Vibronic Calculations in Benzene

Vibronic Calculations in BenzeneAndreas C. Albrecht Citation: The Journal of Chemical Physics 33, 169 (1960); doi: 10.1063/1.1731072 View online: http://dx.doi.org/10.1063/1.1731072 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/33/1?ver=pdfcov Published by the AIP Publishing

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(L E C T RON I C T RAN SIT ION S 169

sary integrals for these distances and the overlap integral, S02 [as I Z(v3"XOl)], may be found in Appendix IV of footnote reference 4. The result is

I2Rhl =I'[0-0.2625+0.2386-0.2746J= -0.2985,

where the units are in (LJe2j ao2) 10-2• (5.201 ev j A for Z=3.18.)

THE JOURNAL OF CHEMICAL PHYSICS

Similar procedures must be followed to obtain the a, {3, and 0 components of IzR and to evaluate the remaining integrals42 Izu, Ilu, and I7U •

42 In evaluating I.u one integral was estimated by extrapolating appropriate available integrals. Only later was it discovered that the estimated integral had in fact also been reported by Liehr. The value obtained by extrapolation was sufficiently correct to cause negligible error in the entries of Table 1.

VOLUME 33, NUMBER 1 JULY, 1960

Vibronic Calculations in Benzene*

ANDREAS C. ALBRECHT

Department of Chemistry, Cornell University, Ithaca, New York

(Received December 21, 1959)

A group-theoretical reduction incorporating normal-coordinate parameters is applied to obtain expressions for the oscillator strengths of forbidden B2u , B1u, and E... transitions in benzene in terms of the minimum number of electronic integrals. The simplification obtained encourages the testing of wave functions within the Herzberg-Teller framework as well as empirical evaluation of the theory. An investigation is made of the possible effect of isotopic substitution on forbidden intensities. The group-theoretical reduction is carried out for 1,4 dideuterobenzene, sym-benzene-da, 2,3,5,6-tetradeuterobenzene, benzene-d., and a hypothetical molecule p-di-X-benzene (where (X)=20 and C-X=1.40 A). It is observed that the B.u

and B1u transitions are very insensitive to mass effects independent of the nature of the electronic functions. On the other hand, E2• transitions mayor may not show a significant sensitivity depending on the details of the wave functions. By way of illustration, oscillator strengths are calculated for all compounds for B2u , B I1" and two &. transitions using conventional benzene ASMO functions and published integrals. The distribution of oscillator strengths among all active normal modes is given for benzene, benzene-d., and p-di-X-benzene. It is found that especially for the E'g transitions the distribution among normal modes is considerably more sensitive to mass effects than is the total oscillator strength.

A general expression is given for the temperature dependence of "forbidden" transition probabilities.

INTRODUCTION

"TTEMPTS to calculate the intensities of "for.ft bidden" transitions in benzenel- 5 have met with various degrees of success. In a critical study of all benzene vibronic calculations Liehr6 reports that with exact solutions of the equilibrium nuclear configuration electronic Schrodinger equation a theory that mixes equilibrium configuration electronic wave functions through vibrational perturbation (Herzberg-Teller7)

should be entirely adequate for predicting forbidden intensities. It is the lack of strictly correct wave functions that has led to at least one important refinementz.4

* Supported in part by the National Science Foundation. 1 D. P. Craig, J. Chern. Soc. 1950, 59. 2 A. D. Liehr, thesis, Harvard University, 1955. a J. N. Murrell and J. A. Pople, Proc. Phys. Soc. A69, 245

(1956). 4 A. D. Liehr, Z. Naturforsch. 13a, 596 (1958). 6 For an extensive discussion and further calculations see A. D.

Liehr, The William E. Moffitt Memorial Session, International Conference on Molecular Quantum Mechanics, University of Colorado, Boulder, Colorado, June 21-27,1959. [Text to be published in the conference proceedings, Revs. Modern Phys. 32 (April, 1960).J

6 A. D. Liehr, Can. J. Phys. 35, 1123 (1957); 36,1588 (1958). 7 G. Herzberg and E. Teller, Z. physik. Chern. (Leipzig) B21,

410 (1933).

and to semiempirical approaches containing a variety of assumptions. In the preceding papers (henceforth referred to as I) the theory for calculating "forbidden" intensity in allowed electronic transitions has been formulated based on the Herzberg-Teller approach. There a group-theoretical reduction of the vibronic problem is presented that incorporates the necessary normal coordinate parameters to express "forbidden" intensities in terms of a minimum number of integrals that alone depend on the choice of electronic wave functions. This simplification not only encourages the testing of wave functions and approximations (within the Herzberg-Teller framework) but it also encourages a search for empirical methods of testing the theory.

The group-theoretical reduction has been carried out for benzene and it is found that only two "one-center" electronic integrals are needed to calculate the intensity of the B2u and the B lu transitions. Four such integrals are required for Ezg transitions. By applying the reduction to several isotopic derivatives and a model benzene derivative a partial study of the effect of mass on forbidden intensities has been carried out.

8 A. C. Albrecht, J. Chern. Phys. 33, 156 (1960), this issue.


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170 ANDREAS C. ALBRECHT

It appeared that a study of intensities in deuterated benzenes might be especially interesting (if effects exist) since the electronic integrals do not change on isotopic substitution. The published9 data on the normal coordinates for benzene and benzene-d6 indicated sufficient mixing among the hydrogen (deuterium) and carbon modes not to discourage such an investigation. Moreover, a larger number of vibrations are vibronically active in the derivatives of lower symmetry. The normal coordinates in terms of local cartesian coordinates have been obtained lO for benzene, 11

1,4-dideuterobenzene, sym-benzene-d3, 2, 3, 5, 6-tetradeuterobenzene, benzene-d6, and a model benzene derivative p-di-X-benzene I2 where X has a mass of twenty and the C-X distance is 1.40 A. The reduction of the vibronic calculations is carried out for "B2u,"

"B1u," and "E2g" transitions for each molecule. It is found that in general the mass effect is essentially insignificant. On the basis of the reduction alone (independent of electronic wave functions), it is predicted that there should be no variation in intensity among the isotopic benzenes. The stability of the B2u and B1u

intensities is maintained even in the p-di-X-benzene (assuming D6h electronic states). In contrast, the Ezg intensity can change markedly in the latter compound from its value in benzene, depending on the details of the D6h wave functions. These observations have important bearing on "forbidden" intensities in D6h or D3h substituted benzenes.

In what follows the general theory is given based on the development in I and a temperature-dependent "forbidden" oscillator strength is obtained. Group theory is then applied to the benzene problem. The reduction is presented in detail (according to active normal modes) for benzene, benzene-d6, and p-di-Xbenzene. The results are summarized for all six molecules. Finally, by way of illustration, conventional ASMO functions are used to calculate oscillator strengths using empirical energies and allowed intensities. This is followed by a discussion.

THEORY

1. General The Herzberg-Teller formalism for calculating "for

bidden" intensities shall be taken directly from I. There an expression has been obtained for the ratio, Rk , of "forbidden" to allowed intensity in the transition from the ground electronic state, g, to the excited electronic state, k. Thus, Eq. (15b) of I reads

Rk =hN°/(S-ff2) L(lIa- 1 cothua) (hksa) 2

B.a

x {(WCg ,NWCg l)2(ilE.kO)-21 (1)

9 D. H. Whiffen, Proc. Roy. Soc. (London) A248,131 (1955). 10 A. C. Albrecht (submitted to J. Mol. Spectroscopy). 11 The force field used is a slightly modified Whiffen force field

involving negligible changes in the normal coordinates. Thus, all previous benzene vibronic work based on the Whiffen normal coordinates remains entirely unaffected.

12 A. C. Albrecht (submitted to J. Mol. Spectroscopy)

where h is Planck's constant, N° is Avogadro's number, Ua = hlla/ (2k T), and lIa is the frequency of the ath normal mode of the ground state. ~Rg,80 and ro1g ,kO are allowed transition moments based on pure electronic wave functions (for ground state equilibrium configuration). ilEskO is the difference in the pure electronic energy of state sand k. The term hksa is the perturbation energy per unit displacement of normal mode a resulting from the coupling of states k and s. The sum goes independently over all electronic states (except state k) and in general over all 3N-6 normal modes, a. Now Rk simply represents the ratio of the second terms to the first in the Herzberg-Teller type expansion of the transition probability, Pg,k [see Eqs. (13) and (14) of 1]. By definition "forbidden" transitions are those for which the first term (WCg ,kO)2, vanishes. By isolating the second terms in Eq. (1) [multiply by (WCg ,kO)2] we have for the temperature-dependent transition probability of a "forbidden" transition:

Pg.k= hN°/ (S-ff2) L (lIa- 1 cothua ) (hk•a ) 2

',a

x {(WCg })2(ilE.kO)-2}. (2)

In keeping with previous practice we shall choose T=OoKI3 and use oscillator strengths instead of transition probabilities. We finally have

fg,k= (hN°/S-ff2) L(va- l ) (hk,a) 2

',a

X [(fg,.o) (EN E,O) (ilE.kO)-2] (3)

for the oscillator strength of a "forbidden" transition at OaK. The quantity in brackets in Eq. (3), involving allowed intensities, fg,.o, and pure electronic energies Eko and E,o, shall normally be regarded as empirically determined. The burden of any calculation then rests on the determination of hks

a for all a and s. Eq. (19) of I reads

where (au /aQa) ° is the coefficient for normal coordinate Qa of the fth local Cartesian coordinate at' nucleus 0'.

The integral is over the space of the pth electron where

P.k(P) =n f 8,o8kodx' (5)

is a one-electron transition density resulting from the one-electron nature of the perturbation operator. 3

8.° and 8kO are pure electronic functions, n is the number of electrons, and the integration is over all

13 While this is not necessary in view of Eq. (2), a thorough investigation of the temperature dependence now underway shows that this assumption is not a critical one at room temperatures and below.


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VIBRONIC CALCULATIONS IN BENZENE 171

space but that of electron p. In Eq. (4) (arp<1ja~/)o is the direction cosine or the vector, rpu, linking electron p with nucleus rr. Xpq is just the coulomb potential between electron p and nucleus rr, (Z.e2jrpu). It proves convenient to symbolize the normal coordinate coefficients by V(a) and the electronic integrals by 1/ (s, k, a) to give

hksa = L~/(a)I/(s, k, a). (6)

i .•

Finally, in Appendix A of I it is shown how Eq. (6) may be greatly simplified through the use of group theory. It is proven that the identical relationships holding among the V (a) hold also among the II (s, k, a) permitting one to write

hksa = L ~q.j*(a)Iq.j*(s, k, a) {L17.i(a, rr*,j*)} (6a)

f*p* .J

where the sum over j* and rr* goes over a minimum number of atomic centers and corresponding cartesian coordinates. The expression in brackets is grouptheoretically determined where

17q/(a, rr*,1*)

=l/(s, k, a)/Iq.I*(s, k, a) =~/(a)j~q*I*(a)'

In any problem where the normal coordinate coefficients, ~q*I*(a), are known it is then possible to obtain the total perturbation energy (squared)

hN°/(87r2) (Va-I) (hk8a) 2,

in terms of the minimum number of electronic integrals Iq .1* (s, k, a). Any set of integrals with the necessary empirical (or calculated) intensities and energies can then lead directly to a calculated oscillator strength [Eq. (3)].

2. GROUP THEORETICAL REDUCTION OF THE VffiRONIC CALCULATIONS IN BENZENE

Speaking generally, "forbidden" transitions to upper states B2u, B lu, and E 2g in benzene can acquire intensity when these states mix with Elu and A 2u states-the only possible upper states in allowed transitions of D6h molecules (assuming a totally symmetric ground state). For the perturbation energy,

not to vanish, the direct product representation of the integrand must contain at least once the totally symmetric representation. SinceI4 BluX Elu and B2u X Elu=E2a while E2aXElu=Elu+Blu+B2U, the representation based on X' must contain E 2g species to permit the mixing of Blu and B2u states with E lu states and it must contain Elu, B Iu , or B2u species to permit the mixing of E2a and Elu states. Similarly it is found

14 The group-theoretic notation used here is that recommended by "Report on notation for the spectra of polyatomic molecules," J. Chern. Phys. 23, 1997 (1955).

that BIg, B2g, and E 2u perturbations are required to mix respectively the B2u, Blu, and E20 states with A 2u states. Now X' must have the same symmetry properties in electron space that the perturbing normal mode has in the configuration space of the nuclei (see I). Thus, the normal modes active in the perturbation are prescribed in each instance. Furthermore, group theory can prescribe the correct form of integrand functions that are the basis of the reduced form of the direct product (rsX r k ) representation. That is, the exact form is given for the electronic functions that interact with each of the normal mode species. For example, consider the case of the mixing of electronic states E2g and Elu. If the pair of degenerate orthogonal components for each state be labeled aI5 and b (E20a, E 2ab, E IUa , E lub ) then one must consider the four possible mixings: aa, ab, ba, and bb, corresponding to E2gaElua, E2aaElub' E2abEIua, and E2ubElub' This set of four functions constitutes the basis of a four-by-four reducible representation which upon reduction contains in block form the irreducible representations of the Elu, B lu, and B2u species. The similarity transformation that accomplishes the reduction also provides the matrix that transforms the original set into four appropriate linear combinations which interact with vibrations elua, elUb' bIU, and b2u. It is interesting to note that this problem is group-theoretically identical to the one of constructing proper excited electronic state functions in benzene from the four configurations which arise upon the promotion of one electron from the filled degenerate elu MO's to the empty e2g MO's. In any case, the exact form of the basis functions thus derived depends on the chosen form (and sign) of the degenerate components. If a common form found in footnote references 9 and 10 is used, then one finds elua perturbing 2-! (bb+aa), elUb perturbing 2-! (ba- ab), bIll perturbing 2-! (bb - aa) and b2u perturbing 2-! (ba+ ab). The case of B2u or B iu mixing with E lu is much more straightforward. The e2ga modes mix the B2u - E lub and Blu -E lua states, while the e2gb modes mix the B2u - EIUa and Blu - E lub states.

In the present investigation we shall consider mixing only with an E1u state since in fact it is the E1u state at 1800 A that is most available for mixing with states of lower energy. (Since no bla vibration exists there is no other possibility for the B2u state and with the AO basis used below there generally can be no interaction with the A2u state.)

Once it is clear how a given normal mode is active in a vibration there remain in general 3N electronic integrals Ii (s, k, a) (rr=l, 2,3"·,, N;f=l, 2, 3) to evaluate for each normal mode. This number is greatly reduced

15 The symbol a has three different meanings in this paper. It is used in conjunction with b to label degenerate components. In the expressions hk'", ~u/ (a) and elsewhere it normally refers to the ath normal mode, one of 3N -6 modes. It is also used to imply the symmetry species label of the ath normal mode. In this sense it is used in f/(s, k, a), ra and fJu/(a, ,,*,j*). It is hoped that in context confusion will not arise.


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TABLE 1. The relative magnitudes of the local cartesian coefficients t.R(a) and t.U(a) and the relative magnitudes of the electronic integrals I.R(s, k,a) and I.U(s, k, a).

A. D6h symmetry

Species of the ath normal mode and type (R or U)

B1u B2u Center

E20 E20a E20b

number R U R U R U R

x 0 0 x x 0 0

2 -x 0 0 -x -ix y (3l/2) x

3 x 0 0 x -ix -y ( -3'/2)x

4 -x 0 0 -x x 0 0

5 x 0 0 x -ix Y (3./2) x

6 -x 0 0 -x -ix -y ( -3'/2)x

B. The Dah group.

The species of the ath normal mode and type (R or U)

E' Center Ea' ED'

number R U R U

x 0 0 (2/3')z

2 -b w (3~/2)y (1/3')w

3 -Ix z ( -3~/2)x (-1/3')z

4 y 0 0 ( -2/3~)w

5 -ix -z (31/2) x (-1/3~)z

6 -b -w ( -3'/2)y (1/31)w

by group-theoretical considerations. In Appendix A of I, it is shown how the minimum number of integrals required may be found and further how all remaining integrals may be simply expressed in terms of this minimum number. Recognizing that the perturbation operator, Je/, in its most general form is the basis of the same representation that the 3N dimensional vector, ~, of cartesian displacements is, then the integrand of the perturbation energy may be analyzed for the number of times r l (totally symmetric representation) appears in r.x rkx r~. This is the minimum number of integrals required. Furthermore, once a suitable local cartesian system has been chosen the minimum number of 1/ for each f may be found by separately analyzing r.xrkxrt, (f=l, 2, 3) for the number of rI'S. Here ~I is an N-dimensional vector of the displacement coordinates of type f.

More to the point, however, once the proper basis functions have been chosen, is the analysisI5 0f raxr~1 for the number of re'S. This gives the minimum number of electronic integrals required for perturbation through normal mode a. For benzene a most convenient choice of local cartesian displacement coordinates consists of (for atom (7) the radial displacement vector RfJ (positive outward), the tangential displacement vector Ug

Elu E1ua E lub

U R U R U

(-2/3~)y x 0 0 (2/3~)y

(1/3')Y !x y ( -3~/2)x (1/3')Y

(1/3') y -ix y ( -3'/2)x (-1/3~)y

( -2/3~)y -x 0 0 ( -2/31)y

(1/3~)y -ix -y (3t!2) x ( -1/3')y

(1/3')y ix -y (3~/2)x (1/3')y

C. The D2h group.

The species of the ath normal mode and type (R or U)

Center A. Ba. B1u B2u

Number R U R U R U R U

1 x 0 0 w x 0 0 ,v 2 Y w x z y w x z 3 Y -U' -x z -y w x -z

4 x 0 0 w -x 0 0 -w

5 y w x z -y -w -x -z

6 y -w -x z y -w -x z

(positive clockwise) and the out-of-plane displacement vector Zo. (We shall consider only carbon motion in the perturbation.) In place of r~/(f=R, U, Z) let us write simply r R , r u , and r z. The characters for these three representations are readily constructed and the number of rr's in raxrR , raXru and raxrz for all relevant a are easily counted. Thus, it is found in the E2g - Elu mixing the Clu perturbation requires only two electronic integrals, one of type lau and one of type laR. The biu perturbation requires one integral of type R which we shall call JaR and the b2u perturbation requires one integral of type U which we shall call Jau. Thus, altogether only four "one-center" integrals, laR, laU, JaR, and Jau, are required to calculate the oscillator strength of an E2g transition. These integrals are the la.l * found in Eq. (6a) where 17* may be somewhat arbitrarily chosen (integrals may vanish on some centers). For the forbidden B2u and Biu transitions, where the C2g modes are active, it is found that each case requires two integrals, an laR and an lau integral, all others being related to these by the group-theoretically determined TJat'S [Eq. (6a)J.

With the minimum number of required integrals established, it is now necessary to specify the TJ'S that relate all other integrals to these few. In Tables lA, B,


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and C appear entries that represent the relative magnitudes of the local cartesian coefficients for relevant types and species of groups DSh, D3h and D2I., respectively. These relationships are group-theoretically determined and were verified in the normal coordinate analysis. lO As a consequence of Appendix A in I these entries also represent the relationships among the electronic integrals for each type and species. This is true provided the same form (and sign) for the degenerate components has been used for the electronic functions as for the symmetry coordinates in the normal coordinate analysis. In the vibronic problem the sign is unimportant since it is the square of the energy that enters [Eq. (2)]. The symbols x, y, z, and w therefore represent either ~u*f*(a) or Iu.f*(s, k, a). For example, the entry in the third row of Table IA under column R of Ezvb means both IaR(s, k, e20b) = (-3~/2)IIR(S, k, ezoJ and ~3R(e2gb) (-3!j2H1R(e20a)' The 1)'S themselves are simply the ratio of a given pair of entries in anyone column, the reference (or starred) entry appearing in the denominator. For example (Table IA), 1)3R(eZIlM

2, R) = -1. Tables IB and C apply to vibronic problems for D3h and DZh molecules. In the present study, these tables are needed only for the normal coordinate coefficients. Under isotopic substitution the pure electronic wave functions see only a D6h potential and the 1)'S derived from Table IA still apply.16

It now requires but a short step to obtain hksu [Eq.

(6a) J by incorporating the ~u*f*(a) but leaving the Iu.f* as parameters. This in tum can be squared and multiplied by hN°j (87fzva ) to give, in reduced form, the complete nonempirical part of the oscillator strength [Eq. (3)]. This has been done for benzene, 1,4-dideuterobenzene, sym-benzene-da, 2,3,5,6-tetradeuterobenzene, benzene-d6, and the hypothetical molecule pdi-X-benzene already described (see also I). The results of the group-theoretical reduction are presented in detail only for benzene and the last two mentioned compounds. Tables IIA, B, and C deal with the Blu

and B2u transitions while Tables IlIA, B, and Care concerned with Eta transitions. A summary for all molecules is given in Tables IVA and B. To illustrate, consider the entries for normal mode 7a in Table IIA. For the e20a species we have, using Table IA and Eq. (6a), hk/

a =3[hR (7a)I1R(s, k, e2oJJ+4UF(7a)IF(s, k, e2oJ]' From Table IIIe of footnote reference 10, ~lR(7a) is found to be -0.05215 (g)-! and ~2U(7a) is -0.00572 (there labeled Ria' and UZa', respectively). Incorporating these values into hk/ a, squaring, and mUltiplying by hN°/(81rzcwa ) (XlOIS to convert to A2) gives (X102) 0.0136 (hR) 2+0.0003 (I2U) 2+0.0040 IIRIP, the entries for normal mode 7a. Here Wa = 3043 cm-I • All cI.)-lculations were in fact done with an IBM 650 computer immediately following the normal coordinate program.

I. This does not contradict the requirement that the same relationships hold among the I af'S as among the tol (a)'5 since all relationships in Tables IB and C are contained in Table IA.

TABLE II. The "ezu" normal mode coefficients for the Bzu and Blu electronic integrals and the calculated oscillator strengths.

Frequencies incm- I

eZYa

7a 3043 Sa 1600 9a 1174 6a 610.8

Total e2aa

~ga

7a 2274 Sa 1546 9a 863.3 6a 575.7

Total e'ga

aq

1 3063 2 1676 3 1286 4 1126 5 805.9 6 399.2

Total au

bag

11 3043 12 1587 13 1304 14 650.0 15 424.0

Total bag

A. Benzene: e2ua modes.

Coefficients for integrals (A)ZX 1()2

(1,R)2 ([2 U)2 (1/1]2U)

0.0136 0.0003 0.0040 0.0452 0.2513 0.2131 0.0359 0.0092 0.0363 0.4344 0.2418 -0.6482 0.5291 0.5026 -0.3946

B. Benzene-d.: e'ga modes

0.0415 0.0047 0.0278 0.0327 0.2736 0.lS92 0.0678 0.0032 -0.0294 0.3781 0.2176 -0.5736 0.5201 0.4991 -0.3860

Oscillator strengths Xl()2

0.001 0.020 0.056 2.025 0.000 0.160 0.252 0.421 0.309 2.626

0.001 0.118 0.067 2.102 0.013 0.007 0.224 0.390 0.305 2.617

C. 1,4-Di-X-benzene: ag and baa modes

0.0043 0.0001 0.0012 0.000 0.006 0.0791 0.1706 0.2324 0.028 1.739 0.0530 0.0492 -0.1022 0.037 0.10S 0.0035 0.0519 0.0271 0.012 0.307 0.1542 0.1456 -0.2996 0.121 0.358 0.1987 0.OS10 -0.253S 0.096 0.102 0.4928 0.4984 -0.3949 0.294 2.620

0.0138 0.0003 0.0041 0.001 0.021 0.0440 0.2223 0.1979 0.056 2.111 0.0056 0.0043 0.0098 0.000 0.084 0.4110 0.1862 -0.5532 0.214 0.269 0.0385 0.0433 -0.0816 0.035 0.120 0.5129 0.4564 -0.4230 0.306 2.605

It can be seen from Tables IA and B how perturbation energies due to "a" and "b" degenerate components are equal. Calculate hk•a for the eZOb component for example. It is seen that with Table IA and Eq. (6a) an expression is obtained identical to the one for the e2ga species.

Finally, Tables II-IV also contain calculated oscillator strengths where the necessary electronic integrals have been evaluated using ASMO functions and the tabulations of Liehr.4 These calculations shall now be discussed.

CALCULATION OF OSCILLATOR STRENGTHS FOR ASMO FUNCTIONS

To evaluate the electronic integrals to be used in conjunction with the data of Tables II-IV, it is necessary first to obtain P8k, which depends on e.o and eko [Eq. (5)], and then to perform the integration indicated in Eq. (4). Whenever e.,o and eko are ultimately based on LCAO type functions (even when they are ASMO with CI), the transition densities will be decomposable into simple component densities such as


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TABLE III. The "e,u", "b,u" and "b2u" normal mode coefficients for the &0 electronic integrals and the calculated oscillator strengths.

Frequencies in cm-'

elua

20a 3078 19a 1480 18a 1035

Total e'ua

13 3067 12 1012

Total b,u

14 1307 15 1148

Total b2u

20a 2287 19a 1324 18a 808.8

Total e'ua

13 2283 12 961.1

Total b,u

14 1285 15 825.1

Total b2u

Frequencies

0.0126 0.0416 0.1220 0.1762

U,R)2

0.0268 0.7512 0.7780

0.0368 0.0665 0.0848 0.1881

U,R)2

0.0790 0.6888 0.7678

in cm-' (I,R)2

18 3071 19 1563 20 1276 21 1026 22 637.6

Total b,u

23 3078 24 1407 25 1285 26 1092 27 331.6

Total b2u

0.0043 0.0664 0.0267 0.0637 0.0226 0.1837

0.0130 0.0591 0.0099 0.0851 0.2698 0.4369

A. Benzene: e'ua, b,u, and b2u modes

Coefficients for integrals (A)2Xl()2

(I2U)2

0.0000 0.0764 0.0118 0.0510 0.2186 0.3578

0.0002 0.1539 0.0190 0.0802 0.1938 0.4471

(I2U)2

0.0002 0.1455 0.0835 0.2292

(J, U)2

0.7112 0.1689 0.8801

(I,RI2U)

0.0029 0.1557 0.2019 0.3605

B. Benzene-d6 : e'ua, blu and b2u modes

0.0020 0.2329 0.0101 0.2450

0.8637 0.0159 0.8796

0.0173 0.2489 0.0585 0.3247

C. p-Di-X-benzene: b,u and b2u modes.

Coefficients for integrals (A)2Xl()2

(I,Rh U) (J,R)2 (J,U)2

0.0010 0.0178 0.1425 0.0393

-0.0355 0.2462 0.1140 0.3140

-0.1405 0.1410 0.0815 0.7583

0.0031 0.0000 0.1907 0.0281 0.0274 0.7992 0.1653 0.0527

-0.4574 0.0008 -0.0709 0.8808


0.026 0.402 0.544 0.972

0.088 0.642 0.232 0.962

0.002 1.080 0.610 1.692

0.014 1.728 0.070 1.812


E2o~ E20!

0.008 0.000 0.528 0.812 0.016 0.118 0.322 0.398 0.060 1.658 0.934 2.986

0.026 0.002 0.512 1.204 0.070 0.142 0.302 0.414 0.072 1.496 0.982 3.258


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TABLE IV. The summed normal mode coefficients for the electronic integrals and calculated oscillator strengths for benzene and five derivatives.

A. B1" and B2" transitions

Total coefficients for electronic integrals Oscillator strengths (A)2X 1()2 X1()2

Molecule (/IB) 2 (/2 U)2 (/IB12U) B2U(obs 0.14)8 Blu(obs 9.4)b

Benzene 1.0582 1.0052 -0.7892 0.618 5.252

1,4-Dideuterobenzene 1.0523 1.0028 -0.7836 0.614 5.248

Sym-benzene-d3 1.0492 1.0015 -0.7806 0.613 5.244

2,3,5,6-Tetradeuterobenzene 1.0461 1.0005 -0.7778 0.612 5.242

Benzene-d6 1.0402 0.9982 -0.7720 0.610 5.234

1,4-Di-X-benzenec 1.0057 0.9548 -0.8179 0.600 5.225

B. E2g transitions

Total coefficients for electronic integrals Oscillator strengths (A)2X 1()2 X1()2

Molecule (11B)2 (/2 U)2 (/IB12U) (J1B)2 (JI U)2 E2g~ (obs 9.4)b E 208 (obs 9.4)b

Benzene 0.3524 0.4584 0.7210 0.7780 0.8801 1.944 3.384

1,4-Dideuterobenzene 0.3621 0.4710 0.6966 0.7707 0.8800 1.940 3.480

Sym-benzene-da 0.3656 0.4756 0.6833 0.7723 0.8799 1.934 3.516

2,3,5,6-Tetradeuterobenzene 0.3694 0.4807 0.6713 0.7712 0.8798 1.930 3.554

Benzene-ds 0.3762 0.4900 0.6494 0.7678 0.8796 1.924 3.624

1,4-Di-X-benzenec 0.6206 0.8049 0.0106 0.7583 0.8808 1.916 6.244

8 H. B. Klevens and J. R. Platt, Technical Reports of the Laboratory of Molecular Structure and Spectra, University of Chicago, (1954-1954). b V. J. Hammond and W. C. Price, Trans. Faraday Soc. 51, 605 (1955). (Neither the E'g state nor the BlU state has been absolutely identified.) C Benzene electronic wave functions assumed for this molecule.

those depicted in Fig. 1 for E2ga and E 1ua species. Here the heavy dot marks the midpoint between a pair of atomic orbitals whose product constitutes a contribution to the transition density of a given type.

It is useful to apply group theory once more, this time to count the number of component transition densities that exist for each species when the electronic wave functions are ultimately based on six equivalent atomic orbitals. It is only necessary to count the number of times the species of interest is contained in the direct product representation rhexagon2• rhexagon is the reducible representation based on a hexagon of six equivalent centers. Counting only one set of signs for each component density, one finds that there are in effect three each of the E2g and E1u type, there are two of the B 1u

type and only one of the B2u type. In addition it is found that there are no Big, B 2g, or E 2u types so that in this rather general case there can be no question of mixing with an A 2u state (see also footnote reference 4).

If Slater 2pz (Z=3.18) atomic orbitals are chosen as a basis then one can use the integrals tabulated by Liehrl to' 6btain energies (per A) for each of the nine basic component itransition densities. Then any choice of electronic functions will specify the coefficients for

each of these densities to give directly the total electronic integral.

To illustrate, we choose the conventional ASMO benzene functions, which have been succinctly presented by Murrell and Pople,3 and are given here in Table V together with empirical energies and intensities. The notation is a mixture of that used in I and

TABLE V. B2", B1u, E.g, and E 1u state functions in benzene, observed energies, and intensities.

Energies Oscillator in ev strengths

State function Symmetry (obs) X102 (obs)

8v=2-·(Xa5-)(2') B2" 4.9 0.148 8u= 2-. (Xa'+X25) B1u 6.2 9.4b 8,a = 2-. (X15 - X26) E.o.(7)

1 8,b= 2-. (Xl'-X36) E.ob(7) 88a = 2-. (XI5+X2S) E.o.(Il) 6.2c 9.4c 88b = 2-. (Xl'+xaS) E20b (ll) J 8x= 2-. (Xl+X2') E 1ua 7.0 44b 8y= 2-l (Xa'-X26) E1"b 7.0 44b

8 H. B. Klevens and J. R. Platt, Technical Report of the Laboratory of Molecular Structure and Spectra, University of Chicago (1953-1954).

b V. J. Hammond and W. C. Price, Trans. Faraday Soc. 51, 605 (1955). c Assumed.


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TABLE VI. E 2ga and E lua transition densities and electronic integrals for ASMO functions.

A. Transition densities and integrals IIR, I 2u The electronic

Integrand of P,k (Eq. 5) integrals (ev / A) (for component a perturbation

only) Symmetry Transition density in terms of component

Transition of P,k densities (see Fig. 1) IIR I 2u

B2u 8v8y E 2ga -1.1027(T,+Ta)+0.2686T2 1.1784 -2.2837

Blu 8 u8x E 2ga -0. 2686 (TI+ Ta) +1.1027T2 1.1030 3.1700

E2Q> 2-i(8~a8x-8~b8y) =2'8~a8x E lua -0.4245 (Sl+ Sa) +1. 1489S2 1.6967 1. 2212

E 20 , 2-!(8 •• 8x-8'b8y) =2!8,a8 x EIU~ -1.1489(S,+Sa) +0.4245S2 0.1274 3.5128

B. IIR and I 2u integrals for each component transition density (see Fig. 1) (ev/A)

-1.3681

2.8290

0.7867

3.5924

0.4912

0.1171

that of footnote reference 3 but should be sufficiently clear. These functions have been chosen without any illusions concerning their suitability.6 The basis MO's are given in I [Eq. (22) ] normalized to include all overlap. In Table VIA the necessary transition densities are indicated including the coefficients of the component transition densities depicted in Fig. 1. Interaction with the ath degenerate component only is considered since the contribution from the b component is identical. In Table VIE the integrals (in ev/A) are given for each of the component densities. The evaluation of a typical integral has been described in Appendix C of I and will not be discussed here. The component integrals in Table VIB are combined as indicated in Table VIA and there tabulated. These total integrals are used directly with the reduced data in Tables II-IV and the empirical factor [bracketed expression in Eq. (3) ] is constructed from the data in Table V to give the reported oscillator strengths.

The present choice of E20 and E lu wave functions has led to the following interesting simplification. The proper transition-density combinations which mix with each of the possible vibrations are, listing the E20

component first, 2-i(aa-bb) (ezoJ, 2-i(ab+ba) (e20b) ,

o. -~-s -s~-s

25

45

-~ -4~-4S

25 25 45

b, S~5 -S~-8

~8

FIG 1. Six of the nine component transition densities arising when the electronic functions have an atomic orbital basis: (a) the &0. densities; (b) the E lua densities. Each dot is located midway between a pair of atomic orbitals whose overlap constitutes a contribution to the transition density. (s= 1/6 and t=!.)

0,1396

3.8926

1.6629

2.5394

0,3640

0,1033

2-! (aa+bb) (bIu), and 2-!(ab-ba) (b2u ). Upon working out the functional form (independent of AO's) for each of these product functions (and for either E2o~ or E 2o ,) it is discovered that aa= -bb and ab=ba. Thus, accidentally, the b2u and blu vibrations cannot be active in the present treatment. However, any breakdown of the configurational interadion as represented in the E2o~ and E 2o , functions will activate the biu and b2u perturbations.

DISCUSSION

Consider the summed coefficients for integrals summarized in Table IV for all six molecules. It is quite apparent that whatever the nature of the electronic functions (and hence electronic integrals) the present theory predicts that there can be no significant variation of intensity for B2u and B lu transitions upon isotopic substitution. From the start it must be emphasized that perturbation due to carbon motion only has been considered throughout. Even when para substituents of mass 20 are introduced there still can be very little change. If a trend is to be discerned, it is seen that all three coefficients become smaller as the total mass of all substituents increases. There is no obvious dependence on symmetry. While sym-benzene-ds has 14 active vibrations and the D6h derivatives have eight, the influence on carbon motion, if any, apparently has more to do with total substituent mass than the formal symmetry.

Analogous trends in coefficients, somewhat stronger and in the reverse direction, are evident in the elu

perturbation of the E20 transition. Here the jump to p-di-X-benzene shows the greatest change of all. Table IIIC shows how these changes originate largely through the "elut-type modes. On the other hand, the coefficients for blu and b2u perturbations, potentially strong, show very little sensitivity to substituent mass and, if anything decrease very slightly upon substitution. These results are understandable in light of the


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extent of "carbon mode" and "substituent mode" mixing in each of the active modes, the elu frequencies showing the greatest extent of mixing. 1O •12

The above observations suggest that if electronic wave functions are such as to emphasize appropriate elu perturbation then there may be a chance of distinguishing the controversial E2g and Blu transitions by mass effects alone in D6h or Dak benzene derivatives. This assumes that one can smoothly extrapolate the experience with p-di-X-benzene to Dah and D6h derivatives as appears to be the case among the isotopic benzenes. On the other hand, relatively significant changes in electronic functions on going from benzene to Dall or D61l derivatives may so alter the transition density that any mass effect becomes obscured. It happens that the E2u , function used here to calculate oscillator strengths gives a mass sensitive intensity that almost doubles on going from benzene to p-di-Xbenzene. Such a change contrasts strikingly with the predicted steadiness of the B lu intensity. It is by no means clear that the E 2g functions chosen here are at all realistic. For example Murrell and McEwenl7 show that doubly excited configurations are very important and Craigl8 reports that a valence-bond calculation by Lyons reveals blu first and elu next in ability to perturb the E 2g state. The latter observation would at best indicate a much weaker mass effect.

Sklarl9 has attempted to isolate the forbidden intensity in the "B2u" bands of a whole series of benzene derivatives by interpolating from "forbidden" intensities in benzene, sym-tri-X-benzene, and hexa-Xbenzene where usually cases with methyl or chloro substituents were considered. A steady increase in intensity with increase in total substituent mass was observed within a given series. This lead to the assumption that formal symmetry is unimportant as far as vibrational perturbation is concerned just as has been suggested here. The change in intensity with increasing total substituent mass, however, is quite large. Our calculations would indicate that this change is entirely due to a change in electronic functions and not to a mass effect. This interpretation for the more than twofold enhancement of intensity upon going from benzene to hexamethylbenzene must be applied with some caution. It may be that the calculated insensitivity to mass breaks down significantly when the effective substituent mass approaches that of carbon. It is hoped that these matters will be better understood after completing a study of vibronic problems in weakly substituted benzenes now in progress.

A second interesting result of the group-theoretical reduction concerns the relative activity of various vibrations. Again certain general conclusions may be

17 J. N. Murrell and K. Lenore McEwen, J. Chern. Phys. 25, 1143 (1956).

18 D. P. Craig, Revs. Pure and App\. Chern. (Australia) 3, 207 (1953).

19 A. L. Sklar, J. Chern. Phys. 10, 135 (1942).

drawn without specifying details of the electronic state functions. The effect of mass is now more apparen t although still it is not strong for the B2u and B lu transitions (Tables IIA-C). The four "carbon mode" frequencies, originally 6a (and 6b) and 8a (and 8b), maintain their dominance throughout although this is somewhat obscured in p-di-X-benzene. In benzene and benzene-d6 the e2u "substituent modes" are sufficiently isolated not to contain much carbon motion and therefore are not influential in the perturbation. In p-di-Xbenzene the separation is no longer so purel2 and several frequencies are active. On the other hand, the elu modes (Tables IlIA-C), which are not so clearly separated into substituent motion and carbon motion, show a distribution that is considerably more mass sensitive. It is seen that the so-called "substituent modes" 18a (and 18b) can contribute very significantly to the perturbation. The blu and b2u frequencies involve relatively distinct separation of motion so that each species has one major contributing vibration with little mass effect.

Apart from the observed intensity itself the most widely used criterion for testing theory is the reported20

exclusive activity of the 6a (and 6b) modes in the B 2u

transition in benzene and the isotopic derivatives. Table IIA reveals how IIR and IF having opposite signs would tend to favor the 6a frequency while with equal signs mode 8a is likely to be more active. Table VIB shows that to obtain integrals with opposite signs the transition density must contain a lot of type TI density. This is an "even" type density in the language of MoffiWI which is to be associated with B2u-Elu

interaction. Although Ta density is also in this category its contribution is not influential. Thus, it is seen that the predominance of the 6a mode in the B2u perturbation lies in the "even" nature of the perturbation and is correctly predicted by the present theory. [Even after the MO's are normalized including all overlap the "even" transition density still dominates (Table VIA).J The Blu state is sensitive to "odd" perturbations21 (see also Table VIA) so that the 8a mode is likely to dominate. The true relative activity of the two frequencies in the B2u transition is obscured by the fact that we ask for the relative activity of ground state frequencies via the sum rule that has been employed in the theory (see I). It is hoped both to test the theory and to directly ascertain the relative activity of the frequencies by trying to match the observed22

temperature dependence of the B2u intensity using an expression based on Eq. (2).

Finally, some comment is necessary concerning the calculated oscillator strengths. The results (Tables II-IV) do serve to illustrate some of the conclusions just reached. However, agreement with experiment is

20 F. M. Garforth, C. K. Ingold, and H. G. Poole, ]. Chern. Soc. 1, 409, 413, 414 (1948).

21 W. Moffitt, J. Chern. Phys. 22, 320 (1954). 22 F. Almasy and R. Laernrnel, Relv. Chim. Acta 34, 462 (1951).


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17S ANDREAS C. ALBRECHT

very poor. It is appropriate to compare these calculations with those by Murrell and Pople3 based on the same wave functions (except for normalization of the MO's). Murrell and Pople used a point-charge approximation to evaluate the electronic integrals and did not include transition densities of types T3 and S3. Further, they considered only "carbon modes." It has been seen that the latter assumption is a serious one for the Clu

perturbation. Otherwise any gross difference in results is probably due to the different methods used for evaluating integrals. Their reported oscillator strengths (X 102) for B2u , B lu , E2g~, and E 2g ; transitions are, respectively, 1.19, 39.S, 5.S1, and lS.5. The results obtained here are consistently lower giving better agreement for the B2u transition but worse for the 2000-A intensity. It must be concluded that either the theory is at fault or that the ASMO functions employed are very poor solutions of the pure electronic SchrOdinger equation. That the latter is no doubt the case is the tenor of the critical analysis given by Liehr.6

SUMMARY

A group-theoretical reduction which incorporates normal coordinate parameters (based on carbon motion only), is presented for Herzberg-Teller type calculations in benzene. The simplification achieved can be useful for the testing of trial electronic functions and the making of assumptions in evaluating integrals. The same reduction has been carried out for five benzene derivatives, four of them isotopic and one a model pdisubstituted derivative with a substituent having a mass of 20. It is found that B2u and B lu transitions are very insensitive to mass changes while E2g transitions mayor may not be somewhat more sensitive depending on the nature of the correct electronic wave functions. For nonisotopic derivatives of D6h or DSh symmetry

there may be a chance of distinguishing E2g and Blu

states by mass effects alone. Furthermore, the results suggest that any marked variation in "B2u" or "Blu"

intensity in such derivatives should be attributed to changes in electronic functions and not mass effects.

It is found how certain vibrations dominate in their influence in a manner relatively independent of substituent mass. In particular, it is seen how the 6a (and 6b) modes perturb B2u states and the Sa (and Sb) modes perturb the Blu states because of the respective "even" and "odd" character of the transition densi-ties. .

Oscillator strengths have been calculated using the conventional benzene ASMO functions. Certain basic integrals are tabulated and the calculation is outlined in a manner permitting the easy testing of any kind of ASMO functions. The results, which agree very poorly with experiment, nevertheless serve to illustrate many of the general conclusions made on the basis of the reduction alone.

No attempt has been made to improve agreement with observation by exploring trial functions. Instead it is hoped that it will be possible to test the basic theory by attempting to fit an observed B2u oscillator strength vs temperature curve using the theoretical expression which after the group-theoretical reduction contains only two electronic integrals as parameters.

ACKNOWLEDGMENTS

The author is very grateful to Professor G. F. Dresselhaus for his criticism and several clarifying remarks concerning group-theoretical problems. He also wishes to thank Dr. A. D. Liehr for a number of helpful discussions concerning his vibronic studies and for his criticism of the manuscript.


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Vibronic Calculations in Benzene