Dipole Moments and Polarizabilities of Electronically Excited Molecules Through the Kerr Effect BY A. D. BUCKINGHAM AND DAVID A. DOWS* Inorganic Chemistry Laboratory, University of Oxford Received 14th January, 1963 Formulae are derived for the Kerr constant of a diatomic gas for frequencies close to those corresponding to electronic transitions. The magnitude of the Kerr constant is proportional to the square of the transition dipole moment and is determined by the dipole moments and polar- izabilities of the relevant electronic states. Close to an absorption line, the rapidly changing Kerr effect may be a million times that for transparent regions of the spectrum. Transitions which are normally forbidden, but which become allowed in the electric field, should also be associated with strong Kerr dispersion.The dispersion of the Kerr effect would seem to offer better possi- bilities than the straightforward Stark effect for measuring the dipole moment and the polarizability of an electronically excited molecule, and may also provide a means of obtaining accurate transition moments. The dipole moment p is one of the very few observables depending solely on an unperturbed molecular wave function. The expectation value (Yi [ p, 1 Y,) of the one-electron operator pz = xeizi is therefore a potentially useful test of the accuracy of approximate wave functions Yn. Dipole moments are accurately known for the ground states of molecules through dielectric constant measurements and through Stark splittings of pure rotational spectra ; for some diatomic molecules (particularly the alkali halides) the slight dependence of p on the vibrational quantum number II has been determined by molecular beam techniques1 However, little is known of the dipole moments of electronically excited molecules ; the explanation of this is that the obvious method of measurement, the Stark effect in electronic spectroscopy, has not yet yielded positive results.The splittings for reasonable field strengths (about 100 e.s.u. = 30 kV cm-1) are very small by ultra-violet and infra-red standards ; for X-X transitions, they may be about 10-2 cm-1 for dipolar molecules (p # 0) and 10-5 cm-1 for non-dipolar molecules. For Il or A states there is a first-order Stark effect in dipolar molecules (provided the Stark splitting is large compared to the A-type doubling) and the splittings may be about 10-1 cm-1.The difficulties of measuring Stark splittings have been discussed by Herzberg.2 Recently, Kopelman and Klemperer 3 showed that earlier failures to detect Stark splittings in the spectra of CO and CO+ indicate that the dipole moments of the appropriate states are less than about 0.1 x 10-18 e.s.u. Another experiment dependent upon the interaction of molecules with an electric field is the electro-optical Kerr effect. It has been shown 49 5 how measurements of the Kerr constant in the vicinity of absorption bands would isolate the con- tribution of the relevant excited state to the induced anisotropy in the refractive index. Since the Kerr effect largely depends on the partial orientation of molecules, its dispersion should also be a convenient tool for determining the polarization of the i * National Science Foundation Senior Postdoctoral Fellow ; permanent address : Department of Chemistry, University of Southern California, Los Angela 7, California.48A. D. BUCKINGHAM AND D. A. DOWS 49 transit.ion with respect to molecule-fixed axes, and hence the symmetry of the excited state. In this paper the theory of the dispersion of the Kerr constant is developed and the potentialities of the experiment for yielding information about excited state dipole moments, electric polarizabilities and transition moments, are assessed. The observable in Kerr's experiment is the phase difference 6 between the electric vector components parallel and perpendicular to an electric field I; = F'z in a beam of light, initially plane polarized at 45" to F, after it has travelled a distance I in the medium in the field.If nx and nz are the refractive indices in the X and 2 directions, where v is the frequency of the light in cm-1, Nn the mean number of molecules in unit volume in the state Yn, and the molecular polarizability tensor (rctJn is 6 hcv;' = Writ- Wn is the energy difference between the n'th and nth eigenstates, and the sum is over all states n ' f n . The frequencies and wave functions in eqn. (3) are those applicable to the molecule in the presence of F . (ntq)$ = -(nr,)Et is the contribution of the n'th excited state to (neq)n. Eqn. (2) and (3) show that there is a strong dispersion of the Kerr constant in the vicinity of the resonant frequency v:', and that for v nearly equal to v::, 6 is domin- ated by the transition Yn+-Yn (provided the population Nn is not negligible).The awkward summation in eqn. (3) is effectively eliminated, and each excited state's contribution to 6 is isolated. The well-known theory of the Kerr effect treats the rotation of the molecules classically and leads to a temperature dependent 6. However, when rotational structure is observed, it is not permissible to use classical mechanics, and the effects of F on the particular states involved must be considered. Actually P splits the degeneracy of the magnetic states, but as these splittings cannot usually be resolved, 6 is determined by the weighted average over all the M states : where z and z' are the internal quantum numbers and JKM and J'K'M' the rotational quantum numbers.Eqn. (2) can be written where the summation in ( 5 ) is over all transitions Ynp+-Yn. It is convenient to distinguish the internal quantum numbers 'I: and the rotational quantum numbers JKM so that Y,. and Yn become Y(z'J'K'M') and Y(zJKM). Then, for frequencies v nearly equal to v:'iK', the phase difference 6 is a sum over all M and M' : and this is the basic equation of the full quantum theory of the Kerr effect.50 POLARIZABILITIES OF ELECTRONICALLY EXCITED MOLECULES Z-Z TRANSITIONS For a linear rotating molecule, the hamiltonian for the interaction of a particular (7) internal state z with the static field F can be written H' = - PLF cos 8 - *(a 11 - al)F2((3 C O S ~ 0 - 3) - ~(CX 11 + 2a,)F2, or, if the internal state undergoes a transition, N' = -pzFZ, ( 7 4 where all and aL are the static polarizabilities parallel and perpendicular to the axis of the molecule in the state z, and 8 the angle between this axis and F.The perturbed energies and wave functions are, to second order in F, +2a,)F2, (8) J 2 + J - 3 M 2 wTJM = wTJ+F2(2J-1)(2J+3) <z"J"M I H' I z J M ) Y $k = Y , j M - C' Y , " j " M + f "J" hcv:jJ'' 1 (z"J"A4 I H' I T J M ) ~ (9) - C' 2 2 T"J" 2 ~ T J M - 2r"J" (vrJ The relevant matrix elements are given in ref. (4), and the full formula for the con- tribution of transitions YT8p--YfJ to S is q;J' = 8n2 IvN, F2 J 2 + J - 3 M 2 ( I l 2 hc(2J + 1) 2. (1 - kT (25 - 1)(2J + 3) 2 hcBJ(J + 1) 3 2hcB'J'(J' + 1)- 3 J 2 + J - 3 M 2 - In the absence of F, there is a selection rule restricting transitions to the P and R branches, with J' = J - 1 and J' = J+ 1.However, the field perturbs the rotational wave functions, and Kerr dispersion should also be detectable in Q, 0 and S branches where J' = J, J - 2 and J+2. For the R(J- 1) and P(J) lines, [L vo-v -kL][.-J+%]] v,+v (I2)A. D. BUCKINGHAM AND D. A. DOWS 51 where - 1 P‘2 P’ 2 (a 11 - al)’J(J + 1) A j = u p [ { 20hc hcB’(2J - l)-hcB(2J+ 1) ) - s ( z J - l ) (‘11 -alV(J-1)}], (13) (2J+ 1) PP‘ 1- P’’ { 8J3 -2J2 - 1GJ + 3 B j = ( j + i ) 120h2~2B’2 J(J+ 1)(2J-1)2 30h2c2BB’j+ (14) 8J3+2J2-16J-3 CXII -a,)’J(J-l) (a], -al)J(J+ 1 ) -- J(J - 1)(2J + 1)2 }-((1OhcBr(2J- 1)2 lOhcB(2J + 1)2 ~ For the induced S(J-2) and O(J) lines, And for the induced Q(J) lines II-I: TRANSITIONS If AfO, the hamiltonian (7) produces a perturbed energy proportional to the first power of P provided pf‘ is much greater than the A-type doubling energy.* For a JJ state pFM p2F2 J 2 - M 2 K,J,K= I,M - W,J,K=I - J(J+ ~ l)+={J(ZJ- 1)(2J+ 1) ( I - $ ) - ( J + I ) ~ - M ~ 1 (CX 11 - ctl)F2(J2 + J - 3M2) (J + 1)(2J + 1)(2J + 3)(’ -m)} - - 3(2J - 1)(2J + 3) * A-type doubling in the lowest J-states, which contribute most of the Kerr dispersion, is generally much smaller than the values of pF which can easily be obtained (ref.(2), p. 254). An exception may occur for the hydrides.52 In Il-E transitions, the full expression for 6:yK=l is complicated by the large number of terms. However, the first-order term in eqn.(21) provides the leading term in 6, which is proportional to (vg-v)-3. The other terms in 6 are pro- portional to (vg -v)-2 and (vg -v)-I and hence normally less important for frequencies close to V O . POLARIZABILITIES OF ELECTRONICALLY EXCITED MOLECULES The leading term is given by and for the Q(J), R(J- 1) and P(J) lines, If the molecules are non-dipolar, the formulae are simpler, and complete ex- pressions for the Q(J), R(J- 1) and P(J) lines are [& vo-v + v,+v L][. - J f %I}> (28) where (J+ 1) (a11 -a,)'(J-1)(J2+3J- 1) - (a11 - [ B'J2(2J - 1)2 E j = ~ lOhcA. D . BUCKINGHAM A N D D. A. DOWS 53 NON-ROTATING MOLECULES In solids, to a first approximation, the rotation of molecules is suppressed and only internal states (electronic and vibrational) can be excited.However, the dipole p can still interact with F, and for a fixed configuration in which the molecular axis makes an angle 8 with F, The leading contribution to 6 near vg is wqq = w, - p~ cos e - +(all - cos2 e - 3) - $(a,, + 2 a , ) ~ ~ . (32) where L(n, E ) = (n2+2)2(E+2)2/81n, (34) approximately represents the effects of long-range interactions on the local field acting on a molecule in a medium of refractive index n and static dielectric constant E ; for a dilute gas L(n, E ) = 1. If the molecule is non-dipolar, INDUCED INTERNAL TRANSITIONS If (z I p I 7') = 0, the transition z ' - ~ is forbidden when F = 0, but the per- turbation of eqn. (7a) can induce a transition probability proportional to F2. If p = p' = (z I p I z'} = 0, there are induced Z-X transitions (e.g., lZg--lCg transi- tions in homonuclear diatomic molecules) with Q, 0 and S branches, and with Kerr dispersions 8 5 5 2 = where (393 and (a): = +(ccII + 2aL):, and z is parallel to the inter-nuclear axis of the molecule. '' Transition polarizabilities " (~II): and (al):' may make small contributions to 6 when (z I p I z'} # 0.DISCUSSION All the above dispersion formulae have been derived on the assumption that the line-width is negligible. In practice 6 would not go to +a as v goes through54 POLARIZABILITIES OF ELECTRONICALLY EXCITED MOLECULES VO. If the absorption line is approximately Lorentzian in shape, with a width at half maximum height of 2A, then (vo-v)-1 must be replaced by (v~-v)/[(v~-v)~+ A2]. The shapes of ((vo-v)/[(vo--v)~+ A2])a near vo for n = 1, 2, and 3 are shown in fig.1. The magnitude of the contributions of the different rotational levels to S generally decreases rapidly as J increases since the perturbation to the more rapidly rotating states is small. For large J values in X-X and If-C transitions, 6 is proportional to J-2 when p#O ; however, in non-dipolar molecules, 6 is proportional to J". In addition the population N,, varies with J. FIG. 1.-Shapes of {(vo-v)/[(vo -v)2+ A2]}n. Except for large J, or else at very low temperatures, the term in (kT)-1, arising from the influence of the field on the populations of the different M states, is negligible. This temperature-dependent orientation normally dominates the Kerr constant of polar fluids in the visible. The well-known term (all - aL)p2/k2T2 in the classical Kerr constant is replaced, in the II-Z case, by (z I p I ~')2p'2/h3~3(vo-v)3, so that the Kerr constant should be greatly enhanced for VWVO ; in X-X transitions, the corresponding factor is (z I p ~')2p2/h3~3B(vo-v)2, which, for (VO-v) = 5 cm-1, is about 106 times the classical expression. Hence very large, and therefore easily detectable, Kerr effects should be obtainable in the vicinity of the absorption bands of polar gases.As an illustration of the magnitude, shape and complexity of the Kerr effect in a vibronic band, fig. 2 shows 6 for a X-X transition calculated with the following constants : p' = p = 10-18 e.s.u., B' = B = 5 cm-1, A = lcm-1,vo = 3 x 104 cm-1, (z I p I T') = 10-18 e.s.u., T = 300"K, I = 1 cm, and p = 0.02 atm.6, in radians, is very large. In this calculation, only the dipolar terms in AJ and BJ were considered for the P and R lines ; terms in the polarizabilities and in kT are negligible. The electric field-induced Q-branch is a prominent feature of the Kerr dis- persion. From the inequality in magnitude of the two peaks at a given R- or P- branch line, the BJ terms (in (vo-v)-~) are seen to be significant. The inequality of the two peaks arises because the BJ term (like the induced Q-branch term) is ofA. D. BUCKINGHAM AND D. A. DOWS 55 odd symmetry about the line centre, and thus contributes in opposite senses to the two peaks arising from the AJ term. Fig. 2 shows a pattern of Kerr dispersion which is antisymmetric about the band origin. This situation arises because the dipole moments and rotational constants of the two electronic states were taken to be equal.Fig. 3 shows the effect of changing p‘ to 2 x 10-18 e.s.u., all other parameters being held constant. The distortion of the Kerr dispersion is remarkable ; the P(1) line has nearly disappeared, all P-branch lines have changed sign, the Q-branch intensity has more than doubled, and the R(1) line has quintupled in intensity. m 0.04-- 0.02,- b Y -0.02 -- It 10) I FIG. 2.-Kerr effect dispersion in a vibronic band. Parameters are given in the text ; 6 is in radians. The magnitude of the Kerr effect near the rotational lines (6 - 0.1 radian for the conditions specified) indicates the feasibility of measurements ; phase differences as small as 10-7 are detectable.7 The polarizability contribution is normally negligible in polar molecules, being about 10-3 of the dipolar term, but in non- polar molecules it should be measurable.The different frequency dependence of the A J and (BJ+ CJ/kT) contributions to 6 permits each to be evaluated. Thus the sum and difference of 6 at the turning points (I VO-v I = A) gives AJ and (BJ+ CJIkT). At the turning points the sample is ab- sorbing light with one-half its maximum absorptivity, and if the sample is too concentrated it will not be possible to measure 6 at these points.* Also, for narrow lines the turning points will not be resolved. It should, however, still be possible to separate the contributions to 6 by addition and subtraction at larger differences from the line centre.For Z-Z transitions, it is possible to determine the dipole moment of the excited state by relative measurements only. The ratio of the AJ terms for cor- responding lines (e.g., R(J- 1) and P(J)) is given by Aj (2J- l ) ~ -(2J+ 1)~‘ A - j ( 2 J + l ) ~ - ( 2 J - l ) ~ ” -- = * For the parameters used in fig. 2, a pressure reduction of perhaps 100-fold would be necessary before the turning points could be measured.where u = p2/B and uf = p’2/B’. Thus, if p, B and B’ are known, p’ is determined from a relative measurement of the AJ factors.* This dependence of the R(J- l)/P(J) ratio on p’ and p is illustrated by comparing fig. 2 with fig. 3, where only p’ has changed. When p’ has been obtained by these relative measurements, the absolute value of 6 determines (z I p I z’), the transition moment for the vibronic band.Oel t R(O1 I R I I ) I FIG. 3.-Kerr effect dispersion in a vibronic band. Parameters are given in the text. The transition moment is thus obtainable from Kerr measurements on X-X bands. However, formulae (23)-(25) for II-X transitions do not allow independent determination of p’ and (z I p IT’); it seems probable, though, that the terms in (vo-v)-2 and (vo-v)-~ might make both accessible, as in X-X transitions. The leading term in the Kerr dispersion formulae for a Z+lI transition, on the other hand, involves only the ground state dipole moment and (z I p I z’) ; the transition moment, therefore, comes from the leading term, and p’ from the other terms. The transition moment (z I p I 7’) corresponds to a vibronic state change, i.e., it is approximately given by the product of the electronic transition moment and the vibrational overlap integral.The Kerr dispersion, which involves electric dipole matrix elements only, is intimately related to the intensity of the electronic transition, and should be smaller by several orders of magnitude for, say, a spin-forbidden transition. However, the amount of sample in the path may in those cases be increased (in principle) by the same factor without absorbing too much light. Thus, Kerr dispersion at forbidden transition frequencies should also be investigated. The exceptionally large phase shifts predicted by this theory suggest that relatively small amounts of sample are necessary for the measurements. It may be possible, therefore, to investigate molecules obtainable only at high temperatures or in reacting systems.Equations for the Kerr dispersion in the vicinity of vibration-rotation bands in the infra-red region have been given before.4 If, in the above equations, p’ is * Implicit in this statement is the assumption that the line widths for R(J-1) and P(J) are the same, which is probably the case.A. D. BUCKINGHAM AND D. A . DOWS 57 put equal to p, and B' equal to B, the vibration-rotation formulae are obtained; * they are complete, and include the small l/kT terms not given in the previous paper. The transition moment (z I p I z'} to be used for the infra-red bands is that con- n'ecting two vibrational states of the ground electronic state ; due to the low intensity of vibrational transitions, the Kerr dispersion will be considerably less than for electronic transitions.If the permanent dipole moment p is substituted for (z I p I TI), and allowance made for the population of excited rotational states, the R and S branch equations then apply to the pure rotational spectrum, and the Kerr dispersion may be of the same order of magnitude there as in the infra-red. Fig. 2 is appropriate to a vibration-rotation band, except that (z I p I Q2, and therefore the scale for 6, would have to be reduced by about three orders of magnitude. In the vibration-rotation region the Kerr dispersion should be closely antisymmetric about the band origin. While the results of this paper concern diatomic molecules only, the extension to polyatomics should involve no fundamentally different principles. Since the selection of stable gaseous diatomics with transitions from the ground state occurring in reasonable spectral regions is limited, it is probable that such extension will soon be necessary. Financial assistance from the D.S.I.R., in the form of a special research grant, * In the earlier work 4 the equation for the Kerr dispersion near the R(J) line was given with is gratefully acknowledged. the wrong sign. 1 Kusch and Hughes, Handbuch der Physik, Band XXXVII/l (Springer-Verlag, Berlin, 1959), p. 1. 2 Herzberg, Spectra of Diatomic Molecules (Van Nostrand, New York, 1950), p. 307. 3 Kopelman and Klemperer, J. Chem. Physics, 1962, 36, 1693. 4 Buckingham, Proc. Roy. Soc. A , 1962, 267,271. 5 Charney and Halford, J. Chem. Physics, 1958, 29, 221. 6 Eyring, Walter and Kimball, Quantum Chemistry (Wiley, New York, 1944), p. 121. 7 Badoz, J. phys. radium, 1956,17, 143A.