CONNECTIONS BETWEEN MOLECULAR STRUCTURE AF?D CERTAIN MAGNETIC EFFECTS IN MOLECULES BY c. H. TOWNES," G. C . DOUSMANIS,* R. L. WHITE,* AND R. F. SCHWARZ Received 25th March, 1955 Several types of magnetic effects in molecules and their relation to molecular structure are discussed. Hyperfine structure in molecules with electronic angular momentum can give experi- mental determination of three independent parameters of the distribution of electronic angular momentum. These parameters are rather simply and directly related to the molecular electronic structure and hence afford critical tests for proposed structures. Fine structure for such molecules can also give useful information on electron distribution. 0 2 , NQ and OH are examples which will be discussed. A systematic investigation has been made of I J interactions in 12 molecules.Al- though these second-order effects are not as easily interpretable as are magnetic inter- actions in molecules with electronic angular momentum, it now appears possible to make an approximate prediction of the magnitude of I J interactions in common types of molecules. The interaction between an external magnetic field and rotation of a 12 molecule may be used to determine the orientation (sign) of the molecule's electric dipole moment. In order to eliminate obscuring effects such as L-uncoupling, it is necessary to measure the molecular g-factor for two isotopic species. Determination of the sign of molecular dipole moments by this method requires rather precise measurement of Zeeman effects, but appears practical for certain molecules.It is our purpose to ctiscuss several types of magnetic effects in molecules which may be observed by the techniques of microwave or radiofrequency spectroscopy, and which can be related to structural parameters of the molecules. These effects involve magnetic fine and hyperfine structure in ,molecules having electronic angular momentum, magnetic hyperfine (I J) interactions in molecules which are in 1C electronic states, and a method for determining the sign or orientation of the electronic dipole moments of a molecule by its interaction with an external magnetic field. 1. INTERPRETATION OF MAGNETIC FINE AND HYPERFINE INTERACTIONS IN MOLECULES Magnetic hyperfine structure in diatomic or linear molecules with electronic angular momentum has been discussed by Frosch and Foley 1 and shown to depend primarily on four parameters of the distribution of electronic angular momentum of the molecule. For a molecule in a 2na state, the interaction has the form IN TERMS OF THEIR ELECTRON STRUCTURE b + c 1 .J d(J+ %).)I* J J+',=.$ a - - _____ ( 2 ) J(J+ 1) * W(J+ 1) The positive sign applies to the A-doubled state of higher energy, and the negative sign to that of lower energy. For a 2ns state * Columbia University. Harvard University. 56C. H . TOWNES, G . C. DOUSMANIS, k . L. WHITE, A N D R . F . SCHWARZ 57 In these expressions, - pop1 3 cos2x + 16niuoc”’p2 (O), 31 b = - (L r3 - ’) I av. c = - ( 3popz 3 COS2 X - I r3 po is the Bohr magneton and p~ and I are the nuclear magnetic moment and spin respectively, @(O) is the density of electron spin (the probability density for an average electron spin of +) at the nucleus whose hyperfine interaction is being considered.X is the angle between the molecular axis rand the radius vector r from the nucleus to the electron which carries net angular momentum. The averages are to be taken over only the electron or electrons which contribute the angular momentum. For coefficients b, c and d, the average is over the net electron spin distribution, whereas for a, the average is over the net orbital angular momentum. The quantity d in (3) differs by a factor of two from that given by Frosch and Foley, but otherwise expressions (1) and (2) can be obtained directly from their work. Expressions equivalent to (1) and (2) for a 3C molecule have already been given.2 In this case the hyperfine structure does not depend on orbital angular momentum and there is no A-type doubling so that only the parameters b and c appear.The parameters a, b, c and d for magnetic hyperfine structure can afford rather precise and definite tests of proposed electronic structures for molecules. They are similar in many ways to the quadrupole coupling constant, eQ32V/3z2, which has already been widely used to examine the electronic structure of molecules. AS does 32V/3z2, they depend primarily on the part of the electronic distribution very near the nucleus. The magnetic interaction constants are, however, still more specific than the quadrupole coupling constant, and can give more detailed information.This is in part because there are three constants which may be experimentally determined rather than one, as in the quadrupole case, and partly because the magnetic hyperfine structure depends on the behaviour of only the electrons which carry angular momentum rather than on the entire charge dis- tribution about the nucleus. Hypesfine structure in 0 2 , due to the magnetic movement o f 0 1 7 has already been discussed.2.3 Experimental result for b and c in this molecule give 3 COG x - 1 ( -rr)av. = - 17.8 x 1024 cm-3, and $2(0) = 1.26 x 1024 cm-3. (4) The most reasonable structure for the O2 molecules involves one unpaired p r electron about each oxygen nucleus, For an electron in a pure atomic orbital of this type, the following values are expected (l/r3)av.= 34.5 x 1024 cm-3, (3 C O S ~ X - I)av, - 2 6, (3 “0”; - = -13.8 x 1024 cm-3, $2(0) = 0. (5) The measured values (4) and the theoretical ones (5) are in good qualitative agree- ment, but clearly show some quantitative disagreement.58 STRUCTURE AND MAGNETIC EFFECTS Consider first the value of $2(0) in (4). This is only 1/40 as large as the #2(0) which would be produced by an atomic 2s electron, and has what may be said to correspond to an admixture of 2.5 % 2s wavefunction. This small amount is not very different from the amount of mixing of configurations in atomic states,4 and hence is probably not a surprising detail of the molecular wavefunction. The discrepancy in the value of (3 coszp, - ljav. appears to be definitely larger than any error in evaluating (l/r3)av.for an atomic wavefunction, and corresponds to a molecular electronic wavefunction which is somewhat more flattened than expected in a plane through the oxygen nucleus and perpendicular to the molecular axis. Addition of pa orbitals or a few more highly excited atomic states would only increase the discrepancy. Overlap effects can increase the estimate of ( 73------) given in (5), but after these have been allowed for, the experi- mental value in (4) is still about 15 % larger than the value obtained from a prr orbital. Hence, although the hyperfine structure of 0 2 indicates that the expected electronic structure of 0 2 is largely correct, it gives good evidence that wave- functions for the unpaired electrons are somewhat more flattened in a plane than expected, and that they have a small amount of s character.A similar behaviour is found in NO, which affords the advantage of determining the value of ( l / ~ 3 ) ~ ~ . directly from experimental results. Hyperfine structure of N14O has been measured for both the n& and ns states,ss697 and that for NlsO in the n+ state.* These results give, from (l), (2) and (3), 3 cos2 x - 1 av. (l/r3)av. = 14.9 x lO24cm-3, 3 cos2 x - 837 + - +(O) = -3.3 x 1024~~-3. ( r3 l)av. 3 It must be remembered that the quantity (l/r3)av, applies to the distribution of orbital angular momentum, while the other two quantities apply to the spin distribution. If it is assumed that the spin and orbital angular momentum have the same value of (l/r3)av., then the first two quantities in (6) can be used in com- bination with the last to give (7) A first approximation to the structure of NO would be a combination of the N = O (4 and -N = O+ (b) where the unpaired electron is in a prr orbit about the N or the 0 atom for structures (a) and (b) respectively.The experimental value (7) of $2(0) indicates that the unpaired electron has at least a small amount of s character. In striking similarity to 0 2 , this corresponds to about 2.5 % of the value of #2(0) for a 2s electron. If the electron is in a pn atomic orbit about N, the value of (1/r3),. should be close to 22.5 x 10% cm-3, while it would be only 0.5 x 1024 cm-3 if the electron were in a pn orbit about the 0 atom. From the experimental value (6) of (l/r3)av., the impaired electron must have a probability of 0.65 of being found on the N atom.This corresponds to 65 % importance for structure (a) if only the two structures (a) and (b) are assumed. This is not unreasonably far from previous estimates 9 of the structure of NO. $2(0) = 0.85 x 1024 cm-3. two structuresc. H. TOWNES, G . C. DOUSMANIS, R . L. WHITE, AND R. F . SCHWARZ 59 From the measured value of (ll143)~~. and ~ ~ ~ x ) a v ~ - sin2X may be evaluated as 0.9 if the variables X and Y are considered separable. For a pn- atomic orbit, sin2 X = 0-8. As with 0 2 , it appears that the molecular electronic wavefunction is not far different from an atomic wavefunction, but is distinctly more flattened in a plane through the nucleus and perpendicular to the molecular axis. Information about the electronic structure of NO from the known quadrupole coupling constant of N14 in this molecule5 is not very precise.However, it is consistent with the structure indicated above. Magnetic hyperfine structure of OH due to the magnetic moment of H has also been studied. In this case it is clear that the electronic wavefunction near the H nucleus is considerably different in the OH molecule than in the H atom. Such a result is not unexpected because of the small charge on the H nucleus. It would be interesting to know the magnetic hyperfine interactions of 017 in both OH and in NO. Another type of magnetic interaction which depends on (l/r3)av. is the fine structure. Its possible usefulness was pointed out to one of the authors by R. S. Mulliken. Atomic fine structure is given by an expression A(W7 where Zi is an effective value of 2 near the nucleus.Since this f i e structure increases so rapidly near the nucleus, one finds that the molecular fine structure for a diatomic molecule is approximately A@*$) = a2Al + (1 - (9) where a2 is the probability for the electron to be in an atomic orbital about the fist atom with fine structure constant A1, and 1 - a2 is the probability that it is on the second atom with fine structure constant A2. Expression (9) assumes, of course, that the molecular wavefunction is a combination of two such atomic wavefunctions. Relation (9) affords by no means as specific or precise information as does magnetic hyperfine structure, partly because the fine structure depends on the atomic orbitals about both nuclei in the molecule.Furthermore, it measures (&/143)~~. rather than (l/r3)av. which is given by hyperhe structure. However, it allows some interesting comparisons with other information. Table 1 shows the type TABLE LATION TI ON BETWEEN ELECTRONIC STRUCTURES OF NO, OH, AND SH AND (The fine structure constant for a neutral 0 or S is multiplied by 1-20 or by 1/1-20 when the atom is positively or negatively charged 10 FTNFl STRUCTURJJ CONSTANTS fine structure constant % g ~ $ p ~ ! ~ ~ ~ ~ observed fine structure fine structure constant in cm-1 electronic structure N = O 868 63 0 - H - 153.9 46 S - H - 4087 56 -N = o+ 185 37 123.8 -0 H+ - 127.5 54 - 139.7 -s H+ - 341 44 - 378.6 of structures for the molecules NO, OH, and SH which predict fine struc- tures in agreement with the observed values.It may be seen that the fine stxucture gives more precise information for NO than for OH or SH because of the greater difference between the fine structure of the two proposed structures.60 STRUCTURE A N D MAGNETIC EFFECTS In addition, the H has such a small nuclear charge that expression (9) may be a rather poor approximation. The percentage importance obtained for the two NO structures is in remarkable agreement with the values obtained from hyperfine s tmcture. 2. SOME OBSERVATIONS ON MAGNETIC HYPERFINE (FJ) INTERACTIONS IN 11 Interactions between the magnetic fields produced by rotation of 12 molecules and the magnetic moments of nuclei in the molecules have been subject to measurements for some time.Their interpretation is rather complex since they depend in part on I-uncoupling or excitation of valence electrons by molecular rotation. A systematic study of this type of interaction indicates, however, that some general rules about its behaviour can be stated. The magnetic field at a given nucleus in a molecule may be considered as the sum of the fields produced by motion of the valence electrons, and that due to rigid rotation of the remaining positive ions considered as point charges at the positions of the appropriate nuclei. These two sources produce fields of opposite sign because of the opposite sign of the charges involved. In a linear molecule, this energy of interaction between the molecular rotation and the magnetic moment of a nucleus may be written 11 MOLECULES where B = the molecular rotational constant, pz = the nuclear magnetic moment, po = the Bohr magneton, Lx = operator for a component of the electronic angular momentum of the valence electrons which is perpendicular to the molecular axis, r = distance from nucleus to valence electron, W, - WO = difference in energy between ground state and nth excited electronic state, qs = net charge on sth atom after removal of valence electrons, rs = distance from nucleus to sth atom, c = velocity of light, J = angular momentum in units of h due to molecular rotation, I = spin of nucleus.The first term in the brackets of (10) is due to the valence electrons and the second to the ions or rigid frame of the molecule. Again, because of the appearance of l/r3, the most important contributions to hyperfine structure are usually due to the electron distribution very near the nucleus. Furthermore, the valence electrons usually produce an effect which is much larger than that of the ions or “rigid frame ” of the molecule.It will be seen from table 2 that the valence electrons dominate in all cases except for hydrogen, where l/r3 is not very large for the valence electrons because the nuclear charge is so small. Expression (10) has the form AW= CTI J, ( 1 1 ) where CZ is a coupling constant which is a measure of the hyperfine energy. One may expect in most cases that for the lowest excited electronic states, the will be approximately the average value of l/r3 for aC . H. TOWNES, G . c. DOUSMANIS, R . L . WHITE, AND R .F. SCHWARZ 61 CII ~BPIPO (1 /r3>av. valence p electron. Therefore the reduced coupling constant CR = should be approximately equal to 2 I (O I Lx and depend on the molecular structure in a somewhat predictable way. Examination of measured values of n wz- wo CI indicates that such an expectation is correct. Table 2 lists the various hyperfine interaction constants CI for linear molecules which have been measured by the authors or by others. With the help of this table the following observations may be made. TABLE 2.-MAG"E HYPERFINE STRUCTURE OF LINEAR 1x MOLECULES DUE TO ROTATION. Theenergyis AW= CII* J. molecule ref. nucleus sign of @I CI (14s) (rigid frame) I-p2 a H + -113.90 - 203 Li7F19 b F19 + 32.9 f 0-1 -2.1 1.87 x 10-44 CsF19 d F19 + 16 + 2 -0.2 6.65 X 10-44 C135F19 f C135 + 22 + 3 -0.8 3.3 x 10-43 cs g s33 + 19 f 15 -0.6 4.1 X 10-43 T1207C135 h ~ 1 2 0 7 + 73 rt 2 + 0.6 6.35 X T1207C135 h C135 + 1.4 &O-1 -O*IO 1-1 x 10-43 017C12S32 f 017 - -40 & 1.5 2.8 X 10-43 016C12S33 f S33 + 2 r t l 1-8 x 10-43 016C12Se79 f Se79 -3.2 f 1.0 3.8 X 10-43 HCN14 i N14 + 10 & 4 3.0 X 10-43 C135CN14 f N14 + 2.5 & 0.8 5.6 x 10-43 CP*CNls f C13S + 3-5 f 0.6 2.0 x 10-43 RbssF19 C F19 4- 11 r t 3 -0.24 3.85 X 10-44 DI e I127 + 140 - 1-03 6.7 X 10-44 a Harrick, Barnes, Bray and Ramsey, Physic.Rev., 1953,90,260 b Schwartz and Trischka, Physic. Rev., 1952, 88, 1085. c Hughes and Grabner, Physic. Rev., 1950,79, 314. d Trischka, Physic. Rev., 1948,74,718. e Burrus and Gordy, Physic. Rev., 1953, 92, 1437. f R. L. White, Thesis (Columbia University, 1954).g Mockler and Bird, Physic. Rev., to be published. h Carlson, Lee and Fabricand, Physic. Rev., 1952,85, 784. i Klein and Nethercot, Quart. Report (Columbia Rad. Lab., Oct. 30, 1953). (i) The predominant contribution to Cz in all cases except for H comes from the valence electrons. (ii) Values of the quantity CR do not differ widely for different types of mole- cules or atoms. The relatively small values of this quantity for the alkali halides is probably connected with the large amount of energy required for electronic excitation of these molecules, (iii) For a given atom it may be assumed that the matrix elements (0 1 LX I n) or (0 I aL, I n) are approximately the same in a series of chemically similar mole- cules. The regular increase in CR - 2 1 (' I Lx ') Iz indicated by the value of CI for F19 in LiF, RbF, and CsF is hence evidently due to the decreasing separations w, - Wo of the energy levels which can be expected for the larger molecules. Another similar series is provided by values of CZ for C135 in CH3C1, SiH3Cl and GeH3C1, which are not listed in the table, but which show the same behaviour, n wn- wo62 STRUCTURE AND MAGNETIC EFFECTS (iv) For 017 and S33 in the same molecule OCS, one would expect CR to be essentially the same, since 0 and S are very similar chemically.The table shows that such an expectation is correct. On the other hand T1 and C1 in the same molecule TlCl and C1 and N in ClCN have very different values of this quantity, since their electronic surroundings are quite dissimilar.These differences between nuclei can give a rough measure of the distribution of orbital angular momentum in the molecule due to I-uncoupling. 3. A POSSIBLE TECHNIQUE FOR DETERMINATION OF THE SIGNS OF MOLECULAR ELECTRIC DIPOLE MOMENTS FROM ZEEMAN EFFECTS Although the orientation or sign of dipole moments in many molecules is believed known from theoretical or indirect reasoning, there seems to be no way by which the sign of a molecular electric dipole moment has been directly measured. A fairly direct determination of the signs of electric dipole moments is poss- ibk from precise measurement of Zeemaii effects and use of the technique described below, Consider as a simple example a linear molecule composed of a series of fixed charges Nse arranged along the x axis, The magnetic moment of such a system when rotating with angular momentum Q about the centre of mass is where A is the moment of inertia and xs the distance of the charge Ns from the centre of gravity.The molecular g-factor, i.e. the ratio of p in nuclear magnetons to 52 in units of h, is (1 3) NsMXs2 g = Z 7 9 S S =?A'-- A' s A' s g ' = C A' where M is the proton mass. is shifted by an amount Ax and A is changed to A', the new g-factor becomes If, now, the mass of one of the charges is changed so that the centre of gravity 2 N . s . (14) NsM(Xs - AX)2 NSMXs2 2MAX cNsxs + M(Ax)2 Since e z N s is the total moiecular charge which is zero, and e z NsXs is the molecular dipole moment Dx, (14) becomes S S gfz7-.--. Ag 2MAxDx, A eA' or NOW, if g and g' are determined by measurement of Zeeman effects, and A, A' and Ax are known from the molecular structure, expression (15) allows a deter- mination of Dx. Basically the same relation as (15) can be proved for an actual molecule.The magnetic moment of a molecule due to rotation may be described in terms of a tensor .A? of second rank. Thus the magnetic moment along a principal axes x of inertia may be written E l 3 px = Ax&% + A x y Q y + =.+?TzQz where dxx, etc., are components of the tensor and Qx, etc., are components of the angular momentum along the principal axes of inertia. Ix, etc., are the prin- cipal moments of inertia and hJx, etc., are the corresponding angular momenta.c. H . TOWNES, G . c. DOUSMANIS, R . L . WHITE, AND R . F . SCHWARZ 63 In many simple molecules the principal axes of dl coincide with the principal axes of inertia, so that expressions (16) can be simplified.The components of &referred to its principal axes have the form 12913 where eZk is the charge on nucleus k, ylc and Zk its co-ordinates with respect to the centre of mass, and m is the electron mass. Lx is the operator for electronic angular momentum and W, - WO the difference in energy between the ground state and some excited state indicated by n. Consider now the change in AXx due to a change in the centre of mass which might be produced by an isotopic substitution in the molecule. The change in 2 ’ (O I I’ may be readily obtained from the fact that the diamagnetic sus- n wn- wo ceptibility X has been shown to be independent of the origin,l4 and X has the form I where c is the velocity of light and other symbols have the same meaning as in (10).The first summation is over all electrons. Letting the new co-ordinates be Y’ = JJ + AY z~ = + aZ, we have from (17) and the invariance of X, - A x x = 2MEZk(YkAYk $. ZkAZk) k 2M - I ( 0 I ~ i A y + ziAz I O)] = - -(DyAy + DzAz), (19) i e where D, and Dz are components of the electric dipole moment due to all charges in the molecule. The electron charge e has, as usual, a negative sign. If the principal axes of & or of A d o not coincide with the principal axes of inertia, a rotation of axes is necessary which transforms these tensors in the usualy way. For simplicity, we shall assume that no such rotation is necessary in the following discussion.From (16) and (17), g, and g,’ may be easily related : This is essentially the same relation as (15), which was derived above from less rigorous assumptions. For the isotopic substitution H20 -+ HDO, changes in g-factors due to the molecular dipole moment are about 0.01, or about 2 % of the total g-factors. This appears to be just slightly larger than errors in the available measurements of the g-factors for H2O and HDO, so that a definite determination of the dipole moment sign from this small effect has not yet been obtained. For a diatomic molecule, the change in g-factor which must be detected in order to determine the dipole moment sign is from (20) approximately (21) where M is the proton mass, MI and M2 are masses of the two atoms, and r the internuclear distance.Consider as an example the molecules C12016 and C12018, for which the dipole moment has the very small value 0.1 D. In this case Ag w 0.0002. The g-factor for CO probably has a magnitude near 0.05, so that SL fractional accuracy near 11500 is needed to detect the desired effect. 2MAMI D (Ml + M2)Mler Ag w64 GENERAL DISCUSSION One may raise the objection that an isotopic substitution in CO will slightly change the intemuclear distance and the electronic wavefunctions, and hence produce another type of change in g which may mask the dipole moment effect. Similarly changes in average internuclear distance may affect the dipole sign determination from the isotopic substitution H20 -+ HDO. This type of change due to a variation in average internuclear distance alone can be measured in simple cases such as CO by determining the g-factor of CO in an excited vibrational state, In the excited vibrational state, the change in average intemuclear distance is much larger than that for isotopic substitution in the ground vibrational state. Hence any such variation in g-factor with isotopic substitution may be determined and taken into account. Measurement of the signs of dipole moments of molecules by the technique described above is not easy, but it appears practical in a number of cases. One of the authors (R. F. S.) is much indebted to Prof. J. H. van Vleck for discussion and aid. 1 Frosch and Foley, Physic. Rev., 1952, 88, 1337. 2 Miller and Townes, Physic. Rev., 1953, 90, 537. 3 Miller, Townes and Kotani, Physic. Rev., 1953, 90, 542. 4 cf. Hartree, Hartree and Swirles, Phil. Trans. Roy. Soc. A, 1939,238, 229. 5 Beringer and Castle, Physic. Rev., 1950, 78, 581. Beringer, Rawson and Henry, 6 Gallagher, Bedard and Johnson, Physic. Rev., 1954,93,729. 7 Burrus and Gordy, Physic. Rev,, 1953,92, 1437. 8 Gallagher, King and Johnson, Bull. Amer. Physic. Soc., 1955,30, no. 2, 28. 10 Dailey and Townes, J. Chem. Physics, 1955, 23, 118. 11 To~nes and Schawlow, Microwave Spectroscopy (McGraw-Hill, New York), to be 12 Eshbach and Strandberg, Physic. Rev., 1952,85,24. 13 R. F. Schwarz, Thesis (Harvard University, 1952). 14 van Vleck, Electric and Magnetic Susceptibilities (Oxford University Press, 1932). Physic. Rev., 1954,94, 343. Pauling, m e Nature of the Chemical Bond (Cornell Univ. Press, Ithica, N.Y., 1945). published.