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Intensities of Transitions in Non-Totally Symmetric Vibrations in the Electronic Spectra of Polyatomic Molecules
JOURNAL OF MOLECULAR SPECTROSCOPY ARTICLE NO.
176, 95–98 (1996)
0065
Intensities of Transitions in Non-Totally Symmetric Vibrations in the Electronic Spectra of Polyatomic Molecules W. L. Smith 2, Pitchcombe Gardens, Coombe Dingle, Bristol BS9 2RH, United Kingdom Received August 30, 1995; in revised form November 8, 1995
The effect of the noncorrespondence of normal coordinates on the Franck–Condon factors governing the appearance of non-totally symmetric vibrations in electronic spectra is examined. In a two-dimensional symmetry species containing vibrations 1 and 2, transitions such as 110210, 110230, etc. are given zero intensity in the usual theory, but it is shown using a simple model that noncorrespondence leads to their appearance with an intensity determined by the elements of the ˚ system of benzene the theory matrix connecting the normal coordinates in the two states. When applied to the 2600A suggests that the transition 61014101510 should have an observable intensity and this leads to a plausible assignment. This work suggests that the usual theory of non-totally symmetric vibrations needs to be modified to recognize that transitions of the sort 110210, and not 140200, are likely to be the most important after the leading member 120200. q 1996 Academic Press, Inc.
/ 1 from £1, £2. Detailed expressions will be given for the predicted intensities in the situation where the symmetry of the molecule is unchanged by the electronic transition and where the change in geometry is small. The theory is applied ˚ spectrum of benzene and the predicted intensity to the 2600A of the 61014101510 transition which involves the b2u vibrational species is calculated. It is shown that this transition should just be detectable and that there is indeed a weak band at a position in the spectrum consistent with this assignment.
INTRODUCTION
The application of the Franck–Condon principle to the electronic spectra of polyatomic molecules has been the subject of many investigations. Ozkan (1) lists many recent papers. For non-totally symmetric vibrations the Franck– Condon factors governing the intensity of vibrational transitions can be derived exactly if one assumes (a) the harmonic approximation, and (b) a one-to-one correspondence between the normal coordinates of the upper and lower states, Q* and Q9 respectively (see, for example, Smith (2)). It is well known that in this approximation (called henceforth the zeroth approximation) there is a selection rule D£a Å 0, 2, 4rrr, where £a is the vibrational quantum number associated with the normal coordinate Qa. However, in general, excited state normal coordinates are linear combinations of ground state coordinates (the Duschinsky effect (3)). The purpose of this paper is to investigate the effect on the intensities of the non-totally symmetric vibrations when the identity of the normal coordinates is not assumed, i.e., when the Duschinsky effect is operating. It will be shown that this leads to the appearance of transitions which have zero intensity in the zeroth approximation. In particular, for a nondegenerate, non-totally symmetric species containing two vibrations 1 and 2, the effective selection rule becomes D(£1 / £2) Å 0, 2, 4rrr. For example, in the zeroth approximation the allowed transitions from the ground state are 120200, 100220, 120220, etc. However, the noncorrespondence of the normal coordinates leads to the appearance of the additional transitions 110210, 110230, etc., with the first much the strongest. To a good approximation, the breakdown of the zeroth approximation leads to the appearance of transitions to the level £1 / 1, £2
THEORY
We shall assume that the symmetries of the upper and lower electronic states are the same and that the position of the principal axes of inertia is also the same in the two states. In this case the normal coordinates Q* and Q9 are related by a linear transformation which for non-totally symmetric vibrations can be written in matrix form, Q* Å AQ9,
[1]
A Å (L*)01L9
[2]
where
and L*, L9 are the usual L-matrices relating the normal coordinates in both states to the (same) symmetry coordinates. L is, of course, related to its transpose L* by the relation LL* Å G.
[3]
From Eqs. [2] and [3] it follows that 95 0022-2852/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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A*A Å L9*(G*)01L9.
H* Å H0 / DH,
[4]
[9]
where
If G* Å G9, then A*A Å AA* Å I;
92 / P292 / l* 92 2H0 Å P192 / l* 1 *Q1 2 *Q2
[5] and
i.e., the coordinates Q* and Q9 are related by an orthogonal transformation. In electronic transitions where the symmetry of the molecule is unchanged, and the change in dimensions is small, Eq. [5] is likely to be a good approximation. The orthogonality of the A matrix will be assumed in the remainder of this paper. Consider a vibrational transition from a lower electronic state where the relevant normal coordinates are Q91 and Q92 with associated frequencies n91 and n92. Then for the upper state, 2H*(Q*1, Q*2) Å P1* / P2* / l*1Q1* / l*2Q2* , 2
2
2
2
[6]
DH Å (a11a12l*1 / a22a21l*2)Q91Q92 Å 4p2c2(n1*2a11a12 / n2*2a22a21)Q91Q92 Å 4p2c2rQ91Q92.
In this formulation the ‘‘unperturbed’’ wave-functions corresponding to H0 are products of normal harmonic oscillator functions C*n(n1*, Q91)C*m(n* 2 , Q9 2), where
[7]
Using Eq. [1] and bearing in mind the orthogonality of A, we can express Eq. [6] in terms of the lower state coordinates Q91 and Q92; we eventually obtain the result 2H*(Q91, Q92) Å P192 / l* 92 / P292 / l* 92 1 *Q1 2 *Q2 / 2Q91Q92(a11a12l* 1 / a21a22l* 2 ),
n* *2a211 / n2*2a221)1/2 1 Å (n1
[11]
n* *2a212 / n2*2a222)1/2. 2 Å (n1
[12]
and
where H* is the vibrational part of the molecular Hamiltonian and l Å 4p2n2c2.
[10]
Although the first order correction to the energy is zero, there is a correction to the wave function. Using as the perturbation function DH as given in Eq. [10], and the wellknown (4) relations for the matrix elements of the harmonic oscillator wave function, the usual methods of first order perturbation theory lead to the following expressions for the perturbed upper state wave function F*, now expressed in terms of the lower state normal co-ordinates Q91 and Q92: F*n,m(n*1, n*2, Q91, Q92) Å C*n(n* 1 , Q9 1)C* m(n* 2 , Q9 2)
[8]
0
∑ [»C*s (n*1 , Q91)C*t (n*2 , Q92)É4p2c2rQ91Q92É s,t
1 C*n(n* 1 , Q9 1)C* m(n* 2 , Q9 2)…C* s (n* 1 , Q9 1)
where the a11, etc. are elements of the matrix A and, for example, 2 2 l* 2 * Å l* 1a12 / l* 2a22.
If l*1 Å l*2 (n*1 Å n*2), then, again remembering the orthogonality of the matrix, the last term in Eq. [8] vanishes, and the coordinates Q91, Q92 are normal coordinates in the upper state as they are in the lower state. This corresponds to the usual theorem that any orthogonal transformation of a pair of degenerate coordinates gives rise to an equivalent pair of normal coordinates. Equation [8] can be put in a form which can be treated by normal first order perturbation theory, i.e.,
01 1 C*t (n* 2 , Q9 2)](Es,t 0 En,m) .
The Franck–Condon factors for a vibrational transition from a lower state (j, k) to an upper state (n, m) will be determined by the matrix element, R Å »F*n,m(n*1, n*2, Q*1, Q*2)ÉC9j,k(n91, n92, Q91, Q92)…,
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[14]
where the lower state wave function is given by C9j,k(n91, n92, Q91, Q92) Å C9j (n91, Q91)C9k (n92, Q92).
Equation [14] then reduces to integrals of the type
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»C*n(n* 1 , Q9 1)ÉC9 j (n9 1, Q9 1)…»C* m(n* 2 , Q9 2)ÉC9 k (n9 2, Q9 2)….
[16]
R thus reduces to combinations of integrals which determine the intensities of transitions in non-totally symmetric vibrations in the zeroth approximation. It is well known (5) that these integrals are nonzero only when the quantum numbers in the two states are either both even, or both odd; i.e., in Eq. [16], when
Using the known values for integrals of the sort »C*2(n* 1, Q91)ÉC90(n91, Q91)… (see, for example, (2)), Eq. [18] reduces eventually to r (n* (n* 2 0 n9 2) 1 0 n9 1) R1,1 Å 0 1/2 2(n* (n* (n* 1 0 n* 2 )(n* 1 n* 2) 2 / n9 2) 1 / n9 1)
H
F
/
These integrals also decrease rapidly from, e.g., j Å n { 2 onward, so long as n* 1 /n9 1 (or n9 1/n* 1 ) is less than about 2, a condition which is met in most practical cases. By inspection of Eqs. [13]–[16] one can see that the Franck–Condon factor R will only be nonzero when n / m Å j / k, j / k { 2, j / k { 4, . . . , and will decrease rapidly after n / m Å j / k { 2, since it will then depend increasingly on integrals of the kind given in Eq. [16] where the vibrational quantum numbers in the two states differ by more than 2. This modification of the selection rule in the quantum numbers m and n leads to the appearance of transitions which have zero intensity in the zeroth approximation; for example, for transitions arising from the vibrationless ground state we not only expect the usual series of 120200, 100220, 120220, 140200, 100240, . . . , but also 110210, 110230, 130210, . . . , which do not appear in the zeroth approximation but are allowed by the noncorrespondence of the normal coordinates in the two states. The 110210 transition is the first member of this forbidden series which is expected to occur. Its intensity can be worked out as follows. From Eq. [13] it follows that
/
/
1/2 (n* 1 / n* 2 )(n* 1 n* 2)
1 »C*0(n* 1 , Q9 1)ÉC9 0(n9 1, Q9 1)….
R0,0 Å »C*0(n* 2 , Q9 2)ÉC9 0(n9 2, Q9 2)… 1 »C*0(n* 1 , Q9 1)ÉC9 0(n9 1, Q9 1)…,
F
/
r 1/2 2(n* 1 / n* 2 )(n* 1 n* 2)
F
1 10
/
(n* 2 / n9 2)(n* 1 / n9 1)
,
R2,0 (n* 1 1 0 n9 1) Å r 1/2 . R0,0 (n* 2 1 / n9 1)
1/2 2(n* 1 0 n* 2 )(n* 1 n* 2)
rC*0(n* 1 , Q9 1)C* 2(n* 2 , Q9 2) q
1/2 2(n* 1 0 n* 2 )(n* 1 n* 2)
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G
(n* 2 0 n9 2)(n* 1 0 n9 1)
[21]
. [17]
[18]
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[22]
The square of Eq. [21] gives the predicted intensity of the transition 110210 relative to the 0 0 0 band; it depends as expected on the vibrational frequencies in the two states and the elements of the matrix A which transforms the coordinates Q* to Q9. APPLICATION
˚ system of benzene provides in The well investigated 2600A principle an opportunity to test the theory outlined above. The
Copyright q 1996 by Academic Press, Inc.
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r (n* (n* R1,1 2 0 n9 2) 1 0 n9 1) Å 0 1/2 R0,0 2(n* (n2* / n92) (n* 1 0 n* 2 )(n* 1 n* 2) 1 / n9 1)
The Franck–Condon factor R1,1 for the transition 110210 is given by
AID
[20]
and hence
q
R1,1 Å »F*1,1(n*1, n*2, Q91, Q92)ÉC90(n91, Q91)C90(n92, Q92)….
[19]
By a similar analysis the Franck–Condon factor for the origin band, 100200, is given by
rC*2(n* 1 , Q9 1)C* 0(n* 2 , Q9 2)
1/2 2(n* 1 / n* 2 )(n* 1 n* 2)
(n* 2 / n9 2)(n* 1 / n9 1)
where r is defined by Eq. [10]. The Franck–Condon factor for the transition 120200 is also modified and we obtain similarly
rC*2(n* 1 , Q9 1)C* 2(n* 2 , Q9 2)
rC*0(n* 1 , Q9 1)C* 0(n* 2 , Q9 2)
GJ
(n* 2 0 n9 2)(n* 1 0 n9 1)
1 »C*0(n* 2 , Q9 2)ÉC9 0(n9 2, Q9 2)…
F*1,1(n*1, n*2, Q91, Q92) Å C*1(n* 1 , Q9 1)C* 1(n* 2 , Q9 2)
0
1/2 2(n* 1 / n* 2 )(n* 1 n* 2)
1 10
k Å m, m { 2, m { 4, rrr.
0
r
F
j Å n, n { 2, n { 4, rrr
G
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TABLE 1 Intensities of Certain Transitions in the Benzene 2600A˚ System Relative to the 610 Band (1104)
symmetry of the molecule is unchanged in the transition and the change in geometry is small; hence the matrix A is likely to be orthogonal to a good approximation. Moreover, there are a number of two-dimensional nondegenerate vibrational symmetry species, and attempts have been made to determine the force field in both electronic states and consequently the elements of the associated transformation matrix A. In particular, Goodman et al. (6) have provided such information for the b2u symmetry species; in this case, n914 Å 1309 cm01,
n*14 Å 1571 cm01,
n915 Å 1150 cm01,
n*15 Å 1150 cm01,
and for the A matrix
F G F Q*14
Q*15
Å
GF G
0.96 0.28 00.28 0.96
Q914
Q915
.
Transitions in this vibrational species are symmetry forbidden in the absence of vibrational excitation, but are allowed by simultaneous transitions in the e2g species, particularly n6. The vibrational structure of the spectrum is similar to an allowed system, but with progressions built up on 610 instead of the 0 0 0 band. It is a reasonable approximation to apply the theory given in this paper to the intensity of transitions like 61014101510 relative to the 610 band. The results of this are given in Table 1. This example is peculiar in that since n915 Å n*15 the zeroth approximation predicts zero intensity in the transition 61014001520. However, Table 1 suggests that the effect of noncorrespondence of the normal coordinates in the upper and lower states is to redistribute the intensity within the species, and in particular to allow the transition 61014101510 to appear. Examination of the relative intensities of bands within this spectrum (see, for example, the early work of Radle and
Beck (7)) suggests that transitions with 4 1 1004 of the intensity of the 610 band should be just about observable. The 61014101510 band is expected to occur at around 41 330 cm01. Radle and Beck give a band at 41 304 cm01, whose intensity is about 73 1 1004 the intensity of 610. On high resolution plates taken by the author there is a weaker band at about 41 330 cm01 which fits better both in position and with the intensity calculations of this paper. However, the point of this example is not so much to propose a minor new assignment within the benzene spectrum as to explore the magnitude of the effects which the theory predicts. It seems likely that weak transitions such as 61014101510 allowed by the breakdown of the zeroth approximation should be observable in the electronic spectra of polyatomic molecules. The intensity of transitions such as 61014101510 relative to 610 is, according to this paper, determined entirely by the A matrix and the relevant vibrational frequencies. There are, however, other mechanisms which can lend intensity to such transitions, i.e., (a) quadratic terms involving Q1Q2 in the expansion of the electronic transition moment; (b) quartic terms involving Q1Q32 and Q31Q2 in the expansion of the potential function. The contributions of these terms are difficult to estimate, but are unlikely to be larger, and will probably be less, than the effects described here (certainly (b) is likely to be small in the case of benzene). In any case calculations based on this paper should provide a first estimate for the intensity of these transitions. In general, as the frequencies n1 and n2 in upper and lower electronic states begin to diverge, so do the forms of the associated normal coordinates. The first transitions which appear as a consequence of these divergences are 120200 and 100220, but as the divergence increases one may expect the appearance of 110210, probably before the next members of the zeroth series, 100240 and 140200. REFERENCES 1. 2. 3. 4.
I. Ozkan, J. Mol. Spectrosc. 139, 147–162 (1990). W. L. Smith, Proc. Phys. Soc. 89, 1021–1042 (1966). F. Duschinsky, Acta Physicochim. URSS 7, 551–566 (1937). E. B. Wilson, J. C. Decius, and P. C. Cross, ‘‘Molecular Vibrations,’’ Appendix III. McGraw–Hill, New York, 1955. 5. G. Herzberg, ‘‘Molecular Spectra and Molecular Structure,’’ Vol. 3, pp. 151–157. Van Nostrand, New York, 1966. 6. L. Goodman, J. M. Berman, and A. G. Ozkabak, J. Chem. Phys. 90, 2544–2554 (1989). 7. W. F. Radle and C. A. Beck, J. Chem. Phys. 8, 507–513 (1940).
Copyright q 1996 by Academic Press, Inc.
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176, 95–98 (1996)
0065
Intensities of Transitions in Non-Totally Symmetric Vibrations in the Electronic Spectra of Polyatomic Molecules W. L. Smith 2, Pitchcombe Gardens, Coombe Dingle, Bristol BS9 2RH, United Kingdom Received August 30, 1995; in revised form November 8, 1995
The effect of the noncorrespondence of normal coordinates on the Franck–Condon factors governing the appearance of non-totally symmetric vibrations in electronic spectra is examined. In a two-dimensional symmetry species containing vibrations 1 and 2, transitions such as 110210, 110230, etc. are given zero intensity in the usual theory, but it is shown using a simple model that noncorrespondence leads to their appearance with an intensity determined by the elements of the ˚ system of benzene the theory matrix connecting the normal coordinates in the two states. When applied to the 2600A suggests that the transition 61014101510 should have an observable intensity and this leads to a plausible assignment. This work suggests that the usual theory of non-totally symmetric vibrations needs to be modified to recognize that transitions of the sort 110210, and not 140200, are likely to be the most important after the leading member 120200. q 1996 Academic Press, Inc.
/ 1 from £1, £2. Detailed expressions will be given for the predicted intensities in the situation where the symmetry of the molecule is unchanged by the electronic transition and where the change in geometry is small. The theory is applied ˚ spectrum of benzene and the predicted intensity to the 2600A of the 61014101510 transition which involves the b2u vibrational species is calculated. It is shown that this transition should just be detectable and that there is indeed a weak band at a position in the spectrum consistent with this assignment.
INTRODUCTION
The application of the Franck–Condon principle to the electronic spectra of polyatomic molecules has been the subject of many investigations. Ozkan (1) lists many recent papers. For non-totally symmetric vibrations the Franck– Condon factors governing the intensity of vibrational transitions can be derived exactly if one assumes (a) the harmonic approximation, and (b) a one-to-one correspondence between the normal coordinates of the upper and lower states, Q* and Q9 respectively (see, for example, Smith (2)). It is well known that in this approximation (called henceforth the zeroth approximation) there is a selection rule D£a Å 0, 2, 4rrr, where £a is the vibrational quantum number associated with the normal coordinate Qa. However, in general, excited state normal coordinates are linear combinations of ground state coordinates (the Duschinsky effect (3)). The purpose of this paper is to investigate the effect on the intensities of the non-totally symmetric vibrations when the identity of the normal coordinates is not assumed, i.e., when the Duschinsky effect is operating. It will be shown that this leads to the appearance of transitions which have zero intensity in the zeroth approximation. In particular, for a nondegenerate, non-totally symmetric species containing two vibrations 1 and 2, the effective selection rule becomes D(£1 / £2) Å 0, 2, 4rrr. For example, in the zeroth approximation the allowed transitions from the ground state are 120200, 100220, 120220, etc. However, the noncorrespondence of the normal coordinates leads to the appearance of the additional transitions 110210, 110230, etc., with the first much the strongest. To a good approximation, the breakdown of the zeroth approximation leads to the appearance of transitions to the level £1 / 1, £2
THEORY
We shall assume that the symmetries of the upper and lower electronic states are the same and that the position of the principal axes of inertia is also the same in the two states. In this case the normal coordinates Q* and Q9 are related by a linear transformation which for non-totally symmetric vibrations can be written in matrix form, Q* Å AQ9,
[1]
A Å (L*)01L9
[2]
where
and L*, L9 are the usual L-matrices relating the normal coordinates in both states to the (same) symmetry coordinates. L is, of course, related to its transpose L* by the relation LL* Å G.
[3]
From Eqs. [2] and [3] it follows that 95 0022-2852/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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A*A Å L9*(G*)01L9.
H* Å H0 / DH,
[4]
[9]
where
If G* Å G9, then A*A Å AA* Å I;
92 / P292 / l* 92 2H0 Å P192 / l* 1 *Q1 2 *Q2
[5] and
i.e., the coordinates Q* and Q9 are related by an orthogonal transformation. In electronic transitions where the symmetry of the molecule is unchanged, and the change in dimensions is small, Eq. [5] is likely to be a good approximation. The orthogonality of the A matrix will be assumed in the remainder of this paper. Consider a vibrational transition from a lower electronic state where the relevant normal coordinates are Q91 and Q92 with associated frequencies n91 and n92. Then for the upper state, 2H*(Q*1, Q*2) Å P1* / P2* / l*1Q1* / l*2Q2* , 2
2
2
2
[6]
DH Å (a11a12l*1 / a22a21l*2)Q91Q92 Å 4p2c2(n1*2a11a12 / n2*2a22a21)Q91Q92 Å 4p2c2rQ91Q92.
In this formulation the ‘‘unperturbed’’ wave-functions corresponding to H0 are products of normal harmonic oscillator functions C*n(n1*, Q91)C*m(n* 2 , Q9 2), where
[7]
Using Eq. [1] and bearing in mind the orthogonality of A, we can express Eq. [6] in terms of the lower state coordinates Q91 and Q92; we eventually obtain the result 2H*(Q91, Q92) Å P192 / l* 92 / P292 / l* 92 1 *Q1 2 *Q2 / 2Q91Q92(a11a12l* 1 / a21a22l* 2 ),
n* *2a211 / n2*2a221)1/2 1 Å (n1
[11]
n* *2a212 / n2*2a222)1/2. 2 Å (n1
[12]
and
where H* is the vibrational part of the molecular Hamiltonian and l Å 4p2n2c2.
[10]
Although the first order correction to the energy is zero, there is a correction to the wave function. Using as the perturbation function DH as given in Eq. [10], and the wellknown (4) relations for the matrix elements of the harmonic oscillator wave function, the usual methods of first order perturbation theory lead to the following expressions for the perturbed upper state wave function F*, now expressed in terms of the lower state normal co-ordinates Q91 and Q92: F*n,m(n*1, n*2, Q91, Q92) Å C*n(n* 1 , Q9 1)C* m(n* 2 , Q9 2)
[8]
0
∑ [»C*s (n*1 , Q91)C*t (n*2 , Q92)É4p2c2rQ91Q92É s,t
1 C*n(n* 1 , Q9 1)C* m(n* 2 , Q9 2)…C* s (n* 1 , Q9 1)
where the a11, etc. are elements of the matrix A and, for example, 2 2 l* 2 * Å l* 1a12 / l* 2a22.
If l*1 Å l*2 (n*1 Å n*2), then, again remembering the orthogonality of the matrix, the last term in Eq. [8] vanishes, and the coordinates Q91, Q92 are normal coordinates in the upper state as they are in the lower state. This corresponds to the usual theorem that any orthogonal transformation of a pair of degenerate coordinates gives rise to an equivalent pair of normal coordinates. Equation [8] can be put in a form which can be treated by normal first order perturbation theory, i.e.,
01 1 C*t (n* 2 , Q9 2)](Es,t 0 En,m) .
The Franck–Condon factors for a vibrational transition from a lower state (j, k) to an upper state (n, m) will be determined by the matrix element, R Å »F*n,m(n*1, n*2, Q*1, Q*2)ÉC9j,k(n91, n92, Q91, Q92)…,
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[14]
where the lower state wave function is given by C9j,k(n91, n92, Q91, Q92) Å C9j (n91, Q91)C9k (n92, Q92).
Equation [14] then reduces to integrals of the type
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»C*n(n* 1 , Q9 1)ÉC9 j (n9 1, Q9 1)…»C* m(n* 2 , Q9 2)ÉC9 k (n9 2, Q9 2)….
[16]
R thus reduces to combinations of integrals which determine the intensities of transitions in non-totally symmetric vibrations in the zeroth approximation. It is well known (5) that these integrals are nonzero only when the quantum numbers in the two states are either both even, or both odd; i.e., in Eq. [16], when
Using the known values for integrals of the sort »C*2(n* 1, Q91)ÉC90(n91, Q91)… (see, for example, (2)), Eq. [18] reduces eventually to r (n* (n* 2 0 n9 2) 1 0 n9 1) R1,1 Å 0 1/2 2(n* (n* (n* 1 0 n* 2 )(n* 1 n* 2) 2 / n9 2) 1 / n9 1)
H
F
/
These integrals also decrease rapidly from, e.g., j Å n { 2 onward, so long as n* 1 /n9 1 (or n9 1/n* 1 ) is less than about 2, a condition which is met in most practical cases. By inspection of Eqs. [13]–[16] one can see that the Franck–Condon factor R will only be nonzero when n / m Å j / k, j / k { 2, j / k { 4, . . . , and will decrease rapidly after n / m Å j / k { 2, since it will then depend increasingly on integrals of the kind given in Eq. [16] where the vibrational quantum numbers in the two states differ by more than 2. This modification of the selection rule in the quantum numbers m and n leads to the appearance of transitions which have zero intensity in the zeroth approximation; for example, for transitions arising from the vibrationless ground state we not only expect the usual series of 120200, 100220, 120220, 140200, 100240, . . . , but also 110210, 110230, 130210, . . . , which do not appear in the zeroth approximation but are allowed by the noncorrespondence of the normal coordinates in the two states. The 110210 transition is the first member of this forbidden series which is expected to occur. Its intensity can be worked out as follows. From Eq. [13] it follows that
/
/
1/2 (n* 1 / n* 2 )(n* 1 n* 2)
1 »C*0(n* 1 , Q9 1)ÉC9 0(n9 1, Q9 1)….
R0,0 Å »C*0(n* 2 , Q9 2)ÉC9 0(n9 2, Q9 2)… 1 »C*0(n* 1 , Q9 1)ÉC9 0(n9 1, Q9 1)…,
F
/
r 1/2 2(n* 1 / n* 2 )(n* 1 n* 2)
F
1 10
/
(n* 2 / n9 2)(n* 1 / n9 1)
,
R2,0 (n* 1 1 0 n9 1) Å r 1/2 . R0,0 (n* 2 1 / n9 1)
1/2 2(n* 1 0 n* 2 )(n* 1 n* 2)
rC*0(n* 1 , Q9 1)C* 2(n* 2 , Q9 2) q
1/2 2(n* 1 0 n* 2 )(n* 1 n* 2)
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G
(n* 2 0 n9 2)(n* 1 0 n9 1)
[21]
. [17]
[18]
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[22]
The square of Eq. [21] gives the predicted intensity of the transition 110210 relative to the 0 0 0 band; it depends as expected on the vibrational frequencies in the two states and the elements of the matrix A which transforms the coordinates Q* to Q9. APPLICATION
˚ system of benzene provides in The well investigated 2600A principle an opportunity to test the theory outlined above. The
Copyright q 1996 by Academic Press, Inc.
JMS 6930
G
r (n* (n* R1,1 2 0 n9 2) 1 0 n9 1) Å 0 1/2 R0,0 2(n* (n2* / n92) (n* 1 0 n* 2 )(n* 1 n* 2) 1 / n9 1)
The Franck–Condon factor R1,1 for the transition 110210 is given by
AID
[20]
and hence
q
R1,1 Å »F*1,1(n*1, n*2, Q91, Q92)ÉC90(n91, Q91)C90(n92, Q92)….
[19]
By a similar analysis the Franck–Condon factor for the origin band, 100200, is given by
rC*2(n* 1 , Q9 1)C* 0(n* 2 , Q9 2)
1/2 2(n* 1 / n* 2 )(n* 1 n* 2)
(n* 2 / n9 2)(n* 1 / n9 1)
where r is defined by Eq. [10]. The Franck–Condon factor for the transition 120200 is also modified and we obtain similarly
rC*2(n* 1 , Q9 1)C* 2(n* 2 , Q9 2)
rC*0(n* 1 , Q9 1)C* 0(n* 2 , Q9 2)
GJ
(n* 2 0 n9 2)(n* 1 0 n9 1)
1 »C*0(n* 2 , Q9 2)ÉC9 0(n9 2, Q9 2)…
F*1,1(n*1, n*2, Q91, Q92) Å C*1(n* 1 , Q9 1)C* 1(n* 2 , Q9 2)
0
1/2 2(n* 1 / n* 2 )(n* 1 n* 2)
1 10
k Å m, m { 2, m { 4, rrr.
0
r
F
j Å n, n { 2, n { 4, rrr
G
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98
W. L. SMITH
TABLE 1 Intensities of Certain Transitions in the Benzene 2600A˚ System Relative to the 610 Band (1104)
symmetry of the molecule is unchanged in the transition and the change in geometry is small; hence the matrix A is likely to be orthogonal to a good approximation. Moreover, there are a number of two-dimensional nondegenerate vibrational symmetry species, and attempts have been made to determine the force field in both electronic states and consequently the elements of the associated transformation matrix A. In particular, Goodman et al. (6) have provided such information for the b2u symmetry species; in this case, n914 Å 1309 cm01,
n*14 Å 1571 cm01,
n915 Å 1150 cm01,
n*15 Å 1150 cm01,
and for the A matrix
F G F Q*14
Q*15
Å
GF G
0.96 0.28 00.28 0.96
Q914
Q915
.
Transitions in this vibrational species are symmetry forbidden in the absence of vibrational excitation, but are allowed by simultaneous transitions in the e2g species, particularly n6. The vibrational structure of the spectrum is similar to an allowed system, but with progressions built up on 610 instead of the 0 0 0 band. It is a reasonable approximation to apply the theory given in this paper to the intensity of transitions like 61014101510 relative to the 610 band. The results of this are given in Table 1. This example is peculiar in that since n915 Å n*15 the zeroth approximation predicts zero intensity in the transition 61014001520. However, Table 1 suggests that the effect of noncorrespondence of the normal coordinates in the upper and lower states is to redistribute the intensity within the species, and in particular to allow the transition 61014101510 to appear. Examination of the relative intensities of bands within this spectrum (see, for example, the early work of Radle and
Beck (7)) suggests that transitions with 4 1 1004 of the intensity of the 610 band should be just about observable. The 61014101510 band is expected to occur at around 41 330 cm01. Radle and Beck give a band at 41 304 cm01, whose intensity is about 73 1 1004 the intensity of 610. On high resolution plates taken by the author there is a weaker band at about 41 330 cm01 which fits better both in position and with the intensity calculations of this paper. However, the point of this example is not so much to propose a minor new assignment within the benzene spectrum as to explore the magnitude of the effects which the theory predicts. It seems likely that weak transitions such as 61014101510 allowed by the breakdown of the zeroth approximation should be observable in the electronic spectra of polyatomic molecules. The intensity of transitions such as 61014101510 relative to 610 is, according to this paper, determined entirely by the A matrix and the relevant vibrational frequencies. There are, however, other mechanisms which can lend intensity to such transitions, i.e., (a) quadratic terms involving Q1Q2 in the expansion of the electronic transition moment; (b) quartic terms involving Q1Q32 and Q31Q2 in the expansion of the potential function. The contributions of these terms are difficult to estimate, but are unlikely to be larger, and will probably be less, than the effects described here (certainly (b) is likely to be small in the case of benzene). In any case calculations based on this paper should provide a first estimate for the intensity of these transitions. In general, as the frequencies n1 and n2 in upper and lower electronic states begin to diverge, so do the forms of the associated normal coordinates. The first transitions which appear as a consequence of these divergences are 120200 and 100220, but as the divergence increases one may expect the appearance of 110210, probably before the next members of the zeroth series, 100240 and 140200. REFERENCES 1. 2. 3. 4.
I. Ozkan, J. Mol. Spectrosc. 139, 147–162 (1990). W. L. Smith, Proc. Phys. Soc. 89, 1021–1042 (1966). F. Duschinsky, Acta Physicochim. URSS 7, 551–566 (1937). E. B. Wilson, J. C. Decius, and P. C. Cross, ‘‘Molecular Vibrations,’’ Appendix III. McGraw–Hill, New York, 1955. 5. G. Herzberg, ‘‘Molecular Spectra and Molecular Structure,’’ Vol. 3, pp. 151–157. Van Nostrand, New York, 1966. 6. L. Goodman, J. M. Berman, and A. G. Ozkabak, J. Chem. Phys. 90, 2544–2554 (1989). 7. W. F. Radle and C. A. Beck, J. Chem. Phys. 8, 507–513 (1940).
Copyright q 1996 by Academic Press, Inc.
AID
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AP: Mol Spec