C. S. Johnson, M. A. Weiner, J. S. Waugh, and D. Seyforth, J . Amer. Chem. SOC.,Y. Arata, H. Shimizu, and S. Fujiwara, J . Phys. SOC. Japan, 1960, 15, 2119.43E. Hiroike, J . Phys. Soc. Japan, 1960, 15, 270.44D. R. Davis, R. P. Lutz, and J. D. Roberts, J . Amer. Chem. SOC., 1961, 83, 246.45 J. Mainwald and A. Lewis, J . Amer. Chem. SOC., 1961, 83, 2769.4'3H. Dreeskamp and E. Sackman, 2. phys. Chem. (Frankfurt), 1961, 27, 136.47 J.R. Holmes and D. Kivelson, J . Amer. Chem. Soc., 1961, 83, 2959.1961, 83, 130660 GENERAL AND PHYSICAL CHEMISTRYa-electrons alone-experimental values tend to be somewhat larger than thecalculated results ( +6.1, 11 -9 c/sec., respectively). Longer-range inter-actions are smaller, but attenuate only slightly with increasing number ofintervening bonds, whether single or multiple. Avalue of 1.1 c/sec. over7 bonds is reported in hexa-2,4-diyn-1-01.~~ This suggests that the n-electrons cause the major contribution JHH(n) to the long-range couplings,and a quantitative valence-bond treatment of this effect in non-aromaticsystems has been given recently by K a r p l ~ s . ~ ~ It arises from c-n exchangeterms, and correspondence between these and the isotropic hyperfine con-stants in related free radicals has enabled JHH(n) to be deduced for severalsystems from empirical values for the latter.Comparison with experi-mental values indicates that coupling constants in a system HC,H, contain-ing unsaturation, are dominated by a-contributions for n = 2, and byn-contributions for n > 2. JHH(n;) across an CiC bond is predicted, andobserved to be about twice as great as that across C:C, and in acetyleneitself accounts for a larger fraction (-Q) of the total coupling (9.1 c/sec.)than in ethylene. Of considerable interest is the prediction of alternatingsigns for JHH(n) for each additional C-C bond,30 and the confirmation ofnegative values for n = 3,'s 259 309 4% a9, 50 and positive values for n = 4,489 51is good evidence for the validity of the conclusions.Furthermore, nearequality of the cis- and trans-interactions in the system H*C:C*CH*~~Y 5% 53in sharp contrast to that for H*C:C-H, is further evidence for dominance ofthe n-coupling mechanism, which is independent of the geometry. Jcis issometimes slightly larger than Jtrans in this system.50, 53 Hoffman andGronowitz 48 have pointed out that the very slight attenuation of JHH(n)expected (with change of sign) on replacing a hydrogen atom by a methylgroup may be used as a measure of the extent of n-coupling and hypercon-jugation. They infer that this occurs in dimethylformamide (see alsoheteroaromatic compounds, below). Extensive investigations of vinyl com-pounds, especially by Briigel et al.,la have enabled substituent effects, notcovered by Karplus's results, to be clearly established.Each JHH withinthe vinyl group *CH:CH* shows a good linear correlation with the electro-negativity (E) of the atom (X) immediately attached to the vinyl group;moreover the plot of each J against E has the same slope (J decreases withincreasing E ) . This was &st noted by Banwell and Sheppard 37 andextended to the exceptionally large couplings in vinyl-lithium (trans 23.9,cis 19-3, gem 7.1 c/sec.).41 J against E for thirteen elements inposition X is given by Waugh and cast ell an^.^^ A very low value of Jcis(2.0 c/sec.) is found in ~is-1,2-difluoroethylene.5~Many data on JHH between ring protons in aromatic compounds haveA plot of c48R.A. Hoffman and S. Gutowsky, Arkiv Kemi, 1960, 16, 471.r S S . Alexander, J . Chem. Phys., 1960, 32, 1700.aoA. A. Bothner-By and C. Naar-Colin, J . Amer. Chem. SOC., 1961, 83, 231.alB. Braillon, J . Chirn. phys., 1961, 58, 495.6a J. H. Richards and W. F. Beach, J . Org. Chem., 1961, 26, 623.6s R. R. Fraser and D. E. McGreer, Canad. J . Chem., 1961, 39, 505, and referencess4 J. S. Waugh and S. Castellano, J . Chem. Phys., 1961, 35, 1900.ssT. D. Coyle, S. L. Stafford, and F. G. A. Stone, J., 1961, 743.therein.BISHOP : NUCLEAR MAGNETIC RESONANCE 61been collected by Hoffman and Gronowitz 56 (which see for earlier refer-ences). Jortho between protons on adjacent carbon atoms is significantlysmaller (1-3 to 5.8 c/sec.) in heteroaromatic rings than in substitutedbenzenes (7.0 to 9.3 c/sec.)y except J3,4, remote from the hetero-atom, inpyridines and quinolines 57, A linear decrease inJOr,,,(H;C*C*Hb) with increase in sum of the inter-bond angles H,CC, CCHb hasbeen reported.59 Coupling again attenuates with increasing number ofbonds between the nuclei, with the exception that it is abnormally low whenthe shortest bond path lies across a nitrogen atom (no values are recordedfor J 2 , 5 in pyrroles).J,,, is not equal to J,,, in furans,21, 6o contrary to earlierinterpretation of their spectra. Few complete determinations of relativesigns have been made, but all coupling constants within each of the systems1,2,4-trichlorobenzeney61 thiophen,62 and 2-furoic acid 25 are of similar sign(assumed positive).A theoretical valence-bond estimate of the n-electroncontribution to JHH in benzenoid systems by McConnell indicates that thisis very small (0.45 c/sec. for Jortho; less for long-range coupling), and isnegative for Jmeta. Both observed magnitudes and signs indicate that theortho- and meta-interactions are dominated by 0-electron coupling, althoughthe n-contribution to Jpara may be appreciable. This is in marked contrastto JHH(n) in aliphatic systems with largely localized double bonds (seeabove). Coupling from side-chain to nucleus is usually undetectable, butinteractions of -1 c/sec. have been found in five-membered heterocyclicrings.48, 6o Comparison of these values with those of corresponding intra-ring couplings indicates that JHH(n) is more important here than in benzene(see earlier)--estimates of 20% of the total J 2 , 3 in thiophens, and 50-100~0in pyrroles and furans, have been made.48 " Cross-ring " coupling of similarmagnitude between protons attached to different aromatic rings has beendetected in the quinoline 57 and thieno[3,2-b Jpyrrole 63 systems,Coupling involving nuclei other than hydrogen is often considerablylarger than that between two protons, and the contact mechanism need notpredominate.It does appear to do so, however, in JHF and JFF across thedouble bond in fluoroethylenes, since calculation by Karplus, similar to thatfor JHH, predicts values close to those observed (again, Jtram > Jcis). Manycoupling constants are tabulated in ref.2, pp. 196-197, and later results arenow discussed relative to these. JHF in the system H*C*C*F shows a similardependence on dihedral angle to that 64 of JHH and longer-range interactionshave also been observed in saturated systems of favourable geometry.44All H,F interactions in vinyl fluoride (gem > trans > cis) have the samesign (assumed positive). l6 JHF in substituted benzenes shows similar(7.0 to 8.3 c/sec.).A A66R. A. Hoffman and S. Gronowitz, Arkiv Kemi, 1961, 16, 562.6 7 F. A. L. b e t , J . Chem. Phys., 1960, 32, 1274.68L. W. Reeves and K. 0. Strrmme, Canad. J . Chem., 1961, 39, 2318.6eR. J. Abraham and H. J. Bernstein, Canad. J . Chem., 1961, 39, 905.soG. S. Reddy and J. H. Goldstein, J .Phys. Chem., 1961, 65, 1539.62R. A. Hoffman and S. Gronowitz, Arkh Kemi, 1960, 15, 45.63 R. J. Tuite, H. R. Snyder, A. L. Porte, and H. S. Gutowsky, J . Phys. Chem.,1961, 65, 187; R. J. Tuite, A. D. Josey, and H. R. Snyder, J . kIr.neT. Chem. SOC., 1960,82, 4360.'j4R. J. Abraham and El. J. Bernstein, Canad. J . Chem., 1961, 39, 39.C. N. Banwell, Mol. Phys., 1961, 4, 26562 GENERAL AND PHYSICAL CHEMISTRYattenuation with increasing number of bonds, to that of JHH;59 a, 18 butwhereas the ortho- and meta-couplings have the same sign as the correspond-ing Jm,5 the para-H-F coupling is of different sign.@ J(31P-H) in thegrouping *CH,-P in phosphate esters and phosphonates is strongly dependenton substitution of alkyl groups, owing mainly to changes in the environmentof the methylene group;65, 66 correlation with the Taft o* resonance para-meter is discussed. Geminal and long-range €€-P coupling constants indiphosphine (186.5; 11.9 c/sec.) are of opposite sign to Jpp and JHH(108.2; -12 c/sec., assumed negati~e).~' JPF in a cyclic triphosphonitriliocompound 68 changes sign between F-P and F-P-N-P (934; 14 c/sec.: thelatter is tentatively assumed to be negative, together with -100 c/sec.for Jpp). J(l1B-l9F) in mixed boron halides 69 shows a very regular andrapid increase on progressive replacement of fluorine in boron trifluorideby less electronegative halogens (from 15 c/sec. in BF, to 108 c/sec. inBB'Br,) .A large number of J(13C-H) values have been obtained by Lauterburfrom the carbon-13 resonance spectra of aromatic compounds.70 These arediscussed in relation to the linear dependence reported earlier 1 upon thes-character of the carbon orbital involved in the bond, and upon the bondlength.Malinowski 71 has demonstrated that J(13C-H) in compoundsCHXYZ can be expressed as the sum of independent constants for eachsubstituent X, Y, and Z, to an internal consistency of &2 c/sec. over theexperimental range 127-206 c/sec. Holmes and Kaesz 7 2 report a similardependence of coupling between Sn and methyl protons in methyltin halidesupon the percentage s-character of the tin atomic orbital in the Sn-Cbond. Thus, a large increase in coupling observed in aqueous solution 72, 73is attributed to rehybridization of tin to give methyltin cations.Examples of long-range J(13CH),74 J(14NH),75 and JFF '13 have in eachcase demonstrated exceptions to the usual principle of attenuation withincreasing number of bonds.This is most pronounced in the last case, whereJ(H*C*C*F) in saturated compounds is nearly zero, whilst J(F*C*C*C*E') is10-17 c/sec. This is rationalized by supposing that most coupling occurs" directly through space," since there is a monotonic decrease in JFF withincreasing nuclear separation when the latter is averaged over all con-figurations. The very large values of JFF(gem) (up to 270.4 c/sec.) alsofit into this scheme. Analysis of the complex ethyl group spectra in com-pounds X(CH,*CH,),, where X is a single magnetically active nucleus andthe carbon atoms are numbered 1, 2 from X outwards, has revealed severaI86 G.D. Dudek, J . Chem. Phys., 1960, 33, 624.66G. Martin and G. M a d , Compt. rend., 1961, 253, 644. ,67 R. M. Lynden-Bell, Tram. Faraday SOC., 1961, 57, 888.68 M. L. Heffernm and R. F. M. White, J., 1961, 1382.60T. D. Coyle and F. G. A. Stone, J . Chem. Phys., 1960, 32, 1892.7OP. C. Lauterbur, J . Amer. Chem. SOC., 1961, 83, 1838; 1846.71E. R. Ivialinowski, J . Amer. Chem. SOC., 1961, 83, 4479.7% J. R. Rolmes and H. D. Kaesz, J . Amp. Chem. Soc., 1961, 83, 3903.73 J. J. Burke and P. C. Lauterbur, J . Amer. Chem. SOC., 1961, 83, 326.74 G. S. Karabatsos, J . Amer. Chem. SOC., 1961, 83, 1230.76 I. D. Kuntz, P. von R. Schleyer, and A. Allerhand, J. Chern. Phys., 1961, 35,76L. Petrakis and C.H. Sederholm, J . Chem. Phys., 1961, 35, 1243.1633BISHOP : NUCLEAR MAGNETIC RESONANCE 63interesting features 76, 77 (for other references, see footnotes in ref. 76). InPEt,, SnEt,, HgEt,, and PbEt,, Jx2 is considerably larger than Jxl, despitethe extra bond in the former interaction, and they are of different sign. Asimilar situation occurs in triethylthallium and mixed thallium alkyls, anddissimilar signs have been established 24 for TlEt, +. Each coupling increasesfairly regularly with increasing atomic number of X, Jxl and Jx2 rangingfrom 0.5 and 13.7 c/sec., respectively, in PEt, to 242.4 and 472-7 c/sec. inTlMe,Et. Anomalous coupling in these compounds has been ascribed totwo-electron terms in the Fermi contact interaction, but this has beencriticized by Stafford and Baldeschwieler.78 They find Jxl > Jx2, and ofsimilar sign, in ethyl fluoride (X = F); and suggest that interactions con-trary to this in the metal ethyls are due to d-electron-bonding effects.Chemical Shi€ts-Intramolecdm Eff ects.-Ramsey ’s general perturbationformula from electronic screening of a nucleus within a molecule includesdiamagnetic and paramagnetic contributions, the latter depending on excitedeigenstates.Use of these can be avoided if the problem is solved by approxi-mate (e.g. , variation) methods, but quantitative interpretation is stillrestricted to relatively simple molecules. Recent theoretical studies havebeen made of proton shielding in heteropolar diatomic molecules:79, 8oGroup VIB hydrides and the CH bond;sO acetylene,sl and molecular hydro-gen.gl, 82 Exact shielding in some two-electron atomic systems has alsobeen considered.83 In more complex systems, chemical shifts are best inter-preted in a semi-quantitat,ive manner as the sum of approximately inde-pendent contributions from local dia- and para-magnetic terms and long-range shielding by other atoms or groups within the molecule.2 This ismore directly related to the practice of nuclear magnetic resonance, sinceattention is then focused on the change of chemical shift within a relatedseries of compounds rather than on its absolute magnitude.The effect ofa substituent group upon the shielding of a remote nucleus may then inprinciple be resolved into two interactions: (1) Alteration of the local elec-tron distribution round the nucleus, e.g., by inductive or mesomeric electronrelay or by altering the state of hybridization. (2) Superposition of a smallmagnetic field due to its own electrons. This ‘‘ long-range ” effect averagesto zero upon rapid molecular orientation in a gas or liquid unless the groupis magnetically anisotropic (“ neighbour anisotropy effect ”).The smalllocal electron density round a proton renders this particularly sensitive tolong-range shielding effects.The most easily recognized case of magnetic anisotropy is the “ringcurrent ” in an aromatic ;z-electron system, which causes deshielding of thering protons l, and, to a smaller extent, of other substituents in the ring.Elvidge and Jackman84 have used this property to estimate aromatic77 P.T. Narasimhstn and M. T. Rogers, J . Chem. Phys., 1959, 31, 1431; 1961, 34,1049.78 S. L. Stafford and J. D. Baldeschwieler, J. Amer. Chem. SOC., 1961, 83, 4473.70 C. W. Kern and W. N. Lipscomb, Phys. Rev. Letters, 1961, 7, 19.soM. Fixman, J. Chern. Phys., 1961, 35, 679.Kurita and K. Ito, J. Amer. Chem. SOC., 1960, 82, 296.82S. K. Sinhe and A. Mukherji, J. Chern. Phys., 1960, 32, 1652.83R. E. Glick, J . Phys. Ghem., 1961, 65, 1871.84 J. A. Elvidge and L. M. Jackman, J . , 1961, 85964 GENERAL AND PHYSICAL CHEMISTRYcharacter in 2-pyridones. By comparing observed shifts with those pre-dicted for non-aromatic or fully aromatic structures, they infer 35 & 5%of aromatic character, as defined by the ability to sustain an induced ringcurrent, relative to benzene.By considering the differential ring-currenteffects upon different proton sites, Hoffman et ~ 1 . 8 ~ conclude that the lateralrings in 2,2’,2’’-terphenyl are roughly perpendicular to the central ringplane. A complete proton spectral analysis of the triphenylcarbonium ionindicates a skew orientation of the rings.86 Proton shifts also provide ameans of investigating the diamagnetic anisotropy of an individual bond,as has been discussed in detail by Jackman.31 When the induced magneticmoment of a bond is approximated to a point dipole, the calculated screen-ing effect upon a remote nucleus has an inverse cube dependence on distance,and an angular dependence, relative to the bond axis, of the form,3 cos2 8 - 1.Hence a magnetically anisotropic bond can cause an increaseor decrease in shielding according to the relative position of the nucleusconsidered. The high shielding of an acetylenic proton is due to the largesusceptibility of CIC along the bond axis. Proton shifts in acrylonitrilehave been interpreted in terms of a similar anisotropy of the CiN bond.87The anisotropy of the C-0 single bond has been estimated from proton shiftsin carbohydrates,88 and that of C=O demonstrated in cc-lumicol~hicine.~~The smaller anisotropy of the C-C bond has also been considered.90# 91Correlations of proton shifts with particular properties of substituentgroups are often uncertain on account of their sensitivity to long-rangeshielding and solvent effects.Unless the change in shielding within a seriesof compounds is dominated by one parameter, the interpretation is ambigu-ous. Spiesecke and Schneider 92-g4 have demonstrated recently that car-bon-13 spectra used in conjunction with proton shifts in organic compoundscan enable far more precise correlations to be made. Carbon-13 shifts areconsiderably larger, and more directly reflect substituent effects, than doproton shifts, and their solvent effects generally account for a smaller pro-portion of the total shift. The major disadvantage is the low naturalisotopic abundance of carbon-13 (-lyo) and ease of signal saturation, butthis can be offset by use of larger samples since lower effective field homo-geneity suEces to achieve similar proportional accuracy in shift measure-ments.Spiesecke and Schneider used isotopically enriched samples onlyfor dilution studies; and they employed a large rotating, spherical sample(1-5 ml.), with provision for an external standard, for the study of un-enriched samples. A flowing-sample method, with some threefold enhance-ment of signal strength, has also been described.95 Proton and carbon-1385R. A. Hoffman, P.-0. Kinell, and G. Bergstrom, Ark& Kemi, 1960, 15, 533.86R. S. Berry, R. Dehl, and W. G. Vaughan, J . Chern. Phys., 1961, 34, 1460.G. S. Reddy, J. H. Goldstein, and L. Mandell, J . Amer. Ghem. Xoc., 1961,83, 1300.88 R. W. Lenz and J. P. Heeschen, J . Polymer Sci., 1961, 51, 247.800. L. Chapman and H.G. Smith, J . Amer. Chem. SOC., 1961, 83, 3914.Qo J. I. Musher, J . Chem. Phys., 1961, 35, 1159.Q1 J. R. Cavanaugh and B. P. Dailey, J . Chem. Phys., 1961, 34, 1099.gaH. Spiesecke and W. G. Schneider, J. Chem. Phys., 1961, 35, 722.03H. Spiesecke and W. G. Schneider, J . Chem. Phys., 1961, 35, 731.D4H. Spiesecke and W. G. Schneider, Tetrahedron Letters, 1961, 468.O K s . Forsen and A. Rupprecht, J . Chem. Phys., 1960, 33, 1888BISHOP : NUCLEAR NAGNETIC RESONANCE 65shifts in methyl and ethyl compounds Me,X and Et,X show a linear correla-tion with the inductive effect of X (single atom or group) as measured by itselectronegativity, allowance being made for the " neighbour anisotropy "effect of the C-X bond.92 The latter causes systematic deviation of certainpoints (notably for X = C1, Br, I) from each of the linear plots, in the sensepredicted from the (3 cos2 8 - 1) angular dependence mentioned earlier :viz., to lower shielding in the cases of alH, PIH, and Pl3C, and to highershielding in the case of a-13C.Linear dependence of proton shift uponelectronegativity is reported also by Cavanaugh and DaileyYg1 who do not,however, find any significant neighbour anisotropy effect. The slopes ofboth proton and carbon-13 plots show greatest variation with X in methylcompounds, slightly less for a-CH, in ethyl compounds (ascribed to theslight difference in electronegativity between H and CH,), and much smallervariation in the p-position (CH, of ethyl group) owing to attenuation of theinductive effect.92 The latter causes a crossing of both proton and carbon-13lines corresponding to the a- and P-positions at an electronegativity value of-2.0, in agreement with the '' reversed " proton-shielding sequence foundin ethyl-metal compounds (see later).Combined study of proton andcarbon-13 spectra in monosubstituted benzenes, C,H,X, has led to a criticalreappraisal of substituent effects upon their chemical shifts. 93 The carbon-13spectrum has the added advantage that it includes also the atom of directattachment to X, where the large substituent effect was adequately explainedby inductive and neighbour anisotropy terms similar to those in the analo-gous a-position in ethyl compounds. These were smaller in the ortho-position (cf. /%position in ethyl compounds), where resonance contributionst o proton and carbon-13 shifts were also appreciable.Shifts in the rneta-position proved difficult to interpret, and covered a very small range forcarbon-13. In the para-position, inductive and anisotropy effects wereabsent. Proton and carbon-13 shifts were parallel, and probably domin-ated by changes in the n-electron density a t the carbon atom. Bothshowed a fairly linear correlation with the substituent Harnmett sigma-factor, although the precise significance of correlations with reactivity para-meters is criticized. In disubstituted benzenes, substituent effects uponthe proton shift in the metu- and para-positions are additive in dilutesolution. 96Linear dependence of both proton and carbon-13 shifts upon the localcarbon n-electron density, other factors being constant, has been confirmedby several workers in the case of homocyclic aromatic ions, where the n-elec-tron density is unambiguous.94, 9 7 p 989 99 Spiesecke and Schneider find thedisplacement of shielding to be +10-6 and +160 p.p.m.for proton andcarbon-13 respectively, per added electron a t the local carbon atom, in goodagreement with earlier values. These enable estimates of charge distribu-tion in other aromatic systems to be made from the observed shifts, withssP. Diehl, Helv. Chi'm. Acta, 1961, 44, 829.97 G. Fraenkel, R. E. Carter, A. McLachlan, and J. H. Richards, J . Amer. Chem.O 8 C . Maclean and E. L. Mackor, Mol. Phys., 1961, 4, 241.O0 P. C . Lauterbur, Tetruhedron Letters, 1961, 274.SOC., 1960, 82, 5846.66 GENERAL AND PHYSICAL CHEMISTRYallowance for other differential shielding effects if necessary, e.g., in azuleneY7Oand the pyridinium 100 and substituted carbonium lo1 ions.para-Substituent effects on fluorine-19 shifts are also parallel to thoseon proton and carbon-13, indicating that the same effect is dominant ineach case.93 Lauterbur 70 has also studied carbon-13 spectra of aromaticcompounds, and finds parallel variations in carbon-13 and fluorine-19 shiftsin both meta- and in para-positions ; fluorine shifts in ortho-substituted deri-vatives are altered by resonance interaction with the substituent.Sub-stituent effects on fluorine-19 shifts have also been discussed in relation toHammett factors and the polarity of the C-F bond.lO2-lO4 Correlationsof oxygen-17 shifts with structure in a large number of organic compoundshave been made by Diehl and his co-w0rliers.10~ Evidence for mesomericand inductive substituent effects in olefins is reviewed by Hoffman;s seealso ref.106.Proton andboron-11 shifts in borazole and substituted derivatives lo8 indicate appre-ciable electron transfer from nitrogen to boron, as would be required foraromaticity. Methylene and methyl proton shifts are approximately equalin B-ethyl derivatives. A cyclic bridge structure is confirmed for 2,4-di-methyltetraborane.109 Tin- 119 spectra in mixtures of tetrahalides SnX,(X = C1, Br, I) have enabled all possible mixed halide species, includingseveral previously unreported, to be detected.Chemical shifts cover arange of 1550 p.p.m. between tin tetrachloride and tetraiodide, and indicateapproximately additive contributions from each halogen atom. They areascribed mainly to changes in the local paramagnetic shielding term.73Some recent studies of molecular conformation have included an estimateof thermodynamic properties for interconversion of conformers of flexiblering systems, from the temperature dependence of the spectrum. The single,sharp proton line observed for cyclohexane at room temperature broadensinto a multiplet pattern, due to non-equivalence of axial and equatorialproton shielding, when the interconversion rate is diminished on cool-ing,110--113 from which an activation energy of 9-10 kcal./mole is found.Harris and Sheppard 112 find a substantial negative entropy of activationBoron-11 shifts have been measured by several workers.1071001. C.Smith and W. G. Schneider, Canad. J . Chem., 1961, 39, 1158.101C. Maclean and E. L. Mackor, J . Chem. Phys., 1961, 34, 2208.102 L. M. Iagupolskii, V. E. Bystrov, and E. Z. Utianskaia, Doklady Akad. Nauk103K. Ito, K. Inukai, and T. Isobe, Bull. Chem. SOC. Japan, 1960, 33, 315.104Y. Yonezawa, K. Fukui, H. Hato, H. Kitano, S. Hattori, and S. Matsuoka,loaH. A. Christ, P. Diehl, H. Schneider, and H. Dahn, Helv. Chim. Acta, 1961, 44,106G. S. Reddy and J. H. Goldstein, J . Amer. Chem. SOC., 1961, 83, 2045.107 T. D. Coyle, S. L. Stafford, and F. G. A. Stone, J., 1961, 3103; H.Landesmanl o s K. Ito, N. Watanabe, and M. Kubo, J . Chem. P h p . , 1960,32,947; 1961,34,1043.l o O I . Shapiro, R. E. Williams, and S. G. Gibbins, J . Phys. Chem., 1961, 65, 1061.1loF. R. Jensen, D. S. Noyce, C. H. Sederholm, and A. J. Berlin, J . Amer. Chem.lllL. W. Reeves and K. 0. S t r m e , Canad. J . Chem., 1960, 38, 1256; Trans.118 R. K. Harris and N. Sheppazd, Proc. Chem. SOC., 1961, 418.llS W. B. Monk and J. A. Dixon, J . Ame~. Chem. SOC., 1961, 83, 1671.S.S.S.R., 1960, 135, 377.Bull. Chem. Soc. Japan, 1961, 34, 707.865.and R. E. Williams, J . Amer. Chem. SOC., 1961, 83, 2663.SOC., 1960, 82, 1256.Faraduy SOC., 1961, 57, 390BISHOP : NUCLEAR MAGNETIC RESONANCE 67(-7.9 & 1 e.u.) in contrast with earlier assumption,l12 and similar to thatin perfluorocyclohexane. 114 The energy barrier is slightly increased onmono- or di-substitution,lll but a low value in the perfluoro-compound isattributed to torsional strain in the stable conformation.l14 The singlepeak of cis-decalin 2 remains sharp at -120",112, 113 confirming the highdegree of flexibility to internal rotation about the C-C bonds.trans-Perfluorodecalin gives distinct signals for equatorial and axial fluorinenuclei, the latter having a relative deshielding of 21-9 p.p.m., whilst rapidinterconversion again occurs in the cis-i~omer.1~~ Interconversion in the1 ,2-dioxan,l16, 117 1,3-dioxan,lls and 1,2-dithian 117 ring systems has alsobeen studied. Abraham and Bernstein have investigated rotational iso-merism in a substituted ethane.64The observation of the non-equivalence of the protons of an aliphaticmethylene group in several compounds is of particular interest.Finegold 119reported a duplication of the usual methylene quartet, with slightly differentchemical shifts and spacings, in the ethyl group resonance of certain diethylcompounds (e.g., diethyl sulphite), and inferred a difference in bonding ofthe methylene groups. This is disproved by later observations (a) that thesame effect occurs in similar compounds with only one ethyl group;120-122and (6) of further weak lines which permit analysis of the overall pattern asan ABC, or ABX, system.120 These were found also on re-examination ofthe diethyl sulphite spectrum,12, and a very recent analysis has shown thateach rnethylene proton is equally coupled to the methyl group, but thatthe interaction is of opposite sign to J,,, between them.39 The protonswithin the same methylene group are hence magnetically non-equivalent ,and this has been ascribed by Shafer et al.to an asymmetric conformationfavoured by restricted rotation. It has been pointed out, how-ever,12o, 121, 122, 124 that non-equivalence can persist in a molecule havingfree internal rotation if the symmetry of substitution is sufficiently low.It appears that non-equivalence of methylene protons should in theory occur,on free rotation, in any molecule in which the plane bisecting the HCHangle and normal to the interproton axis is not a plane of symmetry of themolecule as a whole for any conformation.The presence of an opticallyactive centre is a sufficient but not necessary condition for this, since non-equivalence of methylene protons in a group R should also occur in a com-pound RXR, possessing free internal rotation, such that replacement ofone R by a different group R' would induce optical activity. In practice,l14G. V. D. Tiers, Proc. Chem. Soc., 1960, 389.115 J. Homer and L. F. Thomas, Proc. Chern. SOC., 1961, 139.llsH. Friebolin and W. Maier, 2. Nabwforsch., 1961, 16a, 640.117 G. Claeson, G. Androes, and M. Calvin, J . dmsr. Chem. SOC., 1960, 82, 4428;llsN. Baggett, B. Dobinson, A. B. Foster, J. Homer, and L. F. Thomas, Chern.ll9H. Finegold, Proc. Chem. SOC., 1960, 283; J . Arner. Chem. SOC., 1960, 82, 2641.lZoP.R. Shafer, D. R. Davis, M. Vogel, K. Magarajan, and J. D. Roberts, Proc.lZ1 J. 8. Waugh and F. A. Cotton, J . Phys. Chem., 1961, 65, 662.lzsT. D. Coyle and F. G. A. Stone, J . Amer. Chm. SOC., 1961, 83, 4138.123 J. P. Pritchard and P. C. Lauterbur, J . Arner. Chem. SOC., 1961, 83, 2105.124 J. A. Pople, Mol. Phys., 1958, 1, 1.1961, 83, 4357.and Ind., 1961, 106.Nat. A d . Sci. U.S.A., 1961, 47, 4968 GENERAL AND PHYSICAL CHEMISTRYthe non-equivalence effect can be expected to diminish rapidly with increas-ing separation of the methylene group from the actual or potential centreof optical activity. It appears that all examples in the above referencescan be explained in this manner, without involving restricted rotation. Indiethyl sulphite, the non-equivalence would then depend upon the lack ofcoplanarity of the sulphur bonds.Non-equivalence of the CF, fluorine atomsin CF3*CF,-CFICl l 2 5 accords with this scheme. The non-equivalencereported in thiophosphonates , 66 and some of those in organophosphoruscompounds,126 can be similarly explained.The sensitivity of proton shifts to quite minor changes in molecularstructure has led to increasing application of nuclear magnetic resonance tostructural problems in classes of complex molecules of natural occurrence,and synthetic a,nalogues, where such changes can cause a profound modi-fication of their biological r61e. The success of this method depends largelyon the approximate additivity of long-range shielding effects, as demon-strated in the initial systematic study of steroid spectra by Shoolery a.ndRogers.1 Recent studies have been made of steroids,127, lZ8 bicyclic ter-penes,l29 triterpenes,l30 flavans,131 alkaloids,l32 polypeptides,133 b i ~ i n s , l ~ ~rotenoids,135 pyrimidines and nucleosides,l36 car~tenoids,~~~ and porphy-rins.l3* Abraham 139 has calculated the sign and magnitude of the ring-current effect on protons a t various sites in the porphyrin mo1ecule.l Lumryet ~ 1 .~ ~ 0 have studied the structure and denaturation of hzm-proteins bythe effect upon the proton relaxation time of added water. Balazs et aZ.lP1observed a broadening of the proton signal of water, without change in therelaxation time, in presence of deoxyribonucleic acid. An empiricallymodified form of Karplus's theory of proton spin coupling has been used toIZsL. M.Crapo and C. H. Sederholm, J . Chem. Phyls., 1960, 33, 1583.laa T. H. Siddall, C. A. Prohaska, and W. E. Shuler, Nature, 1961, 190, 903.IZ7 J. S. G. Cox, E. 0. Bishop, and R. E. Richards, J., 1960,5118; R. F. Ziircher andJ. Kalvoda, Helv. Chim. Acta, 1961, 44, 179, 186, 198; R. F. Ziircher, ibid., p. 1380.12sN. R. Trenner, B. H. Arison, D. Taub, and N. L. Wendler, Proc. Chem. SOC.,1961, 214.129 B. A. Arbuzov, Z. G. Isaeba, and Y. Y. Samitov, Doklady Akad. Nauk S.S.S.R.,1961, 13'7, 589.130 R. 0. Mumma, Diss. Abs., 1961, 21, 2485.I31M. M. Bokadia, B. R. Brown, P. L. Kolker, C. W. Love, J. Newbold, G. A.Somerfield, and T. M. Wood, J., 1961, 4663; J. W. Clerk-Lewis and L. M.Jackman,Proc. Chem. SOC., 1961, 165; E. J. Corey, E. M. Philbin, and T. S . Wheeler, TetrahedronLetters, 1961, 429.132 I. R. C. Bick, J. Harley-Mason, N. Sheppard, and M. J. Vernengo, J., 1961,1896.133 D. N. Shygorin, N. M. Pomeraatsev, and L. V. Sumin, Vysokomol. Soedineniya,134 M. S. Barber, A. Hardisson, L. M. Jackman, and B. C. L. Weedon, J., 1961, 1625.136L. Crombie and J. W. Lown, Proc. Chem. SOC., 1961, 299.136 S. Gronowitz and R. A. Hoffman, Arkiv Kemi, 1961,16, 459; J. P. Kokko, J. H.Goldstein, and L. Mandell, J . Amer. Chem. Soc., 1961, 83, 2909; C. D. Jardetzky, ibid.,p. 2919; R. U. Lemieux and M. Hoffer, Canad. J . Chem., 1961, 39, 110.137 J. B. Davis, L. 31. Jackman, P. T. Siddons, and B. C . L. Weedon, Proc. Chem.SOC., 1961, 261.E. D.Becker, R. B. Bradley, and C. J. Watson, J . Amer. Chem. Soc., 1961, 83,3743; R. J. Abraham, A. H. Jackson, and G. W. Kenner, J., 1961, 3468.139R. J. Abraham, MoZ. Phyn., 1961, 4, 145.laoR. Lumry, H. Matsumiya, F. A. Bovey, and A. Kowalsky, J . Phgls. Chem., 1961,65, 837.lgl E. A. Balazs, A. A. Bothner-By, and J. Gergely, J . MoZ. Biol., 1959, 1, 147.1961, 3, 560BISHOP : NUCLEAR MAGNETIC RESONANCE 69estimate bond angles within carbohydrate molecules in aqueous solution.The stereochemistry of reactions in the Kieb’s cycle has also been studied.lP2In organometallic compounds the shielding of a proton is increased ifit is close to the metal atom. The effect is very pronounced when thehydrogen atom is directly bonded to a metal, as in the carbonyl hydridesand related compo~nds,1~~ where displacements of the order of 20 p.p.m.to higher field, outside the usual range in diamagnetic compounds, areobserved. Smaller displacements are found in the vinyl- and ethyl-metalcompounds already discussed in relation to coupling constants.In thelatter, the methylene proton resonance is displaced to slightly higher fieldthan that of the methyl protons, contrary to the usual behaviour in ethylgroups. Olehic n-complexes with transition metals have been studied forstructural determination, in particular by Wilkinson, Pratt, and their co-workers,l4* who find proton spectra considerably Werent from those of theisolated hydrocarbons. Proton shifts are unusually widely spaced in a vinylgroup n-bonded to iron and a-bonded to it second iron atom in a carbonylderi~ative.1~5 Ally1 and substituted allyl complexes of PdCl appear to ben-bonded to palladium in deuterochloroform solution,146 but u-bonded indimethyl s~1phoxide.l~~ In the latter, all methylene protons in ,&substi-tuted allyl derivatives are equivalent. This phenomenon is observed alsoin allyl-lithium 41 in which a-bonding to the metal atom is again inferred.Equivalence of the methylene protons is ascribed to a rapid equilibriumX-CH,*CH:CH, f CH,:CH*CH,-X, as postulated earlier for allylmagne-sium br0mide.l Confirmation of the bridged structure of the dimer oftrimethylaluminium is given by the detection of distinct proton resonancesfrom the bridge and terminal methyl groups at low temperature.148 Protonshielding in tetramethyl compounds of Group IVB elements decreases pro-gressively on replacement of methyl groups by chl0rine.1~~ The structureof the carbon monoxide adduct of mercuric acetate has been determined.150The range of proton shifts in paramagnetic complexes of metal ions is verymuch greater, both to higher and lower shielding, than in diamagneticcompounds. This is attributed 151, 152 to a slight degree of electron spindelocalization transmitted from the paramagnetic metal atom to the ligandcarbon atoms.The direction of a proton shift is then determined by the142 0. Gawron, A. J. Glaid, and T. P. Fondy, J. Amer. Chem. SOC., 1961, 83, 3634.143 E. 0. Bishop, J. L. Down, P. R. Emtage, R.E. Richards, and G. Wilkinson, J.,1959, 2484 (references therein).144R. Burton, L. Pratt, and G. Wilkinson, J., 1961, 594; G. Winkhays and G.Wilkinson, ibid., p. 602; M. A. Bennett, L. Pratt, and G. Wilkinson, ibid., p. 2037;H. H. Hoehn, L. Pratt, K. F. Watterson, and G. Wilkinson, ibid., p. 2738; A. Davison,M. L. H. Green, and G. Wilkinson, ibid., p. 3172; D. Jones and G. Wilkinson, Chem.and Ind., 1961, 35, 1408.145 R. B. King, P. M. Treichel, and F. G. A. Stone, J. AWW. Chem. SOC., 1961, 83,3600.la6H. C. D e b and J. C. W. Chien, J. Amer. Chem. SOC., 1960, 82, 4429.147 J. C. W. Chien and H. C. D e b , Chem. and Id., 1961, 35, 745.14* N. Muller and D. E. Pritchard, J. Amer. Chem. SOC., 1960, 82, 248.lPDM. P. Brown and D. E. Webster, J.Phys. Chem., 1960, 64, 698.150 J. Halpern and S. F. A. Kettle, CheTn. and Ind., 1961, 35, 668.152R. E. Benson, D. R. Eaton, A. D. Josey, and W. D. Phillips, J. Amer. Chm.SOC., 1961, 83, 3714; W. D. Phillips and R. E. Benson, J. Chem. Phys., 1960, 33, 607.D. A. Levy and L. E. Orgel, MoZ. Phys., 1960, 3, 58370 GENERAL AND PHYSICAL CHEMISTRYsign of the spin density on the adjacent carbon atom. The wide range ofshifts (610 p.p.m.) in biscyclopentadienyl complexes has been studiedtheoretically by Levy and 0rgel.l5l Signals of protons are not appreciablybroadened if they are separated from the metal atom by several bonds,permitting nuclear spin-spin multiplets to be resolved and non-equivalentproton sites to be identified. Observed shifts in a series of nickel(@ chelateshave been used to deduce carbon n-electron spin densities, and it is con-firmed that these alternate in sign in conjugated systems.152Isotopic substitution is sometimes employed to simplify a complex spec-trum or to obtain spectral parameters in otherwise symmetrical molecules.To a good approximation, this does not affect other chemical shifts in themolecule, but the high precision of recent results has shown that isotopeshifts can often be detected across two bonds (for papers prior to 1960, seeref.153). In every case, substitution by the heavier isotope causes a dis-placement of the resonance of an adjacent nucleus to higher shielding(positive). Replacement of a hydrogen atom by deuterium causes a shiftof the order of +0.01 p.p.m.in the proton resonance of another hydrogenatom attached to the same ~ a r b o n , l ~ ~ - l ~ ~ and a somewhat higher value(+0.034 p.p.m.) was found in a~et0ne.l~' Slightly smaller effects are foundacross Si, P, and 0 in deuterated silane,26 phosphine,67 and ~ a t e r . 1 0 ~ Thedisplacement of fluorine resonance on deuterium substitution is muchlarger (+0.6 ~ . p . m . ) . l ~ ~ The sign and magnitude of these isotope effectsare explained by the smaller electrostatic deformation of electronic shieldingby the heavier isotope, due to the smaller zero-point vibration ampli-t ~ d e . 1 ~ ~ ~ 15', 158 The effect might reasonably be expected to diminish withincreasing atomic number of the isotopically varied element. Thus, carbonisotope effects upon 1H and 19F, on replacement 12C by 13C, are smaller thanthose on deuteration, and maintain approximately the same ratio.159Isotope effects of carbon-13 upon hydrogen directly bonded to it, have beenrecorded within the range from +0.0011 to 0.0045 ~ .p . m . l ~ ~ , 160 The largereffect on fluorine (+0-1 p.p.m. in 13C-F) has enabled its detection also acrosstwo bonds (W-C-F), where it is attenuated to +0-014 to 0-032 p.p.m.161A small silicon isotope effect upon fluorine has been recorded; replacementof silicon-28 by silicon-29 in hexfluorosilicate(1v) causes a shift of 0.004p.p .m.The principle of spin decoupling has been adapted recently to themeasurement of chemical-shift differences , particularly the very large shiftsbetween nuclei of dissimilar species, with greater accuracy than was hithertoattainable.162153G. V.D. Tiers, J . Inorg. Nuclear Chem., 1961, 16, 363.154M. Saunders, J. Plostnieks, P. S. Wharton, and H. H. Waaserman, J . Chem.lS5 H. Kusumoto, J. Itoh, K. Hirota, and J. Ueda, J . Phys. SOC. Japan, 1960,15,728.lS6 E. B. Whipple, W. E. Stewart, G. S. Reddy, and J. A. Goldstein, J . Chem. Phys.,lS7H. S. Gutowsky, J . Chem. Phys., 1959, 31, 1683.15*T. W. Marshall, MoZ. Phys., 1961, 4, 61.ls9G. V. D. Tiers, J . Phys. Chem., 1960, 64, 373.lsoH. Dreeskamp and E. Samann, 2. phys. Chem. (Prankfurt), 1961, 27, 136.161 G. V. D. Tiers, J . Phys. SOC. Japan, 1960, 15, 354.Phys., 1960, 32, 317.1961, 34, 2136.J. D. Baldeschwieler and E. W. Randall, Proc.Chem. SOC., 1961,304; S . L. ManatBISHOP : NUCLEAR MAGNETIC RESONANCE 71Chemical Shifts-Intermolecular Effects: Solvent Dependent Studies.-In this section are considered those effects which cause the magnetic shield-ing of nuclei to depend on their extramolecular environment. Ideally,such effects should be measured as departures from the chemical shifts inthe gas phase, a reasonable approximation to the condition of isolatedmolecules then being realized. In practice, the experimental difficulties aresuch that shifts have been recorded for only a few simple molecules in thegaseous state. Most studies of intermolecular effects are therefore madeby comparing chemical shifts in solution (“ solvent effects ”) as a functionof solvent, concentration, and temperature. Solvent-dependence of shifts isthen most conveniently referred to shift values extrapolated to infinitedilution in a nonpolar solvent having as nearly isotropic molecular proper-ties (in particular, shape, polarizability, and magnetic susceptibility) aspossible.It is further required that there shall be no specific associationbetween solute and solvent, the latter then being classified as “ inert ”;carbon tetrachloride and cyclohexane are commonly used. By this means,anisotropic interactions between solute molecules are eliminated, and thosebetween solute and solvent are minimized. The residual interactions areconsidered fist.A major cause of difference between the proton shielding in the gasphase and that of a nonpolar solute at infinite dilution in an inert, nonpolarsolvent is the bulk diamagnetic susceptibility of the latter.This effectdepends on the shape of the sample,2, 163 and, since it is not of chemicalinterest, some form of correction for it is applied. If chemical shiffs in acylindrical sample are measured relative to an external standard in a coaxialtube or sealed capillary, the correction can be calculated accurately as the‘‘ infinite cylinder ” condition.2 In this connexion, it has been pointed outrecently that the customary use of the bulk susceptibility for the puresolvent instead of the solution can lead to significant errors even in dilutesolution.16* Use of an internal standard provides an automatic compen-sation for the bulk susceptibility effect, but for comparison of shifts measuredin this way in different solvents it must be noted that the standard itself issubject to other shielding effects which cannot be calculated precisely.These are minimized in proton spectroscopy by use of tetramethylsilane asan internal standard.l This is soluble in all organic solvents but is virtuallyinsoluble in aqueous and ionic media, for which 2,2-dimethyl-Z-silapentane-5-sulphonate (D.S.S.) has been suggested as an alternative standard.165Bothner-By 166 has compared the proton shifts of nonpolar moleculesin the gaseous state with those at infinite dilution in nonpolar, or slightlypolar, inert solvents in cases where anisotropy effects of the solvent moleculesmay be neglected.The earlier observation by Evans is confirmed that, inevery case, the displacement of resonance to lower field on passing from theand D.D. Elleman, J . Amw. Chem. Soc., 1961, 83, 4095; J. A. Glasel, L. M. Jackman,and D. W. Turner. Proc. Chem. SOC.. 1961. 426.163A. D. BucLgham, T. Schaefir, and W. G. Schneider, J . Chenz. Phys., 1960,32. 1227.164R. J. Abraham, MoZ. Phys., 1961, 4, 369.leSG. V. D. Tiers aDd R. I. Coon, J . Org. Chem., 1961, 26, 2097.166 A. A. Bother-By, J . Mol. Spectroscopy, 1960, 5, 5272 UENERAL AND PHYSICAL CHEMISTRYgaseous state to solution is slightly in excess (0.1 p.p.m.) of that calculatedfrom the bulk-susceptibility correction. This is attributed mainly to thefluctuating electric field associated with van der Waals dispersion forcesbetween solute and solvent, with a further contribution from orbital dis-tortion if the solvent molecules contain a permanent dipole.Much largereffects in the same sense were observed by Evans 167 in the spectra offluorine-19 nuclei, where it is supposed that dispersion forces cause a signi-ficant increase in the local paramagnetic shielding term. This effectdiminishes with increase of temperature, as the average molecular separationincreases. Gordon and Dailey 168 have measured the proton shifts insimple nonpolar molecules as a function of pressure in the gaseous state,and of temperature in the pure liquid. The displacement to low field,again corrected for bulk diamagnetic susceptibility, is a similar, linear func-tion of density in both the gas and the liquid state.Turthermore, there isno measurable discontinuity on extrapolating the gas-phase values to theliquid region.Large proton shifts are produced when the solvent molecules are aniso-tropic both in shape and diamagnetic susceptibility. The extreme cases of“ rod-like ” and “ disc-like ’’ molecules are considered in particular byBuckingham et ~ 1 . 1 ~ ~ The most familiar example is the ring current of anaromatic molecule, already discussed in relation to intramolecular shieldinganomalies. Unlike the latter, the “ most effective configuration,” in whicha solute molecule can most closely approach a solvent molecule, is above orbelow the plane of the ring, where it is subject to an increased shielding.Conversely, in linear “ rod-like ” molecules with high magnetic susceptibilityalong the internuclear axis (e.g., acetylenes, carbon disulphide), a netdecrease in solute shielding is observed.Abraham 16* has made a carefulstudy of the solvent anisotropy effect by comparing the shifts, at lowconcentration, of a nonpolar solute in an anisotropic solvent with that of thesame solute in an isotropic solvent of identical bulk susceptibility. Theanisotropy shift for benzene is thereby found to be +0.42 p.p.m. (positivesign indicating a displacement to greater shielding) and, for carbon disulphide,-0.13 p.p.m. Theoretical estimates based on the van der Waals radiiare considered. Solvent anisotropy effects can be largely compensatedfor by use of an internal standard of similar molecular dimensions.I n the case of a polar solute, solvent effects can produce strong differen-tial shifts between protons in the same molecule.An important develop-ment has been the application of a reaction field theory by Buckingham 169to interpret these. The presence of a solute dipole moment causes a polari-zation of neighbouring solvent molecules to an extent determined by thedielectric constant of the medium. This in turn sets up an electric “ reac-tion field ” across the solute molecule, in the opposite direction to its dipole.That component E, along a bond X-H, linking a hydrogen atom to theremainder of the solute molecule, causes a change in the proton shielding bydistortion of the bonding-electron distribution, proportional to the first167D.F. Evans, J., 1960, 877.168s. Gordon and B. P. Dailey, J. Chern. Phys., 1961, 34, 1084.leSA. D. Buckingham, Canad. J. Chern., 1960, 38, 300BISHOP : NUCLEAR MAGNETIC RESONANCE 73power of E,, unless the proton is a t a molecular centre of inversion. Ac-cordingly, both magnitude and sense of the solvent shift depend upon theorientation of the bond X-H with respect to the solute dipole. This haspotential applications in relating spectral lines to proton locations withinthe molecules. Diehl and Freeman 17* have provided an elegant demonstra-tion of the conformation of paraldehyde, where the molecular dipole is per-pendicular to the ring plane. The axial hydrogen atoms have almost thefull reaction field along the C-H bond, and exhibit strong dependence ofproton shift upon solvent dielectric constant, whilst the equatorial methylgroups are scarcely affected.The polar groups are here well distributedover the molecule-it is pointed out that a localized polar group in a largemolecule may give the added complication of a non-uniform reaction field.This may also arise in the case of a molecule with no net dipole moment,but a large electronic quadrupole moment from opposed, well-separateddipoles, e.g., p-dinitroben~ene.~~~Further shielding effects arise if there is specific orientation betweensolute and solvent molecules due to association. Proton shifts are verysensitive to weak interactions which are often difficult t o detect by othermeans, and can be used to investigate them provided that due allowance ismade for the preceding solvent-shift mechanism~.16~, 171 Some examplesare now described.Schaefer and Schneider 1 7 2 have examined a seriesof para-disubstituted benzenes containing a strongly electron-attractinggroup X, in dilute solution in various solvents. They find a high-field dis-placement in benzene relative to an inert solvent, in excess of the solventanisotropy effect as measured by the shifts of p-xylene. The reaction field isexpected to be small, except in media of high dielectric constant, so theyinfer a weak hydrogen-bonding interaction-mainly with the proton metato the group X , since this shows a larger shift. Abraham 164 has recordedshifts for methyl iodide and iodoform in dilute solutions relative to aninternal standard, and finds a linear correlation with the calculated reactionfield except in certain aromatic solvents.The excessively high field shiftis here ascribed to complex formation, presumably of the. n-donor type.Thermodynamic properties of the complexes with toluene were found fromthe temperature dependence of the shifts: the values of AH (1.6, 1.3 kcal./mole) indicate a very weak interaction. Weak complexes involvingn-electron donation from an aromatic ring are also inferred betweenCHX, (X = C1, Br, I) and heteroaromatic compo~nds,~7~ amines and ben-~ e n e , l ~ ~ pyrrole and ~ y r i d i n e , l ~ ~ acetylenes and aromatic andbetween Al,Br, and benzene.177 Hatton and Richards 17* and Kowalewskiand Kowalewski 179 have noted that the proton resonances of the two methyl17*P.Diehl and R. Freeman, Mol. Phys., 1961, 4, 39.171R. E. Richards, Proc. Roy. Soc., 1960, A , 225, 72.172T. Schaefer and W. G. Schneider, J . Chem. Phys., 1960, 32, 1218.173Z. Pajek and F. Pellan, Compt. rend., 1960, 251, 79.174 C. Giessner-Prettri, Compt. rend., 1961, 252, 3238.176 J. A. Happe, J . Phys. Chem., 1961, 65, 72.J. V. Hatton and R. E. Richards, Trans. Paraday Soc., 1961, 57, 28.177D. Janjic, J. Delmau, B. Sum, and G. B6nt5, Compt. rend., 1960, 250, 2889.17* J. V. Hatton and R. E. Richards, Mol. Phys., 1960, 3, 253.179D. E. de Kowalewski and V. J. Kowalewski, Arkiv Kemi, 1961, 16, 37374 GENERAL AND PHYSICAL CHEMISTRYgroups in NN-dimethylformamide and NN-dimethylacetamide cross over ondilution in aromatic, but not in aliphatic, solvents.This provides goodevidence for complex formation in which the methyl group initially a t lowerfield is more directly over the centre of the aromatic ring. Differentialmethyl-group shifts are also reported on dilution of 16-methylcrotonic acidin benzene.lgO Lustig lS1 has applied the same principle to a study ofsyn-anti-isomerism in ketoximes. Two alkyl-group resonances are observedin solutions of symmetrical ketoximes in aromatic solvents, due to complexformation. Unsymmetrical monomethylketoximes again give two methylpeaks, one characteristic of each isomer.The very strong solvent-dependence of the shift of a proton involved inhydrogen bonding was recognized at an early stage.2 The displacementto low shielding on hydrogen bonding is in the sense expected from theelectrostatic nature of the bond, although the magnitude of the effect mustdepend in part upon a lowering of the local electron symmetry, with hin-drance to diamagnetic precession.171 Nuclear magnetic resonance dilutionstudies can provide valuable information about the type and strength ofhydrogen bonding involved.This is well illustrated by recent studies ofalcohols and phen01s.l~~ - l g 5 Dilution in an inert solvent causes progressivedisruption of intermolecular hydrogen bonds to an essentially monomericcondition at low concentration, with a high field displacement of the hydroxylproton resonance of the order of 5 p.p.m. Significantly smaller displace-ments occur in a solute able to form intramolecular hydrogen bonds, sincethis form will be increasingly favoured on dilution (e.g., in chloroethanols,ls2ortho-substituted phenols,l83~ 1859 186 and salicylaldehyde.ls7 Very smallsolvent shifts occur if hydrogen bonding is sterically hindered,lS2, lS8 forexample in triphenylmethanol where this is further conemed by tempera-ture independence.182 This is of help in deciding between molecular con-formations which differ in steric hindrance to hydrogen-bond formation, asin a recent study of ephedrines.l89 In hydroxy-compounds, limited corre-lation is obtained between dilution shifts and changes in intensity of theinfrared hydroxy-group stretching band.182 A better correlation is obtainedbetween hydroxyl proton and infrared shifts for intramolecularly hydrogen-bonded molecules in dilute solution, where the state of hydrogen bondingis more precisely defined.185 Proton resonance of the amino-group inaniline shows negligible displacement on dilution in carbon tetrachloride,suggesting virtual absence of hydrogen bonding, in agreement with infraredspectral data.174 In some of the association studies described in the pre-ceding paragraph, evidence was obtained for conventional hydrogen bonding180 S.Pujiwara, H. Shimizu, Y. Arata, and S. Akahori, Bull. Chem. SOC. Japan,lslE. Lustig, J . Phys. Chem., 1961, 65, 491.lszT. M. Connor and C. Reid, J . Mol. Spectroscopy, 1961, 7, 32.183M. Martin and M. Quilbery, Compt. rend., 1961, 252, 4151.l 8 4 I.Griinacher, Helv. Phys. Ada, 1961, 34, 272.185 L. W. Reeves, E. A. Allan, and K. 0. Strmnme, Canad. J . Chem., 1960,38, 1249.lsaI. Yamaguchi, Bull. Chem. Xoc. Japan, 1961, 34, 451.lS7I. Yamaguchi, Bull. Chem. SOC. Japan, 1961, 34, 353.ls8L. Ebersen and S. Forsbn, J . Phys. Chem., 1960, 64, 767.ls0 J. B. Hyne, Canad. J . Chem., 1960, 38, 135.1960, 33, 428BISHOP : NUCLEAR MAGNETIC RESONANCE 75in competition with n-interactions with aromatic systems. The former isindicated in the interaction of phenylacetylene and pyridine, from the lowfield displacement of the acetylenic proton resonance, and from the infraredspectrum.190 Mavel191 has derived theoretical concentration-dependencecurves for different kinds of complex formation and dissociation, and hasinvestigated the rate of exchange of the hydroxyl-group proton in alcohol-water mixtures.192 On dilution in chloroform, the exchange rate is loweredsuf3Eiciently to enable spin coupling to the alkyl group to be observed.Thisis attributed to hydrogen bonding between chloroform and the alcoholmolecule. l9The influence of solvent upon tautomeric equilibrium may be followedconveniently from changes in the relative intensity of the superimposedspectra of the individual tautomers.2 Several recent studies have beenreported.lg41 lg5 The enol form of acetoacetic ester is suppressed by smallamounts of strong bases.lg5Differential solvent shifts have been used in the interpretation of complexspectra.172 Pyridine may be used advantageously as a solvent for steroids,to separate methyl proton peaks which overlap in chloroform solution.lg6Protonation of weak bases in acid solution can be studied convenientlyby nuclear magnetic resonance, particularly with regard to the site of proto-nation and the rate of proton exchange. These effects are considered in thissection since, although not strictly intermolecular, they are strongly solventdependent. The question of 0- or N-protonation in amides has been anopen one for some time, but recent nuclear magnetic resonance studiesindicate strongly the preponderance of O-protonation, in agreement withother physicochemical methods. References to these are given by Katritzkyand Jones,197 who discuss the evidence in detail. The two methyl peaksof N-methyl-amides often coalesce in acid media.This is attributed tothe presence of a small amount of the N-protonated cation, which will havegreater freedom of internal rotation about the C-N bond. A similar con-clusion is inferred in the analogous case of pyridones.198 Co-ordination ofdimethylformamide to boron trichloride 199 and of dimethylpropionamideto iodine 2oo also occurs via the oxygen atom of the amide. The protonatedcations of water, ethyl alcohol, and acetone have been studied in solutionin hydrogen fluoride saturated with boron trifluoride.201 At low tempera-tures the exchange rate is slowed sufficiently to reveal the spectra of thelDoM. M. Kreevoy, H. B. Charman, and D. R. Vinard, J . Amer. Chem. SOC., 1961,lD1G. Mavel, J .Phys. et Radium, 1960, 21, 37.lD2G. Mavel, Compt. rend., 1960, 250, 1477; J . Phys. et Radium, 1960, 21, 731.lD3 J. Cantacuzhe, J. Gassier, Y . Lhermite, and M. Martin, C m p t . rend., 1960,250, 1474.lD4 G. 0. Dudek and R. H. Holm, J . Amer. Chem. SOC., 1961,83,2099,3914; R. Fillerand S. M. Naqvi, J . Org. Chem., 1961, 26, 2571; I. Griinacher, H. Suhr, A. Zenhaiisern,and H. Zollinger, Helv. Chim. Acta, 1961, 44, 313.lD6 C. Giessner-Prettri, Compt. rend., 1960, 250, 2547.lg6 G. Slomp and F. MacKellar, J . Amer. Chem. Soc., 1960, 82, 999.lS7A. R. Katritzky and R. A. Y. Jones, Chem. and Ind., 1961. 35, 722.lDeA. R. Katritzky and R. A. Y. Jones, PTOC. Chern. SOC., 1960, 313.lDD W. Gerrard, M. F. Lappert, H. Pyszora, and J. W. Wallace, J., 1960, 2144.%OoR. S.Drago and D. Bafus, J . Phys. Chem., 1961, 65, 1066.%OIC. Maclean and E. L. Mackor, J . Chem. Phyg., 1961, 34, 2207.83, 197876 GENERAL AND PHYSICAL CHEMISTRY+protonated forms. Proton shielding in OH is lower than that of OH inthe corresponding neutral hydroxy-compounds. Protonation of azulene intrifluoroacetic acid occurs in the five-membered ring, whilst the positivechange is accommodated in the seven-membered ring.202 At low tempera-ture, protonated mesitylene in hydrogen fluoride-boron trifluoride also showsa separate resonance peak due to ring methylene at the protonation site.It merges with that of the remaining ring protons, but not the hydrogenfluoride peak, at room temperature, indicating an unusual intramolecularhydrogen exchange between different ring sites.101Properties of electrolytes can also be studied by nuclear magneticresonance.1, 203 The temperature coefficient of dissociation of strong acidshas been measured by Hood and Reilly.20* Ionization of nitric acid issuppressed by addition of aluminium nitrate, owing to hydration of thecation and the common-ion effe~t.~O~ Jackson et aZ.206 have found that theoxygen-17 resonance of water in the hydration sphere of a cation can be dis-played as a separate peak by adding a small amount of paramagnetic co++ion, which displaces the resonance of unbound water molecules only. Thisis used to estimate hydration numbers. Proton shifts of water in solutionsof alkali halides vary linearly with concentration up to the point at whichwater molecules begin to be shared between two ions.207 The exchangerates of bromine- and iodine-containing complex ions in aqueous solutionhave been measured by Hertz from the width of the 79Br, 81Br, and 1291resonances.2o8 Rate constants for proton exchange in water 209 andaqueous solutions of ammonium and substituted ammonium ions 2lO havealso been measured from line-widths.Nuclear Magnetic Resonance of Solids.-Brief mention will be made ofsome of the recent applications of broad-line spectra of solids. A few studiesof the angular dependence of proton line shape and width in single, dia-magnetic crystals have been carried out to investigate the location of hydro-gen atoms, for example in thiourea 2 l 1 and Rochelle salt.212 Protonresonance of a single ice crystal is consistent with the Pauling hydrogen-bonding mode1.213 The inter-proton distance and orientation of watermolecules in a number of mono- and di-hydrated salts have been measuredby McGrath and Silvidi,214 and values given by previous workers are col-lected. The mean inter-proton distance in ten compounds is 1.595 A, fromwhich no departures outside the limit of experimental error were found.They infer that the structure and dimensions of water molecules of hydra-203 s. s. Danyluk and W. G. Schneider, J. Amer. Chern. SOC., 1960, 82, 997.203 J. E. Prue, Ann. Reports, 1960, 57, 80, Sect. (c).204G. C. Hood and C. A. Reilly, J. Chem. Phys., 1960, 32, 127.206 R. T. Axtermann; W. E. Shuler, and B. B. Murray, J. Phys. Chem., 1960,64, 57.206 J. A. Jackson, J. F. Lemons, and H. Taube, J. Chem. Phys., 1960, 32, 553.207 B. P., Fabricand and S. Goldberg, J. Chem. Phys., 1961, 34, 1624.2osH. G. Hertz, 2. Elektrochem., 1960, 64, 53; 1961, 65, 20, 36.209 S. Meiboom, J. Chem. Phys., 1961, 34, 375.2loM. J. Emerson, E. Grunwald, and R. A. Kromhout, J. Chem. Phys., 1960, 33,547; E. Grunwald, P. J. Karabatsos, R. A. Kromhout, and E. L. Purlee, ibid., p. 556.211.J. W. Emsley and J. A. S. Smith, Trans. Paraday SOC., 1961, 57, 893.212R. Rlinc and A. Prehesnik, J. Chem. Phys., 1960, 32, 387.213K. Kune and R. Hoshino, J. Phys. SOC. Japan, 1961, 16, 290.214 J. W. McGrath and A. A. Silvidi, J. Chem. Phys., 1961, 34, 322BISHOP : NUCLEAR MAGNETIC RESOPU’ANCE 77tion are not affected by their environment, and that the orientation adoptedis determined by the formation of hydrogen bopds with the nearest electro-negative atoms. The temperature-dependence of line-width in polycrystal-line materials provides information about internal rotations, since theonset of rotation will cause partial averaging-out of the direct dipole inter-action effect and a narrowing of the resonance line. findevidence for rotation of the cyclopentadiene rings in ferrocene, and this isslowed upon substitution in one or both rings. Peterlin and Pintar 216find that free rotation of the methyl group about the C-C axis in aceticacid is prevented in mono-substitution by chlorine. Dunell et aL217 showthat trichloroacetic and tribromoacetic acids are present as hydrogen-bonded dimers in the solid whereas acetic acid forms polymeric chains.The proton resonances of non-stoicheiometric, interstitial metal hydrideshave been studied over a wide range of temperature.21s, 219 A substantialdecrease in line-width on raising the temperature is ascribed to self-diffusionof hydrogen atoms, and the activation energies for this process are found.The lowest value found (2.4 kcal./mole) is that for PdH,.,,.219 Stalinskiet aZ.218 find that the activation energy in TiH, increases slightly withincreasing hydrogen content (x = 1.607 to 1.721). The observation of aproton chemical shift to high field by 0-01-0~032~0, in the opposite senseto the Knight shift in metals, is of interest. An interpretation is given interms of interaction between electrons localized on the hydrogen atom anddelocalized in.the metal conduction band. Bonding of hydrogen in thesecompounds is discussed. Investigation of polymers by proton resonancehas been reviewed by Sauer and W o o d ~ a r d . 2 ~ ~ ~Line shape of the boron-11 resonance signal in polycrystalline samplesdepends on the magnitude and symmetry of the local electric field gradient,which interacts with the nuclear quadrupole moment. In the case oftetrahedral bonding there is no net field gradient (e.g., tetrahydroborateion), but on progressive distortion to trigonal bonding the quadrupoleinteraction causes broadening and asymmetry of the boron-1 1 resonance.The state of co-ordination of boron in compounds of incompletely knownstructure has been investigated in this way.220Nuclear Spin Relaxation Times.-The rate of energy transfer betweena nuclear spin system and its environment is characterized by the spin-lattice relaxation time ( TI), also termed the “ longitudinal ’’ relaxation timesince it involves a change in the net nuclear magnetization vector componentalong H,. The rate of mutual energy transfer between two nuclei is charac-terized similarly by the spin-spin, or “ transverse,” relaxation time ( T2),215 C. N. Mulay, E. G. Rochow, E. 0. Stejskal, and N. E. Weliky, J . Inorg. NucEearChena., 1960, 16, 23.zlsA. Peterlin and M. Pintar, J . Chem. Phys, 1961, 34, 1730.217 B. A. Dunell, L. W. Reeves, and K. 0. Strermme, Trans. Faraday SOC., 1961, 57,218 B. Stalinski, C. K. Coogan, and H. S. Gutowsky, J . Chem. Phys., 1960, 33, 933;219 W. Spalthoff, 2. phys. Chent., 1961, 29, 258.21sa J. A. Sauer and A. F. Woodward, Rev. Mod. Phys., 1960, 32, 88.220 A. H. Bilver and P. J. Bray, J . Chem. Phys., 1960, 32, 288; A. H. Silver, ibid.,p. 959; P. J. Bray, J. 0. Edwards, J. G. O’Keefe, Y . F. ROSS, and I. Tatsuzaki, ibid.,1961, 35, 435.Mulay et372, 351.34, 119178 GENERAL AND PHYSICAL CHEMISTRYwhich involves a change in that component of the net nuclear magnetiza-tion vector which is at right-angles to H,. Only the first of these leads toa net change in spin energy, but both processes limit the lifetime of anucleus in a given spin state, and hence contribute to broadening of theresonance line. Methods used for their measurement are detailed else-where.2 In the case of a liquid, current theory requires that T, shall equalT2 when relaxation time is 222 A recent report by Bonera et aLZ2lconfirms this for a series of simple organic liquids within the limits ofexperimental accuracy, which are set by the T2 measurements. They finddifferent T, values for different groups within the same molecule in thecase of toluene, where the values for the methyl and aromatic protons are8 and 15 sec., respectively, in good agreement with previous measurements.Results by Powles and Cutler 222 are similar, but they find that, in benzene,T2 (11 sec.) is significantly shorter than T, (18 sec.). Possible theories forthis are discussed, and it is suggested that a fluctuation in nuclear spincoupling may be responsible. T, and T,in fluorobenzene (0.7 sec.) are appreciably shorter than in other aromaticsubstituted benzenes. The temperature dependence of T, in butyl alcoholshas been measured by Bernheim et aZ.223 This is linear except a t low tem-peratures, and the inferred activation energies for relaxation increase from3.9 to 7.3 kcal./mole in the sequence n < is0 < s < t-butyl alcohol. Thisdemonstrates that the shape of the molecule exerts a major effect upon themotions governing spin-lattice relaxation. The discrepancy between theobserved low-temperature minima and those calculated on the basis ofrandom molecular re-orientation indicates that rotation is restricted byhydrogen bonding. Spin-lattice relaxation by dipole-dipole interaction iscompounded of contributions from rotational and translational motions,and these are discussed theoretically by Mitchell and E i ~ n e r . ~ ~ ~ An experi-mental estimate of their relative contributions in benzene has been madeby dilution studies in deuteriobenzene.225 The ratio of T, values for protonsand fluorine-19 in fluoroform is over 100 : 1 in the gas phase,226 and it isinferred that spin-rotation interactions are here appreciable. Intermole-cular contributions to T, are sometimes T, and T, of protonsare considerably shortened by the presence of paramagnetic species, andLoembergen and Morgan 228 infer that they are then dominated by interac-tion of nuclear and unpaired electron spins.Both values increase on dilution.E. 0. B.E. 0. BISHOP. C. J. S. M. SIMFSON.T. L. COTTRELL. E. T. STEWART.R. E. RICHARDS. R. L. WILLIAMS.221 G. Bonera, L. Chiodi, G. Lanzi, and A. Rigamonti, Nuovo Cimento, 1960, 17, 198.222 J. F. Powles and D. Cutler, Nature, 1959, 184, 1123.223 R. A. Bernheim, J. D. Mackenzie, and R. C. Millikan, J . Chem. Phys., 1961,224R. W. Mitchell and M. Eisner, J . Chem. Phys., 1960, 23, 86.z2sM. Eisner and R. W. Mitchell, Bull. Arner. Phys. Soc., 1961, 6, 363.236 C. S. Johnson, J. S. Waugh, and J. N. Pinkerton, J . Chem. Phys., 1961,38, 1128.sz7F. A. Bovey, J . Chem. Phys., 1960, 32, 1877.22sN. Loembergen and L. 0. Morgan, J . Chem. Phys., 1961, 35, 842.34, 565