We have used the equations of motion method to calculate the excitation energies and intensities of several transitions in benzene. The ordering of the singlet and triplet states of B_(2u), B_(1u), and E_(1u) symmetry agrees with experiment and the error in the calculated frequencies ranges from 3% to 25%. This error range is reasonable considering the relatively small basis set used. The most extensive calculation included 10 hole and 28 particle states and shows the effect of changes in the sigma core for each transition. The calculated transition moment of 1.74 a.u. for the ^1A_(1g)→^1E_(1u) transition agrees well with the experimental value of 1.61 a.u

McKoy, Vincent

Rose, John B.

Shibuya, Tai-Ichi

English

Caltech Authors - Main

Downloaded 14 Feb 2006 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
Electronic excitations of benzene from the equations of motion 
method* 
John B. Rose t, Tai-lchi Shibuya*, and Vincent McKoy 
Arthur Amos Noyes Laboratory of Chemical Physics, § California Institute of Technology, Pasadena, California 91109 
(Received 22 March 1973) 
We have used the equations of motion method to calculate the excitation energies and intensities of 
several transitions in benzene. The ordering of the singlet and triplet states of B '"' B tu• and E tu 
symmetry agrees with experiment and the error in the calculated frequencies ranges from 3% to 
25%. This error range is reasonable considering the relatively small basis set used. The most 
extensive calculation included 10 hole and 28 particle states and shows the effect of changes in the 
sigma core for each transition. The calculated transition moment of 1.74 a.u. for the 1A 1 ,~ 1 E 1, 
transition agrees well with the experimental value of 1.61 a.u. 
I. INTRODUCTION 
In recent papers 1' 2 we have developed the equations-
of-motion method as an approach specifically designed 
to study the spectroscopic and response properties of 
atoms and molecules. In these methods one directly ob-
tains the transition amplitudes of physical interest in a 
given process, e. g., in photon absorption and in elec-
tron-molecule scattering, thereby avoiding many of the 
difficulties involved in obtaining highly accurate wave-
functions. We have used this method to study the excita-
tion frequencies and transition moments of several mol-
ecules, e. g., N2, CO, C2H4, 
2 H2CO, 
3 and C02 
4 and the 
Born differential cross sections for the scattering of 
electrons by C02• 
4 The results generally agree well 
with experiment. These results indicate that the method 
is practical and is capable of including the major corre-
lation effects on the physical observables in a relatively 
small matrix problem, even for quite large basis sets. 
In this paper we present the results of calculations on 
the excited states of benzene as an application of the 
method to a larger polyatomic molecule. These calcula-
tions were done with a relatively limited basis set and 
include results both with and without sigma-pi coupling. 
The inclusion of two particle-two hOle (2p-2h) excita-
tions also shows the effect of changes in the sigma core 
on each excitation. Sigma-pi coupling affects mainly 
the 1A 1.,- ~ 1• transition where it lowers the oscillator 
strength by about 40%. The ordering of the singlet and 
triplet states of B 2., B 1u, and E 1u symmetry agrees with 
experiment and the error in the calculated frequencies 
ranges from 3% to 25%. This is primarily due to the rela-
tively small basis set used in the calculation. We also 
obtain a l.4 2.(a-7T*) state at 9. 3 eV with an oscillator 
strength of about 0. 03. 
In the next section we give a few equations outlining 
the equations-of-motion method. Section III gives the 
results of the calculations on benzene. 
II. THEORY 
Here we review some essential results of the equa-
tions-of-motion method. The operator Oi. which gener-
ates an excited state I A) from the ground state is exact-
ly a solution of the equation, 5 
(0 I [liO;vH, o;j I 0)= w~(O I [00~, 0~] I O), (1) 
for all variations 60~. The double commutator is de-
fined as 
2[A, H, B] =[A, [H, B]] +[[A, H], B] (2) 
and w~ is the excitation frequency. The amplitudes of 
0~ are elements of the transition density for the excita-
tion /0)- I A). With 0~ restricted to the single particle-
hole (lp-llz) form, Eq. (1) reduces to the matrix equa-
tion for the amplitudes {Y my} and {Zmy}, 
I A B J [Y(A)J [D OJ [Y(A)J L- B* -A* Z(A) = w~ 0 D Z(A) (3) 
The elements of A, B, and Dare given elsewhere. 1 
We have also shown that the equation of motion for 0~ 
including two-particle-two-hole (2p-2h) components can 
be put in the form of Eq. (3). 6 The renormalized ma-
trices now depend on the frequencies, w~, and are hence 
energy dependent. The amplitudes of Eq. (3) determine 
the transition moment 1\11 a~ by the simple relation, 
M a~= .f2 6 [Y~y(A)Mmy+ Z:J;y(A)M my] , (4) 
my 
where M my is a particle-hole matrix element of M. The 
oscillator strength, fa~' is then 
{5) 
where G is the degeneracy factor. The major effect of 
the (2p-2h) components of 0~ on the transition amplitude 
can be included by renormalizing the (lp-lh) ampli-
tudes. 
Ill. RESULTS AND DISCUSSION 
The first step in our calculation is to carry out an 
SCF calculation on the ground state of benzene to obtain 
a suitable particle-hole basis (unoccupied-occupied or-
bitals). The ground state electronic configuration is 
1 a~ 1 1 e iu 1 e i..,. 1 b iu 2 ai..,. 2 e iu 2 e i 1 
x3a~ 1 2bi. 1b~u 3ei. 1a~u 3e~..,.lei..,.. 
In this calculation we used a [ 3s 2p / 1s) Gaussian basis 
contracted from a primitive (7s3p/3s) basis. Table I 
lists the eigenvalues for the 21 occupied orbitals and the 
first 28 virtual {particle) states. The total energy of 
this SCF calculation is - 230. 2184 a. u. 
2700 The Journal of Chemical Physics, Vol. 60, No.7, 1 April 1974 Copyright© 1974 American Institute of Physics 
Downloaded 14 Feb 2006 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
Rose, Shibuya,and McKoy: Electronic excitations of benzene 2701 
For the first set of calculations all 1P-1 h and 2p- 2 h 
pairs that can be generated from the hole states 3a 1.e, 
2b 1u, 1b 1u, 3e 1u, 1a2u, 3e 2 .,, and 1e 1 ~r and all the parti-
cle states in Table I are used to solve the equations of 
motion for transitions to the 1B 2u, ~ 1u, 
1E 1u, 
1E u states 
and their corresponding triplet states. This calculation 
then includes 10 hole states and 28 particle states. The 
truncation of 60 molecular orbitals should still allow the 
major effects described by this basis to display them-
selves. The number of lp-1 h pairs included are 34, 21, 
24, and :34 for theE 1u, B 2u, B Ju• and E 2 , states, respec-
tively. Table II shows the results of this calculation in 
both the lp-lh approximation and the (lp-lh)+ (2p-2h) 
approximation. These results include, among other ef-
fects, the effects of sigma-pi coupling and relaxation of 
the sigma and pi systems. We will discuss these re-
sults in the next paragraph. In Table I we also give the 
excitation frequencies for these transitions in the 7T-
electron approximation which therefore include particle-
hole pairs from the la 2u and le 1, hole states and lbz.e, 
2a 2u, 2e 1,, 2e 2u, and 2b 2 , particle states. In the (1p-
1h)+ (2p-2h) approximation these excitation frequencies 
also include the effect of relaxation of the 1T system in 
the excitation process. 
The excitation energies for the complete calculation, 
i.e., including sigma and pi particle-hole pairs in the 
(lp-1h) + (2p-2h) approximation compare well with ex-
periment. The ordering of the states is correct and the 
error range for the frequencies is reasonable consider-
ing the relatively small basis set, i.e., [3s2p/ls]. The 
differences between the calculated and observed excita-
tion frequencies becomes larger for the higher states 
where double and multiple excitations and a more flexi-
ble basis are expected to be more important. The re-
sults in the 1T approximation agree well with those includ-
ing excitations out of the sigma and pi systems except 
for the 3B lu and 1E 1u states. The increase of '1. 5 eV for 
the 3B 1u excitation energy in going from the (1p-lh) 1T 
TABLE I. SCF orbital eigenvalues for C6H6. 
a 
MO € b I MO €; 
la 11 -11.3218 4at, 0.3327 
lelu -11.3224 4elu 0.4187 
le21 -11.3215 4e2.e 0.5149 
lbtu -11.3214 3btu 0.5242 
2a 11 -1.1945 5e21 0.6441 
2e1u -I. 0625 5elu 0.6447 
2e2.- - o. 8704 1a2.e 0,7572 
3at, -0.7470 2b2u 0.9238 
lb2u - o. 6768 2a2u 0.9897 
2b2u - o. 6731 5a11 0.9988 
3etu - o. 627() 4btu 0,9996 
la2u -0.5561 6e21 1. 0574 
3e21 -0.5336 2e11 1.0859 
1e1., - o. 3889 2e2u 1.1305 
le2u 0.0984 6elu 1.1735 
1b2.- 0.3187 2b2.- 1.1995 
2a21 1.4051 
•rn a [3s2p/ls] basis. Total energy is 
-230.2184 a. u. 
l>rn atomic units. 
TABLE II. Excitation energies of benzene. a 
rrb J:;- Tid 
State (lp-lh) (lp-lh) + (2p-2h) (lp-lh) (lp-lh) + (2p-2h) Obs 
1
Bzu 7.7 5.7 8. 5 5. 5 4.9 
1B1u 8.4 7. 8 8. 7 7.4 6,1 
1
Ezu 10.7 9. 5° 10.5 8. 3' 6.9 
1J:;2& 12.9 11.3 12,8 10.9 ... f 
3Blu 3.0 2.7 4.5 3.8 3.7 
3E1u 6.1 5.2 7. 1 5.2 4.6 
3Bzu 7 .. 3 7.0 8.2 6.7 5.8 
3Ez.e 8. 5 7.9 9.4 8.1 ... f 
arn electron volts. 
bn approximation. See text. 
cThe transition moment for the 1A1.,-
1E 1u transition is 2. 09 in 
this approximation. See text. 
<lsee text. 
"The transition moment for this transition is 1. 74 in this ap-
proximation. The experimental moment is 1. 61. 
fsee text. 
approximation to the (1P-1 h) a-1T theory is presumably 
because this very low energy state is almost exclusively 
a 1T-1T* transition and hence a-a correlation is more im-
portant than a-1T coupling. However, we see that 
changes in the sigma core are important in lowering the 
excitation energy of the 1I lu state by 1. 2-8. 3 eV. The 
calculated excitation energy for the 1E 2 ., state is higher 
than the estimates obtained by configuration interaction 
(CI) studies for this transition. 7•8 We will return to a 
brief comparison of our results with those of CI studies 
later but it does seem that either multiple excitations 
are important in describing this transition or the trunca-
tion of our particle states gives too small a (2p-2h) con-
tribution. 
The only optically allowed transition of those listed in 
Table II is the 1,4 u- 1E lu transition. The transition mo-
ment in the full calculation including a-1T coupling is 
1. 74 a. u. which agrees well with the transition moment 
of 1. 61 a. u. derived from the experimental oscillator 
strength of 0. 88. 9 In the 1T approximation the transition 
moment is 2. 09 a. u. This change in the transition mo-
ment from 2. 09 to 1. 74 a. u. due to a-1T coupling leads 
to a reduction of about 40% in the oscillator strength. 
Our results also give a 1A2u(a-1r*) state at 9. 3 eV with 
a transition moment of 0. 34 a. u. 
The available CI calculations 7 ' 8 on the 1T- 1T* states of 
benzene do not include as many particle and hole states 
as the present calculations. For this and other reasons 
we do not compare our results in any detail with these 
other results. In general, our results are in slightly 
better agreement with experiment. The main difference 
in the predicted excitation energies are for the 1E 1u and 
1Eu states. Our excitation energy for the 1Au- 'E 2 ~: 
transition is about 2 eV larger than the other estimates. 
Part of this difference may be due to the truncation of 
the particle basis which affects the 2P-2h contribution 
for this high singlet state. More extensive calculations 
could clarify this difference. We do not list experimen-
tal values for the 1Eu and 3E 2 ~: states because no con-
clusive experimental assignments 10' 11 have been made. 
Our calculated excitation energy of B. 3 eV for the 1E lu 
J. Chern. Phys., Vol. 60, No. 7, 1 April 1974 
Downloaded 14 Feb 2006 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
2702 Rose, Shibuya, and McKoy: Electronic excitations of benzene 
state is much closer to the experimental value of 6. 9 eV 
than the previous estimates of 9. 4 eV 8 and 10. 6 eV. 7 
This shows the importance of changes in the a core in 
this excitation. No values for the transition moment for 
the 1A 1f-
1E 1u transition are reported in these CI calcu-
lations. 7 ' 8 
IV. CONCLUSIONS 
We have discussed the results of calculations on the 
excited states of benzene as an application of the equa-
tions of motion method to a larger polyatomic molecule. 
The ordering of the singlet and triplet states of B 2", B 1", 
and E 1" symmetry agrees with experiment and the error 
range of 3% to 25% for the calculated frequencies is 
reasonable considering the limited basis set employed. 
These calculations show the effects of changes in the 
sigma core for each transition separately by including 
particle-hole excitations from the 3a 1 ~, 1b 2", 2b 1", 
3e 1., and 3c 2 K core states. The calculations can still 
be done economically at this level. The calculated tran-
sition moment for the 1A 1.,-
1E 1" transition is 1. 74 a. u. 
compared to the value of 1. 61 a. u. derived from the ex-
perimental/ value of 0. 88. 
*work supported by the National Science Foundation. 
tPresent address: CECAM, Batirnent 506, Universite de Paris 
XI, 91-0rsay, France. 
tpresent address: Department of Physics, L'niversity of the 
Pacific, Stockton, CA. 
~Contribution No. 4663. 
1T. Shibuya and V. McKoy, Phys. Hev. A 2, 2208 (1970). 
2J. Hose, T. Shibuya, and V. McKoy, J. Chem. Phys. 58, 74 
(197:3). 
3D. L. Yeager and V. McKoy, "Equations of motion method: 
The excited states of formaldehyde," J. Chem. Phys. (to be 
published). 
4c. W. McCurdy and V. McKoy, "Differential cross sections 
for the scattering of electrons by C02," J. Chem. Phys. (to 
be published). 
5D. J. Rowe, Hev. Mod. Phys. 40, 153 (1968). 
6T. Shibuya, J. Rose, and V. McKoy, J. Chem. Phys. 58, 500 
(1973). 
1R. J. Buenker, J. L. \Vhitten, and J. D. Petke, J. Chem. 
Phys. 49, 2261 (1968). 
8s. Peyerimhoff and R. J. Buenker, Theoret. Chim. Acta 19, 
1 (1970). 
9v. J. Hammond and W. C. Price, Trans. Faraday Soc. 51, 
605 (19.55). 
10B. Katz, M. Brith, A. Ron, B. Sharf, and J. Jortner, Chern. 
Phys. Lett. 2, 189 (1968). 
11 P. R. Monson and W. M. McClain, .J. Chern. Phys. 53, 29 
(1970). 
J. Chern. Phys .• Vol. 60, No. 7, 1 April 1974 


Electronic excitations of benzene from the equations of motion

method

Downloaded 14 Feb 2006 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
Electronic excitations of benzene from the equations of motion 
method* 
John B. Rose t, Tai-lchi Shibuya*, and Vincent McKoy 
Arthur Amos Noyes Laboratory of Chemical Physics, § California Institute of Technology, Pasadena, California 91109 
(Received 22 March 1973) 
We have used the equations of motion method to calculate the excitation energies and intensities of 
several transitions in benzene. The ordering of the singlet and triplet states of B '"' B tu• and E tu 
symmetry agrees with experiment and the error in the calculated frequencies ranges from 3% to 
25%. This error range is reasonable considering the relatively small basis set used. The most 
extensive calculation included 10 hole and 28 particle states and shows the effect of changes in the 
sigma core for each transition. The calculated transition moment of 1.74 a.u. for the 1A 1 ,~ 1 E 1, 
transition agrees well with the experimental value of 1.61 a.u. 
I. INTRODUCTION 
In recent papers 1' 2 we have developed the equations-
of-motion method as an approach specifically designed 
to study the spectroscopic and response properties of 
atoms and molecules. In these methods one directly ob-
tains the transition amplitudes of physical interest in a 
given process, e. g., in photon absorption and in elec-
tron-molecule scattering, thereby avoiding many of the 
difficulties involved in obtaining highly accurate wave-
functions. We have used this method to study the excita-
tion frequencies and transition moments of several mol-
ecules, e. g., N2, CO, C2H4, 2 H2CO, 3 and C02 4 and the 
Born differential cross sections for the scattering of 
electrons by C02• 4 The results generally agree well 
with experiment. These results indicate that the method 
is practical and is capable of including the major corre-
lation effects on the physical observables in a relatively 
small matrix problem, even for quite large basis sets. 
In this paper we present the results of calculations on 
the excited states of benzene as an application of the 
method to a larger polyatomic molecule. These calcula-
tions were done with a relatively limited basis set and 
include results both with and without sigma-pi coupling. 
The inclusion of two particle-two hOle (2p-2h) excita-
tions also shows the effect of changes in the sigma core 
on each excitation. Sigma-pi coupling affects mainly 
the 1A 1.,- ~ 1• transition where it lowers the oscillator 
strength by about 40%. The ordering of the singlet and 
triplet states of B 2., B 1u, and E 1u symmetry agrees with 
experiment and the error in the calculated frequencies 
ranges from 3% to 25%. This is primarily due to the rela-
tively small basis set used in the calculation. We also 
obtain a l.4 2.(a-7T*) state at 9. 3 eV with an oscillator 
strength of about 0. 03. 
In the next section we give a few equations outlining 
the equations-of-motion method. Section III gives the 
results of the calculations on benzene. 
II. THEORY 
Here we review some essential results of the equa-
tions-of-motion method. The operator Oi. which gener-
ates an excited state I A) from the ground state is exact-
ly a solution of the equation, 5 
(0 I [liO;vH, o;j I 0)= w~(O I [00~, 0~] I O), (1) 
for all variations 60~. The double commutator is de-
fined as 
2[A, H, B] =[A, [H, B]] +[[A, H], B] (2) 
and w~ is the excitation frequency. The amplitudes of 
0~ are elements of the transition density for the excita-
tion /0)- I A). With 0~ restricted to the single particle-
hole (lp-llz) form, Eq. (1) reduces to the matrix equa-
tion for the amplitudes {Y my} and {Zmy}, 
I A B J [Y(A)J [D OJ [Y(A)J L- B* -A* Z(A) = w~ 0 D Z(A) (3) 
The elements of A, B, and Dare given elsewhere. 1 
We have also shown that the equation of motion for 0~ 
including two-particle-two-hole (2p-2h) components can 
be put in the form of Eq. (3). 6 The renormalized ma-
trices now depend on the frequencies, w~, and are hence 
energy dependent. The amplitudes of Eq. (3) determine 
the transition moment 1\11 a~ by the simple relation, 
M a~= .f2 6 [Y~y(A)Mmy+ Z:J;y(A)M my] , (4) 
my 
where M my is a particle-hole matrix element of M. The 
oscillator strength, fa~' is then 
{5) 
where G is the degeneracy factor. The major effect of 
the (2p-2h) components of 0~ on the transition amplitude 
can be included by renormalizing the (lp-lh) ampli-
tudes. 
Ill. RESULTS AND DISCUSSION 
The first step in our calculation is to carry out an 
SCF calculation on the ground state of benzene to obtain 
a suitable particle-hole basis (unoccupied-occupied or-
bitals). The ground state electronic configuration is 
1 a~ 1 1 e iu 1 e i..,. 1 b iu 2 ai..,. 2 e iu 2 e i 1 
x3a~ 1 2bi. 1b~u 3ei. 1a~u 3e~..,.lei..,.. 
In this calculation we used a [ 3s 2p / 1s) Gaussian basis 
contracted from a primitive (7s3p/3s) basis. Table I 
lists the eigenvalues for the 21 occupied orbitals and the 
first 28 virtual {particle) states. The total energy of 
this SCF calculation is - 230. 2184 a. u. 
2700 The Journal of Chemical Physics, Vol. 60, No.7, 1 April 1974 Copyright© 1974 American Institute of Physics 
Downloaded 14 Feb 2006 to 131.215.225.176. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
Rose, Shibuya,and McKoy: Electronic excitations of benzene 2701 
For the first set of calculations all 1P-1 h and 2p- 2 h 
pairs that can be generated from the hole states 3a 1.e, 
2b 1u, 1b 1u, 3e 1u, 1a2u, 3e 2 .,, and 1e 1 ~r and all the parti-
cle states in Table I are used to solve the equations of 
motion for transitions to the 1B 2u, ~ 1u, 1E 1u, 1E u states 
and their corresponding triplet states. This calculation 
then includes 10 hole states and 28 particle states. The 
truncation of 60 molecular orbitals should still allow the 
major effects described by this basis to display them-
selves. The number of lp-1 h pairs included are 34, 21, 
24, and :34 for theE 1u, B 2u, B Ju• and E 2 , states, respec-
tively. Table II shows the results of this calculation in 
both the lp-lh approximation and the (lp-lh)+ (2p-2h) 
approximation. These results include, among other ef-
fects, the effects of sigma-pi coupling and relaxation of 
the sigma and pi systems. We will discuss these re-
sults in the next paragraph. In Table I we also give the 
excitation frequencies for these transitions in the 7T-
electron approximation which therefore include particle-
hole pairs from the la 2u and le 1, hole states and lbz.e, 
2a 2u, 2e 1,, 2e 2u, and 2b 2 , particle states. In the (1p-
1h)+ (2p-2h) approximation these excitation frequencies 
also include the effect of relaxation of the 1T system in 
the excitation process. 
The excitation energies for the complete calculation, 
i.e., including sigma and pi particle-hole pairs in the 
(lp-1h) + (2p-2h) approximation compare well with ex-
periment. The ordering of the states is correct and the 
error range for the frequencies is reasonable consider-
ing the relatively small basis set, i.e., [3s2p/ls]. The 
differences between the calculated and observed excita-
tion frequencies becomes larger for the higher states 
where double and multiple excitations and a more flexi-
ble basis are expected to be more important. The re-
sults in the 1T approximation agree well with those includ-
ing excitations out of the sigma and pi systems except 
for the 3B lu and 1E 1u states. The increase of '1. 5 eV for 
the 3B 1u excitation energy in going from the (1p-lh) 1T 
TABLE I. SCF orbital eigenvalues for C6H6. a 
MO € b I MO €; 
la 11 -11.3218 4at, 0.3327 
lelu -11.3224 4elu 0.4187 
le21 -11.3215 4e2.e 0.5149 
lbtu -11.3214 3btu 0.5242 
2a 11 -1.1945 5e21 0.6441 
2e1u -I. 0625 5elu 0.6447 
2e2.- - o. 8704 1a2.e 0,7572 
3at, -0.7470 2b2u 0.9238 
lb2u - o. 6768 2a2u 0.9897 
2b2u - o. 6731 5a11 0.9988 
3etu - o. 627() 4btu 0,9996 
la2u -0.5561 6e21 1. 0574 
3e21 -0.5336 2e11 1.0859 
1e1., - o. 3889 2e2u 1.1305 
le2u 0.0984 6elu 1.1735 
1b2.- 0.3187 2b2.- 1.1995 
2a21 1.4051 
•rn a [3s2p/ls] basis. Total energy is 
-230.2184 a. u. 
l>rn atomic units. 
TABLE II. Excitation energies of benzene. a 
rrb J:;- Tid 
State (lp-lh) (lp-lh) + (2p-2h) (lp-lh) (lp-lh) + (2p-2h) Obs 
1Bzu 7.7 5.7 8. 5 5. 5 4.9 
1B1u 8.4 7. 8 8. 7 7.4 6,1 
1
Ezu 10.7 9. 5° 10.5 8. 3' 6.9 
1J:;2& 12.9 11.3 12,8 10.9 ... f 
3Blu 3.0 2.7 4.5 3.8 3.7 
3E1u 6.1 5.2 7. 1 5.2 4.6 
3Bzu 7 .. 3 7.0 8.2 6.7 5.8 
3Ez.e 8. 5 7.9 9.4 8.1 ... f 
arn electron volts. 
bn approximation. See text. 
cThe transition moment for the 1A1.,-
1E 1u transition is 2. 09 in 
this approximation. See text. 
<lsee text. 
"The transition moment for this transition is 1. 74 in this ap-
proximation. The experimental moment is 1. 61. 
fsee text. 
approximation to the (1P-1 h) a-1T theory is presumably 
because this very low energy state is almost exclusively 
a 1T-1T* transition and hence a-a correlation is more im-
portant than a-1T coupling. However, we see that 
changes in the sigma core are important in lowering the 
excitation energy of the 1I lu state by 1. 2-8. 3 eV. The 
calculated excitation energy for the 1E 2 ., state is higher 
than the estimates obtained by configuration interaction 
(CI) studies for this transition. 7•8 We will return to a 
brief comparison of our results with those of CI studies 
later but it does seem that either multiple excitations 
are important in describing this transition or the trunca-
tion of our particle states gives too small a (2p-2h) con-
tribution. 
The only optically allowed transition of those listed in 
Table II is the 1,4 u- 1E lu transition. The transition mo-
ment in the full calculation including a-1T coupling is 
1. 74 a. u. which agrees well with the transition moment 
of 1. 61 a. u. derived from the experimental oscillator 
strength of 0. 88. 9 In the 1T approximation the transition 
moment is 2. 09 a. u. This change in the transition mo-
ment from 2. 09 to 1. 74 a. u. due to a-1T coupling leads 
to a reduction of about 40% in the oscillator strength. 
Our results also give a 1A2u(a-1r*) state at 9. 3 eV with 
a transition moment of 0. 34 a. u. 
The available CI calculations 7 ' 8 on the 1T- 1T* states of 
benzene do not include as many particle and hole states 
as the present calculations. For this and other reasons 
we do not compare our results in any detail with these 
other results. In general, our results are in slightly 
better agreement with experiment. The main difference 
in the predicted excitation energies are for the 1E 1u and 1Eu states. Our excitation energy for the 1Au- 'E 2 ~: 
transition is about 2 eV larger than the other estimates. 
Part of this difference may be due to the truncation of 
the particle basis which affects the 2P-2h contribution 
for this high singlet state. More extensive calculations 
could clarify this difference. We do not list experimen-
tal values for the 1Eu and 3E 2 ~: states because no con-
clusive experimental assignments 10' 11 have been made. 
Our calculated excitation energy of B. 3 eV for the 1E lu 
J. Chern. Phys., Vol. 60, No. 7, 1 April 1974 
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2702 Rose, Shibuya, and McKoy: Electronic excitations of benzene 
state is much closer to the experimental value of 6. 9 eV 
than the previous estimates of 9. 4 eV 8 and 10. 6 eV. 7 
This shows the importance of changes in the a core in 
this excitation. No values for the transition moment for 
the 1A 1f- 1E 1u transition are reported in these CI calcu-
lations. 7 ' 8 
IV. CONCLUSIONS 
We have discussed the results of calculations on the 
excited states of benzene as an application of the equa-
tions of motion method to a larger polyatomic molecule. 
The ordering of the singlet and triplet states of B 2", B 1", 
and E 1" symmetry agrees with experiment and the error 
range of 3% to 25% for the calculated frequencies is 
reasonable considering the limited basis set employed. 
These calculations show the effects of changes in the 
sigma core for each transition separately by including 
particle-hole excitations from the 3a 1 ~, 1b 2", 2b 1", 
3e 1., and 3c 2 K core states. The calculations can still 
be done economically at this level. The calculated tran-
sition moment for the 1A 1.,- 1E 1" transition is 1. 74 a. u. 
compared to the value of 1. 61 a. u. derived from the ex-
perimental/ value of 0. 88. 
*work supported by the National Science Foundation. 
tPresent address: CECAM, Batirnent 506, Universite de Paris 
XI, 91-0rsay, France. 
tpresent address: Department of Physics, L'niversity of the 
Pacific, Stockton, CA. 
~Contribution No. 4663. 
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be published). 
5D. J. Rowe, Hev. Mod. Phys. 40, 153 (1968). 
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(1973). 
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J. Chern. Phys .• Vol. 60, No. 7, 1 April 1974 


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Electronic excitations of benzene from the equations of motion method

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