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A different approach for calculating Franck-Condon factors including anharmonicity.

08 Jan 2004-Journal of Chemical Physics (American Institute of Physics)-Vol. 120, Iss: 2, pp 813-822

TL;DR: An efficient new procedure for calculating Franck-Condon factors, based on the direct solution of an appropriate set of simultaneous equations, is presented, and both Duschinsky rotations and anharmonicity are included.

AbstractAn efficient new procedure for calculating Franck–Condon factors, based on the direct solution of an appropriate set of simultaneous equations, is presented. Both Duschinsky rotations and anharmonicity are included, the latter by means of second-order perturbation theory. The critical truncation of basis set is accomplished by a build-up procedure that simultaneously removes negligible vibrational states. A successful test is carried out on ClO2 for which there are experimental data and other theoretical calculations.

Topics: Anharmonicity (59%)

Summary (2 min read)

A different approach for calculating Franck–Condon factors including anharmonicity

  • Those phase-space points where the classical Wigner function for the initial state is maximal, subject to a classical energy constraint on the final state, determine propensity rules for the FCF’s.
  • The authors have now begun to develop a rigorous theory for vibrational effects in TPA in order to investigate that situation more thoroughly.

A. General formulation

  • The authors denote the vibrational Hamiltonian, wave functions, and energies of the ground electronic state byĤg, ucng g &, and Eng g and their counterparts for an electronic excited state by Ĥe, ucne e &, andEne e .
  • Then the respective Schrödinger equations for nuclear motion are given by Ĥgucng g &5Eng g ucng g &, ~1! Sngne. ~6! contains the entire set of Franck–Condon overlaps between the initial vibrational wave function of the ground electronic state and all final vibrational wave functions of the excited electronic state.
  • This allows us to solve for the entire set of overlap integrals in which the authors are interested simultaneously.

C. Mechanical anharmonicity

  • Mechanical anharmonicity can be included through a perturbation treatment using the harmonic oscillator Hamiltonian as the zeroth-order approximation.
  • Except that all quantities have a superscript~0!.
  • Once the solution forSngme (1) has been determined, the first-order corrections to the FCF’s are found as Fngme ~1! 52Sngme ~0! Sngme ~1! .
  • Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp.

D. Truncation of the vibrational basis set

  • It is critical to perform the truncation of the vibrational basis set in a way that is efficient and does not create significant error.
  • The authors procedure involves an iterative buildup by increasing the range of vibrational quantum numbers while, simultaneously, removing unimportant states.
  • The next step in the cycle is a screening of the states created in this manner which is based on the difference between the quantum number in each mode and the corresponding quantum number for the FC state.
  • The latter still increases in size more rapidly than desired.
  • It turns out, however, that most of the FC overlaps obtained from Eq.~9! are quite small.

III. COMPUTATIONAL DETAILS

  • Were used to calculate the neutral and cationic force constants, respectively, for the harmonic calculations.
  • The first- and second-order corrections to the wave function are given by Eq.~19!.

IV. RESULTS

  • At the harmonic level their theoretical spectrum is essentially the same as that of Moket al.and thus their geometrical parameters for ClO2 1, obtained from the best match between the simulated and experimental spectrum, are also the same as theirs.
  • Again, in order to compare the two spectra the intensities of the~0,0,1!.
  • In Fig. 3 the authors present the simulated anharmonic spectra calculated by Moket al. and ourselves.

V. CONCLUSIONS

  • In this work a new method to calculate FCF’s taking into account Duschinsky rotations as well as anharmonicity has been developed and implemented.
  • The authors harmonic results are in excellent agreement with those of Mok et al. who used a different procedure and both calculations predict the same geometry for ClO2 1.
  • This computational efficiency is due in large part to the major truncation of the vibrational basis set.

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A different approach for calculating FranckCondon factors
including anharmonicity
Josep M. Luis
a)
Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106
and Department of Chemistry, University of Ottawa, Ottawa, Canada K1N 6N5
David M. Bishop
Department of Chemistry, University of Ottawa, Ottawa, Canada K1N 6N5
Bernard Kirtman
Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106
Received 6 August 2003; accepted 9 October 2003
An efficient new procedure for calculating FranckCondon factors, based on the direct solution of
an appropriate set of simultaneous equations, is presented. Both Duschinsky rotations and
anharmonicity are included, the latter by means of second-order perturbation theory. The critical
truncation of basis set is accomplished by a build-up procedure that simultaneously removes
negligible vibrational states. A successful test is carried out on ClO
2
for which there are
experimental data and other theoretical calculations. © 2004 American Institute of Physics.
DOI: 10.1063/1.1630566
I. INTRODUCTION
Along with the development of experimental high-
resolution vibronic spectroscopies, the problem of analyzing
the observed spectra is receiving increased attention. In the
BornOppenheimer approximation the leading term that
governs the spectral intensity pattern is given by the square
of the vibrational overlap integrals, also known as Franck
Condon factors FCF’s, between the initial and final states.
If the vibrational normal coordinates for the two electronic
states are parallel i.e., if they are the same except for the
shift in equilibrium geometry, then these integrals will sepa-
rate in the harmonic oscillator approximation into a product
of individual oscillator terms. In general, however, this is not
the case and, discounting possible simplifications due to
symmetry, one must evaluate 3N-6 or 3N-5 for linear mol-
ecules dimensional overlap integrals. The difficulty of doing
so is compounded by the fact that the difference in equilib-
rium geometry as well as the anharmonicity of the electronic
potential-energy surfaces must be taken into account.
A variety of methods have been proposed for dealing
with this problem, particularly at the harmonic level.
1
One of
these is based on the generating function approach of Sharp
and Rosenstock,
2
which is an extension of the method intro-
duced by Hutchisson
3
for diatomics. This method has been
further developed by Chen
4
and improved by Ervin et al.
5
in their application to the naphthyl radical. Very recently,
Kikuchi et al.
6
derived a simpler form of the Sharp and
Rosenstock general formula and applied it to SO
2
in the
harmonic oscillator approximation. Another method based on
the generating function approach is due to Ruhoff
7
who de-
rived recursion relations for the calculation of multidimen-
sional FCF’s by generalizing Lerme
´
’s
8
procedure for two-
dimensional FCF’s. Also employing the generating function
method, Islampour et al. derived a closed-form multidimen-
sional harmonic oscillator expression, where the FCF’s were
expressed as sums of products of Hermite polynomials.
9
An alternative procedure, utilizing the recursion rela-
tions of Doctorov, Malkin, and Man’ko,
10
has been employed
for a variety of molecules such as phenol,
11,12
anthracene,
13
and pyrazine.
14
In addition, two different methods for calcu-
lating the FCF’s were developed by Faulkner and
Richardson.
15
The central feature of their first method is a
linear transformation of the normal coordinates in both the
ground and excited electronic states in order to effectively
remove the Duschinsky rotations
16
i.e., the transformation
of coordinates from one electronic state to another. This was
originally restricted to the case where either the initial or
final vibrational wave function is the ground state, but Ku-
lander later removed this restriction.
17,18
The second method
of Faulkner and Richardson is based on a perturbation ex-
pansion of the vibrational wave functions of the excited elec-
tronic state in terms of the ground electronic state vibrational
wave functions.
15
Finally, Malmqvist and Forsberg
19
have
expressed the FCF matrix as the product of lower triangular
and upper triangular matrices which are calculated from re-
cursion formulas.
At this juncture we take note of a very different ap-
proach, developed by Segev, Heller, and co-workers,
20,21
based on considering the transitions in phase space. Those
phase-space points where the classical Wigner function for
the initial state is maximal, subject to a classical energy con-
straint on the final state, determine propensity rules for the
FCF’s. These rules, in turn, provide a way of selecting the
transitions that have substantial intensity and their FCF’s can
be estimated by subsequent phase-space integration. The
truncation of the vibrational basis is a critical aspect in re-
a
Permanent address: Institute of Computational Chemistry and Department
of Chemistry, University of Girona, Campus de Montilivi, 17071 Girona,
Catalonia, Spain.
JOURNAL OF CHEMICAL PHYSICS VOLUME 120, NUMBER 2 8 JANUARY 2004
8130021-9606/2004/120(2)/813/10/$22.00 © 2004 American Institute of Physics
Downloaded 31 Dec 2003 to 130.206.124.176. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

ducing the computational effort of any method. Our own
prescription for doing this is described in Sec. II D.
Although the procedures mentioned above can, in prin-
ciple, include vibrational anharmonicity not much attention
has been paid to this aspect. Iachello’s group has developed a
procedure based on the use of Morse oscillators in a novel
Lie algebraic scheme.
22–24
More recently Mok et al.
25
have
proposed an expansion technique that builds on the earlier
work of Botschwina and co-workers.
26
However, these meth-
ods and other approaches
27–29
to the vibrational anharmonic-
ity problem have only been applied to small molecules or to
two-dimensional model potentials.
30
Reimers has also
described
31
an approximate method for taking into account
the floppy motions of large molecules by means of curvilin-
ear coordinates.
Apart from one-photon absorption and emission FCF’s
figure prominently in two-photon absorption TPA. The vi-
brational contribution to nonlinear optical NLO properties,
including TPA,
32
has occupied the attention of the present
authors for some time.
33–36
As far as nonresonant NLO pro-
cesses are concerned, it is also known that mechanical and
electrical anharmonicities of ordinary as well as floppy
molecules often play a major role.
37
On the basis of very
approximate treatments
38–40
it has been suggested that the
same is true for resonant processes and in particular for TPA.
We have now begun to develop a rigorous theory for vibra-
tional effects in TPA in order to investigate that situation
more thoroughly. In the course of doing so, we have come
across a simple direct way to evaluate FCF’s and it is this
new scheme that is presented here. Effects due to: i
changes in the normal coordinates with electronic state
Duschinsky rotations;
16
ii changes in the equilibrium ge-
ometry with electronic state; and iii mechanical anharmo-
nicities in both electronic states, are all taken into account.
In the next section a general theory, which includes all of
the above effects, is formulated. Then, in Sec. III we discuss
how the resulting equations are solved along with other com-
putational details. This is followed by an example where our
method is used to simulate the He I photoelectron PE spec-
trum of ClO
2
, in order to compare with the work of Mok
et al.
25
Finally, we conclude with a brief discussion of future
plans for incorporating this methodology into our treatment
of TPA for large conjugated molecules.
II. THEORY
The goal of this section is the derivation of a new ana-
lytical procedure to calculate the FranckCondon factors of
polyatomic molecules taking into account both the Duschin-
sky rotations and the mechanical anharmonicity.
A. General formulation
We denote the vibrational Hamiltonian, wave functions,
and energies of the ground electronic state by H
ˆ
g
,
g
g
, and
E
g
g
and their counterparts for an electronic excited state by
H
ˆ
e
,
e
e
, and E
e
e
. Note that g refers to the ground elec-
tronic state and e to an excited electronic state throughout.
In either case the molecule is assumed to be nonrotating and
thus the rotational state is suppressed. Then the respective
Schro
¨
dinger equations for nuclear motion are given by
H
ˆ
g
g
g
E
g
g
g
g
, 1
H
ˆ
e
e
e
E
e
e
e
e
. 2
Multiplication of Eq. 1 by
e
e
and Eq. 2 by
g
g
leads
to
e
e
H
ˆ
g
g
g
E
g
g
S
e
g
, 3
g
g
H
ˆ
e
e
e
E
e
e
S
g
e
, 4
where S
e
g
are the FranckCondon overlap integrals the
wave functions are taken to be real:
S
e
g
S
g
e
e
e
g
g
g
g
e
e
. 5
Subtraction of Eq. 4 from Eq. 3, and using the Hermitian
property of H
ˆ
g
, gives
g
g
H
ˆ
g
H
ˆ
e
e
e
E
g
g
E
e
e
S
g
e
. 6
Since the vibrational eigenfunctions for the excited elec-
tronic state (
e
e
below form a complete set, the left-hand
side of Eq. 6 can be expressed as
g
g
H
ˆ
g
H
ˆ
e
e
e
e
g
g
e
e
典具
e
e
H
ˆ
g
H
ˆ
e
e
e
e
S
g
e
e
e
H
ˆ
g
H
ˆ
e
e
e
. 7
Equation 7 contains the entire set of FranckCondon over-
laps between the initial vibrational wave function of the
ground electronic state and all final vibrational wave func-
tions of the excited electronic state. This allows us to solve
for the entire set of overlap integrals in which we are inter-
ested simultaneously. If the vibrational wave functions of the
electronic excited state had been expanded in terms of the
electronic ground-state vibrational wave functions, then only
one of the desired overlaps would be obtained and the pro-
cess would have to be repeated for each final state. In either
event the properties of both the ground and excited electronic
states are necessary for the calculations.
Combining Eqs. 6 and 7, while taking into account
the fact that the total nuclear kinetic energy operator is the
same in both Hamiltonians (H
ˆ
g
T
ˆ
V
ˆ
g
, H
ˆ
e
T
ˆ
V
ˆ
e
), one
obtains
e
S
g
e
e
e
V
ˆ
g
V
ˆ
e
e
e
E
e
e
E
g
g
e
e
0,
e
, and
g
, 8
where
e
e
is the Kronecker delta.
For a given
g
, Eq. 8 constitutes an infinite set of
homogeneous simultaneous linear equations with an infinite
number of unknowns S
g
e
all
e
). The first step in solving
this set of equations is to truncate to a finite set of
e
and
e
values. The details of the systematic iterative algorithm used
814 J. Chem. Phys., Vol. 120, No. 2, 8 January 2004 Luis, Bishop, and Kirtman
Downloaded 31 Dec 2003 to 130.206.124.176. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

to select the M equations that survive the truncation are
given in the next section. Then after dividing by S
g
e
(
e
is
arbitrary as long as S
g
e
0),
e
M
r
e
e
e
e
V
ˆ
g
V
ˆ
e
e
e
E
e
e
E
g
g
e
e
0,
e
e
,
e
, 9
where r
e
e
is the ratio,
r
e
e
S
g
e
/S
g
e
10
the index
g
is understood in r
e
e
). There are M simulta-
neous equations in Eq. 9 but only M 1 unknown ratios;
hence one of these equations is redundant. Any one can be
omitted assuming the remaining set is nonsingular and our
choice is to remove the equation corresponding to
e
e
.
In order to obtain S
g
e
from the ratios r
e
e
we use the nor-
malization condition,
e
M
S
g
e
2
e
M
g
g
e
e
典具
e
e
g
g
1, 11
which leads to
S
g
e
1
e
M
r
e
e
2
. 12
The remaining S
g
e
are obtained from Eq. 10 as S
g
e
r
e
e
S
g
e
(
e
⫽␭
e
). Finally, the FranckCondon factors
are given by the square of the corresponding Franck
Condon integrals (F
g
e
S
g
e
2
).
B. Duschinsky rotations
In general the equilibrium geometry and the potential-
energy surface PES of the electronic excited and ground
states are not the same. Therefore the respective normal co-
ordinates Q
e
and Q
g
are also different. The relationship be-
tween the two sets of normal coordinates can be obtained
from the corresponding relationship between their mass-
weighted Cartesian displacement coordinates and the formu-
las that connect the normal and Cartesian coordinates. For
the mass-weighted Cartesian displacement coordinates we
have
X
g
X
e
R, 13
where X
g
(X
e
) represents the coordinates of the electronic
ground excited state and R is the vector in mass-weighted
Cartesians obtained by subtracting the ground-state equilib-
rium geometry from that of the excited state. The normal
coordinates are related to the mass-weighted Cartesian coor-
dinates by
Q
g
L
g
X
g
and Q
e
L
e
X
e
, 14
where L
g
and L
e
are unitary matrices see, for example, Ref.
41. Six columns of L
g
and L
e
or five for linear molecules
are associated with translations and rotations, while the re-
mainder correspond to the normal vibrations. By combining
Eqs. 13 and 14 we find that
Q
g
JQ
e
K, 15
where JL
g
L
e
and KL
g
R. The J matrix describes the
Duschinsky rotation between the normal modes of the
ground and excited electronic state, while K is associated
with the change in the normal modes due to the displacement
of the equilibrium geometry between the two electronic
states.
The effect of the Duschinsky rotation and the equilib-
rium geometry displacement on the FranckCondon factors
occurs in the potential-energy difference V
ˆ
g
V
ˆ
e
in Eq. 9
which, for nonlinear states, is given by
V
ˆ
g
V
ˆ
e
V
Q
g
0
g
V
Q
e
0
e
1
2
i1
3N6
2
V
g
Q
i
g
2
Q
g
0
K
i
2
2K
i
j1
3N6
J
ij
Q
j
e
j,k1
3N6
J
ij
J
ik
Q
j
e
Q
k
e
1
2
i1
3N6
2
V
e
Q
i
e
2
Q
e
0
Q
i
e
2
16
in the harmonic approximation.
C. Mechanical anharmonicity
Mechanical anharmonicity can be included through a
perturbation treatment using the harmonic oscillator Hamil-
tonian as the zeroth-order approximation.
42
An alternative
approach is to introduce the anharmonicity by using curvi-
linear coordinates.
31
Expanding Eq. 8 in orders of perturbation theory, we
find that the first-order equation is
e
S
g
e
1
e
e
V
ˆ
g
V
ˆ
e
e
e
E
e
e
E
g
g
e
e
0
e
S
g
e
0
e
e
V
ˆ
g
V
ˆ
e
e
e
E
e
e
E
g
g
e
e
1
0, 17
where the superscripts 0 and 1 indicate the order of per-
turbation theory. The zeroth-order equation is identical to Eq.
8 except that all quantities have a superscript 0. This in-
finite set of equations is truncated to the same finite set
e
e
that is used in the zeroth-order equation. As in
previous work
32
we take the cubic terms in V
g
and V
e
to be
first order. In that event, the first-order corrections to E
e
e
and
E
g
g
vanish. On the other hand, the first-order corrections to
the terms in which the potential-energy difference occurs in
Eq. 17 are given by
815J. Chem. Phys., Vol. 120, No. 2, 8 January 2004 FranckCondon factors
Downloaded 31 Dec 2003 to 130.206.124.176. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

e
e
V
ˆ
g
V
ˆ
e
e
e
1
e
e
1
V
ˆ
g
V
ˆ
e
0
e
e
0
e
e
0
V
ˆ
g
V
ˆ
e
1
e
e
0
e
e
0
V
ˆ
g
V
ˆ
e
0
e
e
1
, 18
where
e
e
1
⫽⫺
e
e
M
e
e
0
V
ˆ
e
1
e
e
0
e
e
0
E
e
e
0
E
e
e
0
19
and
V
ˆ
g
V
ˆ
e
1
1
6
i,j,k 1
3N6
3
V
g
Q
i
g
Q
j
g
Q
k
g
Q
g
0
K
i
K
j
K
k
3K
i
K
j
l1
3N6
J
kl
Q
l
e
3K
i
l,m1
3N6
J
jl
J
km
Q
l
e
Q
m
e
l,m,n1
3N6
J
il
J
jm
J
kn
Q
l
e
Q
m
e
Q
n
e
1
6
i,j,k 1
3N6
3
V
e
Q
i
e
Q
j
e
Q
k
e
Q
e
0
Q
i
e
Q
j
e
Q
k
e
.
20
The column vector S
v
g
(1)
with components S
g
e
(1)
, can be
written as
S
v
g
1
S
v
g
1
S
v
g
1
S
v
g
1
fS
v
g
0
, 21
where S
v
g
(1)
is the component of S
v
g
(1)
orthogonal to S
v
g
(0)
and
S
v
g
(1)
is the component of S
v
g
(1)
parallel to S
v
g
(0)
. The first term
on the left-hand side of Eq. 17 vanishes if we substitute
S
v
g
(0)
for the first-order eigenvector S
v
g
(1)
cf. Eq. 8兲兴. There-
fore S
v
g
(1)
is a solution of Eq. 17 for any arbitrary f.We
choose f equal to zero so that S
v
g
(1)
is orthogonal to S
v
g
(0)
and
thereby satisfies the first-order normalization condition
2
e
S
g
e
1
S
g
e
0
2S
v
g
1
S
v
g
0
0. 22
One easy procedure to solve the set of simultaneous equa-
tions 17 is to transform to a basis consisting of the vector
S
v
g
(0)
and an arbitrary set of M1 vectors perpendicular to
S
v
g
(0)
. Then we only need to solve the reduced set of M1
inhomogeneous equations in the subspace orthogonal to
S
v
g
(0)
. Once the solution for S
g
e
(1)
has been determined, the
first-order corrections to the FCF’s are found as
F
g
e
1
2S
g
e
0
S
g
e
1
. 23
A similar procedure may be followed for the second-
order correction which is obtained by solving
e
M
S
g
e
2
e
e
V
ˆ
g
V
ˆ
e
e
e
E
e
e
E
g
g
e
e
0
e
M
S
g
e
2
e
e
V
ˆ
g
V
ˆ
e
e
e
E
e
e
E
g
g
e
e
0
e
M
S
g
e
1
e
e
V
ˆ
g
V
ˆ
e
e
e
E
g
e
E
g
g
e
e
1
e
M
S
g
e
0
e
e
V
ˆ
g
V
ˆ
e
e
e
E
e
e
E
g
g
e
e
2
0, 24
where we have written S
v
g
(2)
for the component of S
v
g
(2)
or-
thogonal to S
v
g
(0)
and S
v
g
(2)
for the component of S
v
g
(2)
parallel
to S
v
g
(0)
. Again the parallel component is given by S
v
g
(2)
fS
v
g
(0)
where the multiplicative constant f is chosen to sat-
isfy the normalization condition which, in second-order, is
0 2S
v
g
2
S
v
g
0
S
v
g
1
S
v
g
1
2S
v
g
2
S
v
g
0
S
v
g
1
S
v
g
1
2 f S
v
g
1
S
v
g
1
25
or
f⫽⫺0.5S
v
g
1
S
v
g
1
. 26
In this case V
e(2)
and V
g(2)
contain the quartic terms in the
expansion of the vibrational potential in terms of normal
coordinates:
V
ˆ
g
V
ˆ
e
2
1
24
i,j,k,l1
3N6
4
V
g
Q
i
g
Q
j
g
Q
k
g
Q
l
g
Q
g
0
K
i
K
j
K
k
K
l
4K
i
K
j
K
k
m1
3N6
J
lm
Q
m
e
6K
i
K
j
m,n1
3N6
J
km
J
ln
Q
m
e
Q
n
e
4K
i
m,n,p1
3N6
J
jm
J
kn
J
lp
Q
m
e
Q
n
e
Q
p
e
m,n,p,q1
3N6
J
im
J
jn
J
kp
J
lq
Q
m
e
Q
n
e
Q
p
e
Q
q
e
1
24
i,j,k,l1
3N6
4
V
e
Q
i
e
Q
j
e
Q
k
e
Q
l
e
Q
e
0
Q
i
e
Q
j
e
Q
k
e
Q
l
e
. 27
816 J. Chem. Phys., Vol. 120, No. 2, 8 January 2004 Luis, Bishop, and Kirtman
Downloaded 31 Dec 2003 to 130.206.124.176. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

Then S
v
g
(2)
is obtained by solving Eq. 24 for S
v
g
(2)
in a basis
orthogonal to S
v
g
(0)
and adding S
v
g
(2)
fS
v
g
(0)
with the value of
f determined by Eq. 26. Finally, the second-order correction
to the FranckCondon factors is given by
F
g
g
2
2S
g
e
0
S
g
e
2
S
g
e
1
S
g
e
1
. 28
D. Truncation of the vibrational basis set
It is critical to perform the truncation of the vibrational
basis set in a way that is efficient and does not create signifi-
cant error. Our procedure involves an iterative buildup by
increasing the range of vibrational quantum numbers while,
simultaneously, removing unimportant states.
We begin by identifying an initial guess for the vibra-
tional state associated with the vertical FC transition to the
excited electronic state based on energy and geometry con-
siderations. This gives a starting set of vibrational quantum
numbers for all modes. Next, an initial basis set is formed
which contains all vibrational wave functions wherein the
quantum number for each mode differs by less than two units
from the corresponding quantum number in the vertical FC
state. Equation 9 is solved in this basis to yield an initial set
of FC overlaps S
v
g. Augmentation of the basis set is, then,
carried out iteratively. In each iterative cycle we, simulta-
neously, increase by one unit the maximum quantum number
of all modes where the previous two augmentations pro-
duced one or more states that have a non-negligible FC over-
lap i.e., an overlap larger than 10
6
). An exactly analogous
procedure is applied at the same time to the minimum quan-
tum number except, of course, that the minimum cannot be
reduced below zero. The next step in the cycle is a screening
of the states created in this manner which is based on the
difference between the quantum number in each mode and
the corresponding quantum number for the FC state. If the
sum over modes of the absolute value of these differences for
any given state is larger than a threshold value, then that state
is removed. The threshold is taken to be the largest differ-
ence between the maximum and minimum quantum numbers
in any one mode considering all states. Mok et al. employed
a similar screening criterion to reduce their basis sets.
25
Us-
ing this reduced basis Eq. 9 is solved and a new set of FC
overlap integrals S
v
g is obtained.
Although the algorithm described above limits the
growth of the basis set, the latter still increases in size more
rapidly than desired. It turns out, however, that most of the
FC overlaps obtained from Eq. 9 are quite small. Therefore
the cycle is completed by setting all S
v
g smaller than a pre-
set threshold (10
6
) equal to zero, and the corresponding
states are marked for exclusion in subsequent cycles. They
are retained, however, for the purpose of augmentation. This
simple procedure drastically reduces the growth of the basis
set thereby leading to a major improvement in efficiency.
The overall process is converged when a complete cycle
leads to no augmentation of the basis set.
We tested our algorithm in several different ways for
ClO
2
. Thus the calculations were repeated separately with:
i the FC overlap threshold for expanding the range of quan-
tum numbers systematically decreased from 10
4
to 10
9
,
FIG. 1. Simulated first band of the ClO
2
He I PE spectrum using harmonic FCF’s obtained from the QCISD PES. The dashed and solid lines represent our
work and that of Mok et al. Ref. 25, respectively. The geometry of ClO
2
is the experimental one Refs. 51 and 52 and the geometrical parameters of the
cation are R
Cl–O
1.410 Å and
O–Cl–O
121.8°.
817J. Chem. Phys., Vol. 120, No. 2, 8 January 2004 FranckCondon factors
Downloaded 31 Dec 2003 to 130.206.124.176. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

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References
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Abstract: By use of the commutation relations and sum rules, the Darling-Dennison vibration-rotation hamiltonian for a non-linear molecule is rearranged to the form: The order of the factors in the first term is immaterial, on account of the relation: A simple expansion is given for the μαβ tensor in terms of the normal coordinates.

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Abstract: Publisher Summary The purpose of this chapter is to provide information on the recent developments in perturbation theory. In recent years, there is a great increase of interest in the application of perturbation theory to the fundamental problems of quantum chemistry. Perturbation theory is designed to deal systematically with the effects of small perturbations on physical systems when the effects of the perturbations are mathematically too difficult to calculate exactly, and the properties of the unperturbed system are known. The new applications have been mainly to atoms where the reciprocal of the atomic number, l/Z, provides a natural perturbation parameter. These may be divided into two groups. The first consists of calculations of energy levels, and is a natural outgrowth of Hylleraas's classic work on the 1/Z expansion for two-electron atoms. The applications in the second group are to the calculation of expectation values and other properties of atoms and molecules, and are of much more recent origin. There are two principal reasons for the success of these new applications: (1) sufficient accuracy is frequently obtained from knowledge of a first-order perturbed wave function, and (2) a great advantage of perturbation theory is that the functional form of the perturbed wave function is shaped by the perturbation itself.

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Abstract: While use of curvilinear coordinates such as bond lengths and bond angles is common in accurate spectroscopic and/or scattering calculations for triatomic and other small molecules, their use for large molecules is uncommon and restricted. For large molecules, normal-mode analysis is feasible but gives sensible results only if the dynamical or spectroscopic process being considered involves changes in angular coordinates, including ring deformations, which are so small that the motion can be approximated by its tangential component. We describe an approximate method by which curvilinear normal-mode-projected displacements and hence Franck–Condon factors, reorganization energies, and vibronic coupling constants, as well as Duschinsky (Dushinsky, Duschinskii) rotation matrices, can be evaluated for large systems. Three illustrative examples are provided: (i) to understand the nature of the first excited state of water, illustrating properties of large-amplitude bending motions; (ii) to understand the nature of the “boat” relaxation of the first excited state of pyridine, illustrating properties of large-amplitude torsional motions; and (iii) to understand the coupling of vibrational modes to the oxidation of bacteriochlorophyll-a, a paradigm with many applications to both chemical and biological electron transfer, illustrating properties of macrocyclic deformations. The method is interfaced to a wide variety of computational chemistry computer programs.

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