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Born-Oppenheimer Breakdown
Spectroscopic Effects of a Breakdown of the Born-Oppenheimer Approximation S. J. Strickler Department of Chemistry, University of Colorado,Boulder, Colorado 80309
(Received February 17, 1976)
Publication costs assisted by the National Science Foundation
Some effects of a breakdown of the Born-Oppenheimer approximation on vibronic intensities in electronic spectra are discussed. The treatment starts from “true Born-Oppenheimer” electronic wave functions which follow the nuclei in their motion. A simple conceptual picture is derived relating the displacement of the electron distribution to that of the nuclei during a vibration. This picture is used to predict qualitative spectroscopic consequences. A calculation based on CNDO/S wave functions is presented for the lowest singletsinglet transition of naphthalene. The results are in a t least qualitative agreement with experiment.
Introduction The Schroedinger equation for a molecule may be written:
In their pioneering work on vibronic effects in the spectra of polyatomic molecules, Herzberg and Teller3 made use of a so-called “crude Born-Oppenheimer” basis, starting from electronic wave functions at the equilibrium position and expanding the wave functions at other configurations in terms of them:
x * ( x , X ) = E*(x,X)
(1) Here x; represents a Cartesian coordinate of one of the n electrons and X , a Cartesian coordinate for one of the N nuclei, having mass M,. Born and Oppenheimerl (BO) showed that it is often possible to write a wave function
* ( X , X ) = J/(.,X)4(X)
(2)
a product of an electronic wave function IC/ and a nuclear wave function 4, which is a good approximation to the true wave function. They gave a prescription for finding this wave function; it involves solving the electronic part of the problem for nuclei fixed at each geometry, X, thus obtaining J / ( x , X ) and an energy V ( X ) ,and then solving for the motion of the nuclei in the potential V ( X )to get +(X). The coordinates X cover vibrational, rotational, and translational motions of the molecule. This paper will be concerned only with vibrational motions. It is possible to transform from the Cartesian coordinates to vibrational normal coordinates, Q (including rotational and translational coordinates for completeness), by the methods of normal coordinate analysk2 The result is to make the replacement in the nuclear kinetic energy part of the Hamiltonian operator: (3)
We will write the wave functions as J/(x,Q)and 4(Q),thinking of them in this paper as functions of vibrational normal coordinates. When the nuclear kinetic energy terms (3) in the Hamiltonian operate on a product function ( 2 ) the result is three types of terms: second derivatives of 4, second derivatives of +, and products of first derivatives of 4 and The essence of the BO approximation is to neglect the latter two terms since the electronic wave function varies only slowly with Q. I t is these two types of terms which make ( 2 )only an approximate solution to the Schroedinger equation and which must be considered in dealing with a breakdown of the Born-Oppenheimer approximation.
+.
J
The coefficients Cij (Q)are generally obtained by perturbation theory. Herzberg and Teller showed that, unless the electronic states are close in energy, mixing coefficients due to BO breakdown are much smaller than those due to ordinary vibronic mixing. The ratio is of the order of the vibration frequency divided by the electronic energy separation. Most subsequent treatments of vibronic effects and BO breakdown have been based on expansions such as (4). However, these expansions are often slow to converge because it takes many terms to express an electron distribution about an atom in one position in terms of orbitals centered about another position. The use of an expansion truncated after a few terms can lead to serious difficulties. Recently it has become feasible to do electronic wave function calculations in some suitable approximations at a series of nuclear configurations. Although the results are approximate and at only a finite set of configurations, they may be called “true BornOppenheimer” wave functions in the sense that they allow direct calculation of vibrational-electronic interactions without requiring expansion (4). Extensive calculations of vibronic intensities in naphthalene* and other molecules5have been carried out by Robey et al. using a CNDO/S calculation.6 The purpose of this paper is to point out some advantages of starting with true BO electronic wave functions in considering breakdown of the BO approximation. The emphasis will be on the effects on vibronic intensities in electronic spectra. Some simple conceptual pictures will be presented relating the displacement of the electron cloud to the nuclear displacements when BO breakdown mixes a vibrational level of one electronic state with higher electronic states. This is the usual situation where vibronic intensities can be studied in detail. The pictures help to visualize the specttoscopic consequences expected. Calculation of BO breakdown may be carried out quite readily starting from the true BO wave functions in the CNDO approximation. Some calculations on the mixing of the first and second excited singlet states of The Journal of Physical Chemistry, Vol. 80, No. 20, 1976
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S.J. Strickler and
(9) (C)
-l-
c-~++-m-rtl-+
Figure 1. Wave functions for visualizing the Born-Oppenheimer breakdown: (a)$(x,O) solid lines and $(x,Q) dashed lines; (b) d$/dQ or A* wave functions;(c)4oand d$,/dQ; (d) 4, and d&/dQ. I and m indicate regions where the electron cloud is displaced less or more than
expected from nuclear displacements.
naphthalene and the spectroscopic consequences are presented as an example.
A Picture of Born-Oppenheimer Breakdown Let us consider a particular vibrational level a of a particular electronic state,1. The BO approximation to its wave function is +l(x,Q)+la(Q).If we go beyond the BO approximation we must consider that other vibronic wave functions may be mixed with this one. Perturbation theory gives the result that, to first order, the corrected wave function is
where the sum is over all other BO wave functions. To evaluate the changes in *la we must examine the matrix element of the Hamiltonian 3f between BO wave functions. For the purposes of this paper we shall be interested in matrix elements involving different electronic states. Consider the matrix element between state la and another particular state 2b belonging to another electronic state, and for simplicity consider only one vibration Q as mixing the two levels. The results are easily generalized to several vibrations. There are two sorts of terms in the Hamiltonian matrix elements:
and
Now consider what happens to state l a when we mix all the vibrational levels of $2 with it according to eq 5 with matrix elements as in eq 9. *la
= $1(x,Q)4la(Q)
X $z(x,&)4zb(Q) (11)
(6)
This equation can be simplified if we make the approximation that the individual denominators E 2 b o - Elaocan all be replaced by an average or effective separation between the electronic states, E2 - El. We can then make use of a sum rule that
(7)
which holds because the &(Q) form a complete set of functions. Then eq 11 becomes
arising from the two terms neglected in the BO approximation. Here the subscripts on the brackets indicate the integration coordinates in the matrix element. The electronic matrix elements are still functions of Q, and should properly be inside the vibrational brackets as written. However, it should frequently be satisfactory to remove them from the vibrational brackets; this amounts to neglecting higher order vibronic effects. We can rewrite the two terms as
The Journal of Physical Chemlstry, Vol. 80, No. 20, 1976
Now the electronic matrix element in (8) vanishes unless the two electronic states are of the same symmetry. It is generally considered to be small for mixing of different electronic states. The term in (9) can mix states of the same or different symmetry (depending on the symmetry of Q), and we shall examine it in more detail. Use of a simple example is helpful in visualizing the effects of BO breakdown. The following discussion refers to Figure 1. Let $1 be a simple one-electron wave function in a linear triatomic molecule, represented as a linear combination of Is orbitals on each nucleus, schematically indicated by the circles in Figure la. The solid circles represent $l(x,O), the orbitals in the equilibrium configuration, the dashed circles represent $1(x,Q), the orbitals when the molecule is distorted along the bending normal coordinate Q. That particular distortion will be taken as a positive Q. The major effect in distortion is that the orbitals follow the nuclei. It is easily seen that d$lldQ N [$(x,Q) - $(x,O)]/Q will be a function resembling a p orbital on each atom (although the radial dependence is not identical with a p orbital). This function is schematically shown in Figure lb. It bears a resemblance to a molecular orbital which we would call a A* orbital, and Figure l b could also be taken to represent that A* orbital. We would then expect a positive and nonzero overlap between A* and dlC/l/aQ.In other words, if we let $2 be the A* electronic state, the matrix element
*la
= +l(X,Q)@la(Q)
We are now in a position to translate this mathematical expression into a conceptual picture of what should happen to the electron distribution because of BO breakdown. Let us suppose that state 2 lies above state 1,so the level of interest is interacting with higher levels; this will be the usual case of
Born-Oppenheimer Breakdown interest. Then E2 - El > 0 and the entire mixing coefficient in eq 13 is positive. We can write
where c is a small positive coefficient. If a is the u = 0 level of state 1,the result may be visualized with reference to Figure IC,where the solid line represents $la and the dashed line d$l,/dQ. Now pick a particular positive Q such as that indicated by the dashed. vertical line. Here &a and a$la/dQ have opposite signs. Inspection of eq 14 then shows that, at this instantaneous configuration, the electronic wave function is $l(x,Q) with a small amount of $2(x,Q) mixed in with negative sign (i.e., with coefficient c d$ldQ < 0). The mixing of a bit of 2p character into the 1s orbital on each atom gives a hybrid orbital which has the electron density predominantly on one side of the nucleus. Reference to Figures l a and l b , with careful attention to the signs of all terms, will show that the electron density is higher on the side closer to the equilibrium position. Similar considerations when Q is negative will show that the electron density is again on the side toward the equilibrium position. The conclusion is that for the u = 0 level the electrons are distorted from the equilibrium POsition less than expected from the nuclear positions. Similar considerations can be applied to the u = 1 level shown in Figure Id. It is found that in the region labeled m near the equilibrium position the electrons will be displaced more than expected, while for the regions labeled 1toward the classical turning points the electrons will be displaced less than expected for the BO wave functions. Higher vibrational levels will show alternating regions where the electrons are displaced more or less than expected; as above the outermost regions will always have the electrons displaced less than expected. It would be noted that this whole treatment was based on a BO state interacting with higher electronic states. If a level were interacting primarily with a lower state, the sign of E2 - E would be negative and the conclusions would be just the opposite-regions 1 and m would be interchanged. Furthermore, the picture cannot be applied at all to radiationless transitions where the initial level interacts with nearly isoenergetic levels of a lower electronic state. In that case an average energy and the sum of eq 12 cannot be used and the vibrational levels must be considered individually. Nevertheless, the case discussed above is one of frequent interest. The conceptual pictures presented above can be used to aid in understanding the effects of BO breakdown on vibronic intensities in electronic spectra. For example, a lack of mirror image symmetry between absorption and emission can be caused by BO breakdown. This has been discussed by Geldof, Rettschnik, and Hoytink' and by Orlandi and Siebrand.8 Another example in the spectrum of naphthalene will be considered below. This will illustrate the qualitative behavior expected from the conceptual pictures. Application to Naphthalene The lowest excited singlet state of naphthalene has symmetry B2u, and the second state B1,. The transition from the ground state to S1 is allowed for light polarized along the long axis, and to Sz allowed polarized along the short (in-plane) axis. However, the transition to SI is extremely weak due to nearly exact cancellation of transition moments coupling the ground state to the two major configurations making up S1. In fact, most of the intensity of the S1 band is short-axis polarized, made allowed vibronically by B3, vibrations. A good spectrum of this transition is given by Knight, Selinger, and
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Ross? The dominant band in the spectrum is the band having one quantum of vibration 8B3, excited in the upper state. When the molecule is distorted along a B3, coordinate its symmetry is reduced from D 2 h to CZh, and s1and s 2 belong to the same irreducible representation in that group. Therefore BSg vibrations can mix SI and S2. Since mode 8B3, (the lowest frequency of the eight vibrations of that symmetry) induces the most intensity, it is likely to mix the states most strongly. We first qualitatively consider what the effect of BO breakdown should be. For a symmetry forbidden transition like the short-axis polarized component of SI, the transition moment is zero at the equilibrium position but becomes nonzero when the molecule is distorted, in this case along 8B3,. Of course, the important thing is how far the electron cloud is distorted rather than how far the nuclei are distorted. Thus we see that BO breakdown may affect intensities. Mixing of S1 with S2 due to BO breakdown will affect the extent to which the electron cloud is distorted from the equilibrium distribution during the 8B3, vibration. One way this could show up is in a lack of mirror image relationship between absorption and e m i s s i ~ n . A ~ , related ~ quantity which could be checked experimentally is the relative. intensities of the dominant (8B3,); band and the (8B3,): hot band in the absorption spectrum. In this notation the right superscript and subscript are the quantum numbers of this vibration in the upper and lower states, respectively. In the absence of BO breakdown the relative intensities can be calculated readily if the vibrations are assumed harmonic and the transition moment assumed to vary linearly with displacement. Both assumptions are probably good. It is also assumed that there is no rotation of normal coordinates between the two states, i.e., no Duschinsky effect.1° Now BO breakdown involving S1should have the following effects. In the u = 0 level the electrons are always distorted less than expected from the nuclear displacements, and this results in a lower intensity for (8B3,):. On the other hand, in the u = 1 level the electrons are distorted more than the nuclei in the region near equilibrium (region m in Figure Id), and it is just this region which overlaps well with the u = 0 level of the ground state, so the (8B3,); band should be increased in intensity by BO breakdown. Qualitatively then, the ratio of intensities of (8B3,): to (8B3,); should be decreased relative to that calculated from the vibrational wave functions and Boltzmann factors. Another observable effect should be the following: if the (8B3,); band is excited by a narrow band of exciting light in a low pressure gas, a single vibronic level (SVL) emission can be observed from the u = 1 level. The emission should go primarily to the v = 2 and u = 0 levels of the ground state, Le., the (8B3,): and (8B3,); bands should be predominant in the SVL emission spectrum. The ratio of the intensities of these two bands should be about 2 to 1on the same assumptions as above. The exact ratio can be calculated from vibrational wave functions and frequencies. However, BO breakdown should make the (8B3,); stronger for the reasons mentioned above. On the other hand, the u = 2 level of the ground state will overlap most with the regions called 1 in Figure Id, where the electrons are distorted less than expected from the nuclear displacements, and this will result in a lower intensity for (8B3,);. BO breakdown will then lower the intensity ratio of (8B3,); to (8B3,); from the expected value near 2 to 1to some smaller value. These two qualitative effects are predicted on the basis of the simple conceptual pictures presented above. The next The Journal of Physical Chemistry, Vol. 80, No. 20, 1976
S.J. Strickler
2152
section describes a calculation of the expected magnitude of the effects based on CNDO/S wave functions. Mixing of S1 with Szdue to BO breakdown caused by the 8B3, vibration will be considered. The results will then be comDared with some experiments already in the literature.
CNDO Calculations The calculations described here are based on wave functions for the ground and excited singlet states of naphthalene obtained by Robey et aL4 These are done in the CNDO/S approximation.6 In this method all valence electrons are treated, and in the ground state these completely fill the molecular orbitals up to number 24. The excited states are described by fairly extensive configuration interaction (CI) between singly excited configurations eSt(x,Q):
h(x,Q) =
ci8t(Q)eS,(x,Q)
(15)
st
Each 0,t is a sum of two antisymmetrized products of MO wave functions 0 with an electron promoted from orbital s to an originally unfilled orbital t. Each MO 0j is in turn given by a linear combination of atomic orbitals x; ej(x,Q) = C aj,(Q)x,(x,Q)
(16)
P
At the equilibrium geometry, the first excited singlet state $l(x,O) is a linear combination of configurations (24,26), (23,251,(24,291,and (22,28) while the second state $z(x,O) is a combination of (24,25), (23,26), (22,27), and (23,29). All of these promotions involve only P orbitals. When the molecule is distorted along the 8B3, coordinate, three things happen within the BO approximation: (a) First and most importantly, the orbitals follow the nuclei. (b) The coefficients uj, change, so that the MO's have a slightly different form as expressed in eq 16. (c) The CI coefficients Clst defining the states by eq 15 change. In particular, there is a mixing of the configurations of $2 into $1 and vice versa. The distorted molecule has lower symmetry, and one would expect a nonzero short-axis transition moment for the ground state to S1 transition. In fact, each of the three effects contribute to this transition moment. It happens that the first effect contributes very little in this case; most of the short-axis transition moment comes from the changes of coefficients. Effects (b) and (c) produce transition moments which partially cancel, with the predominant contribution being (c),the change in CI coefficient^.^ We wish to consider the mixing of S1 with Sz due to BO breakdown, so we must look a t the matrix element ($21HI $ I ) ~which , will reduce to terms such as eq 8 and 9. The terms of eq 8 will vanish in first order because the two states have different symmetry. We will consider the term from eq 9 which will involve the derivative of $1 with respect to Q(8B3,). We must take account of each of the three types of changes described above. Consider first changes of type (a). Because all of the configurations in SI and S2 are P,K*,we need only consider the 2pa type atomic orbitals. The derivative of such an orbital when the nucleus moves in the molecular plane will resemble a d orbital on the atom. This will have zero overlap with the original p orbital on the same center. It may overlap with p orbitals on other centers, but such terms are neglected in the CNDO approximation. Thus matrix elements between S1 and Sz involving the motions of the orbitals in following the nuclei will be taken to be zero. Terms of types (b) and (c) will both contribute to the electronic matrix element. Some straightforward algebra, making The Journal of Physical Chemistry, Vol. 80, No. 20, 1976
use of the orthogonality of wave functions and neglecting overlap between orbitals on different atoms shows that the whole matrix element reduces to a simple expression involving coefficients and their derivatives:
+E
, aQ +
CzstClrt C a s p
r,s,t
CzstClBu 8,t.u
C atoda,, u
aQ
(17)
Here the first sum involves terms of type (c) and the other two sums terms of type (b). The second sum involves configurations where in the two states electrons are promoted from different filled orbitals r and s to the same empty orbital t, while the third sum involves promotions from the same filled orbitals s to different empty orbitals t and u. All these coefficients and derivatives can be evaluated from the CNDO wave functions obtained by Robey4 at Q = 0 and a t the zero point rms value of Q, Q* = (&)1/2 = ( h / 8 ~ % l r ) ~ / ~ , where 5 is the frequency of the vibration, taken to be 509 cm-l for 8B3,. One must be careful to use a consistent set of phases for all wave functions, since these phases are arbitrary and the CNDO program often chooses them differently at Q = Q* and a t Q = 0. The derivative of a coefficient is taken to be aC/& = [C(Q*) - C(O)]/Q*. The calculation does give good linearity over the small displacements u ~ e d .The ~ ? value ~ of the matrix element is found to be
We now wish to make use of this value in eq 12 to get corrected vibronic wave functions for the (8B3,)' and (8B3g)1 levels of SI. To do so, we must pick a suitable electronic energy separation. It is probably best to use the separation of the absorption band maxima which is about 4850 cm-1. I t also simplifies the equation to make use of the harmonic oscillator relationship
a$,/aQ
= (P~lr/h)l/2{v'&-~- V ' Z T $ ~ + ~(19) I
where 5 is the vibration frequency, 438 cm-l for 8B3, in state
s1.
The results for the wave functions are *[Si, (8B3,)OI = $ ~ $ ( B B ~ , ) o 3.092 X 10-3$~$(8~,,)1 (20)
*[si,(8B3g)1] = $ I $ ( B B ~ ~ ) ~ 3.092 x 10-3$2 ($(8Bsg)0 - d $ ( 8 B 3 , ) 2 ) (21) The vibrational wave functions $ in these expressions all refer to states of S1with 0, 1,or 2 quanta of the 8B3, vibration excited. The wave functions (20) and (21) can be used to compute transition moments between various vibrational levels of the ground state and these two levels of the S1 state. In each case, one can calculate a transition moment in the absence of BO breakdown from the first term in the wave function. Inclusion of the second term then gives important interference effects which may increase or decrease the transition moment. Even the small component of Szcan have a large effect because the electronic transition moment to S2 is much larger than that to Si.The absolute value of the transition moment is given only approximately by the CNDO cal~ulation,~ but the relative values produced by vibronic effects and BO breakdown should be reasonable. It is necessary to take account of the difference in vibration frequencies in the two states, as this plays a role in the relative intensities of bands.
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Born-Oppenheimer Breakdown
In absorption, the integrated intensity, J c dg, is proportional to the frequency multiplied by the square of the transition moment. For bands of similar shape and width, the relative peak height would be proportional to the relative values of aM2. If transitions originate in different vibrational levels, the relative populations must also be taken into account. In emission, the integrated intensity of a band, measured in photons per unit time, is proportional to C3M2. Calculated transition moments for three vibronic bands are shown in Table I. The column headed Mo gives the calculation in the absence of BO breakdown, while column M gives the value including the latter. Actually, these values correspond to the sum of all transitions with these particular changes in 8B3, and all possible changes in A, modes. However, the relative values should apply to the particular bands where no other vibrations change. As indicated above, the ratio of the (8B3,): to (8B,); bands in absorption is one place where BO breakdown should show up. Table I shows clearly that M for the first band is decreased while that for the second is increased by BO breakdown. The Boltzmann factor for (8B3&: at 296 K is 0.084 24. We then obtain a ratio of intensities of 0.0950 in the absence of BO breakdown and 0.0810 including BO breakdown. This is not a large change and is on the borderline of detectability. The prediction can be compared with the spectrum of Knight et al.9 The ratio of band intensities (measured from the large spectrum in a preprint of their paper) is about 0.07 f 0.01, in reasonable agreement with the present estimate. Another test for BO breakdown is the ratio of intensities of (8B3g)ito (8B3,); in the fluorescence spectrum. Table I shows that the transition moment for the former decreases while that for the latter increases due to BO breakdown, in accord with the earlier qualitative picture. Use of the moments and frequencies in Table I gives a ratio of intensities of 2.25 in the absence of BO breakdown and 1.93 when it is taken into account. Again the predicted effect (a 14% decrease in ratio) is significant but not large. A single-vibronic-level fluorescence spectrum from level (8B3,)l has been published recently by Stockburger, Gattermann, and Klusman.ll There are experimental difficulties because the (8B3,); band is subject to self-absorption and to interference by scattered exciting light. However, their reported ratio, corrected for these effects, is 1.16, which is a great deal smaller than calculated. They recognize that this' is lower than expected, but mistakenly conclude that this cannot be due to BO breakdown, so they attribute it to anharmonicity in the ground state. Both of the above experiments suggest that our calculation has underestimated the BO breakdown. In particular, the second experiment would require that the BO breakdown matrix element be larger by a factor of 4.34. If that were the case, the ratio of (8B3,): to (8B& in absorption should be about 0.0476 which seems too low to be consistent with the absorption experiment. I t seems possible that there are still scattered light problems in the emission spectrum. A few possible sources of error in the calculation should be mentioned. Of course, the wave functions are only approximate. The neglect of overlap between the derivative of an atomic orbital and another orbital could be a problem. However, it would be quite inappropriate to include such terms when the wave functions were obtained by neglecting overlap. Improvement of this approximation would require a different sort of wave function. We have also neglected any rotation of normal coordinates between the ground and excited state. This would affect vibronic intensities without the BO breakdown. We have assumed the vibrations to be harmonic, in-
TABLE I: Frequencies and Transition Moments Calculated for Vibronic Bands in the Ground to SI Transition of Naphthalene u,
Transition
cm-l
(8B3,E
31 500 32 470 31 450
(8B3JB
(sB&
MO, eA
0.049 69 0.046 10 0.072 52
M,
e%, 0.047 86 0.048 07 0.070 04
cluding no Fermi resonance or mixing of different normal modes. Parmenter et al. have shown that the latter effect plays a significant role in vibronic intensities in benzene,12 but it probably has less influence in naphthalene where most of the oscillator strength is in the bands with no change in totally symmetric vibrations. The average energy separation of 4850 cm-l was chosen fairly arbitrarily. Use of the separation of the origins of S1 and S2 would about double the effect calculated, but this would surely be too small a value to use for the average separation. Any of these approximations may affect the accuracy of the calculations, although all seem unlikely to produce a very large effect. In any case, the absorption experiment suggests that the calculations are not too far off. In summary, we have reviewed a method of treating breakdown of the BO approximation, starting from a true BO basis. We have shown how the treatment can lead to a simple conceptual picture of the effects in certain cases. This picture was based on the terms we have called type (a), emphasizing the importance of the orbitals following the nuclei. But terms of types (b) and (c) work in analogous ways. If a wave function varies with Q by changes of coefficients, i.e., changes of electron distribution among atoms, then BO breakdown will make the changes greater or smaller in different regions of the vibration just as it can make the electron displacements greater or smaller. (It should be mentioned that it is possible to imagine a situation where the effects on transition moment are in the opposite direction from that predicted. This could happen if say the major transition moment came from terms of type (a), while a nearby mixing state affected only terms of type (b) or (c) and these were in the opposite direction.) The conceptual picture was used to help understand the expected spectral effects of BO breakdown. A method of calculation of these spectral effects using CNDO wave functions was summarized, and applied to the first singlet transition of naphthalene. Finally the results were compared with experiment and found to be in at least qualitative agreement. Acknowledgments. This work was supported in part by the National Science Foundation. The author also wishes to thank the Council on Research and Creative Work of the University of Colorado for the grant of a Faculty Fellowship during which this work began, and to thank the Australian National University for their hospitality during that time. Special thanks are due to Dr. M. J. Robey and Professor I. G. Ross for many discussions and for providing the CNDO calculations used in this study. References a n d Notes (1) M. Born and R. Oppenheimer, Ann. Phys., 84, 457 (1927). (2) E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, "Molecular Vibrations", McGraw-Hill, New York, N.Y., 1955. (3) G. Herzberg and E. Teller, Z. Phys. Chem. (Leipzig), 821, 410 (1933). (4) M. J. Robey, I. G. Ross, R . Southwood-Jones, and S. J. Strickler, to be submitted for publication. (5) M. J. Robey, Ph.D. Thesis, The Australian National University, Canberra, A.C.T., 1974. (6) J. Del Bene and H. H. Jaffe, J. Chem. Phys., 48, 1807 (1968). (7) P. A. Geldof, R. P.H. Rettschnik, and G. J. Hoytink, Chem. Phys. Lett., 10,
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549 (1971).
(8)G. Orlandi and W. Siebrand, Chem. Phys. Lett., 15, 465 (1972). (9) A. E. W. Knight, B. K. Selinger,and I. G. Ross, Aust. J. Chem., 26, 1159 (1973). (IO) F. Duschinsky, Acta Physicochim. URSS, 7, 551 (1937). (11) M. Stockburger,H. Gattermann,and W. Klusman, J. Chem. Phys., 63,4519 (1975). (12) C . S. Parmenter, private communications.
Discussion R. M. HOCHSTRASSER.How does this work compare with the recent work done by Metz? S. J. STRICKLER. I believe that most of his calculations-in fact, most of the calculations that have been done-have been based on the crude Born-Oppenheimer basis set, where one starts with wave functions only a t the equilibrium geometry and expands the wave functions for distorted geometry in terms of them. Then a perturbation treatment is used to obtain the distorted wave functions followed by a perturbation treatment to get the Born-Oppenheimer breakdown. There are cases where that sort of expansion is very slow to converge. In some of the cases that you are looking at, naphthalene really is an example, the particular coupling between SIand Sp does not really contribute to making those orbitals follow the nuclei, and so one may be able to get by with it. If the particular coupling that you are interested in does contribute to making the orbitals follow the nuclei, then the expansion is slow to converge. One gets by with it if the terms that he is looking at are very small anyway. This, of course, certainly can give trouble, although one can probably reasonably calculate this particular case because the S2,Sl mixing doesn’t contribute to making the orbitals follow the nuclei.
L. GOODMAN.Rotation of the normal modes on going from the ground to the excited state could by itself cause a change in the coupling mode transition probability in absorption compared to emission. Would you comment on how you distinguish between this effect and corrections to the Born-Oppenheimer approximation, especially for a case like naphthalene where the observed intensity change is only a few percent?
S. J. STRICKLER.I have not worried about a rotation of normal coordinates. In other words, I have assumed in everything I have said here that the normal coordinate in the ground state was parallel to that in the excited state, but I have corrected for the difference in frequencies. Well, the rotation is something that one has to worry about. I’m not sure that I can prove this, but I believe that looking a t that ratio of the two things in the single vibronic level emission is probably less sensitive to rotation than looking at the mirror image relationship between absorption and emission. That’s one reason I think that’s probably a better check. C. PARMENTER. The exploration of the S(bl,)@: fluorescence intensity ratios in naphthalene to test the proposition concerning non-BO behavior is reminiscent of a study recently completed by Dr. K. Tang a t Indiana on a similar question in benzene. He found that analogous intensity ratios involving the inducing mode in benzene depart from the BO expected values, but he was able to show that the effect was dominantly (if not entirely) due to Fermi resonances involving levels more than 100 cm-’ apart. His results were consistent with the discussion of these effects by Fischer, Scharf, and Parmenter in Molecular Physics, 1975. I wonder if such effects could occur also in naphthalene to complicate the search for non-BO behavior?
S. J. STRICKLER. Yes, I think that that’s the thing that’s most likely to affect it-the
W. R. MOOMAW. If I understood you correctly, you calculate the changes in the intensity based upon the root mean square displacement rather than by integrating over the actual displacements (which might include anharmonicity effects). How does the error due to this approximation compare in magnitude with the other approximations in your formalism and the use of CNDO wave functions? did were S. J. STRICKLER. The calculations that Ross and based only on the value a t the root mean square displacement; however, they did in various cases check to see that the variation appeared to be linear, and it does. As long as it’s linear, then that’s good. W. R. MOOMAW.I was wondering how much of that might account for discrepancies compared to using inaccurate wave functions or some of the other approximations which you discussed on the first slide.
S. J. STRICKLER.The linearity looks pretty good in this case. The errors must be due to other factors. R. KOPELMAN.What is the precise nature of your zero-order wavefunctions? What are the exact nuclear configurations for these? S. J. STRICKLER. This is a CNDO calculation which just starts from atomic orbitals, 2s and 2p on the carbon atoms plus the Is orbitals on the hydrogens, but being a CNDO calculation it’s semiempirical and doesn’t really use the forms of the atomic orbitals on the nuclei in the calculation. It uses semiempirical sorts of electronic repulsion integrals, and so on, in doing the SCF and CI calculations.
Fermi resonance interaction between the levels.
S. P. MCGLYNN.It appears to me that you have analyzed CNDO results a t a root mean square geometry in terms of zero-order results (defined at the ground state geometry and, also, by the CNDO algorithm). This may or may not have relevance to fact (i.e., experiment). I would believe such relevance if the f number a t the root mean square geometry obtained by CNDO/CI was in good correspondence with respect to polarization? How good is the numerical agreement? S. J. STRICKLER.I am analyzing the effect relative to calculated short axis polarized intensity induced by the vibrations. In other words, a t the equilibrium position there is a calculated long-axis polarized intensity but no calculated short-axis polarized intensity. As one distorts the molecule along that mode, one obtains some short axis intensity. One also gets a’calculated Born-Oppenheimer breakdown, which does things to the intensity which I have taken relative to the calculated intensity because they are calculated from the same wave function. So I think that even if the wave functions are some what inadequate, probably the ratios that I calculate are what the BornOppenheimer breakdown does compared to the intensity that is induced by that distortion. The ratio is probably reasonable.
S. P. MCGLYNN.In other words, what you are doing, is that you are partitioning the calculated root mean square geometric f value into contributions added to the zero-order or ground state conformational oscillator strength. Now my question simply is how does the root mean square calculated oscillator strength compare with the actual known oscillator strength for this (transitions). In other words, are you partitioning yourself way off relative to experiment or something that’s right on.
R. KOPELMAN.But for what nuclear configuration?
S. J. STRICKLER.The equilibrium position, I guess, is an x-ray structure, and the normal coordinate analysis was actually done with two different force constant sets which led to somewhat different normal coordinates. The best results are from a potential function derived by Kydd. (R. A. Kydd, Ph.D. Thesis, University of British Columbia, Vancouver, B.C. 1969). They look quite good in terms of the normal coordinate analysis.
The Journal of Physical Chemistry, Vol. 80, No. 20, 1976
S. J. STRICKLER.The calculated short-axis polarized intensity is somewhat low in the calculation-maybe a factor of 2 or something of that order. It comes from a very interesting mix of the orbitals following the nuclei, changes in A 0 coefficients in the MO’s, and the changes in coefficients in the CI,,and it turns out for this particular mode the CI is the dominant thing. For other modes which are calculated to be much weaker, the CI is not always the dominant thing.
