JOURNAL OF CHEMICAL PHYSICS
VOLUME 112, NUMBER 6
8 FEBRUARY 2000
Calculation of the vibrational wave function of polyatomic molecules
Per-Olof Åstrand
Condensed Matter Physics and Chemistry Department, Riso” National Laboratory, POB 49,
DK-4000 Roskilde, Denmark
Kenneth Ruud and Peter R. Taylor
San Diego Supercomputer Center and Department of Chemistry and Biochemistry, University of California,
San Diego, 9500 Gilman Drive Dept. 0505, La Jolla, California 92093-0505
共Received 2 August 1999; accepted 9 November 1999兲
A modified perturbation approach for the calculation of the vibrational wave function of polyatomic
molecules is discussed. It is demonstrated that if the expansion point of the potential is determined
variationally, the leading first-order term in the perturbation expansion of the vibrational wave
function vanishes. Furthermore, the new expansion point is a very good approximation to the
vibrationally averaged molecular geometry. The required third derivatives of the potential energy
with respect to geometrical distortions have been calculated by numerical differentiation. Two
approaches are discussed, one based on the differentiation of the molecular Hessian and the other on
the molecular gradient. Results are presented for the averaged molecular geometry of a large set of
molecules, including studies of electronically excited states and effects of electron correlation. The
largest molecule included is butane with a total of 14 atoms. © 2000 American Institute of Physics.
关S0021-9606共00兲30905-9兴
functional,24,46,50–52 although analytical calculations have
been presented at the Hartree–Fock level.53,54 The former
approach requires little programming effort since only the
molecular energy is needed. However, selecting the appropriate energy points needed for the fitting process is a delicate matter, and the numbers of points needed may be very
large.26 Furthermore, fitting is a process in which there are
several possible solutions, and ensuring that the correct solution is found may be difficult. Indeed, only in rare instances has this approach been applied to molecules with
more than four atoms.31,50,55,51
A few years ago, a method to calculate the intermolecular vibrational frequencies of bimolecular complexes was introduced, in which the potential was expanded around a
variationally determined expansion point instead of the equilibrium geometry.56 For diatomic molecules, it has been
demonstrated that this effective geometry corresponds to the
vibrationally averaged molecular geometry to second order
in the order parameter of the perturbation expansion,57 and
furthermore that a perturbation expansion around this effective geometry gives accurate results for molecular properties
of diatomic molecules even when only the harmonic term in
the expansion of the property surface is included.57–61 The
reason for this improved convergence is that the leading
first-order term in the perturbation expansion of the vibrational wave function vanishes when the variationally determined expansion point is used.57 It is therefore of interest to
extend this approach also to vibrational averages in polyatomic molecules. We here present the necessary extensions
of the theory to polyatomic molecules and calculate effective
geometries and expansion coefficients for the molecular vibrational wave function for a set of molecules ranging from
simple triatomic molecules to molecules containing up to 14
atoms.
I. INTRODUCTION
Vibrational contributions to molecular properties are often substantial and cannot normally be neglected when comparing calculated and experimental molecular properties. The
magnitude of the vibrational effects is determined by the
anharmonicity of the potential—that is, the ratio between the
third derivative of the potential with respect to a geometrical
distortion 共the cubic force constant兲 and the harmonic frequency. Consequently, vibrational contributions to molecular
properties will be more important for large molecules and
molecular complexes than for small molecules since larger
systems normally contain modes with smaller harmonic frequencies. The vibrational part of the molecular wave function can, however, only be accurately calculated accurately
for small molecules 共see, for example, Refs. 1–10兲. For
larger molecules, a Taylor expansion of the potential around
the equilibrium geometry is normally carried out, followed
by a perturbation analysis, but convergence problems have
been noted in this approach even for diatomic molecules.11
The molecular vibrational energy levels determine both
the vibrational spectrum and the vibrationally averaged molecular properties and the possibility of accurately calculating vibrational energy levels have been exploited extensively. However, even if progress has been substantial this is
a difficult task. Of the many approaches developed for treating this problem, the most important contributions have used
perturbation expansions to obtain vibrational frequencies and
vibrationally averaged molecular properties12–43 or different
variational approaches.44–49
Earlier attempts at calculating vibrational effects in polyatomic molecules have determined the higher potentialenergy derivatives by fitting ab initio calculated molecular
energies at different geometries to a potential energy
0021-9606/2000/112(6)/2655/13/$17.00
2655
© 2000 American Institute of Physics
Downloaded 22 Jan 2001 to 149.156.71.29. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html.
2656
J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
Åstrand, Ruud, and Taylor
II. THE MOLECULAR VIBRATIONAL WAVE FUNCTION
The theory for a variation-perturbation approach to molecular vibrations using a variationally determined expansion
point has been discussed for diatomic molecules in Ref. 57.
The extension to polyatomic molecules is in principle
straightforward, although additional computational considerations need to be made. Our presentation here will follow
quite closely that of Kern and Matcha,24 but we will consider
a general expansion point instead of restricting ourselves to
an expansion around the equilibrium geometry. Particular attention will be given to two expansion points, the equilibrium geometry 共as also studied by Kern and Matcha兲 and a
variationally optimized expansion point 共vide infra兲.
The potential energy may be expanded in a Taylor expansion around an arbitrary expansion point q exp as
N
(0)
V 共 q 1 ,q 2 , . . . ,q N 兲 ⫽V exp
⫹
1
⫹
6
兺
i⫽1
(1)
V exp,i
q i⫹
1
2
N
(2)
q 2i
兺 V exp,ii
i⫽1
N
兺
i jk⫽1
1
⫹
24
V (3)
exp,i jk q i q j q k
N
兺
i jkl⫽1
⫹••• ,
V (4)
exp,i jkl q i q j q k q l
共1兲
where q i is the deviation of normal coordinate i from the
expansion point r exp , (q i ⫽r i ⫺r exp,i). N is the number of
(n)
is the nth derivative of the
normal modes and V exp,i
1 i 2 •••i n
potential with respect to the normal coordinates. If massជ
weighted coordinates (rជ mw
k ⫽ 冑m k r k ) are used, the zerothorder Hamiltonian can be written as
1
H (0) ⫽
2
N
兺
i⫽1
冋
册
⳵2
2
⫺ 2 ⫹V (2)
ii q i ,
⳵ qi
共2兲
which is the Hamiltonian for a harmonic oscillator. It has the
well-known eigenvalues
N
E (0) ⫽
兺
i⫽1
冉 冊
n i⫹
1
␻ ,
2 i
共3兲
where ␻ i ⫽ 冑V (2)
ii , and where n i specifies the vibrational
state. The corresponding eigenfunctions are products of Hermite functions
N
⌿ (0) ⫽
兿
i⫽1
␺ ni,
(0)
(0)
Ẽ (0) ⫽V exp
⫹ 具 ⌿ (0) 兩 H (0) 兩 ⌿ (0) 典 ⫽V exp
⫹
(1)
V eff,
j⫹
1 2
共5兲
with ␰ i ⫽ 冑␻ i q i , N n i a normalization constant and H n i ( ␰ i ) a
Hermite polynomial. Normally, the expansion of the potential is carried out around the equilibrium geometry where the
molecular gradient is zero. Here we will in addition consider
an expansion point that is variationally determined from a
minimization of the energy functional56
兺 ␻i ,
共6兲
i⫽1
1
4
N
兺
(3)
V eff,ii
j
i⫽1
␻i
⫽0,
共7兲
is fulfilled for this choice of expansion point by differentiation of Eq. 共6兲 with respect to r exp,j . The remaining terms in
the expansion of the potential in Eq. 共1兲 can be considered to
be perturbations to H (0) with
N
H
(1)
⫽
兺
i⫽1
(1)
V exp,i
q i⫹
1
6
N
兺
i jk⫽1
共8兲
V (3)
exp,i jk q i q j q k ,
where we note that also the gradient of the potential is included, and the quartic term
H (2) ⫽
␺ n i ⫽N n i H n i 共 ␰ i 兲 e ⫺ 2 ␰ i ,
N
with respect to the expansion point r exp,i . The second term
on the right-hand side of Eq. 共6兲 is the zero-point vibrational
energy which thus is regarded as an additional potential energy. It is instructive to discuss this ansatz in terms of the
Born–Oppenheimer approximation. Since the motion of the
electrons is much faster than the nuclear motion, it is assumed in the Born–Oppenheimer approximation that only an
average of the electronic motion 共the potential energy兲 has to
be considered when the nuclear motion is studied. Furthermore, if one of the nuclear motions is much slower than the
other a similar separation of the nuclear motion can be carried out and an average of the nuclear motion of all modes
apart from that one can be regarded as an extra potential
energy term. For example, in variational transition state
theory the reaction coordinate is regarded as the slow nuclear
motion and the zero-point vibrational energy of the other
modes are treated on an equal footing as the electronic potential energy.62 In general, it is, however, difficult to identify a mode with a much slower nuclear motion and it is then
reasonable to include all nuclear motions in Eq. 共6兲.56 To
include all nuclear motions may be regarded as analogous to
a system with only electrons. If we assume that each electron
only interacts with an average potential of the other electrons
it is possible to write the energy as a sum of contributions,
one for each electron. In that sense, the ansatz in Eq. 共6兲
where we add the zero-point vibrational energy for each
mode may also be regarded as a kind of mean-field approach.
We will denote the variationally determined expansion
point by r eff,i , and it may be shown that the condition
共4兲
where
1
2
1
24
N
兺
i jkl⫽1
共9兲
V (4)
exp,i jkl q i q j q k q l .
By applying standard Rayleigh–Schrödinger perturbation
theory,
具 ⌿ 兩 H⫺E 兩 ⌿ 典 ⫽ 具 ⌿ (0) ⫹␭⌿ (1) ⫹␭ 2 ⌿ (2) ⫹••• 兩 共 H (0) ⫺E (0) 兲
⫹␭ 共 H (1) ⫺E (1) 兲 ⫹␭ 2 共 H (2) ⫺E (2) 兲
⫹••• 兩 ⌿ (0) ⫹␭⌿ (1) ⫹␭ 2 ⌿ (2) ⫹••• 典 ⫽0,
共10兲
Downloaded 22 Jan 2001 to 149.156.71.29. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html.
J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
Vibrational wave functions
that can be minimized with respect to the trial function ⌿̃ (1) .
Expanding the trial function in the harmonic oscillator
eigenfunctions24
and solving this equation for each order of the order parameter ␭, it is easy to show that the first-order contribution to
the energy
E (1) ⫽ 具 ⌿ (0) 兩 H (1) 兩 ⌿ (0) 典 ,
共11兲
1
24
⫹
N
兺
i jkl⫽1
V (4)
exp,i jkl 具 q i q j q k q l 典 00⫹
⬁
兺兺
i⫽1 r⫽1
冋
兺
m⫽1;m⫽i
册
⫹
N
(3)
(1)
(1)
(1)
r ␻ i a r,i
⫹
关 V exp,imm
兴 ⫹a r,i
具 q m2 典 00具 q i 典 0r a r,i
冋
⬁
N
兺
i, j⫽1;i⫽ j r,s⫽1
(1)
b rs,i
j ␾ rs,i j
⬁
兺
(1)
兺 c rst,i
jk ␾ rst,i jk ,
i, j,k⫽1;i⫽ j⫽k rst
共13兲
1 (3)
(1)
(1)
(1)
2V exp,i
⫹ V exp,iii
具 q i 典 0r a r,i
具 q 3i 典 0r a r,i
3
N
⫹
兺
⬁
with ␾ rs•••t,i j•••k ⫽ ␺ r,i ␺ s, j ••• ␺ t,k ,i⫽ j⫽k, we can express
the second-order energy as
共12兲
N
兺兺
i⫽1 r⫽1
N
(1)
a r,i
␾ r,i ⫹
N
Ẽ (2) ⫽ 具 ⌿ (0) 兩 H (2) 兩 ⌿ (0) 典 ⫹2 具 ⌿ (0) 兩 H (1) ⫺E (1) 兩 ⌿̃ (1) 典
Ẽ (2) ⫽
⬁
N
⌿̃ (1) ⫽
is zero since H (1) in Eq. 共8兲 always is odd with respect to at
least one geometrical displacement q i . The second-order energy can be regarded as an energy functional,24
⫹ 具 ⌿̃ (1) 兩 H (0) ⫺E (0) 兩 ⌿̃ (1) 典 ,
2657
兺
⬁
兺
i, j⫽1;i⫽ j r,s⫽1
(1)
(3)
2
(1)
b rs,i
j 关 V exp,ii j 具 q i 典 0r 具 q j 典 0s ⫹b rs,i j 共 r ␻ i ⫹s ␻ j 兲兴
册
1
(1)
V (3) 具 q 典 具 q 典 具 q 典 ⫹c (1) 共 r ␻ i ⫹s ␻ j ⫹t ␻ k 兲 ,
兺
兺 c rst,i
jk
3 exp,i jk i 0r j 0s k 0t rst,i jk
i, j,k⫽1;i⫽ j⫽k r,s,t⫽1
共14兲
where we have used the shorthand notation 具 H 典 rs ⫽ 具 ␺ r 兩 H 兩 ␺ s 典 . Minimizing the second-order energy with respect to the
(1)
(1)
(1)
, b rs,i
expansion coefficients a r,i
j , and c rst,i jk , we get
(1)
V exp,i
具 q i 典 0r ⫹
(1)
⫽⫺
a r,i
1 (3)
1
V
具 q 3典 ⫹
6 exp,iii i 0r 2
N
兺
m⫽1;m⫽i
(3)
V exp,imm
具 q m2 典 00具 q i 典 0r
,
r␻i
1 (3)
2
2 V exp,ii j 具 q i 典 0r 具 q j 典 0s
(1)
b rs,i j ⫽⫺
,
r ␻ i ⫹s ␻ j
b (1)
21,i j ⫽⫺
r,s⫽1, . . . ,⬁,
r,s,t⫽1, . . . ,⬁,
(1)
a 3,i
,
共17兲
(1)
b 21,i
,
and
The only nonzero contributions are
(1)
due to the symmetry properties of the integrals 具 q n 典 rs .
c 111,i
The only difference between an expansion around the equilibrium geometry and around the effective geometry defined
(1)
since this is the only term where the
in Eq. 共7兲 occurs for a 1,i
molecular gradient contributes. When expanding the poten(1)
is zero—which may
tial around the effective geometry, a 1,i
be shown using Eq. 共7兲—whereas for an expansion around
the equilibrium geometry it is24
N
兺
4 冑2 ␻ 3/2 m⫽1
i
(3)
V e,imm
␻m
,
i⫽1, . . . ,N.
共18兲
(1)
⫽⫺
a 3,i
(3)
冑3V exp,iii
36␻ i 5/2
,
i⫽1, . . . ,N,
共19兲
i⫽ j⫽1, . . . ,N,
24冑␻ i ␻ j ␻ k 共 ␻ i ⫹ ␻ j ⫹ ␻ k 兲
共15兲
共20兲
,
共21兲
which is consistent with the results of Kern and Matcha.24 It
has thus been shown that the leading first-order contribution
(1)
to the wave function, a 1,i
, vanishes if r eff is chosen as the
(1)
is a sum of
expansion point. For polyatomic molecules, a 1,i
N terms, one term for each mode of the molecule. It can
(1)
becomes larger for large moltherefore be expected that a 1,i
ecules, in particular for molecules with large couplings be(1)
tween the nuclear modes. The importance of a 1,i
vanishing
at the effective geometry will therefore be more important
for larger molecules.
The vibrational average of a molecular property can be
calculated as
具 P典⫽
For the other terms, we obtain the general expressions
,
冑2V (3)
exp,i jk
i⫽ j⫽k⫽1, . . . ,N,
i⫽ j⫽k⫽1, . . . ,N.
(1)
,
a 1,i
1
4 ␻ i 冑␻ j 共 2 ␻ i ⫹ ␻ j 兲
c (1)
111,i jk ⫽⫺
1 (3)
6 V exp,i jk 具 q i 典 0r 具 q j 典 0s 具 q k 典 0t
(1)
⫽⫺
,
c rst,i
jk
r ␻ i ⫹s ␻ j ⫹t ␻ k
i⫽1, . . . ,N,
V (3)
exp,ii j
共16兲
i⫽ j⫽1, . . . ,N,
(1)
a 1,i
⫽⫺
r⫽1, . . . ,⬁,
具⌿兩 P兩⌿典
,
具⌿兩⌿典
共22兲
where the property surface may be expanded in a Taylor
expansion
Downloaded 22 Jan 2001 to 149.156.71.29. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html.
2658
J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
P 共 q 1 ,q 2 , . . . ,q N 兲 ⫽
N
兺m
(0)
P m ⫽ P exp
⫹
1
2
⫹
Åstrand, Ruud, and Taylor
1
⫹
6
兺
具 r j 典 ⫽r e, j ⫺
(1)
P exp,i
qi
i⫽1
N
兺
兺
P (3)
exp,i jk q i q j q k ⫹
i, j,k⫽1
...,
and we may write the expansion of the averaged molecular
property as24
具 P 典 ⫽ 兺 具 P m(n) 典 ,
共24兲
mn
冋兺
冋
⬁
具 P m(n) 典 ⫽
具 ␭ k ⌿ (k) 兩 P m 兩 ␭ n⫺k ⌿ (n⫺k) 典
k⫽0
⬁
⫻ 1⫹
册
⬁
兺 兺 共 ⫺1 兲 m共 具 ␭ l ⌿ (l)兩 ␭ l ⌿ (l) 典 兲 m
m⫽1 l⫽1
册
.
共25兲
The normalization of the wave function is written as a Taylor
expansion around ⌿⫽⌿ (0) , and it is noted that the leading
term from the normalization only contributes to second order
in ␭. The zeroth-order term in ␭ gives
(0)
具 P (0) 典 exp⫽ 具 P (0)
0 典 exp⫹ 具 P 2 典 exp⫹•••
(0)
⫽ P exp
⫹
1
4
N
兺
i⫽1
(2)
P exp,ii
␻i
⫹ ...
共26兲
and the first-order term is
(1)
具 P (1) 典 exp⫽ 具 P (1)
1 典 exp⫹ 具 P 3 典 exp⫹••• ,
共27兲
where
具 P (1)
1 典 exp⫽ 冑2
N
兺
(1)
(1)
P exp,i
a 1,i
共28兲
冑␻ i
i⫽1
兺
m⫽1
(3)
V e,
jmm
␻m
共30兲
.
具 r j 典 ⫽r eff, j
N
共23兲
where
4 ␻ 2j
N
If we instead carry out the expansion around the effective
geometry, we obtain
P (2)
exp,i j q i q j
i, j⫽1
1
具 P (1)
1 典 eff
共31兲
(1)
a 1,i
⫽0
becomes zero since
for an expanbecause
sion around r eff . The effective geometry, r eff , defined by Eq.
共7兲 thus corresponds to the averaged geometry to the second
order in the order parameter ␭. This is in line with the observation that if the expansion point is chosen as the vibrationally averaged molecular geometry, the contribution from
the gradient of P in Eq. 共23兲 will vanish since 具 q i 典 ⫽ 具 r i
⫺ 具 r i 典典 trivially becomes zero.63 Instead of minimizing Eq.
共6兲, we can thus adopt Eq. 共30兲 to obtain r eff . The seemingly
difficult task of minimizing the energy functional in Eq. 共6兲
has thus been reduced to a determination of the second and
third derivatives of the potential energy at the equilibrium
geometry. Finally, we mention that a perturbation approach
for calculating the vibrational average of a molecular property will converge faster for an expansion around r eff than
around r e . This can be realized from Eq. 共29兲, where the
(1)
leading terms to 具 P (1)
3 典 can be seen to stem from a 1,i , and
(2) 57
this is also the case for the second-order term, 具 P 典 .
To summarize this section, a standard perturbation approach has been applied together with the ansatz in Eq. 共6兲.
The consequences of this ansatz are
共1兲 Eq. 共6兲 defines an effective geometry which fulfills the
condition in Eq. 共7兲.
共2兲 The condition in Eq. 共7兲 makes the leading term in the
perturbation expansion of the wave function vanish.
共3兲 As a consequence, the effective geometry is the vibrationally averaged geometry to second order in ␭ and can
be calculated from a perturbation expansion of the potential surface around the equilibrium geometry using
Eq. 共30兲.
共4兲 If averaged molecular properties are calculated from an
expansion of the potential and property surfaces around
the effective geometry, the leading term arising from the
anharmonicity of the potential vanishes.
and
具 P (1)
3 典 exp⫽
1
6
冑
1
⫹
4
⫹
3
2
N
兺
i⫽1
N
(3)
P exp,iii
␻ i 3/2
共冑
(1)
3a 1,i
⫹
In the remaining part of this paper, we will focus on
point 3, whereas point 4 is considered in the subsequent
paper.64
冑
(1)
2a 3,i
兲
P (3)
exp,ii j
(1)
⫹2b (1)
共 冑2a 1,i
兺
21,i j 兲
i, j⫽1,i⫽ j ␻ 冑␻
i
1
6 冑2
N
兺
i, j,k⫽1,i⫽ j⫽k
III. COMPUTATIONAL ASPECTS
j
(1)
P (3)
exp,i jk c 111,i jk
冑␻ i ␻ j ␻ k
.
共29兲
Considering the vibrational average of the molecular geom(1)
etry, 具 r 典 the only contributions are 具 P (0)
0 典 exp and 具 P 1 典 exp .
(2)
Furthermore, 具 P 典 does not contain any contribution from
the gradient of the molecular property and will therefore not
contribute to an averaged molecular geometry.57 An expansion of the molecular geometry around the equilibrium geometry thus gives
As discussed in the preceding section, the effective geometry is, to second order in the order parameter ␭, determined by the anharmonicity of the potential in an expansion
around the equilibrium geometry. Although third derivatives
of the potential with respect to geometrical distortions energy in principle can be rather easily implemented analytically for simpler wave functions like the Hartree–Fock
approximation,53 implementing analytical higher derivatives
for correlated wave functions is a much more laborious task.
For this reason we have chosen to address the evaluation of
the third derivatives using numerical methods. This will al-
Downloaded 22 Jan 2001 to 149.156.71.29. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html.
J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
low us to extend the approach presented here to any correlated wave function for which analytical Hessians 共or even
only gradients兲 are available.
We here determine the third derivatives of the potential
energy by numerical differentiation. The most numerically
stable approach, as well as the most general, is to calculate
the entire third-derivative tensor by numerical differentiation
of the full molecular Hessian in Cartesian coordinates, transforming the Cartesian third-derivative tensor to normal coordinates when the numerical differentiation is complete. This
allows the effective geometry of any isotopically substituted
molecule to be obtained in a single analysis step. However,
this approach would in general require the calculation of
6N⫹1 (N being the number of atoms in the molecule兲 Hessians, a task that becomes unwieldy for the larger molecules
that ultimately are our goal. Suffice it to mention here that
the benzene molecule would require the calculation of 73
molecular Hessians 共if the molecular symmetry is not utilized兲, a time-consuming task for highly correlated wave
functions. In addition, for such large molecules, doing the
numerical differentiation in normal coordinates would gain
us little, but would restrict us to one isotopomer.
In fact, we note that in general we do not need the full
(3)
third-derivative tensor, but only the V imm
elements. This
means that we should be able to determine these thirdderivative tensor elements in normal coordinates by taking
the second derivative of the molecular gradient. For a general polyatomic molecule, this approach would require the
evaluation of one Hessian 共to determine the normal modes兲,
and 6N⫺11 gradients. For variational wave functions such
as the Hartree–Fock and the multiconfigurational selfconsistent-field 共MCSCF兲 wave functions, the gradient can
be evaluated as an expectation value, making the most timeconsuming step the initial determination of the normal
modes from the molecular Hessian.
However, evaluating the cubic force fields as second derivatives requires that care must be taken to ensure that the
calculated derivatives are numerically stable and do not contain significant contributions from higher-order terms. In
Table I we have collected the results of Hartree–Fock calculation on water using the cc-pVDZ basis set of Dunning. In
all calculations, the wave function was converged to 1
⫻10⫺8 in the orbital gradient. Also reported in this table is
the results obtained by numerical differentiation of the molecular Hessian in Cartesian coordinates, in which the response equations needed for the Hessian was converged to
1⫻10⫺5 in the residual of the response vectors.
Table I clearly demonstrates the superior numerical stability of the cubic force constants calculated with the full
Cartesian Hessian, whereas the gradient-derived cubic force
constants are much more sensitive to the lengths of the steps
taken in the numerical differentiation. We note in particular
the symmetry breaking that occurs for very short steplengths in the gradient-based approach. However, in the interval 0.01a 0 to 0.0025a 0 , the gradient-derived effective geometries are very stable and for all practical purposes
identical to the effective geometries obtained by numerical
differentiation of the molecular Hessian. On the basis of
these results, in the calculations presented in the next section
Vibrational wave functions
2659
TABLE I. Effective bond length and bond angle as obtained using different
step-lengths in the numerical differentiation. Comparison is made between
using the Hessian 共in Cartesian coordinates兲 or the gradient 共in normal coordinates兲 to determine the third-derivative of the potential. Bond lengths in
Ångström.
⬔ eff
r eff
Step length
0.075
0.05
0.025
0.01
0.0075
0.005
0.0025
0.001
0.00075
0.0005
0.00025
0.0001
Gradient
Hessian
Gradient
Hessian
0.959 627
0.959 602
0.959 588
0.959 584
0.959 584
0.959 584
0.959 589
0.959 610/0.959 606
0.959 640/0.959 634
0.959 747/0.959 738
0.960 271/0.960 241
0.963 344/0.963 277
0.959 599
0.959 590
0.959 585
0.959 583
0.959 583
0.959 583
0.959 583
0.959 583
0.959 583
0.959 583
0.959 583
0.959 583
104.650
104.652
104.654
104.654
104.654
104.654
104.654
104.647
104.641
104.624
104.547
103.726
104.670
104.661
104.656
104.654
104.654
104.654
104.654
104.654
104.654
104.654
104.654
104.654
we have used a step-length of 0.001a 0 when using the full
Hessian, and a step length of 0.0075a 0 when using only the
gradients.
IV. CALCULATION OF EFFECTIVE GEOMETRIES
In this section we present the results of a number of
calculations to investigate the performance of the approach
described above for calculating zero-point vibrationally averaged molecular geometries. Our purpose here is not to obtain highly accurate vibrationally averaged geometries, but
rather to demonstrate the applicability of our approach. We
will also investigate, for selected systems, the importance of
electron correlation for determining the vibrationally averaged geometries. These calculations are performed for the
series H2 O, H2 S and H2 Se. We also demonstrate the applicability of the approach for studying electronically excited
states. Finally, we present Hartree–Fock results for some
larger molecules.
In all calculations reported here we have used the atomic
natural orbital 共ANO兲 basis sets of Widmark et al.65 These
basis sets have been used in a 关 3s2 p 兴 contraction for hydrogen, 关 4s3 p2d 兴 contraction for second-row elements,
关 5s4 p3d 兴 contraction for sulphur and a 关 6s5 p4d 兴 contraction for selenium. The only exception to this is H2 CO, for
which a slightly larger set was used in order to more properly
describe the more diffuse nature of the excited states. In
practice this has been achieved by adding a complete shell of
functions to each atoms—that is, 关 1s1 p 兴 for hydrogen and
关 1s1 p1d 兴 for carbon and oxygen. For butane, we have used
a slightly smaller basis set, the carbon basis has been contracted to 关 4s3 p1d 兴 and the hydrogen basis to 关 3s1p 兴 .
Electron correlation has been treated using MCSCF
wave functions.66 In the case of the H2 X series 共X⫽O,S,Se兲,
a complete active space SCF 共CASSCF兲 wave function with
10 active electrons and 13 correlating orbitals were employed. While this wave function cannot be expected to obtain results close to the FCI limit, it provides results with an
accuracy between MP2 and CCSD for the ten-electron sys-
Downloaded 22 Jan 2001 to 149.156.71.29. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html.
2660
J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
Åstrand, Ruud, and Taylor
We have used the Dalton program package70 in all our
calculations. The molecular gradients and Hessians has been
calculated as described in Ref. 71. The geometry optimizations have been carried out using a first-order method as
described in Ref. 72 using a model Hessian.73 For molecules
where it is required, the atomic labels are given in Fig. 1.
A. H2 O, H2 S, and H2 Se: Electron correlation effects
FIG. 1. Atomic labels of some of the molecules.
tems studied here.67 For all molecules, the core electrons
were kept inactive. In the study of H2 CO, we have adopted
the restricted active space 共RASSCF兲 method.68,69 In the case
of formaldehyde, the 1s electrons were kept inactive
whereas up to quadruple excitations are allowed from the
remaining six occupied orbitals into eight unoccupied orbitals. For simplicity, all calculations have been done without
imposing molecular point group symmetry.
Our SCF and MCSCF results for the H2 O, H2 S, and
H2 Se molecules are collected in Table II. In addition to the
r e and r eff geometries, we also report in Table II the shift in
the bond length and bond angle at the SCF and MCSCF
levels in order to more directly compare the effects of electron correlation on the vibrational corrections to the geometry. We also give the harmonic frequencies and the vibrational wave function parameters defined in Eqs. 共18兲–共20兲.
Note that some of the vibrational wave function parameters
are zero by symmetry.
For all these molecules, the main effects of zero-point
vibration is an elongation of the bond length and a reduction
of the bond angle, the only exception being the bond angle in
water at the Hartree–Fock level, where the bond angle actually increases when zero-point vibrational motion is included. In all cases, the effect of electron correlation is to
increase the bond elongation and the reduction of the bond
angle. This is related to the inverse dependence of the effective geometry on the vibrational frequencies 关see Eq. 共30兲兴,
which can thus be seen to dominate compared to the electron
correlation effects on the third-derivative tensor elements.
The importance of the electron correlation effects increase as
we go down in the periodic table, the relative change in the
bond length being 11.6% , 14.5%, and 16.3% for H2 O, H2 S,
and H2 Se, respectively.
It is interesting to note that despite the large differences
TABLE II. Bond lengths and bond angles in the H2 X series 共X⫽O,S,Se兲 obtained with an SCF and a 共6331兲
CASSCF wave function. Both the r e and r eff geometries are reported. Bond lengths in Ångström, bond angles
in degrees, and vibrational frequencies in cm⫺1 . Harmonic frequencies and coefficients of the vibrational wave
function have been calculated at the equilibrium geometry.
H2 O
re
⬔e
r eff
⬔ eff
⌬r
⌬⬔
␻1
␻2
␻3
a (1)
1,2
a (1)
1,3
a (1)
3,2
a (1)
3,3
b (1)
21,12
b (1)
21,13
b (1)
21,23
b (1)
21,32
H2 S
SCF
H2 Se
SCF
MCSCF
MCSCF
SCF
MCSCF
0.940 429
106.162
0.958 656
104.876
1.326 778
94.101
1.348 673
93.023
1.453 305
93.143
1.479 841
91.764
0.953 506
106.205
0.013 077
0.043
4234.51
4135.38
1760.36
⫺0.14262
⫺0.002 92
⫺0.021 37
⫺0.008 35
⫺0.037 00
0.005 70
0.001 07
0.011 95
0.973 250
104.792
0.014 594
⫺0.084
3972.07
3854.35
1667.80
⫺0.15289
0.005 58
⫺0.022 78
⫺0.008 18
⫺0.038 67
0.006 76
0.002 03
0.010 99
1.341 235
94.070
0.014 457
⫺0.031
2874.90
2865.01
1323.58
⫺0.13186
0.001 39
⫺0.019 02
⫺0.001 96
⫺0.033 04
0.002 29
0.000 00
0.006 07
1.365 231
92.931
0.016 558
⫺0.092
2672.97
2654.61
1210.31
⫺0.14514
0.006 22
⫺0.020 85
⫺0.001 80
⫺0.035 86
0.003 03
0.000 34
0.005 83
1.467 835
93.135
0.014 530
⫺0.008
2591.96
2585.43
1163.65
⫺0.12694
0.001 81
⫺0.018 45
⫺0.001 92
⫺0.031 87
0.002 23
0.000 08
0.006 41
1.496 745
91.710
0.016 904
⫺0.054
2389.98
2376.08
1057.03
⫺0.14207
0.006 90
⫺0.020 46
⫺0.001 48
⫺0.035 06
0.002 88
0.000 28
0.005 83
Downloaded 22 Jan 2001 to 149.156.71.29. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html.
J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
Vibrational wave functions
2661
TABLE III. Effect of isotopic substitution on the effective geometries as obtained at the MCSCF level with the basis set and geometries described in the text.
Bond lengths in Å ngström and bond angles in degrees.
Theory
H2
0.958 656
O
16
16
HD
O
O
O
D2
16
H2
18
HD
32
S
S
H2 34S
D2
0.970 14
0.969 28
1.348 673
1.365 231 共1.3528兲
1.363 917
32
r e /⬔ e
78
Se
HD 78Se
D2 78Se
H2
0.973 25
0.971 91
1.361 385
1.360 562 共1.3481兲
1.365 216
1.479 841
1.496 83
1.495 43
1.492 81
1.491 94
a
104.52 /103.9b/
104.51008c
104.50b
0.9687b
93.023
92.931 共91.97兲
92.940
92.945 共92.98兲
92.932
91.764
91.692
91.702
91.708
b
0.9572 /0.9587 /
0.957 6257共11兲 c
0.9724b
104.792
104.798
104.789
104.797
1.3362d
1.3518d
104.35b
92.06d
92.13d
1.3474d
1.459 09共77兲e
⬔
r DX
r XH
a
104.876
0.973 20
re /⬔ e
H2 32S
⬔
r XD
rXH
r e /⬔ e
Experiment
92.11d
90.958共11兲e
a
Reference 22.
Reference 74.
c
Reference 76.
d
Reference 75.
e
Reference 80, as obtained for H2
b
80
Se.
in the vibrational frequencies for the three molecules, only
minor differences exist in the change in the geometry induced by the zero-point vibrational motion, a fact that is
related to the increasing cubic force constant for the bending
motion as we go down in the periodic table.
For these molecules, the determination of the effective
geometry were done using the full analytical Hessian, and
the effects of isotopic substitution can thus also easily be
investigated. In Table III we have collected the effective geometries obtained at the MCSCF level for various isotopic
substitutions of the molecules. Also included in this table are
the available experimental data for the equilibrium and zeropoint averaged geometries.
Although a detailed comparison of the theoretical results
is not warranted because of the small basis set and somewhat
restricted active space used, it is interesting to note that our
results for the zero-point averaged bond length of water is
within 0.01 Å of that determined by Cook, De Lucia, and
Helminger74 whereas the agreement for the bond angle is
less satisfactory. In particular we are unable to reproduce the
change in the bond angle upon deuterium substitution. However, since no such effect is observed for H2 S,75 we are inclined to ascribe this to an experimental artifact. Our equilibrium bond length coincides with that of Cook et al., but
this is likely to be accidental. Furthermore, there appears to
be a problem in the analysis of Cook et al., since the two
most recent determinations of the equilibrium geometry22,76
are generally recognized as being very good,77,78 but do not
agree well with the results of Cook et al. The agreement
between our results and those of Cook et al. thus also illustrates the inadequate description of the basis set and electron
correlation effects in our calculations.
For H2 S, the agreement between our results and experi-
ment is much less satisfactory, although in this case we reproduce the deuterium shift in the bond angle more satisfactorily, being small but in opposite direction to experiment.75
However, the main reason for the observed discrepancy is
likely to be due to the poor description of the equilibrium
geometry, most likely because of inadequacies in the basis
set and the treatment of electron correlation, and also some
small relativistic effects. Using the experimental equilibrium
geometry, we obtain the vibrationally averaged geometries
given in parentheses, which now also are within 0.01 Å of
the experimental vibrationally averaged bond lengths,
whereas the agreement for the bond angle still is not satisfactory.
In comparison with experiment, H2 Se behaves like H2 S.
The MCSCF equilibrium bond length is slightly longer than
the experimental result and the bond angle is slightly larger.
The equilibrium geometry obtained here for H2 Se is in a
reasonably good agreement with a previous investigation
employing a considerably larger basis set and orbital space
(r HSe⫽1.474 Å , ⬔ HSeH⫽90.0°) 79 and with experiment.80
It is noted that both the bond lengths and the correspond(1)
ing a r,2
parameters increase in magnitude when electron correlation is added, which is to be expected since a restricted
Hartree–Fock wave function cannot describe a dissociation
process, whereas an MCSCF wave function gives a more
balanced description of the wave function at all bond dis(1)
parameter is for the
tances. For all molecules, the largest a 1,i
symmetric stretching mode (i⫽2) and its magnitude is more
(1)
than an order larger than for the bending mode, a 1,3
. The
effects of electron correlation are rather small for the
(1)
a 1,2
-parameters, but consistently the magnitude increases
slightly when electron correlation is added. Correlation al-
Downloaded 22 Jan 2001 to 149.156.71.29. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html.
2662
J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
TABLE IV. Equilibrium and vibrationally averaged geometries of formaldehyde as obtained at the Hartree–Fock level. Basis set described in the
text. Bond lengths in Ångström, bond angles in degree, and vibrational
frequencies in cm⫺1 . Harmonic frequencies and coefficients of the vibrational wave function have been calculated at the equilibrium geometry.
Equilibrium
r CO
r CH
1.179 955
1.091 602
⬔ HCH
Mode
2
3
4
116.305
␻i
3101.12
1985.11
1647.49
Effective
1.183 016
1.103 956
TABLE V. Equilibrium and vibrationally averaged geometries of the
ground state of formaldehyde as obtained at the MCSCF level. Basis set
described in the text. Bond lengths in Ångström, bond angles in degree, and
vibrational frequencies in cm⫺1 . Harmonic frequencies and coefficients of
the vibrational wave function have been calculated at the equilibrium geometry. Experimental results from Ref. 85 given in parentheses.
Change
0.003 061
0.012 354
116.221
⫺0.084
(1)
a 1,i
(1)
a 3,i
⫺0.1160
⫺0.0539
⫺0.0261
Åstrand, Ruud, and Taylor
⫺0.0206
⫺0.0139
0.0017
␻ 1 ⫽3171.20 cm⫺1 , ␻ 5 ⫽1366.61 cm⫺1 , ␻ 6 ⫽1333.73 cm⫺1
(1)
b (1)
21,12⫽⫺0.0371, b 21,62⫽0.0136
(1)
c (1)
⫽0.0019,
c
111,145
111,135⫽⫺0.0012
(1)
ters, however, a 1,3
drastically. For the water molecule, it
(3)
changes sign when correeven changes sign because V e,333
lation is included. This difference also explains the sign
change for the zero-point vibrational effects on the bond
angle when adding electron correlation. For H2 S and H2 Se,
(1)
become 3–4 times larger when
the bending parameters a 1,3
electron correlation is added, which also is reflected in the
change of the bond angles. Even if the vibrational effects on
the bond angles are small, they are much more affected by
correlation than the bond lengths, and this may explain the
less satisfactory agreement with experiment for the bond
(1)
parameters, the effects of electron correangle. For the a 3,i
lation is small.
For the b (1)
21,i j parameters, the same trends are found for
all three molecules. The coupling between the asymmetric
(1)
(1)
dominates, but also b 21,32
and symmetric stretchings b 21,12
(1)
gives important contributions. Indeed, the bilinear b 21,12
(1)
terms are more important than the linear a 3,i terms. We finally note that the c (1)
111,i jk terms are zero for these molecules
because of symmetry.
B. H2 CO: Electronic ground and excited states
The averaged geometry of formaldehyde has been calculated at the SCF level 共Table IV兲 and its electronic ground
state (S 0 ) and lowest excited singlet state (S 1 ) have been
studied with the RASSCF method 共Tables V and VI兲. The
ground state is planar (C 2 v point group兲, whereas the S 1 state
is nonplanar (C s point group兲. If we first consider the ground
state, the zero-point vibrational effects on the C–H bond
length show the same trends as for the X–H bond lengths in
the H2 X series. At the SCF level, the C–H bond length increases by 0.012 Å and the effect is slightly larger when
electron correlation is included. The effects on the CvO
bond is, however, considerably smaller. At the SCF level, the
bond length is increased by 0.003 Å, increasing slightly at
the MCSCF level to 0.004 Å. As for the H2 X series, the
(1)
and b (1)
change in the bond angle is less than 0.1°. The a r,i
21,i j
parameters also follow the pattern observed for the H2 X molecules. For formaldehyde, we have also presented the c 111,i jk
Equilibrium
r CO
r CH
⬔ HCH
Mode
2
3
4
1.209551
1.115221
116.152
␻i
Effective
1.213526共1.208兲
1.129645共1.105兲
116.103共116.3兲
(1)
a 1,i
2852.17
1778.43
1530.82
Change
0.003975
0.014424
⫺0.049
(1)
a 3,i
⫺0.1307
⫺0.0622
⫺0.0331
⫺0.0225
⫺0.0153
0.0009
␻ 1 ⫽2913.48 cm⫺1 , ␻ 5 ⫽1271.95 cm⫺1 , ␻ 6 ⫽1191.38 cm⫺1
(1)
b (1)
21,12⫽⫺0.0405, b 21,62⫽0.0146
c (1)
⫽⫺0.0017,
c (1)
111,145
111,135⫽0.0013
parameters larger than 1.0⫻10⫺3 . It is found that the largest
c 111,i jk parameters are less than 2.0⫻10⫺3 , an order of mag(1)
and b (1)
nitude smaller than the largest a r,i
21,i j parameters.
Since the S 1 state of formaldehyde is nonplanar, its behavior is quite different from the ground state. The frequencies of the modes mainly describing the C–H bonds ( ␻ 1 and
␻ 2 ) are slightly higher for the S 1 state than for the ground
state. Since the corresponding zero-point vibrational effects
on the C–H bond is reduced by a third for the excited state,
it implies that the corresponding cubic force constants are
also reduced. Furthermore, for the CvO bond stretch
共mainly ␻ 3 ), both the harmonic frequency and the vibrational contribution are reduced for the S 1 state, which implies that the corresponding cubic force constants have been
even further reduced. However, the zero-point vibrational
contributions to the bond angles are about an order of magnitude larger for the S 1 state than for the ground state. It is
TABLE VI. Equilibrium and vibrationally averaged geometries of the S 1
state of formaldehyde as obtained at the MCSCF level. Basis set described
in the text. Bond lengths in Ångström, bond angles in degree, and vibrational frequencies in cm⫺1 . Harmonic frequencies and coefficients of the
vibrational wave function have been calculated at the equilibrium geometry.
Experimental results from Ref. 86 given in parentheses.
Equilibrium
r CO
r CH
⬔ OCH
⬔ HCH
Mode
2
3
4
6
1.356 254
1.101 707
113.798
117.699
␻i
2956.91
1370.02
1143.32
791.11
Effective
Change
1.358 383共1.323兲
1.111 370共1.098兲
0.002 129
0.009 663
114.477
118.366共118.40兲
(1)
a 1,i
0.0959
⫺0.0166
⫺0.0209
0.1216
0.679
0.667
(1)
a 3,i
0.0221
0.0006
⫺0.0175
0.0419
␻ 1 ⫽3052.19 cm⫺1 , ␻ 5 ⫽979.19 cm⫺1
(1)
(1)
(1)
b (1)
21,62⫽⫺0.0357, b 21,12⫽0.0394, b 21,52⫽⫺0.0184, b 21,64⫽0.0133
(1)
c 111,246⫽0.0015
Downloaded 22 Jan 2001 to 149.156.71.29. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html.
J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
Vibrational wave functions
TABLE VII. Equilibrium and vibrationally averaged geometries of methane
as obtained at the Hartree–Fock level. Basis set described in the text. Bond
lengths in Ångström, bond angles in degree, and vibrational frequencies in
cm⫺1 . Harmonic frequencies and coefficients of the vibrational wave function have been calculated at the equilibrium geometry.
Equilibrium
r CH
Mode
4
1.081 711
␻i
3158.68
Effective
1.093 953
(1)
a 1,i
(1)
a 3,i
0.1682
TABLE IX. Equilibrium and vibrationally averaged geometries of propane
as obtained at the Hartree–Fock level. Basis set described in the text. Bond
lengths in Ångström, bond angles in degree and vibrational frequencies in
cm⫺1 . Harmonic frequencies and coefficients of the vibrational wave function have been calculated at the equilibrium geometry.
Change
0.012 242
0.0132
␻ 1,2,3 ⫽3253.58 cm⫺1 , ␻ 5,6⫽1672.28 cm⫺1 , ␻ 7,8,9 ⫽1456.06 cm⫺1
(1)
(1)
b (1)
21,14⫽b 21,24⫽b 21,34⫽0.0247
r CC
r CHip
r CHop
r CHm
⬔ CCC
⬔ Hm CHm
⬔ CCHip
⬔ CCHop
⬔ Hop CHop
Mode
mainly due to the mode with the lowest frequency, which in
the nonplanar case has become totally symmetric.
A comparison with available experimental data for the
vibrationally averaged geometries shows a rather good
agreement with experiment for the ground state, although the
vibrational corrections brings our results further away from
experiment. However, we note that recent CCSD calculations of the equilibrium geometry indicate a slightly shorter
CH bond and slightly longer CO bond81 which would, when
combined with our vibrational corrections, bring the vibrationally averaged geometry closer to experiment. For the excited state, agreement with experiment is less satisfactory.
However, this is another manifestation of our inadequate description of the equilibrium geometry—because we have
used small basis sets and because we have an incomplete
recovery of the electron correlation effects—since the vibrational corrections are small and because EOM–CCSD calculations give an equilibrium geometry for the excited state in
much better agreement with experiment.81
C. Larger molecules
To demonstrate that the approach presented in this paper
allows vibrationally averaged structures to be determined for
TABLE VIII. Equilibrium and vibrationally averaged geometries of ethane
as obtained at the Hartree–Fock level. Basis set described in the text. Bond
lengths in Ångström, bond angles in degree and vibrational frequencies in
cm⫺1 . Harmonic frequencies and coefficients of the vibrational wave function have been calculated at the equilibrium geometry.
Equilibrium
r CC
r CH
1.525 582
1.083 483
⬔ CCH
Mode
5
11
15
111.152
␻i
3169.28
1549.68
1042.33
Effective
1.535 871
1.089 270
Change
0.010 289
0.005 787
111.279
0.127
(1)
a 1,i
(1)
a 3,i
⫺0.0940
⫺0.0117
0.1309
⫺0.0112
0.0035
0.0146
␻ 1,2⫽3230.12 cm⫺1 , ␻ 3,4⫽3202.58 cm⫺1 , ␻ 6 ⫽3162.27 cm⫺1 ,
␻ 7,8⫽1623.74 cm⫺1 , ␻ 9,10⫽1619.34 cm⫺1 , ␻ 12⫽1522.29 cm⫺1 ,
␻ 13,14⫽1322.50 cm⫺1 , ␻ 16,17⫽880.18 cm⫺1 , ␻ 18⫽328.99 cm⫺1
b (1)
21,18;5 ⫽0.0771
2663
2
6
8
9
12
14
19
23
25
Equilibrium
Effective
Change
1.526 002
1.083 497
1.084 468
1.085 022
112.920
106.371
111.265
111.038
107.683
1.535 017
1.083 348
1.085 353
1.092 545
113.021
106.216
111.393
111.285
107.613
0.009 015
⫺0.000 149
0.000 885
0.007 523
0.101
⫺0.155
0.128
0.247
⫺0.070
(1)
a 1,i
(1)
a 3,i
␻i
3218.55
3163.31
3156.06
1633.59
1611.90
1544.08
1271.76
915.09
390.58
⫺0.0070
0.0412
0.0580
⫺0.0001
0.0100
⫺0.0306
⫺0.0504
⫺0.1525
⫺0.0704
⫺0.0111
0.0103
0.0107
0.0004
⫺0.0015
⫺0.0026
⫺0.0017
⫺0.0080
⫺0.0062
␻ 1 ⫽3220.70 cm⫺1 , ␻ 3 ⫽3213.77 cm⫺1 , ␻ 4 ⫽3204.13 cm⫺1 ,
␻ 5 ⫽3170.92 cm⫺1 , ␻ 7 ⫽3156.21 cm⫺1 , ␻ 10⫽1627.39 cm⫺1 ,
␻ 11⫽1616.19 cm⫺1 , ␻ 13⫽1609.76 cm⫺1 , ␻ 15⫽1532.89 cm⫺1 ,
␻ 16⫽1484.30 cm⫺1 , ␻17⫽1421.14 cm⫺1 , ␻ 18⫽1316.65 cm⫺1 ,
␻ 20⫽1110.44 cm⫺1 , ␻ 21⫽1002.56 cm⫺1 , ␻ 22⫽975.40 cm⫺1 ,
␻ 24⫽801.92 cm⫺1 , ␻ 26⫽293.64 cm⫺1 , ␻ 27⫽232.97 cm⫺1
(1)
(1)
b (1)
21,27;6 ⫽⫺0.0953, b 21,26;6 ⫽⫺0.0595, b 21,27;8 ⫽0.0545
larger polyatomic molecules, we have optimized the structures and determined the effective geometries of ten larger
molecules at the Hartree–Fock level. The molecules are
methane, ethane, propane, butane, ethene, butadiene, ethyne,
nitroethene, formic acid and formamide, and the results of
these calculations are collected in Tables VII–XVI. For most
of these molecules, we have used the normal coordinate
scheme for determining the effective geometry. As an example of the time needed for a complete determination of the
effective geometry, we note that less than 11 hours of CPU is
required for the determination of the effective geometry of
formamide 共6 atoms, requiring 1 Hessian and 25 gradient
calculations兲, whereas the determination using the full Hessian required almost 52 CPU hours on an IBM 590. For each
molecule, we present the equilibrium and effective geometries as well as the difference between them. We also
(1)
present the nonvanishing linear expansion coefficients a r,i
and the corresponding harmonic frequencies. At the end of
each table, the remaining harmonic frequencies and the largest bilinear expansion coefficients b (1)
21,i j are given.
In Tables VII–X, results are presented for a series of
n-alkanes: methane, ethane, propane, and butane. For methane, we find a change in the C–H bond length of 0.012 Å ,
which is considerably larger than for the other alkanes which
exhibits shifts in the C–H bond length of less than 0.008 Å .
For propane and butane, a clear difference is found between
the ⫺CH3 end groups and the central – CH2 – groups. The
Downloaded 22 Jan 2001 to 149.156.71.29. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html.
2664
J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
TABLE X. Equilibrium and vibrationally averaged geometries of butane as
obtained at the Hartree–Fock level. Basis set described in the text. Bond
lengths in Ångström, bond angles in degree, and vibrational frequencies in
cm⫺1 . Harmonic frequencies and coefficients of the vibrational wave function have been calculated at the equilibrium geometry.
Equilibrium
Effective
Change
r C1C1
r C1C2
r C1H
r C2Hip
r C2Hop
⬔ C1C1C2
⬔ HC1H
⬔ C1C1H
⬔ C1C2Hip
⬔ C1C2Hop
⬔ Hop C2Hop
1.528 889
1.527 962
1.088 284
1.085 597
1.086 675
113.153
106.301
109.206
111.238
111.104
107.685
1.537 245
1.530 584
1.093 820
1.083 371
1.084 114
113.636
106.340
108.857
111.827
110.804
107.670
0.008 356
0.002 622
0.005 536
⫺0.002 226
⫺0.002 561
0.483
0.039
⫺0.149
0.589
⫺0.200
⫺0.015
Mode
␻i
(1)
a 1,i
(1)
a 3,i
3
6
10
12
16
17
19
24
25
29
32
3215.54
3157.22
3141.11
1619.75
1605.03
1536.79
1519.10
1257.93
1126.32
891.44
449.18
⫺0.0057
⫺0.0086
0.0835
0.0011
⫺0.0133
⫺0.0221
⫺0.0241
⫺0.0555
0.0225
⫺0.1197
⫺0.1342
⫺0.0120
⫺0.0098
0.0102
⫺0.0007
0.0015
0.0017
0.0008
⫺0.0021
0.0017
⫺0.0045
⫺0.0045
␻ 1 ⫽3218.89 cm⫺1 , ␻ 2 ⫽3216.92 cm⫺1 , ␻ 4 ⫽3210.91 cm⫺1 ,
␻ 5 ⫽3178.22 cm⫺1 , ␻ 7 ⫽3156.46 cm⫺1 , ␻ 8 ⫽3153.52 cm⫺1 ,
␻ 9 ⫽3147.81 cm⫺1 , ␻ 11⫽1628.05 cm⫺1 , ␻ 13⫽1614.45 cm⫺1 ,
␻ 14⫽1612.94 cm⫺1 , ␻ 15⫽1606.13 cm⫺1 , ␻ 18⫽1533.60 cm⫺1 ,
␻ 20⫽1436.52 cm⫺1 , ␻ 21⫽1431.95 cm⫺1 , ␻ 22⫽1393.00 cm⫺1 ,
␻ 23⫽1308.43 cm⫺1 , ␻ 26⫽1082.77 cm⫺1 , ␻ 27⫽1041.79 cm⫺1 ,
␻ 28⫽1028.59 cm⫺1 , ␻ 30⫽865.96 cm⫺1 , ␻ 31⫽781.96 cm⫺1 ,
␻ 33⫽276.23 cm⫺1 , ␻ 34⫽275.62 cm⫺1 , ␻ 35⫽239.21 cm⫺1 ,
␻ 36⫽125.07 cm⫺1
(1)
(1)
b (1)
21,35;6 ⫽ 0.0898, b 21,34;6 ⫽ 0.0674, b 21,36;6 ⫽ 0.0482
r CC
r CH
1.316 325
1.073 438
⬔ CCH
Mode
3
5
7
121.588
␻i
3304.05
1819.85
1476.36
Effective
Change
1.321 896
1.080 753
0.005 571
0.007 315
121.599
0.011
(1)
a 1,i
(1)
a 3,i
0.0934
⫺0.0673
⫺0.0623
TABLE XII. Equilibrium and vibrationally averaged geometries of butadiene as obtained at the Hartree–Fock level. Basis set described in the text.
Bond lengths in Ångström, bond angles in degree, and vibrational frequencies in cm⫺1 . Harmonic frequencies and coefficients of the vibrational wave
function have been calculated at the equilibrium geometry.
r C1 C2
r C1 C1
r C1 H
r C2 Ht
r C2 Hc
⬔ C1C1C2
⬔ C2C1H
⬔ C1C2Ht
⬔ C1C2Hc
Equilibrium
Effective
1.320 924
1.463 773
1.075 038
1.074 017
1.072 110
123.933
116.512
121.559
121.386
1.321 718
1.472 503
1.079 878
1.077 269
1.072 717
124.003
116.337
121.663
121.481
␻i
Mode
2
4
6
7
9
12
13
19
22
3377.24
3308.35
3288.54
1863.23
1594.74
1419.99
1310.86
949.19
548.84
Change
0.000 794
0.008 730
0.004 840
0.003 252
0.000 607
0.070
⫺0.175
0.104
0.095
(1)
a 1,i
⫺0.0081
⫺0.0531
⫺0.0022
0.0153
⫺0.0059
⫺0.0316
0.0373
0.0847
0.0720
(1)
a 3,i
⫺0.0061
⫺0.0143
⫺0.0048
0.0054
⫺0.0018
⫺0.0027
0.0039
0.0052
0.0036
␻ 1 ⫽3377.24 cm⫺1 , ␻ 3 ⫽3310.48 cm⫺1 , ␻ 5 ⫽3293.41 cm⫺1 ,
␻ 8 ⫽1785.52 cm⫺1 , ␻ 10⫽1526.93 cm⫺1 , ␻ 11⫽1427.38 cm⫺1 ,
␻ 14⫽1146.18 cm⫺1 , ␻ 15⫽1105.26 cm⫺1 , ␻ 16⫽1081.86 cm⫺1 ,
␻ 17⫽1060.69 cm⫺1 , ␻ 18⫽1055.80 cm⫺1 , ␻ 20⫽847.44 cm⫺1 ,
␻ 21⫽582.02 cm⫺1 , ␻ 23⫽320.09 cm⫺1 , ␻ 24⫽171.58 cm⫺1
(1)
(1)
b (1)
21,24;12⫽0.0364, b 21,24;7 ⫽⫺0.0291, b 21,34⫽⫺0.0278
TABLE XI. Equilibrium and vibrationally averaged geometries of ethene as
obtained at the Hartree–Fock level. Basis set described in the text. Bond
lengths in Ångström, bond angles in degree, and vibrational frequencies in
cm⫺1 . Harmonic frequencies and coefficients of the vibrational wave function have been calculated at the equilibrium geometry.
Equilibrium
Åstrand, Ruud, and Taylor
0.0135
⫺0.0083
⫺0.0027
␻ 1 ⫽3382.45 cm⫺1 , ␻ 2 ⫽3352.71 cm⫺1 , ␻ 4 ⫽3280.63 cm⫺1 ,
␻ 6 ⫽1590.42 cm⫺1 , ␻ 8 ⫽1344.40 cm⫺1 , ␻ 9 ⫽1137.95 cm⫺1 ,
␻ 10⫽1097.81 cm⫺1 , ␻ 11⫽1081.38 cm⫺1 , ␻ 12⫽892.07 cm⫺1
(1)
(1)
(1)
b (1)
21,23⫽ 0.0251, b 21,13⫽0.0246, b 21,12;3 ⫽ ⫺0.0245, b 21,43⫽0.0238
C–H bonds of the – CH2 – groups are stretched by 0.008 Å
for propane and 0.006 Å for butane, in line with the results
for ethane. In contrast, most of the C–H bonds in the ⫺CH3
groups are slightly contracted. If the C–C bond lengths are
compared, the shift in ethane, propane and the central bond
in butane is about the same, 0.008–0.010 Å, whereas the
other two C–C bonds in butane are changed by only 0.003
Å. For the bond angles, the zero-point vibrational contributions are in the range 0.01–0.6°. The largest effects are
found for the in-plane angles of butane, the C1 C1 C2 angle
TABLE XIII. Equilibrium and vibrationally averaged geometries of ethyne
as obtained at the Hartree–Fock level. Basis set described in the text. Bond
lengths in Ångström, bond angles in degree, and vibrational frequencies in
cm⫺1 . Harmonic frequencies and coefficients of the vibrational wave function have been calculated at the equilibrium geometry.
Equilibrium
r CC
r CH
Mode
1
3
1.183 332
1.054 076
␻i
3673.14
2211.81
Effective
1.187 740
1.051 532
(1)
a 1,i
⫺0.0393
⫺0.0539
Change
0.004 408
⫺0.002 544
(1)
a 3,i
0.0165
⫺0.0113
␻ 2 ⫽3561.16 cm⫺1 , ␻ 4,5⫽857.60 cm⫺1 , ␻ 6,7⫽786.53 cm⫺1
(1)
(1)
(1)
(1)
b (1)
21,41 ⫽ b 21,51 ⫽ ⫺0.0394, b 21,61 ⫽ b 21,71 ⫽ ⫺0.0385, b 21,21 ⫽ 0.0316
Downloaded 22 Jan 2001 to 149.156.71.29. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html.
J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
Vibrational wave functions
TABLE XIV. Equilibrium and vibrationally averaged geometries of nitroethene as obtained at the Hartree–Fock level. Basis set described in the text.
Bond lengths in Ångström, bond angles in degree, and vibrational frequencies in cm⫺1 . Harmonic frequencies and coefficients of the vibrational wave
function have been calculated at the equilibrium geometry.
Equilibrium
Effective
Change
r C1 C2
r C1N
r N1O1
r N1 O2
r C1 H1
r C2 H2
r C2 H3
⬔ C2C1N
⬔ C2C1H1
⬔ C1C2H2
⬔ C1C2H3
⬔ C1NO1
⬔ C1NO2
1.308 226
1.458 491
1.188 663
1.188 095
1.065 760
1.070 881
1.069 675
120.819
127.325
119.491
121.406
115.516
119.048
1.308 775
1.466 838
1.190 901
1.191 503
1.066 032
1.072 925
1.068 285
120.966
127.301
119.641
121.386
115.450
119.080
0.000 549
0.008 347
0.002 238
0.003 408
0.000 272
0.002 044
⫺0.001 390
0.147
⫺0.024
0.150
⫺0.020
⫺0.066
0.032
Mode
␻i
1
2
3
4
5
6
7
8
9
12
14
15
17
3435.18
3423.07
3331.99
1858.92
1758.72
1613.30
1525.03
1402.48
1178.17
1004.93
726.10
601.53
357.46
(1)
a 1,i
⫺0.0102
⫺0.0124
⫺0.0038
⫺0.0087
0.0045
⫺0.0322
0.0017
⫺0.0443
⫺0.0881
⫺0.0627
⫺0.0568
⫺0.0667
⫺0.0008
TABLE XV. Equilibrium and vibrationally averaged geometries of formic
acid as obtained at the Hartree–Fock level. Basis set described in the text.
Bond lengths in Ångström, bond angles in degree, and vibrational frequencies in cm⫺1 . Harmonic frequencies and coefficients of the vibrational wave
function have been calculated at the equilibrium geometry.
Equilibrium
r CO1
r CO2
r CH1
r O2 H2
⬔ O1 CO2
⬔ O1CH1
⬔ CO2H2
1
2
3
4
5
6
8
␻ 10⫽1138.72 cm⫺1 , ␻ 11⫽1074.15 cm⫺1 , ␻ 13⫽941.77 cm⫺1 ,
␻ 16⫽599.40 cm⫺1 , ␻ 18⫽120.99 cm⫺1
(1)
(1)
b (1)
21,18;4 ⫽⫺0.0511, b 21,18;8 ⫽⫺0.0474, b 21,18;2 ⫽⫺0.0470
being altered by 0.48° and the C1 C2 Hip angle with 0.59°. All
other changes in the bond angles of the n-alkanes are smaller
than 0.25°.
(1)
we make two
For the linear expansion coefficients a r,i
(1)
-term is in the
main observations. The size of the largest a 1,i
range 0.10–0.20 for all the n-alkanes, whereas the largest
(1)
-terms are in the range 0.010–0.015, about an order of
a 3,i
magnitude smaller. All n-alkanes give very similar results
because they all have a large degree of symmetry and all the
totally symmetric modes are stretching or bending of bonds
which both are local in character and not strongly coupled to
other modes in the molecule. Secondly, in propane and bu(1)
terms occur for the modes with relatane the largest a 1,i
tively low frequencies, whereas the modes with highest fre(1)
terms. This aspect is important
quency have the largest a 3,i
(1)
since each a 1,i term has a contribution from every mode of
the molecule and it thus describes a coupling between the
different modes.
Turning to the largest b (1)
21,i j terms, they are considerably
(1)
terms. For ethane, propane
larger than the corresponding a 3,i
and butane, the size of the largest b (1)
21,i j parameter is in the
(1)
term is
range 0.07–0.10 whereas the size of the largest a 3,i
(1)
0.015. Furthermore, for all the large b 21,i j terms the quadratic
Effective
1.177 404
1.321 171
1.082 060
0.946 599
1.179 977
1.328 102
1.091 187
0.944 539
124.926
124.445
108.915
␻i
Mode
(1)
a 3,i
⫺0.0145
0.0117
⫺0.0189
0.0069
0.0003
⫺0.0054
0.0019
⫺0.0026
⫺0.0042
⫺0.0031
⫺0.0047
⫺0.0099
⫺0.0002
2665
4095.16
3264.52
1984.51
1533.40
1431.35
1254.87
696.36
Change
0.002 573
0.006 931
0.009 127
⫺0.002 060
124.917
124.449
109.503
⫺0.009
0.004
0.588
(1)
a 1,i
(1)
a 3,i
0.0154
0.0584
⫺0.0425
⫺0.0031
⫺0.0706
0.0691
⫺0.0611
⫺0.0305
0.0289
⫺0.0134
⫺0.0001
⫺0.0077
0.0104
⫺0.0041
␻ 7 ⫽1188.21 cm⫺1, ␻ 9 ⫽692.37 cm⫺1
(1)
b (1)
21,91⫽0.0886, b 21,72⫽⫺0.0223
term in q 共subscript i) is always a mode with a low frequency and the linear term in q 共subscript j) is a totally
symmetric mode with a high harmonic frequency. The b (1)
21,i j
terms do not contribute to the vibrationally averaged molecu-
TABLE XVI. Equilibrium and vibrationally averaged geometries of formamide as obtained at the Hartree–Fock level. Basis set described in the text.
Bond lengths in Ångström, bond angles in degree, and vibrational frequencies in cm⫺1 . Harmonic frequencies and coefficients of the vibrational wave
function have been calculated at the equilibrium geometry.
Equilibrium
r CN
r CO
r CH1
r NH2
r NH3
⬔ NCO
⬔ NCH1
⬔ CNH2
⬔ H2NH3
Mode
1
2
3
4
5
6
7
9
11
1.346 004
1.189 533
1.089 577
0.991 264
0.988 535
125.000
112.724
119.505
119.492
␻i
3964.40
3822.89
3157.60
1948.27
1768.63
1538.93
1359.70
1148.81
619.27
Effective
1.362 055
1.190 106
1.099 275
0.976 194
0.955 581
Change
0.016 051
0.000 573
0.009 698
⫺0.015 070
⫺0.032 954
124.816
112.992
117.704
122.226
⫺0.184
0.268
⫺1.801
2.734
(1)
a 1,i
(1)
a 3,i
0.1202
⫺0.2469
⫺0.0628
0.0282
⫺0.1703
⫺0.0101
⫺0.0826
⫺0.1171
⫺0.0026
⫺0.0072
0.0205
⫺0.0297
⫺0.0123
⫺0.0040
0.0011
⫺0.0084
⫺0.0062
⫺0.0021
␻ 8 ⫽1180.45 cm⫺1 , ␻ 10⫽668.19 cm⫺1 , ␻ 12⫽148.38 cm⫺1
(1)
(1)
b (1)
21,12;2 ⫽⫺0.3844, b 21,12;5 ⫽⫺0.1957, b 21,12;1 ⫽0.1884
Downloaded 22 Jan 2001 to 149.156.71.29. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html.
2666
J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
lar geometry considered in this work, but they may be important for other molecular properties.
To compare with the alkanes, ethene and butadiene have
been included 共Tables XI and XII兲. The most striking observation is that the effects on the CvC double bonds are much
smaller than for the C–C single bonds. The shift in ethene is
approximately half of the shift in ethane 共0.006 Å compared
to 0.010 Å兲, and the double-bonds in butadiene are only
stretched with about a tenth of the stretching of the central
single-bond 共0.0008 Å compared to 0.009 Å兲. The effects on
the C–H bonds are approximately the same for the alkenes
and alkanes studied here. For the bond angles, the vibrational
contributions are much smaller for ethene compared to
ethane 共0.01° and 0.13°, respectively兲 and for butadiene
compared to butane. This is also reflected in the size of the
(1)
wave function coefficients. The largest a 1,i
and b (1)
21,i j parameters are much smaller for ethene and butadiene than for the
(1)
alkanes, whereas the a 3,i
parameters are of the same
magnitude.
For comparison with ethane and ethene, we have also
included ethyne 共Table XIII兲. The stretching of the C⬅C
bond is about 0.004 Å , which is less than for ethene and
ethane 共0.006 Å and 0.010 Å , respectively兲. For the C–H
bond, we find, however, a contraction of the bond length of
⫺0.003 Å , which is in line with a recent CCSD共T兲 study
where a contraction of ⫺0.006 Å was obtained.82 The difference in results are due to the different accuracies of the potential surfaces. For example, our SCF value of r e for the
C–H bond is 1.0541 Å, whereas the more accurate CCSD共T兲
value is 1.0621 Å.
For further comparison, we have also included the nitroethene molecule 共Table XIV兲. The shift of the CvC double
bond is about a factor of 10 smaller in nitroethene than in
ethene 共0.0005 Å compared to 0.006 Å兲 and the C–H bond
for the carbon bonded to the nitrogen is stretched much less
than the other two C–H bonds in nitroethene 共0.0003 Å compared to 0.002 and ⫺0.001 Å , respectively兲. Thus, a neighboring functional group such as the nitro group has a very
strong influence on the molecular motion in the rest of the
molecule.
In Tables XV and XVI, SCF results are presented for
formic acid and formamide. These results may be compared
to the previous SCF results of formaldehyde in Table IV. As
in the comparison of ethene and nitroethene, we note that
changing the neighboring functional group alters the vibrational effects on the remaining molecular fragment, in this
case the CvO bond. Whereas the CO bond is stretched with
0.0031 Å in formaldehyde, the vibrational contributions are
0.0026 Å for formic acid and only 0.0006 Å for formamide.
For formic acid, the largest effects are found for the CO2 H2
angle, because the largest anharmonicity in the molecule
stems from the intramolecular hydrogen bond,
O2 ⫺H2 •••O1 .
In the series of molecules studied in this work, the largest zero-point vibrational effects on a molecular geometry
are found for formamide. As in formic acid, it has an intramolecular hydrogen bond, but at the SCF level we also
found a very low harmonic frequency of 148 cm⫺1 , which is
mainly the out-of-plane rotation of the ⫺NH2 group. Indeed,
Åstrand, Ruud, and Taylor
initial calculations on the MCSCF level indicates that the
equilibrium geometry of formamide may not be planar, but
that will be investigated in more detail in a future work. The
dependence of the planarity of formamide on the theoretical
level have also been noted previously,83 and this sensitivity
on the theoretical model is also present in acetamide.84 The
small harmonic frequency associated with the out-of-plane
rotation gives a relatively large vibrational effect on the C–N
bond length, 0.016 Å, and on the CNH2 and H2 NH3 bond
angles, ⫺1.8° and 2.7°, respectively. We also find significantly larger wave function coefficients for formamide than
for the other molecules in this study. In particular, the term
(1)
, has a large value
describing the N–H stretching mode, a 1,2
(1)
of ⫺0.25 and the b 21,12;2 parameter, which describes a coupling between the N–H stretch and the NH2 out-of-plane
rotation, has a value of ⫺0.38.
V. CONCLUSION
In this work, we have discussed a method for calculating
the vibrational wave function of large molecules. It is based
on a perturbation approach and it has shown that the expansion converges faster if the expansion point is determined
variationally. The applicability of the approach has been
demonstrated with calculations of the averaged molecular
geometry on molecules with up to 14 atoms.
ACKNOWLEDGMENTS
K.R. has been supported by the Norwegian Research
Council through a postdoctoral fellowship from the Norwegian Research Council 共Grant No. 125851/410兲 and through
a grant of computer time from the Program for Supercomputing. The research was supported by the National Science
Foundation 共USA兲 through Cooperative Agreement DACI9619020 and by Grant No. CHE-9700627, and by a grant of
computer time from SDSC. Fruitful comments from Dr. J.
Lounila are acknowledged.
1
D. Sundholm, J. Gauss, and A. Schäfer, J. Chem. Phys. 105, 11051
共1996兲.
2
D. Sundholm and J. Gauss, Mol. Phys. 92, 1007 共1997兲.
3
P. Jensen, I. Paidarova, V. Spriko, and S. P. A. Sauer, Mol. Phys. 91,
319–332 共1997兲.
4
O. Bludský and P. Jensen, Mol. Phys. 91, 653 共1997兲.
5
G. Osmann, P. R. Bunker, P. Jensen, and W. P. Kraemer, Chem. Phys.
225, 33 共1997兲.
6
P.-Å. Malmqvist, R. Lindh, B. O. Roos, and S. Ross, Theor. Chim. Acta
73, 155 共1988兲.
7
S. Carter, N. Pinnavaia, and N. C. Handy, Chem. Phys. Lett. 240, 400
共1995兲.
8
S. Carter and N. C. Handy, J. Chem. Phys. 110, 8417 共1999兲.
9
H. Y. Mussa and J. Tennyson, J. Chem. Phys. 109, 10885 共1998兲.
10
O. L. Polyanski and J. Tennyson, J. Chem. Phys. 110, 5056 共1999兲.
11
L. L. Sprandel and C. W. Kern, Mol. Phys. 24, 1383 共1972兲.
12
H. H. Nielsen, Rev. Mod. Phys. 23, 90 共1951兲.
13
R. C. Herman and K. E. Shuler, J. Chem. Phys. 22, 481 共1954兲.
14
E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, Molecular Vibrations
共McGraw-Hill, New York, 1955兲.
15
D. R. Herschbach and V. W. Laurie, J. Chem. Phys. 37, 1668 共1962兲.
16
Y. Morino, K. Kuchitsu, and T. Oka, J. Chem. Phys. 36, 1108 共1962兲.
17
M. Toyama, T. Oka, and Y. Morino, J. Mol. Spectrosc. 13, 193 共1964兲.
18
Y. Morino, K. Kuchitsu, and S. Yamamoto, Spectrochim. Acta 24, 335
共1968兲.
19
A. D. Buckingham, J. Chem. Phys. 36, 3096 共1962兲.
Downloaded 22 Jan 2001 to 149.156.71.29. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html.
J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
A. D. Buckingham and J. E. Cordle, Mol. Phys. 28, 1037 共1974兲.
A. D. Buckingham and W. Urland, Chem. Rev. 75, 113 共1975兲.
22
A. R. Hoy, I. M. Mills, and G. Strey, Mol. Phys. 24, 1265 共1972兲.
23
N. J. D. Lucas, Mol. Phys. 23, 825 共1972兲.
24
C. W. Kern and R. L. Matcha, J. Chem. Phys. 49, 2081 共1968兲.
25
W. C. Ermler and C. W. Kern, J. Chem. Phys. 55, 4851 共1971兲.
26
B. J. Krohn, W. C. Ermler, and C. W. Kern, J. Chem. Phys. 60, 22 共1974兲.
27
W. C. Ermler and B. J. Krohn, J. Chem. Phys. 67, 1360 共1977兲.
28
B. J. Krohn and C. W. Kern, J. Chem. Phys. 69, 5310 共1978兲.
29
L. B. Harding and W. C. Ermler, J. Comput. Chem. 6, 13 共1985兲.
30
L. O. Hargiss and W. C. Ermler, J. Phys. Chem. 92, 300 共1988兲.
31
H. C. Hsieh and W. C. Ermler, J. Comput. Chem. 9, 95 共1988兲.
32
W. C. Ermler and H. C. Hsieh, in Advances in Molecular Electronic
Structure Theory: Calculation and Characterization of Molecular Potential Energy Surfaces, edited by T. H. Dunning, Jr. 共JAI, Greenwich, 1990兲.
33
J. M. Herbert and W. C. Ermler, Comput. Chem. Eng. 22, 169 共1998兲.
34
Q. Zhang, P. N. Day, and D. G. Truhlar, J. Chem. Phys. 98, 4948 共1988兲.
35
D. G. Truhlar and A. D. Isaacson, J. Chem. Phys. 94, 357 共1991兲.
36
J. Lounila, R. Wasser, and P. Diehl, Mol. Phys. 62, 19 共1987兲.
37
P. W. Fowler and W. T. Raynes, Mol. Phys. 43, 65 共1981兲.
38
P. W. Fowler and W. T. Raynes, Mol. Phys. 45, 667 共1982兲.
39
W. T. Raynes, Mol. Phys. 49, 443 共1983兲.
40
P. W. Fowler, Mol. Phys. 51, 1423 共1984兲.
41
W. T. Raynes et al., Mol. Phys. 64, 143 共1988兲.
42
D. M. Bishop and J. Pipin, J. Chem. Phys. 98, 522 共1993兲.
43
D. M. Bishop and S. M. Cybulski, J. Chem. Phys. 101, 2180 共1994兲.
44
G. D. Carney, L. L. Sprandel, and C. W. Kern, Adv. Chem. Phys. 37, 305
共1978兲.
45
R. J. Whitehead and N. C. Handy, J. Mol. Spectrosc. 55, 356 共1975兲.
46
G. J. Sexton and N. C. Handy, Mol. Phys. 51, 1321 共1984兲.
47
N. C. Handy, Int. Rev. Phys. Chem. 8, 275 共1988兲.
48
Z. Baĉić and J. C. Light, Annu. Rev. Phys. Chem. 40, 469 共1989兲.
49
S. E. Choi and J. C. Light, J. Chem. Phys. 97, 7031 共1992兲.
50
J. M. L. Martin, T. J. Lee, P. R. Taylor, and J.-P. Francois, J. Chem. Phys.
103, 2589 共1995兲.
51
A. J. Russel and M. A. Spackman, Mol. Phys. 84, 1239 共1995兲.
52
J. Vaara, J. Lounila, K. Ruud, and T. Helgaker, J. Chem. Phys. 109, 8388
共1998兲.
53
P. E. Maslen et al., J. Chem. Phys. 95, 7409 共1991兲.
54
G. Klatt et al., J. Mol. Spectrosc. 176, 64 共1996兲.
55
B. J. Persson, P. R. Taylor, and J. M. L. Martin, J. Phys. Chem. A 102,
2483 共1998兲.
56
P.-O. Åstrand, G. Karlström, A. Engdahl, and B. Nelander, J. Chem. Phys.
102, 3534 共1995兲.
57
P.-O. Åstrand, K. Ruud, and D. Sundholm. Theor. Chem. Acc. 共in press兲.
58
P.-O. Åstrand and K. V. Mikkelsen, J. Chem. Phys. 104, 648 共1996兲.
59
K. Rund, P.-O. Åstrand, T. Helgaker, and K. V. Mikkelsen, J. Mol.
Struct.: THEOCHEM 388, 231 共1996兲.
60
P.-O. Åstrand, K. Ruud, K. V. Mikkelsen, and T. Helgaker, Chem. Phys.
271, 163 共1997兲.
20
21
Vibrational wave functions
2667
61
P.-O. Åstrand, K. Ruud, K. V. Mikkelsen, and T. Helgaker, J. Chem.
Phys. 110, 9463 共1999兲.
62
D. G. Truhlar, B. C. Garrett, and S. J. Klippenstein, J. Phys. Chem. 100,
12771 共1996兲.
63
V. H. Smith, Jr. et al., J. Chem. Phys. 67, 3676 共1977兲.
64
K. Ruud, P.-O. Åstrand, and P. R. Taylor, J. Chem. Phys. 112, 2668
共2000兲, following paper.
65
P.-O. Widmark, P.-Å. Malmqvist, and B. O. Roos, Theor. Chim. Acta 77,
291 共1990兲.
66
B. O. Roos, Adv. Chem. Phys. 69, 399 共1987兲.
67
K. Ruud, T. Helgaker, and P. Jo” rgensen, J. Chem. Phys. 107, 10599
共1997兲.
68
J. Olsen, B. O. Roos, P. Jo” rgensen, and H. J. Aa. Jensen, J. Chem. Phys.
89, 2185 共1988兲.
69
P.-Å. Malmqvist, A. Rendell, and B. O. Roos, J. Phys. Chem. 94, 5477
共1990兲.
70
T. Helgaker, H. J. Aa. Jensen, P. Jo” rgensen, J. Olsen, K. Ruud, H. Ågren,
T. Andersen, K. L. Bak, V. Bakken, O. Christiansen, P. Dahle, E. K.
Dalskov, T. Enevoldsen, B. Fernandez, H. Heiberg, H. Hettema, D. Jonsson, S. Kirpekar, R. Kobayashi, H. Koch, K. V. Mikkelsen, P. Norman,
M. J. Packer, T. Saue, P. R. Taylor, and O. Vahtras. DALTON, release 1.0.
an ab initio electronic structure program, 1997.
71
T. U. Helgaker, J. Almlöf, H. J. Aa. Jensen, and P. Jo” rgensen, J. Chem.
Phys. 84, 6266 共1986兲.
72
V. Bakken and T. Helgaker 共unpublished兲.
73
R. Lindh, A. Bernhardsson, G. Karlström, and P.-Å. Malmqvist, Chem.
Phys. 251, 141 共1996兲.
74
R. L. Cook, F. C. De Lucia, and P. Helminger, J. Mol. Spectrosc. 53, 62
共1974兲.
75
R. L. Cook, F. C. De Lucia, and P. Helminger, J. Mol. Struct.:
THEOCHEM 28, 237 共1975兲.
76
O. L. Polyanski, P. Jensen, and J. Tennyson, J. Chem. Phys. 101, 7651
共1994兲.
77
J. M. L. Martin, J.-P. Francois, and R. Gijbels, J. Chem. Phys. 96, 7633
共1992兲.
78
T. Helgaker, J. Gauss, P. Jo” rgensen, and J. Olsen, J. Chem. Phys. 106,
6430 共1997兲.
79
P.-O. Åstrand, J. Chem. Phys. 108, 2528 共1998兲.
80
J.-M. Flaud et al., J. Mol. Spectrosc. 167, 383 共1994兲.
81
J. E. Del Bene, S. R. Gwaltney, and R. J. Bartlett, J. Phys. Chem. A 102,
5124 共1998兲.
82
J. M. L. Martin, T. J. Lee, and P. R. Taylor, J. Chem. Phys. 108, 676
共1998兲.
83
R. D. Brown, P. D. Godfrey, and B. Kleibömer, J. Mol. Spectrosc. 124, 34
共1987兲.
84
S. Samdal, J. Mol. Struct.: THEOCHEM 440, 165 共1998兲.
85
K. Takagi and T. Oka, J. Phys. Soc. Jpn. 18, 1174 共1963兲.
86
P. Jensen and P. R. Bunker, J. Mol. Spectrosc. 94, 114 共1982兲.
Downloaded 22 Jan 2001 to 149.156.71.29. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html.