rp 0 corresponds to the point of reversal (BT of Fig 1.8b) of the lQ
1.7 THE STRUCTURE OF THE EXCITED STATE
An approach to the
a prioricalc u l a t i o n of the geometries of aromatic molecules in any electronic state was put forward by M c Coy and Ross  and was successfully applied to the ground states of benzene, n a p h t h a l e n e and anthracene. A similar a p p roach by Coulson and Looyenga
 proved successful for the ground states of single-ring nitrogen heterocycles. An excited state geometry of n aphthalene [Innes, Parkin, Ervin
et at.1965] and one in azulene and azulene-ds [McHugh and R.oss 1968] were also predicted by the method, and the inertial constants conformed well w i t h experiment. In this p r o cedure bond lengths are computed from bond orders by the widely used empirical relation [Coulson 1939],
r ______ (s - d)______
1 + 0. 765(1 - p)/p (1.20)
Ad = 1.330 A .
The quantities s and d are the best values of the carbon-carbon bond- lengths in ethane and ethylene respectively, and p is the bond order In a pplications to excited states, the geometry of a m o l ecule in its ground electronic state is assumed to represent the strain-free structure in respect of angle deformations. Upon electronic excitation the bond
lengths in the m o l e c u l e change in a ccordance w i t h the n e w bond orders, and the angles are assumed to change in order to mi n i m i z e the strain imposed by the altered bond lengths. The potential energy change upon electronic exc i t a t i o n was assumed to have the simplest a c c e ptable form:
dV = d ö.2 + k ' 2 2 dO.0. , (1.21)
i l l j
w h e r e dö^ is the small change in the i bond angle and the double sum is over those adjacent angles sharing a bond common to two rings. M c Hugh and Ross  found that the one a d j u s t a b l e parameter in the expression for dV, n a m e l y k ’/k adopted the value of 1-1.0 for the best fit to the ground state of azulene. In that w ork the geometry of the molecule obtained in a m i n i m i z a t i o n of dV was rela t i v e l y insensitive to k'/k, pr o v i d i n g its v a l u e was b e t w e e n +3 and -3. (In three-ring compounds such as anthracene, this is not so.)
There are four conditions upon the geometry of IBF which must be met in any small distortion, assuming C ^ symmetry to be maintained. Two are angle redundancy conditions and are given in Eqs. (1.22) and
(1.23), w h e r e the differentials indicate small changes.
d a j + d a2 + da 3 = 0 (1.22)
d ß i + d ß2 + ^ d ß3 = 0 .
The other two conditions ensure ring closure:
— ----2-- r 1 (cos aj - d a2 sin a^) + r 2 (cos 03 - d a2 sin 03) = 0
2 r 3 (cos ßi - dßi sin ß^) - r 4
. ß 3 , ß 3 8 3"
sin — + cos 0 . (1.25)
The p r o b l e m is thus to minimize dV subject to these four conditions, and find the resultant values of daj, d a2 ..., etc. The m e thod of L a g r a n g i a n undetermined multipliers results in a series of eight s i m u ltaneous nonlinear equations in the six variables (da^, dß_^, Xj, A 2 ). The A Ts are Lagrangian multipliers. These equations may be treated as linear by setting the small cross terms, such as Ajdoti,to zero in a first approximation. The values of A^ obtained in the solution may
then be reinserted in the cross term for the next iteration, and so on, until convergence is achieved.
The CNDO/CI calculation for the first excited state of IBF, mentioned in an earlier section, predicted that the state could be described by an MO configuration which is almost entirely unmixed with other configurations. Bond orders in this configuration were obtained from the coefficients of the occupied MO's and converted to bond lengths by Eq. (1.20). These bond lengths, together with the ground state bond angles (a^, 3^), were used in the calculation of angles, as described above. Convergence to the solution was attained within three iterations. Table 1.8 gives the results of calculations obtained with 1.5 for the value adjustable parameter k ’/k.
Calculated geometry of isobenzofuran in the ground and first excited singlet states
Bond lengths (Ä) Bond angles (degrees)
Bond^ Ground state Change on excitation Angle a Ground state Change on excitation r i 1.4620 -0.0226 <*1 118.00 +0.28 r 2 1.3530 +0.0432 a 2 121.39 +0.27 r 3 1.3544 +0.0244 <*3 120.61 -0.55 r 4 1.3818 +0.0271 3i 106.11 +0.57 r m 1.4590 -0.0090 32 110.77 -0.14 r e 1.4540 -0.0386 33 106.24 -0.87
F i g . 1 . 1 6 : The g e o m e t r y c h a n g e a cc o m p a n y i n g t h e B2 M t r a n s i t i o n o f
i s o b e n z o f u r a n . Arrows a r e q u a n t i t a t i v e amo un ts by w h i c h t h e at oms
move t o a d j u s t t o t h e e x c i t e d s t a t e g e o m e t r y . The s c a l e by w h i c h t h e y a r e e n l a r g e d i s 15; an a r r o w t h e l e n g t h o f t h e c e n t r a l bond r e p r e s e n t s a d i s p l a c e m e n t o f 0 . 0 9 7 3
A.The s h a p e c h a n g e a s s o c i a t e d w i t h t h e e l e c t r o n i c e x c i t a t i o n i s i l l u s t r a t e d i n F i g . 1 . 1 6 . As was found i n a z u l e n e , t h e r e s u l t s a r e n o t s e n s i t i v e i n any w o r t h w h i l e way t o t h e v a l u e c h o s e n f o r k ’ / k , s o l o n g a s i t l i e s b e t w e e n +3 and - 3 . T a b l e 1 . 9 g i v e s t h e c h a n g e s i n t h e i n e r t i a l f c o n s t a n t s f rom t h e g r o un d t o t h e e x c i t e d s t a t e s f o r t h i s c a l c u l a t i o n . F o r t h e e x c i t e d s t a t e , a l l CH b o n d l e n g t h s i n t h e s i x - m e m b e r e d r i n g c o n t r a c t by 0 . 0 1
At o 1 . 0 7
A,a s i n b e n z e n e [ C n r f o r t h , J n g o l d and P o o l e 1 9 4 8 ] . A l l CH b on d s w er e as sumed t o be t h e e x t e r n a l b i s e c t o r s o f t h e a p p r o p r i a t e r i n g a n g l e s .
The change in the inertial constants in isobenzofuran upon electronic excitation to the lowest singlet state
AA +0.0012(4) +0.0009(1)
AB -0.0012(2) -0.0009(6)
AC -0.0005(2) -0.0004(4)
For comparison the changes found from the band contour method are also presented. The agreement between the two sets of figures is surprisingly good. One notices particularly that the rather unusual positive change in the long axis inertial constant A, found from the band contour
analysis, is predicted with about the correct magnitude. The positive value of AA indicates that the molecule is contracted about the long axis by electronic excitation. An examination of Fig. 1.16 indicates the rearrangement which the molecule undergoes upon electronic excitation. This primarily involves a contraction of the bonds and and
expansion of r? , r3 and rit. The effect in the f.ive-membered ring is to elongate the whole ring and to reduce the COC angle.
When a molecule undergoes electronic excitation with a consequent geometry distortion, then the vibrations which can best
accommodate it to the new molecular parameters will be most active in the Franck-Condon way. For the purpose of determining which vibrations
should be active in IBF, the mass-weighted cartesian normal coordinates of the totally symmetric modes of the molecule in its ground electronic
state were computed and are plotted in Fig. 1.17. Table 1.8 indicates that the principal effect of electronic excitation is bond deformation rather than angle-deformation (judged by the yardstick of zero-point amplitudes). The vibration 11a± at 1422 cm-1 contains most of the
elements of the excited state geometry, except that it clearly compresses the CO bonds instead of extending them, and opens the COC angle instead of closing It (little attention should be paid to the comparison of the
See Chapter 3 for details of Lhe method, and Sec. 1.5.2 for details of Lhe force field.
The molecular geometry is that of Fig. 1.5. H and C displacements
are reduced by the factors (m^/m^)2 and (m^/m^) 2, respectively. All displacement vectors are enlarged by a factor of 80, thus a vector the length of the central bond represents a displacement of
0.01824 Ä. The symbols beneath each figure represent in order: the label, the molecule type and the frequency in cm-1. The three high frequency stretches are omitted.
f / ,lfll IBP'. 520 2fll IBP.719 5 R 1 IBP.975 4 R 1 IBP.966 701 IBP,1223 6ft1 IBP.1189 901 IBP.1396 801 IBP.1354 1101 IBP.1558 1CR1 IBP.1422
hydrogen atom motion in these figures). The other vibration which has some of the elements of Fig. 1.16 is 10ai at 1396 cm *. The other modes appear to have less in common with the distortions produced by the
electronic excitation, primarily because none of them simultaneously compress r^ and expand r?_ by sufficiently large amounts. The strongest two vibrational bands in IBF (see Table 1.1) are at 1479.7 cm-1 and 1353.1 cm-1 respectively. This is in excellent agreement with the
+ qualitative findings above.