Quantum-Mechanical Study of H + H, Reactive Scattering*BY M. KARpLust and K. T. TANGSDept. of Chemistry and Physics and IBM Watson Laboratory, Columbia University,New York, New York and Department of Chemistry, Harvard University,Cambridge, Mass.Received 22nd June 1967The results of an approximate quantum-mechanical treatment of the H+ H2 reaction are reported.Two limiting models in the distorted wave formulation (DWB) are compared ; in one, the moleculeis unperturbed by the incoming atom and in the other the molecule adiabatically follows the incomingatom. For thermal incident energies and the semi-empirical interaction potential employed previouslyfor a quasi-classical trajectory analysis of the reaction, the adiabatic model seems to be more appro-priate. To examine the total and differential cross section for a range of energies, a simplified(linear) version of the DWB method is used.Most of the results are similar to those obtainedquasi-classically. However, the quantum total cross section has a significantly higher thresholdthan does the classical cross section. In agreement with the quasi-classical treatment, the differentialcross section is strongly backward peaked at low energies and shifts in the forward direction as theenergy increases. The calculated variation of the total cross section with final rotational state raisesquestions concerning the standard procedure for determining para-H, to ortho-H, conversion rateconstants.The reactive or rearrangement scattering of hydrogen atoms by hydrogen mole-cules (H+H,-+H,+H) plays a fundamental role in the theory of chemical kinetics.lExact quasi-classical calculations of the cross sections and other reaction attributeshave been made with a realistic, though approximate, potential surface for the H3~ystem.~ Since it has been suggested that quantum effects are important for thisrea~tion,~ it is of interest to have direct estimates of their nature and magnitude.Inthis paper we report some results of an approximate quantum-mechanical treatmentwhich employs the same interaction potential as the classical calculation.Other quantum studies of H3 have been made by G ~ l d e n , ~ Yasumori and Sato,6Mortenson and Pitzer,' Nyeland and Bak,s M i ~ h a , ~ and Marcus.lo They alldiffer from the present work both in the choice of potential and in the simplifyingassumptions which reduce the problem to tractable form.METHODOLOGYWe consider the reaction A+ BC-+AB + C, where A, B, and C are structurelessparticles (except for nuclear spin) that interact through a known potential energysurface V.This surface may be either the exact or an approximate Born-Oppenheimer solution of the electronic Schroedinger equation for the real atomscorresponding to A, B, and C.The differential scattering cross section for rearrangement from the reactantchannel a to the product channel /? can be written in the centre-of-mass co-ordinate* supported in part by a contract with the U.S. Atomic Energy Commission. t present address : Department of Chemistry, Harvard University, Cambridge, Massachusetts.$ present address : Department of Physics, Pacific Lutheran University, Tacoma, Washington.This work is mainly from a Ph.D.thesis submitted by K. T. Tang to the Faculty of Pure ScienceColumbia University (1965).5M. KARPLUS A N D K . T . TANG 57system l2Here a corresponds to the quantum numbers for a particular rotation-vibration stateof molecule BC and to momentum ka of A relative to BC, the reduced mass (A, BC)being P a ; the index B has the same connotation for the final channel. The quantityTsa is the transition or scattering matrix (T matrix) defined by the two equivalentexpressionsTpa = <QB I YB I %"> = Vj-' I v a I @a>, (2)where $"'a, Vp are initial and final state interaction potentials ; i.e., Va is the part ofV that goes to zero as the initial atom-molecule relative co-ordinate goes to infinityand VB is correspondingly defined for the final channel. In terms of these, thetotal system Hamiltonian H i s writtenwhere K is the total (centre-of-mass system) kinetic energy operator, and Sa andZB are the non-interacting initial and final state Hamiltonians, which have solutions@a and 'DB (normalized to unit density) :Here R is the relative co-ordinate, r the molecular co-ordinate, and qBC(r), the mole-cular (rotation-vibration) wavefunction for the initial channel ; the quantities S, s,and vAB(~)B are defined correspondingly for the final channel.The functions Y$*),Yg* are, respectively, initial and final channel eigenfunctions of the entire Hamiltonian&' with energy E and outgoing (+) or incoming (-) spherical-wave boundaryconditions ; e.g., the Yi*) satisfy the Lippmann-Schwinger equations,12Z = K +V" = Z,+V", = Xp+ Vp, (3)@a = exp (ika * R)qBc(r)a, @p = exp (ikp S)vAB(s)p* (4)1with the positive infinitesimal E introducing the appropriate asymptotic behaviour.The functions ma, mB, Va, and Vfi being known, the difficulty in obtaining TBa, andfrom it the differential cross section Cafi(iB), occurs in the determination of Yp) orYb-).Since their exact evaluation, which is equivalent to solving the three-bodySchroedinger equation with appropriate boundary conditions, is not feasible atpresent, approximations must be introduced at this point. The simplest is to replaceYL') or Y$-' in TBa by d i a or d i p , respectively ; i.e.,which is the well-known Born approximation.12 Although it may be useful forsome atomic rearrangement problems (e.g., high energy, low activation barrier,)TBa(B) would yield incorrect results for the H+H2 reaction in the thermal region.To account in part for the strong repulsive interaction (high activation barrier) betweenthe approaching or receding atom and the molecule, it is appropriate to separateV", and Yp into two parts l2where rtrz(R) and V;(S) are distortion potentials that cannot produce rearrangement.They are chosen to account for the interaction as completely as possible, subject tothe condition that the Hamiltonians Se+Yz and x),+-y.,O have solutions xL*)TBa Tpa(B) = (@p I YB I @a) = <@p I v a I @a>, (6)Ya = v,O+ctr,:, VD = "Ir;+v-; (758 QUANTUM MECHANICS OF H 4- H2 REACTIONand xp)¶ respectively, which can be evaluated exactly or, at least, to a high degree ofapproximation.Here the xi*) satisfy the equationand a corresponding equation exists for xi*). Introduction of xi-) and use of eqn.(7) and (8) in the first (post) form of T’, [eqn. (2)] yields l2If now Yy) is approximated by x?), the so-called distorted wave Born approximation(DWB) is obtained; i.e.,T’, = (xj-1 I “yj I Yi”).TB, s T,,(DWB) = (xi-’ I V’i I xr)>.(9)(10)As compared with TBa(B) [eqn. (6)], eq. (10) should be considerably better forH+H2 because it includes distortion of the relative motion wavefunctions in boththe initial and final channels.However, the replacement of Yy) by x$+) in eqn. (9)to obtain (10) is generally a serious approximation, the accuracy of the resultingscattering matrix T’,(DWB) depending both on the nature of the problem and on thejudicious choice of Vz and Vi. The present work uses distorting potentials obtainedfrom two different approximations, which represent limiting cases as far as theresponse of the molecule to the incoming atom is concerned. In the first (I), thedistorting potentials, called VIO,(R) and V-&(s>, were calculated with the assumptionthat the molecule is unperturbed by the atom; i.e., the potential “t’, (and similarlyVfi) is expanded-tr,(R,r) =Cvan(R,r))P,(cos Y>, (11)(12)nwhere y is the angle (R, r ) and Yia(R) is writtenwith qBc(r), the vibrational wavefunction for the isolated molecule ; in the H3 system,the average over the ground vibrational state in eqn.(12) is very close to equilibriumvalue V,,(R, re), so that the latter was used for V;a(R). Thus, in case I (free moleculeapproximation), the wavefunctions xf:) and xfj-) can be writtenwith Frh+)(R) and Fb-)(S) determined by a standard partial wave expansion l2 for thespherically symmetric potentials VL(R) and V$(S), respectively. The secondapproach (II), which is in the spirit of the perturbed stationary state approximation,12bdetermines the distortion potentials Via@) and V&(S) with the assumption thatthe molecule adjusts adiabatically to the presence of the incoming atom ; e.g., V&(R)is the eigenvalue of the Schroedinger equationvia(R) = <VBdr)a I vaO(R?r> I VBdr)or),xf,“(R,r) = F‘,,+)(R)VBC(% X W ¶ 9 = F‘,i)(S)?*B(S)’, (13)where the eigenfunction qBc(R, r)= is the one which goes to qBc(r), as R+m.Tosimplify the solution of eqn. (14), only the first two harmonics in Y(R, r ) were includedand it was assumed that qBC(R, r), can be written for each value of the parameter Rin the separable form(16)V(RYr) = G ( R , r ) + ~2(R,r)Pz(cos y), (15)?!BC(R,r)a = (llr)zlBC(R?r)ar(R, cos y)aM. KARPLUS AND K . T. TANG 59Here (1 / r ) qBC (R, r), is a solution of the radial Schroedinger equation with the potentialV',,(R, r), and T(R, cos y) is a solution of the speroidal equation l3 obtained bysubstituting eqn.(15) and (16) into (14), multiplying by (l/r)qBc(R, r):, and inte-grating over r. Thus, for case I1 (adiabatic perturbation approximation), the initialand final state wavefuncticns arewhere F&:)(R) and Fhp)(S) are obtained from partial wave expansions with thepotentials "r/&(R) and Y&(S), respectively ; the unperturbed molecular wavefunctionqAB(~)B is used in xf,) to retain the form given in eqn. (10) for T,,(DWB).Although determination of the TP,(DWB) matrix elements by either of the pro-cedures (I or 11) outlined above is possible, the required integral evaluation is sotime-consuming that only a small number of such calculations were carried out.To permit a more general exploration of the nature of the reaction cross section, anadditional simplification was introduced.This was to assume, as justified in partby the DWB results, that reaction occurs only when the three atoms are in the neigh-bourhood of the linear geometry. The matrix element T',(DWB) is then approxi-mated by the expressionwhere y' is the angle (S, s) and A is the &function " strength " parameter. AlthoughA would be expected to vary as a function of the energy and of the initial and finalstates, a fixed value of A was chosen by comparison with a TB,(DWB) result. Becauseof the arbitrariness in A, all of the work with the DWBL model was based on thesimpler free molecule (case I) approximation for the distorted waves associated withthe relative motion.In the formulation outlined in this section, the particles (A, B, and C) are treatedas distinguishable and only the reaction A+BC+AB+C is considered.Since thereaction A + BC-+AC + B yields exactly equivalent results for the H + H2 system,the reported total cross sections include this factor of two. The symmetry conditionson the initial and final state wavefunctions due to the indistinguishability of theparticles are not explicitly introduced. This is permissible because the necessarysymmetrization can be applied to the T matrix elements obtained from the un-symmetrized calculation.14* l5 The details of the computations, which were pro-gramed for the IBM 7094 primarily in FORTRAN 11, will be given s~bsequent1y.l~dtk = F\tk(R)qBC(R,r)a, x\;b' = F$~)(~~AB(s)fi, (17)Tp,(DWBL) = A(x$-' I V;s(r' - n) I x:") (1 8)RESULTS AND DISCUSSIONIn this section we report some of the results obtained by the methods outlinedabove for the H+Hz reaction with the initial and final molecule in the groundvibrational state.We consider first the DWB approximation and then turn to themore extensive studies by the DWBL model.DISTORTED WAVE BORN APPROXIMATIONFor the free molecule (case I) and the adiabatically perturbed molecule (case 11)approximations, the effective potentials V&(R) and .rY;",,(R) are shown in fig. 1.Although the two potentials are similar, the difference in the region between 1-8and 3.5 a.u. is important for the reaction. That v & ( R ) is a " softer "potentialin this region results from both vibrational and rotational distortions of the moleculeby the atom; i.e., in the presence of the atom, the molecule has a smaller effectivezeropoint energy and its average orientation is shifted toward the linear geometry o60 QUANTUM MECHANICS OF H + H2 REACTIONminimum-energy .The elastic differential cross sections obtained with the twopotentials at an incident energy of 0.5 eV are shown in fig. 2. The difference betweenthem is relatively small. However, it is not negligible particularly for large1.0 2.0 3.0 4.0 5:O 6.0 7.0R (a.u.)potential Y&(R) corresponding to the adiabatically perturbed molecule.FIG. 1 .-Two-body potentials : -, potential .Yfa(R) corresponding to the free molecule ;0 30 60 90 I20 150 1808 (degrees)FIG. 2.-Differential elastic cross sections from two-body potentials as a function of the scatteringangle 8 for an incident energy of 0.5 eV : - from potential V&(R) corresponding to the freemolecule, - - - from potential V&(R) corresponding to the adiabatically perturbed molecule.scattering angles, which correspond to the small orbital angular momenta or impactparameters that contribute to reactive scattering.For both potentials the corres-ponding classical elastic cross sections are identical with the quantum results exceptin the small angle region (scattering angle O;55°).1M. KARPLUS AND K . T. TANG 61With the distorted wavefunctions from the effective two-body potentials, theDWB scattering matrix and cross sections were evaluated. Only the reaction withboth the initial and final molecule in the lowest rotational state (J = 0, J’ = 0) wasconsidered.The total cross sections Sap obtained by integrating ca(ia) over allangles and multiplying by two are listed in table 1 for a few incident energies. AtTABLE TO TOTAL CROSS SECTION (J = 0, J’ = 0) BY DWB APPROXIMATIONfree molecule perturbed moleculerelative energy,a approximation (I) approximation (11)eV a.u. a.u.0-5 0.009 0.200.33 - 0.0270-21 -a The barrier height is 0.396 eV.0-oO0103eV there is a profound difference between the two approximate models, theadiabatic perturbation of the molecule by the incoming atom yielding a 20-foldincrease in S,, over that corresponding to an unperturbed molecule. From thecollision time for this relatively low energy, it appears that the adiabatically perturbed0 30 60 90 120 150 180FIG.3.-Differential reaction cross section as a function of the scattering angle 8 for an incidentenergy of 0.5 eV : - from adiabatic model (11) for (J = 0, J’ = 0) ; - - - from exact quasi-classicalcalculation for (J= 0, all S).model should be the better approximation. Certainly, this is true for the vibrationaldistortion, although it is more questionable for the rotational reorientation.The differential cross section (in arbitrary units) obtained from the adiabaticmodel (11) at an energy of 0.5 eV is shown by the solid line in fig. 3. It correspondsto “ backward scattering ” in the centre-of-mass system ; i.e., the incoming atomstrikes the molecule, picks up an atom, and the newly formed molecule goes back0 (degrees62 QUANTUM MECHANICS OF H + H2 REACTIONdominantly in the direction from which the atom came. The differential crosssection for the free molecule model (I) is similar in shape to the adiabatic resultalthough the magnitude is much smaller.Also shown in fig. 3 by a dotted line isthe differential cross section (in arbitrary units) determined from the quasi-classicaltrajectory treatment at 0.5 eV ( J = O-+all J’). The form is again almost identicalto the model I1 result. Such strongly backward peaked cross sections are expectedwhen the quantum-mechanical wavefunction or classical path of the incoming atomis strongly distorted by a repulsive barrier. They contrast sharply with the Bornapproximation, which yields an oscillating cross section with its maximum in theforward direction.By an expansion of the total reaction cross section in terms of Contributions fromindividual partial waves 2, a comparison with the classical impact parameter (b)dependence can be made.For J = 0 to J’ = 0 and E = 0.5 eV the final state (exit)I values, which are essentially the same as those for the initial state, were found tomake contributions that decrease smoothly with increasing 2 and approach zero forZr lO(6 z2). This behaviour is very similar to that of the classical reaction proba-b i l i t ~ , ~ which goes to zero at b = 1-85 a.u.2 0 6 - &04 -0 2 -0 30 60 90 120 150 1.50I1.0 -0 8 -////0 2 - /04-I l l l t l l l l l l l lT (degrees)FIG. 4.-Fractional contribution of H3 configurations to the total reactive cross section (J = 0,J’ = 0) for model (I) at 0.5 eV.For each value of7, all configuration anglesy less than7 are included(see text).To obtain an idea of the configurations of the three nuclei which make the dominantcontributions to reaction, we used model I and considered a quantity Tpa(DWB, z)defined bywith H(x) the Heaviside function [H(x) = 0, x <O ; H(x) = I, x> 01 and OSz <n ;thus, Tpa(DWB, n) = Tpa(DWB). The corresponding total cross section Sap(T)provides a semi-classical measure of the contribution to reaction for atom, moleculeorientations with y in the range between 0 and 2. The quantity S’aa(z)/Sab. for theJ = 0 to J’ = 0 reaction at an energy of 0-5 eV is plotted as a function of z in fig.4.Only small angles contribute ; i.e., 80 % of the cross section is obtained with y < 40”.In the quasi-classical calculation,2 the average value of y is 24” at the same relativeenergy. It is these results, and the molecular reorientation in the adiabatic treatment,which supply some justification for the linear model discussed belowM. KARPLUS A N D K . T. TANG 63LINEAR APPROXIMATIONThe linear model [eqn. (IS)] has the constant A as a strength parameter for the&function. This was chosen so that T,,(DWBL) = TB,(DWB) at 0.5 eV for the(J = 0, J‘ = 0) reaction and was kept fixed at the same value for all of the calculations.For J = 0 to J’ = 0, the total cross section in the energy range between 0.25 and3.3 eV is shown in fig.5. Fig. 6a and 6b show the total cross sections for J = 0to J’ = 0, 1, and 2 in the threshold and the intermediate energy regions. In fig. 7are given the sum of the linear model cross sections (J = 0-d’ = 0, 1,2) and for COWL-parison the quasi-classical result (J = O-+all J’). The most important point is thatthe quantum calculations yield a significantly higher effective threshold than doesthe quasi-classical treatment. This is reasonable when one considers that the sameI0 I. 0 2.0 3.0incident energy (eV)FIG. 5.Total reaction cross section for (J = 0, J’ = 0) and incident energies between 0-25 and3.3 eV from the linear model.initial molecular zero-point energy is present in both approaches, but that the quantumconstraints in the saddle-point region may provide a limit on the vibrational energyavailable for crossiog the barrier which does not exist for the classical trajectories.The much higher values reached by the quantum cross sections at large energies mayresult from the breakdown of the adiabatic approximation, which invalidates anenergy-independent choice of the strength parameter.Also, since the DWB methodis a perturbation procedure, it becomes less valid as the magnitude of the total crosssection increases.16Of interest in fig. 6a and b are the relative values of the cross sections for differentfinal rotational states. While at threshold the (J = 0, J’ = 0) cross section is largest,at most energies the ratios are (J = 0, J’ = l ) > ( J = 0, J’ = 2)>(J = 0, J’ = 0).From the quasi-classical calculations at energies below 0.75 eV, the result (J = 0,J’ = 1)>(J = 0, J’ = 2)>(J = 0, J’ = 0) has been obtained by making a corres-pondence between the final state angular momentum (in units of fi) and the nearestinteger value of J.Since the nuclear spin and rotational states of the H2 moleculeme coupled by the Pauli principle, the variation in the cross sections for differen64 QUANTUM MECHANICS OF H -I- H2 REACTIONrotational states is significant for an appropriately symmetrized calculation of thepara ( t 5. ) to ortho ( t t ) hydrogen conversion in the H+H, exchange reaction.In particular, the present results raise questions concerning the essentially classicalassumption l7 of a simple (3 : 1) ratio for the (para-ortho) against (para+para)rate constants.1.20.20-3 0.4 0.5 0.6incident energy (eV)0.3 04 0.5 06 0.7 0.8 0.9 1-0 1.1incident energy (eV)FIG. 6.-Total reaction cross sections as a function of incident energy from the linear model:- corresponds to J = 0, J’ = 0, -.-.corresponds to J = 0, J’ = 1 ; - .. - corresponds to J = 0,J’ = 2. (a) threshold energy region ; (b) intermediate energy regionM. KARPLUS AND K. T. TANG 6586c? e,..4 8 + 0g 4El22 2v).Mc,00.2 0-4 0.6 0-8incident energy (eV)FIG. 7.-Total reaction cross sections as a function of incident energy : - from linear model forJ = 0 to J’ = 0, 1 and 2 ; - - - from quasi-classical calculation for J = 0 to all J’.0 30 60 90 120 150 1808 (degrees)FIG.8.-Differential reaction cross section as a function of the scattering angle 8 from the linearmodel at an incident energy of 0.5 eV : - corresponds to J = 0, J’ = 0 ; -.-. corresponds toJ = 0 S = 1 ; -..- corresponds to J = 0, J’ = 2.66 QUANTUM MECHANICS OF H -I- Hz REACTIONThe differential cross sections at an incident energy of 0.5 eV for J = 0 to J' =0, I, 2 are shown in fig. 8. All the curves are similar to each other and to the completeDWB result (fig. 3). More interesting is fig. 9 which presents a plot of the variationwith incident energy of the form of the (J = 0, J' = 0) differential cross section (inarbitrary units). As the energy increases from 0.4 to 1.5 eV, the primary peak in thecross section gradually shifts in the forward direction and a secondary peak appears." 0 30 60 90 120 150 1800 (degrees)FIG.9.-Differential reaction cross section as a function of the scattering angle 0 for J = 0, S = 0at a series of incident energies from the linear model.Thus, the incoming atom is strongly repelled by the barrier for low incident energies,but tends to '' remember " its initial direction of motion as the incident energy becomesgreater than the barrier height (0-396 eV). This trend can be obtained from a simplesemi-classical model which relates the elastic and reactive scattering. Although ithas not yet been observed in the molecular case, exactly corresponding results arefound for the (d, p) stripping reaction in a Coulomb field as the energy increases froma few MeV to a few hundred MeV.18CONCLUSIONSA distorted wave Born (DWB) approximation has been utilized to examinerearrangement scattering in the H + H2 system.The large difference in the total crosssection between the two limiting cases which have been considered (i.e., the moleculeis unperturbed or adiabatically perturbed by the incoming atom), show thatmolecular perturbations and their effect on the two-body potential for relativemotion can play a significant role in reactions with an activation barrier.The results for a variety of reaction attributes (i.e., energy dependence of differentialcross section, cross-section variation with final rotational state, geometry of atoms inreactive region) suggest by their similarity to the available quasi-classical calculationswith the same interaction potential that quantum effects are not very important.However, the energy dependence of the total reaction cross section is different inthe quantum and the classical treatments; in particular the effective threshold ishigher in the quantum calculation.To determine whether the difference is real oM. KARPLUS AND K. T. TANG 67is the consequence of the approximations in the quantum formulation will requirefurther study.The differential cross section is found to be strongly backward peaked at lowenergies in correspondence with expectations for a strongly repulsive barrier. Asthe energy increases and the effect of the barrier decreases, the peak in the differentialcross section moves forward and secondary maxima appear.This shows thatmeasurement of the differential cross section as a function of energy would yieldinformation concerning the potential surface.The dependence of the total reaction cross section on the final rotational state(J’ = 0, 1, 2) has been examined for molecules initially in the ground state (J = 0).The results question the usual procedure for obtaining para-H2 to ortho-€3, conversionrate constants. However, more extensive and reliable cross section calculations areneeded before an unequivocal conclusion is possible.We thank Dr. Keiji Morokuma for assistance with some of the calculations.We are grateful to the Columbia University Computing Centre for their co-operationin making available the machine time used in the project.(a) S.Glasstone, K. J. Laidler and H. Eyring, TIze Theory of Rate Processes (McGraw-HillBook Company, Inc., New York, 1941) ; (b) H. S . Johnston, Gas Phase Reaction Rate Theory(Ronald Press, New York, 1966) ; (c) K. J. Laidler and J. C. Polanyi, Prog. Reaction Kinetics,1965, 3, 3.M. Karplus, R. N. Porter and R. D. Sharma, J. Chem. Physics, 1964,40,2033 ; 1965,43,3259.R. N. Porter and M. Karplus, J. Chem. Physics, 1964, 40, 1105.see ref. (l), particularly (l(b)), for a discussion of this point.S. Golden, J. Chem. Physics, 1954,22, 1938.I. Yasumori and S. Sato, J. Chem. Physics, 1951,22, 1938.C. Nyeland and T. Bak, Trans. Faraday SOC., 1965,61, 1239.D. Micha, Ark. Fysik, 1965, 30,425, 437.’ E. Mortenson and K. S. Pitzer, J. Chem. Soc., Spec. Publ. 1962, 16, 57.lo R. A. Marcus, J . Chem. Physics, 1966, 45,4493.l1 The work of Micha is most closely related to certain parts of the present treatment. Inparticular, the general model used by Micha is similar, in principle, to one of the models(linear model) described by us ; however, the additional approximations introduced by Michain the actual calculation raise serious doubts concerning the results.l2 (a) M. L. Goldberger and K. M. Watson, ColZision Theory(John Wiley and Som,Inc., New York,1964); (b) T. Y. Wu and T. Ohmura, Quantum Theory of Scattering (Prentice-Hall, Inc.,Englewood Cliffs, New Jersey, 1962) ; (c) A. Messiah, Quantum Mechanics (John Wiley andSons, Inc., New York, 1962), chap. XIX.l3 J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little and F. J. Corbato, Spheroidal WaveFunctions (John Wiley and Sons, Inc., New York, 1956).l4 see ref, (12), particularly chap. 4 of 12(a).K. T. Tang and M. Karplus, to be published.l6 A. C. Allison and A. Dalgarno, Proc. Physic. SOC., 1967, 90, 609.l7 A. Farkas, Orthohydrogen, Parahydrogen, and Heavy Hydrogen (Cambridge University Press,1935).J. R. Erskine, W. W. Buechner, H. A. Enge, Physic. Rev., 1962, 128, 720; L. C. Biedenharn,K. Boyer and M. Goldstein, P/zysic. Rea., 1956, 104, 383