JOURNAL OF CHEMICAL PHYSICS
VOLUME 112, NUMBER 6
8 FEBRUARY 2000
An efficient approach for calculating vibrational wave functions
and zero-point vibrational corrections to molecular properties
of polyatomic molecules
Kenneth Ruud
San Diego Supercomputer Center and Department of Chemistry and Biochemistry, University of California,
San Diego, 9500 Gilman Drive MC-0505, La Jolla, California 92093
Per-Olof Åstrand
Condensed Matter Physics and Chemistry Department, Riso” National Laboratory, POB 49, DK-4000
Roskilde, Denmark
Peter R. Taylor
San Diego Supercomputer Center and Department of Chemistry and Biochemistry, University of California,
San Diego, 9500 Gilman Drive MC-0505, La Jolla, California 92093
共Received 10 August 1999; accepted 9 November 1999兲
We have recently presented a formalism for calculating zero-point vibrational corrections to
molecular properties of polyatomic molecules in which the contribution to the zero-point vibrational
correction from the anharmonicity of the potential is included in the calculations by performing a
perturbation expansion of the vibrational wave function around an effective geometry. In this paper
we describe an implementation of this approach, focusing on computational aspects such as the
definition of normal coordinates at a nonequilibrium geometry and the use of the Eckart frame in
order to obtain accurate nonisotropic molecular properties. The formalism allows for a black-box
evaluation of zero-point vibrational corrections, completed in two successive steps, requiring a total
of two molecular Hessians, 6K – 11 molecular gradients, and 6K – 11 property evaluations, K being
the number of atoms. We apply the approach to the study of a number of electric and magnetic
properties—the dipole and quadrupole moments, the static and frequency-dependent polarizability,
the magnetizability, the rotational g tensor and the nuclear shieldings—of the molecules hydrogen
fluoride, water, ammonia, and methane. Particular attention is paid to the importance of electron
correlation and of the importance of the zero-point vibrational corrections for obtaining accurate
estimates of molecular properties for a direct comparison with experiment. © 2000 American
Institute of Physics. 关S0021-9606共00兲31005-4兴
I. INTRODUCTION
properties in polyatomic molecules is made difficult by the
importance of the anharmonicity of the potential energy surface, which requires knowledge of the cubic force constants.
Furthermore, in order to obtain vibrationally averaged properties the second derivatives of the properties with respect to
geometrical distortions are needed, which requires fourth derivatives of the energy to be calculated in the case of secondorder molecular properties such as the nuclear magnetic
shielding constants and the molecular polarizability. As the
number of vibrational degrees of freedom increases as the
molecule grows larger, the task of calculating cubic force
fields and second derivatives of the molecular properties becomes increasingly more prohibitive for highly correlated
electronic wave functions.
One of the first systematic attempts at describing the
effects of zero-point vibrational contributions to molecular
properties in polyatomic molecules was presented in a series
of papers by Kern and co-workers,12–14 and was applied to
the study of molecular properties of various diatomic molecules as well as the water molecule. The perturbation analysis we will adopt here will to a large extent follow the same
analysis as the works of Kern et al. A similar approach was
presented by Lounila, Wasser, and Diehl,15 and has in recent
With the development of more accurate treatments of the
electron correlation problem, it is becoming increasingly obvious that in order to be able to compare theoretically calculated molecular properties directly with experimental observations, zero-point vibrational effects, as well as temperature
effects due to molecular rotation and population of excited
vibrational states, need to be taken into account. Indeed, recent studies of the rovibrational dependence of nuclear magnetic resonance 共NMR兲 parameters have demonstrated that
the effects of rotation and vibration may be as large as the
effects of electron correlation,1–7 an observation that questions the inclusion of electron correlation in theoretical calculations of NMR parameters without at the same time taking into account the effects of molecular vibration and
rotation. Studies of nonlinear optical properties of conjugated systems have also shown that vibrational effects may
be very large, and in some cases even dominate the calculated nonlinear optical properties, although most of these effects are due to the so-called pure vibrational
contributions.8–10 Still, zero-point vibrational corrections
may be substantial, see for instance Ref. 11.
The theoretical study of vibrational motion on molecular
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© 2000 American Institute of Physics
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J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
years been applied to the study of the magnetic properties of
several triatomic molecules.3,16–18 Important contributions
have been presented by Raynes and co-workers on the rovibrational contributions to nuclear magnetic resonance parameters such as the nuclear shieldings, indirect spin–spin coupling constants and the magnetizabilities.19–22 Špirko and coworkers have also done a number of studies of the
rovibrational contributions to a wide variety of electric and
magnetic properties of polyatomic molecules.23–26 Recently,
Russell and Spackman presented a method for the routine
calculation of zero-point vibrational contributions to the molecular properties of polyatomic molecules27 and applied it to
a series of different molecules.28,29 However, these approaches appear to require substantial efforts by the user before the final, rovibrationally averaged result can be
extracted.
Our goal here is to be able to calculate zero-point vibrationally averaged properties for large molecules, even for
rather highly correlated wave functions, in a reasonable
amount of time. As additional requirements, we want the
calculations to require as little user intervention as possible,
yet obtain results for the vibrational corrections that are of
comparable accuracy to the accuracy of the wave function
used. The approach presented here allows us to fulfill all
these goals by requiring only two calculations to be performed: 共1兲 At the optimized molecular geometry we determine an effective geometry30 by calculating gradients along
the normal coordinates 共this step requires 1 molecular Hessian and 6K – 11 molecular gradients, where K is the number
of atoms in the molecule兲. 共2兲 At the effective geometry we
calculate the second-derivative of the property 共properties兲 of
interest along the normal coordinates 共this step requires one
molecular Hessian and 6K – 11 property calculations兲.
Although we will only apply the method here to molecules with five or fewer atoms, the approach can very easily
and cost-effectively be applied to larger systems, and the first
step described above has been applied to systems as large as
butane and nitroethene in an earlier paper.30 Other applications for larger system will be presented elsewhere.
The rest of this paper is divided into five sections. Section II summarizes the theory behind our approach, and Sec.
III discusses certain implementational aspects of the theory.
In Secs. IV and V we present our results for different electric
and magnetic properties of the ten-electron hydrides. In Sec.
VI we summarize our findings and give some concluding
remarks.
Vibrational wave functions
2669
in the order parameter of the perturbed vibrational wave
function. It was demonstrated that the most important term in
the perturbation expansion of the vibrational wave function
vanishes if the expansion is carried out around this effective
geometry, which means that each normal mode can be accurately represented by a single Gaussian function and still
give vibrationally averaged results of a quality comparable to
those obtained when the anharmonicity of the potential is
included in an expansion around the equilibrium geometry.
A detailed comparison of the different expansion contributions was discussed for diatomics in Ref. 31, where we also
demonstrated that the use of the harmonic contribution
around the effective geometry for selected diatomic molecules in general recovers more of the total vibrational correction for a range of molecular properties than the terms
normally included when expanding around the equilibrium
geometry.
We will not repeat the complete perturbation analysis
here, but rather focus on the main feature of the approach we
use. Since our previous paper focused on the determination
of the effective geometry by a perturbation expansion around
the equilibrium geometry,30 certain theoretical and computational issues arising when molecular properties are calculated
at a nonequilibrium geometry were not treated in that paper,
and we will consider these points at the end of this section.
We will consider an expansion of the potential energy
around an arbitrary expansion point, r exp
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 a mass-weighted displacement of the nuclei from
the effective geometry along normal coordinate i, N is the
number of normal coordinates, of which there are 3K – 6
(3K – 5 for linear molecules兲, K being the number of atoms
(n)
is the nth derivative of the
in the molecule. Vexp,i
1 ,i 2 , . . . ,i n
potential energy at the expansion point with respect to the
normal coordinates. An effective geometry is chosen such as
to minimize the energy functional
II. THEORY
In a recent paper30 we presented a generalized analysis
of the perturbation expansion of the vibrational wave function of polyatomic molecules, similar to the approach discussed by Kern and Matcha.12 In that paper, we considered
the expansion of the vibrational wave function around an
arbitrary expansion point, instead of restricting ourselves to
an expansion around the equilibrium geometry as was done
by Kern and Matcha. Particular attention was given to two
expansion points; the equilibrium geometry, and an effective
geometry that was shown to be equivalent to the zero-point
vibrationally averaged molecular geometry to second-order
(0)
⫹ 具 ⌿̃ (0) 兩 H (0) 兩 ⌿̃ (0) 典 ,
Ẽ (0) ⫽V exp
共2兲
with respect to the expansion point. The zeroth-order Hamiltonian is the ordinary harmonic oscillator Hamiltonian,
which can be written as
H (0) ⫽
1
2
N
兺
i⫽1
冋
⫺
⳵2
⳵ 2q i
册
2
⫹V (2)
ii q i .
共3兲
Writing the trial function as a product of eigenfunctions to
the harmonic oscillator problem
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2670
J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
Ruud, Åstrand, and Taylor
N
兿
⌿̃ (0) ⫽
i⫽1
␺ ni ,
共4兲
with each harmonic oscillator eigenfunction being
␺ n i ⫽N n i H n i 共 ␰ i 兲 e
1 2
⫺ 2 ␰i
具 P 典 ⫽ 兺 具 P m(n) 典 ,
1
2
共5兲
,
N
N
兺 ␻i .
共6兲
i⫽1
兺
V (3)
eff,i j j
␻j
j⫽1
⫽0,
H (1) ⫽
1
(1)
V eff,i
q i⫹
6
兺
i⫽1
共7兲
N
兺
i jk⫽1
(3)
V eff,i
jk q i q j q k ,
共8兲
where it is noted that the gradient of the potential is included
and
H (2) ⫽
1
24
N
兺
i jkl⫽1
共9兲
(4)
V eff,i
jkl q i q j q k q l ,
and do a perturbation expansion of the vibrational wave
function with respect to these perturbations.
The vibrational average of a molecular property may be
determined from the expectation value of the property with
respect to a vibrational wave function
具⌿兩 P兩⌿典
.
具 P典⫽
具⌿兩⌿典
共10兲
Expanding the property surface in a manner similar to the
expansion of the potential energy in Eq. 共1兲 we get
P 共 q 1 ,q 2 , . . . ,q N 兲
⫽
兺m
N
(0)
P m ⫽ P eff
⫹
1
⫹
6
兺
i⫽1
(1)
P eff,i
q i⫹
1
2
N
兺
i j⫽1
(2)
P eff,i
jq iq j
N
兺
i jk⫽1
k⫽0
具 ␭ k ⌿ (k) 兩 P m 兩 ␭ n⫺k ⌿ (n⫺k) 典
⫻ 1⫹
where eff denotes that the expansion is carried out around the
effective geometry. We now consider the remaining terms in
Eq. 共1兲 as perturbations to the zeroth-order Hamiltonian in
Eq. 共3兲. More specifically, we define
N
冋兺
冋
⬁
At the effective geometry the gradient of Ẽ (0) is zero,
and a differentiation of the right-hand side of Eq. 共2兲 with
respect to the expansion point gives the following relationship at the effective geometry
1
4
where
具 P m(n) 典 ⫽
In Eqs. 共5兲 and 共6兲 we have introduced the harmonic frequencies ␻ i ⫽ 冑V (2)
ii , the Hermite polynomials H n i ( ␰ i ), the normalization constants N n i and ␰ i ⫽ 冑␻ i q i .
(1)
⫹
V eff,i
共12兲
mn
we can rewrite Eq. 共2兲 as
(0)
⫹
Ẽ (0) ⫽V exp
tation value expression Eq. 共10兲, we may expand the property expectation value in orders of the property surface and
potential energy surface derivatives as
(3)
P eff,i
jk q i q j q k ⫹••• .
共11兲
Inserting this expansion of the property surface in the expec-
⬁
⬁
册
兺 兺 共 ⫺1 兲 m共 具 ␭ l ⌿ (l)兩 ␭ l ⌿ (l) 典 兲 m
m⫽1 l⫽1
册
.
共13兲
The second pair of brackets here represents a Taylor expansion of the normalization around ⌿⫽⌿ (0) . The terms that
only depend on the unperturbed vibrational wave function
can now be written
(0)
具 P (0) 典 eff⫽ 具 P (0)
0 典 eff⫹ 具 P 2 典 eff⫹•••
0
⫽ P eff
⫹
1
4
N
兺
i⫽1
(2)
P eff,ii
␻ eff,i
⫹ ...,
共14兲
and the contributions that include the first-order terms in the
order parameter of the perturbed vibrational wave function
are
(1)
具 P (1) 典 eff⫽ 具 P (1)
1 典 eff⫹ 具 P 3 典 eff⫹••• .
共15兲
(1)
Detailed expressions for 具 P (1)
1 典 eff and 具 P 3 典 eff can be found
(1)
in Ref. 30. Here we only note that 具 P 1 典 eff vanishes because
of the condition Eq. 共7兲. The major contribution to the zeropoint vibrationally averaged property from the anharmonicity of the potential as calculated at the equilibrium geometry
thus vanishes when the effective geometry is used as an expansion point, and the correction to a molecular property
from zero-point vibrational motion can to a good approximation be calculated from a knowledge of the zeroth-order vibrational wave function alone30,31
0
⫺ P 0e 兲 ⫹
具 P 典 0,0⫽ 共 P eff
1
4
N
兺
i⫽1
(2)
P eff,ii
␻i
(0)
(0)
⫽ 共 具 P (0)
0 典 eff⫺ 具 P 0 典 e 兲 ⫹ 具 P 2 典 eff ,
共16兲
具 P (0)
0 典e
where
indicate the property calculated at the equilibrium geometry for a particular choice of wave function.
Equation 共16兲 does not imply that there is no anharmonicity
of the potential included in the calculation of vibrationally
averaged properties. Instead, the anharmonicity is included
through the use of the effective geometry instead of the equilibrium geometry as an expansion point for the vibrational
wave function.
Although conceptually simple, Eq. 共16兲 does contain
some aspects that need consideration. One point is the definition of normal coordinates at a nonequilibrium geometry,
since Eq. 共16兲 involves harmonic frequencies at the effective
geometry, which differs from those at the equilibrium geometry. At a nonequilibrium geometry, the forces are nonvanishing, which means that there will be a mixing of the overall
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J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
Vibrational wave functions
rotational forces into the internal degrees of motion. Pulay
has suggested that in such situations one should use a nonredundant set of internal coordinates for the evaluation of the
molecular Hessian, since such a set of coordinates will not
contain any rotational forces.32 However, there is no unique
definition of such a set of nonredundant internal coordinates.
We have instead chosen to apply a projection operator to
remove the translational and rotational components from the
molecular Hessian. This will provide us with a diagonalizable matrix in which six 共five in the case of linear molecules兲
of the eigenvalues will be exactly zero for the translational
and rotational degrees of freedom. Since the definition of
normal coordinates and harmonic frequencies at a nonequilibrium geometry is not well-defined, this serves as a useful
starting point for a perturbation expansion using this definition of the normal coordinates. Furthermore, our perturbation
expansion can in principle be performed along any set of
well-defined internal vibrational modes.
A second point that requires some consideration is the
evaluation of molecular properties at nonequilibrium geometries, and in particular the assignment of the components of
the property tensors to their counterpart at the equilibrium
geometry. This is particularly important for nonsymmetric
geometries of highly symmetric molecules, an example being a nonsymmetric displacement of the hydrogens in ammonia, where the symmetric-top symmetry of the ammonia
molecule will be destroyed. For most properties, the small
displacements we use in our numerical differentiation
scheme will ensure that the errors introduced are small.
However, for properties that depend on a well-defined
moment-of-inertia axis system—such as the rotational g
tensor—or in the case of anisotropic properties in highly
symmetric system, the errors may be substantial. To avoid
this problem, it is important to define a coordinate system
that is centered at the molecule and rotates with the molecule
rather than a space-fixed axis system. This is achieved by the
use of the Eckart axes.33–35
The use of the Eckart frame in the calculation of zeropoint vibrational corrections to molecular properties have
been discussed several times before 共see for instance Refs. 3
and 36兲, and we will only briefly recapitulate the main points
here. The conditions that fix the Eckart frame can be derived
by requiring that the angular momentum of the nuclei relative to the molecule-fixed axes vanishes when the nuclei are
in their equilibrium positions, that is, we require
兺K m K rKe ⫻rK ⫽0,
共17兲
where the summation runs over all the nuclei in the molecule, m K is the mass of nucleus K, and rKe and rK is the
position of the nucleus K at the equilibrium and instantaneous geometry, respectively. The Eckart frame is placed at
the instantaneous center-of-mass and oriented such that Eq.
共17兲 is fulfilled. It follows from Eq. 共17兲 through differentiation that
兺K m K rKe ⫻ṙK ⫽0.
共18兲
2671
Since the vibrational angular momentum of a molecule for a
rigid molecule close to its equilibrium geometry is to a good
approximation given by
Jvib⬇
兺K m K rKe ⫻ṙK ,
共19兲
the Eckart frame is thus seen to ensure that the coupling
between rotational and vibrational motion is small, which is
important for the estimation of vibrational contributions. For
more details we refer to Ref. 34.
III. COMPUTATIONAL ASPECTS
We apply the approach presented above to the study of
zero-point vibrational corrections 共ZPVCs兲 to a wide range
of different molecular properties for the ten-electron hydrides hydrogen fluoride, water, ammonia, and methane. Our
purpose is to investigate the importance of electron correlation in the calculation of zero-point vibrational corrections,
as well as the relative importance of ZPVCs in polyatomic
molecules. We will only consider the most abundant species
of each molecule, and we do thus not consider isotope effects. This also means that we will not consider properties
such as nuclear quadrupole coupling constants and spinrotation constants, since the nuclear magnetic and quadrupole moments are zero for most of the nuclei considered. We
will study the dipole moment, the quadrupole moment, static
and frequency-dependent polarizabilities, magnetizabilities,
rotational g tensors and nuclear shielding constants. We have
deliberately avoided the indirect spin–spin coupling constants, although feasible, because of their special basis-set
requirements.7
Finding a basis set that is suitable both for the calculation of properties dependent mainly on the near-nucleus region, such as the nuclear shieldings, and properties that depend mainly on the outer-valence regions, such as the
polarizabilities and magnetizabilities, is difficult. We have
chosen to use the ANO basis sets of Widmark and
co-workers,37 which have been designed to accurately treat
electronically excited states and ions of the atom, and which
have been demonstrated to accurately describe different
magnetic properties in a number of papers.6,38,39 The contraction we use is 关 6s5 p4d3 f 兴 for the second-row elements, and
关 5s4 p3d 兴 for hydrogen. This basis set should be large
enough to treat all the correlation effects recovered with our
complete active space self-consistent-field 共CASSCF兲 wave
function,40 and will also include enough diffuse functions to
describe the quadrupole moment and the polarizability with a
reasonable degree of accuracy.
To describe the electron correlation effects, we will use a
CASSCF wave function in which we keep the 1s-electrons
uncorrelated. All valence electrons are correlated in a space
consisting of 13 orbitals 共6 of A 1 , 3 of B 1 , 3 of B 2 and 1 of
A 2 symmetry in the C 2 v point group, and 9 in A ⬘ and 4 in A ⬙
in the C s point group兲. Since numerical differentiation with
respect to nuclear distortions by necessity will involve configurations of the nuclei that do not possess the full molecular point-group symmetry, all calculations have been done
without the use of point-group symmetry.
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2672
J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
Ruud, Åstrand, and Taylor
TABLE I. The numerical accuracy of the calculated nuclear shielding and dipole moment vibrational contribution from the second derivative of the molecular property 关Eq. 共16兲兴 in hydrogen fluoride. In atomic units.
Step length
␮z
F
␴ xx
F
␴ zz
␴F
H
␴ xx
H
␴ zz
␴H
0.075
0.05
0.025
0.01
0.0075
0.005
0.0025
0.001
0.000 777
0.000 775
0.000 774
0.000 774
0.000 774
0.000 774
0.000 774
0.000 773
⫺4.2726
⫺4.2736
⫺4.2787
⫺4.3139
⫺4.3464
⫺4.4395
⫺4.9421
⫺8.4603
0.0420
0.0419
0.0419
0.0419
0.0419
0.0419
0.0419
0.0419
⫺2.8344
⫺2.8351
⫺2.8385
⫺2.8120
⫺2.8836
⫺2.9457
⫺3.2808
⫺5.6262
0.2327
0.2324
0.2317
0.2276
0.2238
0.2129
0.1540
⫺0.2578
0.4423
0.4420
0.4419
0.4419
0.4419
0.4419
0.4419
0.4419
0.3026
0.3023
0.3018
0.2990
0.2965
0.2892
0.2500
⫺0.0737
Fit from Ref. 31
0.000 774
We do not claim that our active space will give highly
accurate results. In most cases, our choice of active space
gives results for the molecular properties of the molecules
investigated here that are of MP2 quality, although they often are somewhat closer to a CCSD共T兲 result than are the
MP2 results.41 For many of the properties investigated here,
in particular for ammonia and methane, we believe that our
results still are the most accurate to date, although an extension to more elaborate electron correlation schemes such as
the coupled-cluster method42 clearly would be of interest.
We will discuss the accuracy of our results in some more
detail in the next section.
The choice of molecules is the same as that of a recent
study of zero-point vibrational corrections to molecular electric properties by Russell and Spackman,27 and includes hydrogen fluoride, water, ammonia and methane. In addition to
studying the same electric properties as Ref. 27, we also
consider the polarizability at a finite frequency of 441.8 nm,
and several magnetic properties; the magnetizability, the rotational g tensor and the nuclear shielding constants. As Russell and Spackman, we will not consider the pure vibrational
contribution to the polarizability, although we note that it
may be of substantial magnitude.43
In conventional finite basis set calculations of magnetic
properties, the magnetic properties show an unphysical dependence on the choice of gauge origin.7 This artificial dependence on the gauge origin also means that the calculations will not be size extensive and thus not be suited for a
study involving nuclear distortions. This can be circumvented by using local gauge origins, and all the magnetic
properties presented here have been obtained using 共rotational兲 London orbitals,44,45 using the methods described in
several recent papers.45–48
All calculations reported here have been done with a
local version of the Dalton quantum chemistry program,49 in
which the scheme presented above for the calculation of
zero-point vibrational corrections have been implemented in
a black-box manner. The calculation proceeds in two steps.
共1兲 Determine the effective geometry as described
previously.30 Since we here only will consider the most
abundant isotopic species, we have only determined the
cubic force constants along the normal coordinates, calculating the third derivatives as second derivatives of
analytically calculated gradients. As recommended in
⫺2.8355
0.3023
our previous paper, a step length of 0.0075 bohr have
been used in the numerical differentiation.
共2兲 At the effective geometry, we calculate the molecular
properties and determine the contributions from 具 P (0)
2 典 eff
关see Eq. 共16兲兴. This is done by calculating the second
derivatives of the properties along the normal coordinates. For all properties we have used the Eckart frame
to determine the anisotropic components of the different
molecular properties.
To optimize the molecular geometry, we have used the
new first-order geometry optimizations routines implemented
in the Dalton program by Bakken and Helgaker,50 using the
model Hessian of Lindh et al.51
In determining the second derivatives of the properties
by numerical differentiation, numerical accuracy and stability is a key issue. The necessary requirements for the reliable
determination of the effective geometries were investigated
in detail in our previous paper,30 leading to the recommended
step length of 0.0075 bohr we have used here. However, a
similar analysis has not been undertaken for the second derivatives of the molecular properties. In Tables I and II we
have collected the vibrational contributions from the 具 P (0)
2 典 eff
term to the dipole moment and nuclear shieldings of hydrogen fluoride and water using a modest-sized ANO basis
关 4s3 p2d/3s2p 兴 and a SCF wave function. We have chosen
to focus on the nuclear shieldings since this property is, from
our experience1,6 as well as that of the literature,2,4 one of the
molecular properties that show the biggest geometry dependency. Indeed, looking at the results of Tables I and II, we
note that the results vary quite significantly, and that it is
probably not possible to get more than three digits accuracy
for the nuclear shielding derivatives, whereas this is not a
concern in the case of the dipole moment derivatives.
For hydrogen fluoride we may compare our results with
a fit to the property surface31 determined using the same
wave function as employed here. Although the results for
various step lengths scatter quite significantly, there appears
to be reasonable numerical stability for step lengths of 0.05
bohr and longer, and when comparing with the results of
Ref. 31, we have chosen to use, in the calculations presented
here, a step length of 0.05 bohr. We notice that this step is
somewhat smaller than that we have used in earlier studies of
diatomic molecules.6 We also note that for most of the other
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J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
Vibrational wave functions
2673
TABLE II. The numerical accuracy of the calculated nuclear shielding and dipole moment vibrational contribution from the second derivative of the molecular property 关Eq. 共16兲兴 in water. In atomic units.
Step length
␮x
O
␴ xx
␴ Oy y
O
␴ zz
H
␴ xx
␴ Hy y
H
␴ zz
0.075
0.05
0.025
0.01
0.0075
0.005
0.0025
0.001
⫺0.001 81
⫺0.001 81
⫺0.001 81
⫺0.001 81
⫺0.001 81
⫺0.001 81
⫺0.001 84
⫺0.002 00
⫺2.1331
⫺2.1344
⫺2.1473
⫺2.2433
⫺2.3326
⫺2.5868
⫺3.9594
⫺13.5757
⫺5.5063
⫺5.5046
⫺5.5051
⫺5.5087
⫺5.5113
⫺5.5197
⫺5.5640
⫺5.8914
⫺5.3318
⫺5.3366
⫺5.3606
⫺5.7036
⫺5.9201
⫺6.5224
⫺9.7446
⫺32.2943
0.0733
0.0729
0.0725
0.0715
0.0706
0.0681
0.0543
⫺0.0413
0.0830
0.0831
0.0840
0.0909
0.0973
0.1156
0.2144
0.9063
0.0366
0.0363
0.0349
0.0209
0.0062
⫺0.0360
⫺0.2655
⫺1.8716
properties 共not reported兲, the numerical stability is much
more satisfactory than that observed for nuclear shieldings
and the closely related nuclear spin–rotation constants.
IV. ZERO-POINT VIBRATIONAL CORRECTIONS
In this section we will present and discuss our results for
the zero-point vibrational corrections to the different molecular properties as obtained with the approach described in Sec.
II. Particular attention will be paid to the relative importance
(0)
(0)
the 具 P (0)
0 典 eff⫺具P0 典e and 具 P 2 典 eff contributions and the importance of electron correlation. Comparison with other studies of zero-point vibrational corrections will be made when
such data are available. These comparisons will be restricted
to correlated studies unless only SCF studies are available.
Comparison with experimental observations will be deferred
to the next section. We will not discuss the calculated force
fields or the vibrationally averaged geometries since this was
the topic of Ref. 30, but we report for completeness the most
important geometric parameters obtained with the SCF and
MCSCF wave functions in Tables III and IV, respectively.
A. Hydrogen fluoride
Since hydrogen fluoride is a diatomic molecule, the importance of vibrational corrections to most properties of this
molecule has been extensively studied in the
literature,2,27,52–55 also by ourselves using MCSCF wave
functions with much larger active spaces and better basis set
than that employed here.1,6 However, we include this molecule for completeness and because the availability of results
of an accuracy higher than that obtained here allows us to
assess the quality of our wave function and thus help in
judging the merit of our approach for the polyatomic molecules, in particular since some of the vibrational corrections
calculated for hydrogen fluoride have been obtained using
numerical methods.2,53,54
Our results, as well as the findings of other recent correlated studies, are collected in Table V. For the dipole and
quadrupole moments and the isotropic polarizability, our results are in very good agreement with the MP2 results of
Russell and Spackman, with differences of at most 6% in the
case of the quadrupole moment, and much less for both the
dipole moment and the polarizability. The differences are
probably dominated by the greater recovery of the correlation effects in our study, although it may, in the case of the
quadrupole moment, also indicate a possible inadequacy in
our basis set for the study of this property.
The correlation effects on the ZPVCs are substantial,
amounting to more than 15% for the dipole moment and the
isotropic polarizability, and clearly cannot be neglected if
accurate results are of interest. For the polarizability anisotropy, the agreement with Russell and Spackman is far from
satisfactory. Comparing with the SCF results of Russell and
Spackman, 0.1534 a.u., it appears that much of the difference
is due to the lack of any correlation effects in their MP2
result, a result that we find surprising for the polarizability
anisotropy. In contrast, our result is in excellent agreement
with the recent CCSD共T兲 result of Christiansen, Hättig, and
Gauss.56 We note that also Bishop and Cybulski have pre-
TABLE III. Geometric parameters for the molecules studied as obtained with the SCF wave function described
in the text. Only totally symmetric modes included. Frequencies and perturbation coefficients calculated at the
optimized geometry.
HF
H2 O
NH3
CH4
re 共Å兲
⬔ e 共degrees兲
0.897 154
0.939 747
106.324
0.997 812
108.158
1.081 447
reff 共Å兲
⬔ eff 共degrees兲
0.911 411
0.952 782
106.365
1.008 564
108.595
1.09360
␻ i 共cm⫺1 )
4474.60
4130.25
3691.06
3149.40
(1)a
0.1136/0.0309
⫺0.1421/⫺0.0213
⫺0.1427/⫺0.0167
⫺0.1667/0.0142
a (1)
/a
共atomic
units兲
1i
3i
␻ j 共cm⫺1 )
a 1(1)j /a 3(1)a
j 共atomic units兲
1747.73
1093.65
⫺0.0028/⫺0.0083 ⫺0.0471/⫺0.0320
a
The wave function coefficients of the first-order perturbed vibrational wave function, see Refs. 13 and 30.
Downloaded 22 Jan 2001 to 149.156.71.29. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html.
2674
J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
Ruud, Åstrand, and Taylor
TABLE IV. Geometric parameters for the four molecules studies as obtained with the MCSCF wave function
described in the text. Only totally symmetric modes included. Frequencies and perturbation coefficients calculated at the optimized geometry.
HF
H2 O
NH3
CH4
re 共Å兲
⬔ e 共degrees兲
0.915 565
0.957 962
104.807
1.013 446
106.926
1.092 281
reff 共Å兲
⬔ eff 共degrees兲
0.930 872
0.972 523
104.719
1.025 886
107.200
1.105 756
␻ i 共cm⫺1 )
4186.53
3848.86
(1)a
0.1180/0.0321
0.1524/0.0227
a(1)
1i /a3i 共atomic units兲
␻ j 共cm⫺1 )
a1(1)j /a3(1)a
j 共atomic units兲
3466.66
2990.73
⫺0.1573/⫺0.0178
0.1802/0.0141
1657.39
1050.64
0.0058/⫺0.0081 ⫺0.0273/⫺0.0302
a
The wave function coefficients of the first-order perturbed vibrational wave function, see Refs. 13 and 30.
sented calculations of vibrationally corrected polarizabilities
for HF at the MP2 level.53 However, they only quote their
total polarizability tensor and that calculated at the experimental equilibrium geometry. Their total value of 1.227 a.u.
is in reasonable agreement with the total values we obtain
共see Table X兲.
Turning our attention to the vibrational corrections to the
magnetic properties, we notice that the MP2 character of our
CASSCF wave function 共as discussed in Sec. III兲 gives
poorer agreement with the more elaborate MCSCF and
CCSD共T兲 results previously published,1,2,6 whereas fair
agreement is obtained with available MP2 data.54 Interestingly, correlation does not appear to play an important role
for the zero-point contributions to the magnetic properties,
with the exceptions of the magnetizability anisotropy and the
rotational g tensor. Indeed, it appears to be very difficult to
converge with respect to the treatment of electron correlation
effects for the rotational g tensor, as have been observed
previously as well.6,17,41
The change in geometry about which we do the perturbation expansion accounts for most of the zero-point corrections for the majority of properties considered, an exception
being the magnetizability anisotropy. However, there does
not appear to be any easily discernible pattern as to whether
(0)
(0)
the two contributions 具 P (0)
0 典 eff⫺具P0 典e and 具 P 2 典 eff enhance
TABLE V. The contributions to the zero-point vibrational contributions to the properties of hydrogen fluoride. Basis sets and wave functions described in the
text. All properties reported in atomic units except the magnetizability which is reported in units of 10 ⫺30 JT⫺2 .
SCF
Literature
具 P (0)
2 典 eff
具 P 典 0,0
(0)
具 P (0)
0 典 eff⫺ 具 P 0 典 e
⫺0.0109
⫺0.0008
⫺0.0117
⫺0.0099
0.0000
⫺0.0099
⌰ zz
␣
0.0454
0.0671
0.0084
0.0181
0.0539
0.0852
0.0481
0.0815
0.0067
0.0187
0.0547
0.1002
⌬␣
0.1142
0.0464
0.1607
0.1262
0.0504
0.1767
0.0694
0.0197
0.0891
0.0848
0.0207
0.1055
Property
␮z
␣ 441.8
⌬␣
nm
441.8 nm
␰
⌬␰
0.1183
0.0502
0.1685
0.1308
具 P (0)
2 典 eff
0.0547
具 P 典 0,0
具 P 典 0,0
⫺0.0103 共MP227兲
⫺0.0092 共ACCD52兲
⫺0.009 共CCSD共T兲56兲
0.0579 共MP227兲
0.1014 共MP227兲
0.09 共CCSD共T兲56兲
0.1573 共MP227兲
0.18 共CCSD共T兲56兲
0.1856
⫺0.48
0.25
⫺0.23
⫺0.53
0.30
⫺0.22
0.06
⫺0.30
⫺0.24
0.02
⫺0.36
⫺0.34
g
⫺0.0067
⫺0.0019
⫺0.0086
⫺0.0080
⫺0.0027
⫺0.0107
␴F
⫺5.75
⫺2.91
⫺8.66
⫺5.84
⫺2.91
⫺8.75
8.54
4.43
12.97
8.67
4.42
13.09
␴H
⫺0.64
0.29
⫺0.35
⫺0.61
0.30
⫺0.31
⌬␴H
⫺0.81
0.23
⫺0.59
⫺0.86
0.23
⫺0.63
⌬␴F
a
MCSCF
(0)
具 P (0)
0 典 eff⫺ 具 P 0 典 e
⫺0.18 共MCSCF6兲
⫺0.32 共MP254兲
⫺0.41 共MCSCF6兲
⫺0.36 共MP254兲
⫺0.0122 共MCSCF6兲
⫺0.0095 共MP254兲
⫺10.3 共MCSCF1兲
⫺10.0 共CCSD共T兲2兲
15.3 共MCSCF1兲
15.58 共CCSD共T兲2兲a
⫺0.48 共MCSCF1兲
⫺0.323 共CCSD共T兲2兲
⫺0.68 共MCSCF1兲
⫺0.75 共CCSD共T兲2兲
Estimated for a temperature of 300 K.
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J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
Vibrational wave functions
2675
TABLE VI. The contributions to the zero-point vibrational contributions to the properties of water. The water molecule is placed in the xy-plane, with the
x-axis along the dipole axis of the molecule. Basis sets and wave functions described in the text. All properties reported in atomic units except the
magnetizability which is reported in units of 10 ⫺30 JT⫺2 .
SCF
MCSCF
(0)
具 P (0)
0 典 eff⫺ 具 P 0 典 e
具 P (0)
2 典 eff
具 P 典 0,0
⫺0.0039
⫺0.0064
0.0386
⫺0.0322
0.1561
0.2576
0.0627
0.0019
0.0046
0.0068
⫺0.0114
0.0971
0.1514
0.0220
nm
0.1643
nm
␣ 441.8
yy
441.8
␣ zz
Literature
(0)
具 P (0)
0 典 eff⫺ 具 P 0 典 e
具 P (0)
2 典 eff
具 P 典 0,0
⫺0.0020
⫺0.0018
0.0454
⫺0.0436
0.2533
0.4090
0.0846
⫺0.0033
0.0006
0.0351
⫺0.0357
0.1952
0.2909
0.0979
0.0030
0.0051
0.0047
⫺0.0098
0.0939
0.1437
0.0236
⫺0.0003
0.0057
0.0399
⫺0.0455
0.2891
0.4346
0.1214
0.1055
0.2698
0.2069
0.1035
0.3104
0.2685
0.1644
0.4329
0.3044
0.1573
0.4617
0.0663
0.0241
0.0904
0.1075
0.0278
0.1353
␰
⌬␰1
⌬␰2
g xx
gyy
g zz
␴O
⫺1.11
⫺0.14
0.32
⫺0.0055
⫺0.0096
⫺0.0050
⫺6.51
0.10
⫺0.37
0.22
⫺0.0057
⫺0.0115
0.0003
⫺4.54
⫺1.02
0.23
0.54
⫺0.0112
⫺0.0211
⫺0.0047
⫺11.05
⫺1.22
⫺0.16
0.30
⫺0.0069
⫺0.0119
⫺0.0066
⫺6.59
0.23
0.29
0.25
⫺0.0065
⫺0.0133
⫺0.0011
⫺4.34
⫺1.00
0.13
0.55
⫺0.0134
⫺0.0252
⫺0.0077
⫺10.93
⌬␴O
␴H
0.80
⫺0.56
⫺1.66
0.06
⫺0.86
⫺0.50
0.02
⫺0.57
⫺1.98
0.08
⫺1.96
⫺0.49
⌬␴H
⫺0.70
⫺0.22
⫺0.91
⫺0.78
⫺0.21
⫺0.99
Property
␮x
⌰ xx
⌰yy
⌰ zz
␣ xx
␣yy
␣ zz
441.8
␣ xx
nm
or cancel each other. Electron correlation does, however, appear to affect both contributions to a similar extent.
B. Water
It appears that only a limited number of correlated studies have been undertaken on vibrational corrections to the
molecular properties of the water molecule, and our comparison will therefore be restricted to a comparison with the MP2
results of Russell and Spackman for the electric properties,27
and results previously obtained by one of us using larger
multiconfigurational SCF expansions than that employed
here for the magnetic properties3,17 and finally a CASSCF
study of the nuclear shieldings by Wigglesworth et al.57 using the same active space we use here, but employing an
experimental force field. All these result are collected in
Table VI.
Considering first the electric properties, we note that the
differences to the results of Russell and Spackman for this
molecule are much larger than were observed in the case of
hydrogen fluoride. There are several possible reasons for
this, and it is difficult to properly discriminate between these.
One difference is the treatment of electron correlation, and
this is undoubtedly an important factor since the agreement
between our SCF results and those of Russell and Spackman
in general is better than at the correlated level. This applies
in particular to the dipole moment, where our SCF zero-point
vibrational correction to the dipole moment 共⫺0.0020 a.u.兲 is
in almost perfect agreement with the SCF correction found
by Russell and Spackman 共⫺0.0021 a.u.兲.
具 P 典 0,0
⫺0.0011
0.0068
0.0440
⫺0.0508
0.3070
0.4236
0.1458
共MP227兲
共MP227兲
共MP227兲
共MP227兲
共MP227兲
共MP227兲
共MP227兲
⫺0.99 共MCSCF17兲
0.10 共MCSCF17兲
0.30 共MCSCF17兲
⫺0.019共MCSCF17兲
⫺0.027共MCSCF17兲
⫺0.014共MCSCF17兲
⫺11.68 共MCSCF3兲
⫺9.86 共MCSCF57兲
⫺2.35 共MCSCF3兲
⫺0.523 共MCSCF3兲
⫺0.484共MCSCF57兲
⫺1.04 共MCSCF3兲
The second source of differences arises from the difference in choice of basis set, as already discussed in the previous subsection. Finally, the different perturbation expansions may differ in their ability to recover the zero-point
vibrational corrections. As regards this latter source of errors
共affecting the comparison both at the SCF and correlated
level兲, we demonstrated in a recent paper that our approach,
including the anharmonicity of the potential by expanding
around an effective geometry, was able to recover a greater
portion of the exact zero-point vibrational correction in diatomic molecules than the approach used by Russell and
Spackman in comparison to the exact result obtained using
numerical integration.31 In order to distinguish between basis
set differences and differences in the convergence of the perturbation expansion of the vibrational wave function, we
have collected in Table VII the vibrational corrections obtained at the SCF level for the electric properties using the
ANO basis described in Sec. III and the polarized basis set of
Sadlej58,59 which was used in Ref. 27. Included also are the
results of Russell and Spackman, and we can see that in most
cases the change of basis set leads to a greater change in the
ZPVC than does the change of perturbation expansion. The
exception is ␣ y y where the basis set changes ␣ y y only to a
very small extent, whereas the two perturbation expansions
differ by almost 4%. The y-direction is the in-plane component perpendicular to the dipole axis. Since for diatomic
molecules we have demonstrated that our perturbation expansion is able to recover more of the exact 共numerical兲
ZPVC than does the expansion around the equilibrium geometry, we find it reasonable to assume that such differences
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2676
J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
TABLE VII. Comparison of zero-point vibrational corrections to the properties of water obtained at the SCF level with different basis sets and different perturbation expansions of the vibrational wave function 共see text兲.
The water molecule is placed in the xy-plane, with the x-axis along the
dipole axis of the molecule. All properties reported in atomic units.
This work
Property
␮x
⌰ xx
⌰yy
⌰ zz
␣ xx
␣yy
␣ zz
Ref. 27
ANO basis
Sadlej basis
Sadlej basis
⫺0.0020
⫺0.0018
0.0454
⫺0.0436
0.2533
0.4090
0.0846
⫺0.0021
⫺0.0023
0.0481
⫺0.0458
0.2591
0.4121
0.0887
⫺0.0021
⫺0.0021
0.0477
⫺0.0456
0.2544
0.3982
0.0887
in ZPVC recovery will be enhanced as the number of vibrational modes increases, and thus believe that the differences
in the SCF results obtained by Russell and Spackman and
those obtained in this study are due to a more complete recovery of the ZPVCs in this work.
Turning to the magnetic properties, we make the same
observations as for hydrogen fluoride: The zero-point vibrational correction to the isotropic magnetizability is small and
independent of electron correlation whereas the magnetizability anisotropy shows a much stronger dependence on
electron correlation. Electron correlation is also important
for the rotational g tensor, and we in particular note the difficulty in recovering the electron correlation effects, as for
instance only about half of the zero-point vibrational contribution to g zz 共the out-of-plane component兲 is recovered compared to the bigger MCSCF results previously published.17
Before discussing the results for the nuclear shielding,
we must make one important remark. Our results for the
oxygen shielding refers to an isotope of oxygen 16O that does
not have a magnetic moment and thus cannot be observed in
a NMR experiment. Our results cannot therefore be directly
compared to experiment, nor to the other theoretical studies
which both studied the isotopic species H2 17O.3,57 However,
these differences should, considering the relatively minor
difference in the total mass of the system, be rather small.
Interestingly, only the shielding anisotropy of the oxygen atom shows any significant correlation effect on the
zero-point vibrational corrections, although the importance
of electron correlation on the oxygen shielding itself is well
established.48,60–62 As expected, our zero-point vibrational
corrections are in good agreement with the CASSCF results
of Wigglesworth et al.57 since we use the same wave function, the differences being due mainly to the difference in
force field and perturbation expansion of the vibrational
wave function and to some extent also because of the different isotopic species investigated. It is also worth noticing
that our CASSCF wave function is much more successful in
recovering the correlation effects on the ZPVCs to the
nuclear shieldings than was the case for the ZPVC to the
rotational g tensor.
As regards the relative importance of the shift in the
geometry and the expectation value of the second-derivative
of the property, much of the same trend observed for hydro-
Ruud, Åstrand, and Taylor
gen fluoride applies to water as well; in most cases the shift
in the geometry dominates, once again the magnetizability
anisotropy being an exception, although the oxygen shielding anisotropy and two of the rotational g tensor components
also deviate from this behavior. Correlation is in most cases
important, although an interesting exception to this appears
to be the isotropic hydrogen shielding. As we will see also
for ammonia and methane, the ZPVC to the isotropic hydrogen shielding appears to be independent of electron correlation for all four molecules studied here, which is in accordance with the observations made for the isotropic hydrogen
shielding itself.63 Although further studies are needed, it
seems like the main correction to SCF hydrogen shieldings
are vibrational corrections, and that these may be accurately
determined from SCF calculations alone.
C. Ammonia
There have been some previous investigations of vibrational corrections to the molecular properties of ammonia, in
particular the polarizability, much of this spurred by the special umbrella-inversion mode of the ammonia molecule.
Since here we only consider the vibrational ground state, we
expect our approach to be able to accurately model also this
highly anharmonic vibrational mode of the molecule, although errors from the truncation of the perturbation expansion of the vibrational wave function may be somewhat more
pronounced for this molecule than the other molecules of this
study.
We have collected our results in Table VIII together
with other theoretical results. An important observation to
make is that for this molecule the harmonic oscillator contribution to the ZPVC often is of equal or greater importance
than the shift in the geometry for the electric properties,
whereas the shift in geometry dominates for most of the
magnetic properties.
Comparing our results for the dipole and quadrupole moments and the static polarizability with the results of Russell
and Spackman, the agreement is not very satisfactory. Part of
the reason for these discrepancies is undoubtedly due to the
treatment of electron correlation, since the SCF ZPVC corrections for the dipole moment and the isotropic polarizability are in very good agreement, whereas the agreement is less
satisfactory for the polarizability anisotropy and the quadrupole moment, indicating also some potential differences in
basis set quality. Comparing all the theoretical results for the
isotropic and anisotropic polarizability, the scatter in the results is striking. Russell and Spackman indicated that the
lack of agreement with the MP2 results of Wormer et al.64
may be due to their neglect of all vibrational modes but the
two totally symmetric modes. However, they could not explain the large difference to the results of Špirko, Jensen, and
Jo” rgensen for the isotropic polarizability.26 Our ZPVC for
the isotropic polarizability is even further away from the results of Špirko et al. and does not help resolve this discrepancy. We may note that the active space used in this study is
much larger than that used in Ref. 26. Clearly, further studies
of the ZPVC to the polarizability of ammonia are needed.
In contrast, both our SCF and MCSCF ZPVCs to the
nuclear shieldings are in excellent agreement with the recent
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J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
Vibrational wave functions
2677
TABLE VIII. The contributions to the zero-point vibrational contributions to the properties of ammonia. Basis sets and wave functions described in the text.
Magnetizabilities in units of 10⫺30 JT⫺2, other properties in atomic units.
SCF
Property
(0)
具 P (0)
0 典 eff⫺ 具 P 0 典 e
MCSCF
具 P (0)
2 典 eff
具 P 典 0,0
(0)
具 P (0)
0 典 eff⫺ 具 P 0 典 e
Literature
具 P (0)
2 典 eff
具 P 典 0,0
具 P 典 0,0
0.0160 共MP227兲
⫺0.0409 共MP227兲
0.5715 共MP227兲
0.801共MCSCF26兲
0.189 共MP264兲
⫺0.1937 共MP227兲
⫺0.193共MCSCF26兲
⫺0.045 共MP264兲
␮z
⌰ zz
␣
0.0107
⫺0.0591
0.2351
0.0097
0.0126
0.2591
0.0204
⫺0.0465
0.4941
0.0077
⫺0.0506
0.2805
0.0105
0.0128
0.2457
0.0181
⫺0.0378
0.5262
⌬␣
⫺0.1314
⫺0.1586
⫺0.2900
⫺0.1144
⫺0.1182
⫺0.2325
␣ 441.8 nm
⌬ ␣ 441.8 nm
␰
⌬␰
g⬜
g储
␴N
⌬␴N
␴H
⌬␴H
0.2505
⫺0.1275
⫺1.56
0.30
⫺0.0026
⫺0.0039
⫺4.05
0.47
⫺0.48
⫺0.41
0.2826
⫺0.1685
⫺0.90
⫺0.85
⫺0.0070
0.0005
⫺3.16
3.76
⫺0.12
⫺0.38
0.5330
⫺0.2960
⫺2.46
⫺0.55
⫺0.0097
⫺0.0034
⫺7.21
4.23
⫺0.60
⫺0.79
0.3036
⫺0.0998
⫺1.64
0.17
⫺0.0053
⫺0.0050
⫺4.15
0.62
⫺0.50
⫺0.47
0.2732
⫺0.1168
⫺0.65
⫺0.78
⫺0.0082
⫺0.0012
⫺2.65
3.51
⫺0.10
⫺0.40
0.5769
⫺0.2166
⫺2.29
⫺0.60
⫺0.0135
⫺0.0062
⫺6.80
4.13
⫺0.61
⫺0.87
SCF results of Fukui et al.,4 who also used London atomic
orbitals to ensure gauge origin independence of their results.
We would expect that the use of London atomic orbitals
would ensure that the results for the nuclear shieldings are
less dependent on the basis set than are the polarizability.
Still, this cannot explain the good agreement observed for
the nuclear shieldings which stands in marked contrast to the
scatter in the results for the polarizability. As was the case
for hydrogen fluoride, the ZPVC to the isotropic and anisotropic shieldings are in most cases independent of electron
correlation, the exception being the hydrogen shielding
anisotropy.
It is worth noticing that we experienced some problems
with numerical stability in the calculation of the second derivatives of the nitrogen shielding, and we had to increase the
step length to 0.115 bohr to ensure numerically stable results
for the nitrogen shielding. All the other results were obtained
with the usual step length of 0.05 bohr.
Considering finally the magnetizability and rotational g
tensor elements, the agreement with the work of Sauer,
Špirko and Oddershede65 is modest at best. There are several
⫺6.25
⫺0.66
⫺0.0128
⫺0.0149
⫺7.0
5.1
⫺0.56
共RPA65兲
共RPA65兲
共RPA65兲
共RPA65兲
共SCF4兲
共SCF4
共SCF4兲
reasons for these discrepancies. In particular, perturbation
dependent basis sets were not used in the calculation of the
magnetizabilities and rotational g tensors, and considering
the strong basis set dependence of these properties47,45,66 this
is likely to be an important source of errors in their investigation. Furthermore, only the symmetric stretching and inversion modes were considered in their work, an approximation which, based on the comparison with Wormer et al. for
the polarizability, may not be well justified.
D. Methane
Turning our attention to methane 共see Table IX兲, we note
that hardly any correlated studies appear to have been undertaken for the properties of this molecule. The exceptions are
the indirect spin–spin coupling constants, for which a lot of
work has been done on rovibrational corrections to these
properties.21,67 Our result for the static polarizability is in
reasonable agreement with the MP2 results of Russell and
Spackman.27 In contrast to ammonia, our SCF ZPVC to the
shieldings compare less favorable with the results of Fukui
TABLE IX. The contributions to the zero-point vibrational contributions to the properties of methane. Basis
sets and wave functions described in the text. All properties reported in atomic units except the magnetizability
which is reported in units of 10 ⫺30 JT⫺2 .
SCF
Property
␣
␣ 441.8
␰
g
␴C
␴H
⌬␴H
nm
(0)
具 P (0)
0 典 eff⫺ 具 P 0 典 e
具 P (0)
2 典 eff
MCSCF
具 P 典 0,0
(0)
具 P (0)
0 典 eff⫺ 具 P 0 典 e
0.3754
0.3968
⫺1.75
⫺0.0041
⫺2.55
0.5257
0.9011
0.5747
0.9715
⫺2.03
⫺3.78
⫺0.0020 ⫺0.0062
⫺0.53
⫺3.07
0.3895
0.4123
⫺1.76
⫺0.0052
⫺2.69
⫺0.42
⫺0.22
⫺0.18
⫺0.15
⫺0.44
⫺0.26
⫺0.60
⫺0.36
具 P (0)
2 典 eff
Literature
具 P 典 0,0
具 P 典 0,0
0.4487
0.8382
0.9040共MP227兲
0.4917
0.9040
⫺1.59
⫺3.35
⫺6.38 共SCF20兲
⫺0.0034 ⫺0.0086
⫺0.51
⫺3.20
⫺3.591 共SCF20兲
⫺4.7 共SCF4兲
⫺0.16
⫺0.60
⫺0.87 共SCF4兲
⫺0.11
⫺0.37
Downloaded 22 Jan 2001 to 149.156.71.29. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html.
2678
J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
Ruud, Åstrand, and Taylor
TABLE X. Zero-point vibrationally corrected molecular properties of hydrogen fluoride. Basis sets and wave functions described in the text. All properties
reported in atomic units except the magnetizability which is reported in units of 10 ⫺30 JT⫺2 .
SCF
Property
␮z
⌰ zz
␣
⌬␣
␣ 441.8 nm
⌬ ␣ 441.8 nm
␰
⌬␰
g
␴F
⌬␴F
␴H
⌬␴H
MCSCF
P expt
P expt⫹ 具 P 典 0,0
P expt
P expt⫹ 具 P 典 0,0
⫺0.7562
1.7272
4.8443
1.3564
4.9171
1.3781
⫺172.60
8.83
0.7626
⫺0.7679
1.7811
4.9295
1.5171
5.0062
1.5466
⫺172.83
8.59
0.7540
⫺0.7220
1.7243
5.3259
1.2602
5.4193
1.2720
⫺176.38
8.65
0.7584
⫺0.7319
1.7790
5.4261
1.4369
5.5248
1.4576
⫺176.60
8.31
0.7477
414.26
101.08
28.22
23.79
405.60
114.05
27.87
23.20
423.17
87.39
28.66
23.38
414.42
100.48
28.35
22.75
Literature
⫺0.707 67(⫾0.000 14兲a, ⫺0.718 47b
1.75(⫾0.02兲c
5.60d
1.31(⫾0.13兲e
⫺171f
8.92g
0.7392(⫾0.0050兲,0.741 04(⫾0.000 15兲,
0.741 599(⫾0.000 005兲h
410(⫾6兲i
110(⫾4兲j
28.5(⫾0.2兲h
19(⫾3兲i
Molecular beam electric resonance experiment for v ⫽0,J⫽1 共Ref. 69兲 scaled by 0.984 70 to account for the more recent OCS dipole moment measurement
共Ref. 89兲.
b
Molecular beam experiment from Ref. 70.
c
Molecular beam electric resonance quadrupole moment for v ⫽0,J⫽1 共Ref. 71兲.
d
Extrapolated from refractive index data in Ref. 72.
e
Static molecular beam electric resonance result 共Ref. 70兲 corrected for the pure vibrational contribution of 0.18 a.u. 共Ref. 56兲.
f
Quoted in Ref. 90.
g
Molecular-beam experiments from Ref. 71.
h
Molecular beam results taken from, respectively, Refs. 91, 71, and 73.
i
Derived from experimental spin–rotation constants in Ref. 92.
j
Molecular beam experiments from Ref. 73.
a
et al. for methane,4 and the reason for this is not clear to us.
We note, however, that the surfaces used by Fukui are incomplete as they have not included all shielding derivatives
contributing to the hydrogen shieldings, whereas all derivatives have been included for the heavy-atom shieldings.68
Our SCF ZPVC to the carbon shielding is in fair agreement
with the results of Raynes et al.20 This is not the case for the
magnetizability, although this is most likely due to the lack
of perturbation-dependent basis sets in Ref. 20.
V. COMPARISON WITH EXPERIMENT
From the discussion in the previous section, where it was
seen that even small changes in the molecular geometry may
lead to substantial changes in the molecular properties, it is
clear that for an accurate comparison with experiment, good
equilibrium geometries are required. This has also been
noted in other investigations, see for instance the ‘‘Note
added in proof’’ in Ref. 3, as well as the prediction of vibrationally averaged geometries in our previous paper.30 Furthermore, in a recent work where the rovibrationally averaged fluorine shielding of hydrogen fluoride was refined, it
was found that the improvement was gained from a more
accurate description of the potential surface rather than the
shielding surface.6 Whereas we believe our wave functions
provide reasonably accurate molecular properties for a given
molecular geometry,7 the molecular geometries themselves
are too inaccurate.30 This applies in particular to strongly
geometry dependent properties such as the nuclear shielding.
For this reason we follow the approach used by Russell
and Spackman,27 in which we combine our zero-point vibrational corrections in Tables V—IX with the molecular prop-
erties calculated at the experimentally determined equilibrium geometries. The geometries used are the same as those
used by Russell and Spackman and can be found in the footnote to Table 4 in their paper.27 Since we use this combination of experimental data with pure ab initio estimates, we
will in this section restrict ourselves to a comparison with
experimental observations.
A. Hydrogen fluoride
We have collected our SCF and MCSCF results for HF
at the experimental equilibrium geometry along with the estimated vibrationally corrected results 共adding the vibrational
corrections described in the previous section to the value
calculated at the experimental equilibrium geometry兲 in
Table X. The agreement with experiment is in most cases
satisfactory, in particular considering the approximations
made in the treatment of electron correlation in this study.
Still, the rather poor agreement with experiment for the dipole moment69,70 is somewhat discouraging. Comparing our
result with the recent accurate results of Christiansen, Hättig,
and Gauss56 it becomes apparent that almost all the discrepancy is due to the overestimation of the equilibrium dipole
moment. For the other electric properties, the agreement with
the experimentally derived numbers70–72 are in general good,
with the zero-point vibrational corrections bringing the results closer, or to the other side of, the experimental number.
Turning to the magnetic properties, only the experimental results for the rotational g tensor can be considered to be
of high accuracy, in particular the latest result of Bass, DeLeon, and Muenter.73 The other data have either been derived from experimental data using certain theoretically de-
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J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
Vibrational wave functions
2679
TABLE XI. Zero-point vibrationally corrected molecular properties of water. The water molecule is placed in the xy-plane, with the x-axis along the dipole
axis of the molecule. Basis sets and wave functions described in the text. All properties reported in atomic units except the magnetizability which is reported
in units of 10⫺30 JT⫺2 .
SCF
Property
␮x
⌰ xx
⌰yy
⌰ zz
␣ xx
␣yy
␣ zz
441.8
␣ xx
␣ 441.8
yy
441.8
␣ zz
␰
⌬␰1
⌬␰2
g xx
gyy
g zz
␴O
⌬␴O
␴H
⌬␴H
nm
nm
nm
MCSCF
P expt
P expt⫹ 具 P 典 0,0
P expt
P expt⫹ 具 P 典 0,0
⫺0.7789
⫺0.1046
1.8961
⫺1.7915
8.4563
9.1726
7.8094
8.6573
9.3580
8.0453
⫺231.09
⫺2.36
3.90
0.7335
0.6820
0.6625
328.15
57.16
30.58
20.82
⫺0.7809
⫺0.1064
1.9415
⫺1.8351
8.7096
9.5816
7.8940
8.9271
9.7909
8.1357
⫺232.11
⫺2.13
4.44
0.7223
0.6609
0.6578
317.10
56.30
30.08
19.91
⫺0.7447
⫺0.1043
1.9234
⫺1.8191
9.2260
9.6554
8.9243
9.4823
9.8654
9.2811
⫺235.17
⫺1.90
3.21
0.7314
0.6750
0.6566
341.76
48.71
30.62
20.76
⫺0.7450
⫺0.0986
1.9633
⫺1.8646
9.5151
10.0900
9.0457
9.7927
10.3271
9.4164
⫺236.17
⫺1.77
3.76
0.7180
0.6498
0.6489
330.83
46.75
30.13
19.77
Literature
⫺0.71856(⫾0.000 16兲a
⫺0.097(⫾0.022兲b
1.955(⫾0.015兲b
⫺1.859(⫾0.015兲b
9.91(⫾0.02兲c
10.31(⫾0.08兲c
9.55(⫾0.08兲c
⫺1.7(⫾0.4兲d
3.4(⫾0.2兲d
0.718(⫾0.007兲/0.7145(⫾0.002兲e
0.657(⫾0.001兲/0.6650(⫾0.002兲e
0.645(⫾0.006兲/0.6465(⫾0.002兲e
344.0(⫾17.2兲f
30.052(⫾0.015兲g
28.5(⫾0.1兲/34.2(⫾0.1兲/34(⫾4兲h
Molecular beam electric resonance experiment from Ref. 74 scaled by 0.984 79 to account for a more recent OCS dipole moment measurement 共Ref. 89兲.
Molecular beam maser experiment from Ref. 75.
c
Polarizability components derived from rotational Raman, Rayleigh light scattering and refractive index data at 514.5 nm 共Ref. 93兲.
d
Data recorded for D16
2 O in beam-maser experiments in Ref. 75.
e
The first number is from a beam-maser experiment reported in Ref. 75, the second from a microwave Zeeman experiment of Ref. 76. However, both results
are in part derived from data for D2 O using a relation that appear not to be valid for a vibrating molecule,17,79 see discussion in text.
f
Gas-phase measurement from Ref. 77. However, it has been proposed that a more appropriate value is 324.0(⫾1.5兲 共Ref. 3兲.
g
Derived from gas- and liquid-phase data in Ref. 78.
h
All these data have been obtained from ice and are included here mainly for completeness. The data are taken from Refs. 94–96, respectively.
a
b
termined numbers, as for instance nuclear shieldings have
been derived from nuclear spin–rotation constants using theoretical estimates of the diamagnetic shielding, or have been
based on less accurate data, as is the case for the magnetizability anisotropy.71 The agreement with experiment is in
general, however, reasonably good, and almost within the
experimental error bars. However, for the shieldings and
magnetizabilities, the purely theoretical data in Refs. 2 and 6
are likely to be more accurate than the experimental numbers.
B. Water
As was the case for the hydrogen fluoride molecule, our
vibrationally corrected dipole moment for water is in very
poor agreement with experiment74 共see Table XI兲. Although
there is some disagreement between the MP2 result of Russell and Spackman and our MCSCF result for the ZPVC
correction to the dipole moment, the ZPVC is in both cases
small, and the observed deviation to experiment is thus also
here due to an overestimation of the equilibrium dipole moment. In contrast, the agreement with experiment for the
quadrupole moment and the polarizability 共when considering
the dispersion effects兲 is striking. Since it is not clear that our
choice of basis set is optimal for these two properties, it may
be that this excellent agreement is due to a fortuitous cancellation of errors from basis set incompleteness and incomplete
treatment of electron correlation.
Turning to the magnetic properties, the situation is much
the same as for hydrogen fluoride, most of the experimental
magnetic properties are derived quantities involving some
theoretical analysis.75–77 Indeed, it has recently been proposed on the basis of theoretical calculations which included
rovibrational corrections3 that the oxygen nuclear shielding
constant of water is 324.0(⫾1.5兲 ppm, rather than the originally derived result of 344.0(⫾17.2兲 ppm.77 Our result does
not favor either of these results. However, the ZPVC to the
isotropic oxygen shielding is remarkably independent of
electron correlation, and once again the discrepancy, in this
case with respect to the theoretical estimate,3 is due to the
poorly estimated equilibrium value. For the isotropic hydrogen shielding, where the experimental result is of good
quality,78 our results are in good agreement with experiment.
This is due to the fact that both the ZPVC and the equilibrium value are correlation independent. The lack of electron
correlation effects for hydrogen shieldings gives some hope
that it should be possible to accurately determine ZPVCs to
hydrogen shieldings in larger molecules. This is an important
finding because the ZPVCs we have found for the isotropic
hydrogen shieldings in HF, H2 O, NH3 , and CH4 is in the
range ⫺0.30–⫺0.60 ppm—that is, almost 5% of the normal
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2680
J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
Ruud, Åstrand, and Taylor
TABLE XII. Zero-point vibrationally corrected molecular properties of ammonia. Basis sets and wave functions described in the text. All properties reported
in atomic units except the magnetizability which is reported in units of 10⫺30 JT⫺2 .
SCF
Property
␮z
⌰ zz
␣
⌬␣
␣ 441.8 nm
⌬ ␣ 441.8 nm
␰
⌬␰
g⬜
g储
␴N
⌬␴N
␴H
⌬␴H
MCSCF
P expt
P expt⫹ 具 P 典 0,0
Pe
P expt⫹ 具 P 典 0,0
⫺0.6367
⫺2.1429
12.9089
0.4431
13.3544
0.7615
⫺287.57
16.76
0.5793
0.5058
262.03
⫺39.21
31.55
16.24
⫺0.6163
⫺2.1894
13.4030
0.1531
13.8874
0.4655
⫺290.03
16.21
0.5696
0.5024
254.82
⫺34.98
30.95
15.45
⫺0.6165
⫺2.2044
13.8027
1.6095
14.3891
2.2097
⫺291.19
16.81
0.5792
0.5120
273.79
⫺42.22
31.39
16.48
⫺0.5984
⫺2.2422
14.3289
1.3770
14.9660
1.9931
⫺293.48
16.21
0.5657
0.5058
266.99
⫺38.09
30.78
15.61
Literature
⫺0.5791(⫾0.0003兲a
⫺2.17(⫾0.06兲b
14.573,14.58c
1.75(⫾0.17兲,1.90(⫾0.07兲d
⫺271(⫾13兲e
⫺16.4(⫾1.0兲f
0.5654(⫾0.0007兲g
0.5024(⫾0.0005兲g
264.54h
31.2(⫾1.0兲h
Laser-microwave double resonance measurements 共Ref. 80兲.
Molecular beam maser Zeeman result 共Ref. 97兲.
c
Quadratic extrapolation of refractive data using, respectively, five wavelengths in the range 325–633 nm 共Ref. 98兲 and eight wavelengths in the range
480–671 nm 共Ref. 99兲.
d
Rayleigh light-scattering experiments at 632.8 nm in Refs. 100 and 101, respectively.
e
Gas-phase measurement by Barter et al. 共Ref. 83兲. However, it has been proposed on the basis of theoretical calculations 共Ref. 85兲 that these results contain
a calibration error and should be scaled by 1.07, giving a magnetizability of ⫺290(⫾13)⫻10⫺30 JT⫺2 .
f
Microwave spectroscopy measurements from Ref. 102.
g
Microwave spectroscopy measurements from Ref. 25.
h
Derived from spin–rotation data obtained in microwave spectroscopy experiments in Ref. 84.
a
b
hydrogen shielding range, an effect that clearly cannot be
neglected, although it may be that the vibrational effects partially cancel in the relative chemical shifts that most often
are of interest.
For the rotational g tensor, the agreement with
experiment75,76 appears to be very good. However, as discussed in a recent paper,17 the experimental numbers were in
part derived from data obtained for D2 O, assuming that a
relationship between the change in the center of mass of the
molecule from an isotopic substitution and the molecular dipole moment holds. However, this relationship is valid only
for a rigid molecule, and the errors introduced by neglecting
the effects of molecular vibrations may be large.17,79 Our
calculations are unable to recover enough of the correlation
effects to be able to reproduce the probably more accurate
theoretical values of g xx ⫽0.640(⫾0.003), g y y ⫽0.709
(⫾0.003) and g zz ⫽0.637(⫾0.001).17
C. Ammonia
As observed also for the previous two molecules, the
dipole moment of ammonia is in unsatisfactory agreement
with experiment80 共see Table XII兲. We believe that most of
this discrepancy is due to the poor equilibrium dipole moment, and if we instead combine our ZPVC with the recent
accurate CCSD共T兲 value of Halkier and Taylor81 共⫺0.603
a.u.兲, we get a vibrationally averaged result 共⫺0.5850 a.u.兲
which is much closer to experiment although still not within
the experimental error bars. It is worth noticing that by combining our ZPVC with the experimentally derived equilibrium dipole moment of ⫺0.614,82 the agreement with the
experimental result for the vibrationally averaged dipole mo-
ment is much less satisfactory, supporting the proposal by
Halkier and Taylor that the experimentally deduced equilibrium dipole moment of ammonia is too large. Considering
the rest of the electric properties, the agreement is seen once
again to be very satisfactory, with our results within or
slightly outside the experimental error bars.80 However, as
for water, this agreement may be somewhat fortuitous.
For the magnetic properties, the agreement with experiment is in most cases good with the exceptions of the isotropic magnetizability,83 the parallel component of the g
tensor,25 and the isotropic hydrogen shielding constants.84
However, by taking the calibration error in the experimental
determination of the isotropic magnetizability and scaling the
experimental result to correct for this as suggested in Ref.
85, the agreement for this property becomes very good. For
g 储 , it is difficult to explain the reason for the large discrepancy, although we note that electron correlation effects are
large both for the equilibrium value and the ZPVCs, and that
our choice of active space may be, as was the case for water,
unable to recover all of the correlation effects. The error bars
on the isotropic hydrogen shielding ensures that our result is
well within the experimental observation,84 and considering
the small corrections arising from electron correlation on the
isotropic hydrogen shieldings, we are inclined to believe that
our result is of higher accuracy than that of experiment.
D. Methane
Turning finally to the methane molecule 共see Table
XIII兲, we once again find the agreement with experiment
satisfactory for most of the properties, in the case of the
isotropic magnetizability only with the more recent experi-
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J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
Vibrational wave functions
2681
TABLE XIII. Zero-point vibrationally corrected molecular properties of methane. Basis sets and wave functions described in the text. All properties reported in atomic units except the magnetizability which is reported
in units of 10 ⫺30 JT⫺2 .
SCF
Property
␣
␣ 441.8
␰
g
␴C
␴H
⌬␴H
nm
MCSCF
P expt
P expt⫹ 具 P 典 0,0
P expt
P expt⫹ 具 P 典 0,0
15.9552
16.4113
⫺313.59
0.3031
194.66
31.59
9.94
16.8563
17.3829
⫺317.37
0.2969
191.59
30.99
9.58
16.1362
16.6093
⫺318.14
0.3276
201.69
31.40
10.36
16.9744
17.5133
⫺321.49
0.3190
198.49
30.80
9.99
Literature
17.258, 17.27a
⫺310,b⫺289(⫾13兲,c⫺306d
198.7e
30.61f
a
Quadratic extrapolations from refractive data from Refs. 103 and 104 using five wavelengths in the range
325–633 nm and 436–644 nm, respectively.
b
References 105 and 106.
c
Gas-phase measurement by Barter et al. 共Ref. 83兲. However, it has been proposed on the basis of theoretical
calculations 共Ref. 85兲 that these results contain a calibration error and should be scaled by 1.07, giving a
magnetizability of ⫺309(⫾13)⫻10⫺30 JT⫺2 .
d
Gas-phase measurement from Ref. 86.
e
Gas-phase measurements from Ref. 107.
f
Gas-phase measurements quoted in Ref. 78.
mental result,86 or with the results of Barter et al.83 scaled by
a factor of 1.07.85 To our knowledge, these are the first correlated results presented for ZPVCs to the nuclear shieldings
and magnetizabilities for this molecule.
VI. CONCLUSIONS
We have calculated zero-point vibrational corrections to
a large number of different molecular properties using a recently proposed scheme in which the anharmonicity is introduced in the calculation through the use of an effective geometry as expansion point for the vibrational wave
function.30 This effective geometry corresponds to the vibrationally averaged geometry to second order in the order parameter of the perturbed vibrational wave function. In this
way, the vibrational wave function can be accurately represented by a single Gaussian function for each mode of the
molecule.
This approach to ZPVCs has been outlined and implemented in a black-box manner for SCF and MCSCF wave
functions in the Dalton quantum chemistry program.49 The
complete determination of the ZPVCs can be done in a twostep procedure: 共1兲 Determining the effective 共vibrationally
averaged兲 geometry by a calculation of parts of the cubic
force field. 共2兲 At the effective geometry, determine the harmonic force field and calculate the second derivatives of the
molecular properties of interest using numerical differentiation along the normal coordinates.
We believe that many of the ZPVCs for the polyatomic
molecules studied here are the most accurate to date, in particular for ammonia and methane. Comparisons with more
accurate results available for hydrogen fluoride and for the
magnetic properties of water indicate that our choice of wave
function is able to recover a large fraction of the electron
correlation effects to the ZPVCs, although the extent varies
for the different properties. In particular for the rotational g
tensor it appears difficult to recover all of the electron correlation effects, and for this property more accurate results
are probably needed.
Certain interesting observations can be drawn on the basis of this systematic study of the ZPVCs in these four molecules. In particular, whereas most nuclear shielding constants are strongly dependent on electron correlation,7 the
only well-known exception being the isotropic hydrogen
shieldings,7,63 the ZPVCs do not appear to be strongly dependent on electron correlation. Although this conclusion
needs to be verified for a larger class of molecules, it clearly
lends some hope to the prospect of being able to estimate
quite accurately ZPVC to nuclear shieldings using the
scheme presented in this article at the SCF level, and thus
making it possible to calculate ZPVCs to the shieldings of
large molecules. Indeed, it has recently been proposed that
part of the discrepancy observed between theory and experiment for the shieldings of the inner and outer protons of
关18兴-annulene may be due to vibrational motion,87 and it
would clearly be of interest to test this hypothesis.
Isotropic magnetizabilities are known to be largely independent of electron correlation,85,88 and this appears also to
be the case for the zero-point vibrational corrections to this
property as well. Furthermore, the ZPVCs never exceed
1.2% in these molecules, being thus of the same order of
magnitude as the correlation effects.41,54 Both correlation and
ZPVC in general make the molecule more diamagnetic
共more negative magnetizability兲, and the combined effect of
electron correlation and ZPVCs to SCF magnetizabilities
thus appear to support the proposed scaling of SCF magnetizabilities by a factor of 1.02.85
In contrast, the rotational g tensors are difficult to converge with respect to the treatment of electron
correlation,17,41 and this appears to carry over to the ZPVCs
to these properties as well. Thus, we strongly recommend a
reinvestigation of the rotational g tensor of ammonia, as it
appears unlikely that our results are converged with respect
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2682
J. Chem. Phys., Vol. 112, No. 6, 8 February 2000
to the treatment of electron correlation in this system. For
hydrogen fluoride and water, more accurate results are already available.6,17
For the electric properties, our results support the conclusions drawn by Russell and Spackman,27 and although we
believe that our MCSCF wave function is better at recovering the electron correlation effects than the MP2 wave function they used, our basis set is probably somewhat inferior. It
is therefore difficult to say which of the results are the more
accurate in the case of the quadrupole moment and the polarizability.
ACKNOWLEDGMENTS
K.R. would like to acknowledge helpful discussions with
Professor David Bishop. 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 DACI-9619020 and by Grant No. CHE9700627, and by a grant of computer time from SDSC.
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