Electric field dependence of magnetic properties

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Electric field dependence of magnetic properties: Multiconfigurational
self-consistent field calculations of hypermagnetizabilities and nuclear
shielding polarizabilities of N2 , C2H2 , HCN, and H2O
Antonio Rizzo
Istituto di Chimica Quantistica ed Energetica Molecolare del Consiglio Nazionale delle Ricerche,
Via Risorgimento 35, I-56126 Pisa, Italy
Trygve Helgaker and Kenneth Ruud
Department of Chemistry, University of Oslo, P.O.B. 1033 Blindern, N-0315, Oslo, Norway
Andrzej Barszczewicz and Michal” Jaszuński
Institute of Organic Chemistry, Polish Academy of Sciences, 01 224, Warszawa, ul. Kasprzaka 44, Poland
Poul Jo” rgensen
Department of Chemistry, Aarhus University, DK-8000, Aarhus C, Denmark
~Received 23 December 1994; accepted 24 February 1995!
Multiconfigurational self-consistent field ~MCSCF! response is used to study the electric field
dependence of magnetizabilities and nuclear shielding constants for N2 , C2H2 , HCN, and H2O.
London perturbation-dependent atomic orbitals are used to ensure gauge origin independence. The
computed magnetizabilities and shielding derivatives show a strong electron correlation
dependence. The N2 results confirm the conclusions of previous ab initio studies. For the other
molecules, this is the first study of the above magnetic properties beyond the SCF
approximation. © 1995 American Institute of Physics.
I. INTRODUCTION
Recent developments in theoretical and computational
methods have enabled ab initio studies of a wide variety of
molecular electric, magnetic, and optical properties. In particular, properties arising from the nonlinear response of the
molecule to a combination of electric and magnetic fields can
now be calculated. The theory needed to describe these properties has been known for many years ~see, for example, the
reviews by Buckingham,1 Buckingham and Orr,2 Bishop,3
and Raynes4!. In practice, however, the calculations have
been hampered by a strong dependence on such aspects as
the choice of the gauge origin, the choice of the basis set,
and the description of electron correlation.
In this work we study the electric field dependence of
magnetic properties. The problems encountered in the calculation of magnetic properties, such as magnetizabilities and
NMR shielding constants, are reflected in the unphysical dependence of the calculated values on the chosen gauge origin. Even at the SCF level, it is practically impossible to
ensure gauge origin independence of the results for a polyatomic molecule, unless methods specially formulated for
this type of calculations are used. In this work, we apply
atomic basis sets that are explicitly dependent on the external
magnetic field, the so-called London atomic orbitals5 ~LAOs,
or GIAOs, gauge invariant atomic orbitals6!. Efficient methods that enable practical use of these orbitals have only recently been formulated, first for SCF functions7 and more
recently for arbitrary approximate wave function.8 The
theory of linear response for multiconfigurational selfconsistent field ~MCSCF! wave functions9 has been successfully combined with the use of LAOs.10–13 Nuclear shieldings using London orbitals have also been calculated at
second, third, and partly fourth14 –16 order in perturbation
J. Chem. Phys. 102 (22), 8 June 1995
theory and using coupled cluster singles and doubles wave
functions.17
There is one aspect of the theory we shall discuss in
some detail in this work. For electric field derivatives it is
necessary to use a new formulation of the approach,18,19 applying what is called the natural connection for perturbationdependent basis sets. This is needed in order to avoid numerical problems arising when large basis sets are used and,
at the same time, high accuracy of the computed magnetic
properties is required. In this work the electric field dependence of the magnetic properties is analyzed performing
finite-field calculations. Numerical differentiation of the
magnetizabilities and shielding constants with respect to the
strength of the applied electric field requires high accuracy.
The final third and fourth order properties are thus obtained
in a mixed analytical-numerical scheme.
The experimental quantity that may give an estimate of
the magnetizability polarizability ~hypermagnetizability, h!
is the Cotton–Mouton constant, related to the Cotton–
Mouton effect20—the birefringence of light in gases in a constant magnetic field. Isotropic substances show weak birefringence when radiation passes through a sample in a
direction orthogonal to a strong magnetic field. This effect,
which is analogous to the electric Kerr effect, is commonly
expressed by experimentalists through the relation21
Dn5n i 2n' 5C CMlB 2
linking the observed birefringence ~anisotropy of the refraction index n! to the magnetic field flux B and the wave
length l through the so-called Cotton–Mouton constant
C CM . Experimentally, one measures the ellipticity gained by
polarized light passing through the sample in a very strong
0021-9606/95/102(22)/8953/14/$6.00
© 1995 American Institute of Physics
8953
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Rizzo et al.: Electric field dependence of magnetic properties
8954
and uniform magnetic field. The resulting magneto-optic effect is often extremely weak, making its accurate observation
very difficult.
The shielding polarizabilities provide an approximate
description of the effects of inter- and intramolecular electric
fields on the shielding. They can be used to estimate the role
of solvent effects ~see, for example, Ref. 22! or the interaction with a distant part of the molecule ~see, e.g., Ref. 23!.
We refer to some recent reviews4,24,25 for a more detailed
discussion of the calculations of shielding polarizabilities,
their applications and a bibliography of numerous earlier
works.
We first briefly review the definitions of the studied
properties. Next, we present the formulas used to compute
these properties within the LAO approach. We then present
some computational details, and finally analyze the results
for the electric field derivatives of the magnetizabilities and
shielding constants.
II. THEORY
A. Definition of magnetizability and shielding
polarizabilities
The quantities whose dependence on the electric field is
studied in this paper are the molecular magnetizability26 x
and the nuclear magnetic shielding s (K), defined as
x 52
] 2 e ~ B,m!
] 2B
s ~ K ! 511
U
~1!
,
B50
] 2 e ~ B,m!
] B ] mK
U
~2!
,
B5mK 50
where e~B,m! is the molecular energy in the presence of the
external magnetic field induction B and the nuclear magnetic
moments m. mK is the nuclear magnetic moment of the Kth
nucleus.
Switching on the electric field perturbation E, the elements of the molecular magnetizability tensor xab may be
written as
E
x ab
5 x ab 1 j ab , g E g 1 21 h gd , ab E g E d 1•••
,
~3!
where the hypermagnetizability tensors jab,g and hab,gd are
given as
j ab , g 5
E
]x ab
,
]Eg
h ab , gd 5
B
E
] 2 a ab
] 2 x gd
5
.
] B g] B d ] E a] E b
~4!
E
8 ,g~ K ! E g
s ab
~ K ! 5 s ab ~ K ! 1 s ab
1
1
2
9 , gd ~ K ! E g E d 1••• ,
s ab
where the so-called shielding polarizabilities
E
]s ab
~K!
8 ,g~ K ! 5
s ab
]Eg
In the first definition of hab,gd the electric dipole polarizability tensor aab is introduced. The second definition, taken
from Eq. ~3! above, is directly related to the technique used
in this work to compute hab,gd , in the sense that hab,gd is
obtained by numerical differentiation of analytically calculated magnetizabilities.
Similarly, the shielding tensor for nucleus K may be expanded as
~7!
and
9 , gd ~ K ! 5
s ab
E
] 2 s ab
~K!
] E g] E d
~8!
describe the effect of the electric field on the magnetic
shielding tensor. In the following the index K will be
dropped when no confusion can arise.
The theory for the calculation of magnetizabilities and
nuclear magnetic shielding constants has been thoroughly
described in previous papers ~see, e.g., Refs. 12 and 13!.
Here, we compute electric field derivatives of these observables. The theory has some new aspects as discussed in the
following.
B. Calculation of magnetizability and shielding
polarizabilities
The Hamiltonian for a molecule in the presence of an
external magnetic field and nuclear magnetic moments can
be written as ~atomic units!
H5
1
2
( p 2i 2 (
i
i,K
ZK 1
1
r i,K 2
(
iÞ j
1
,
rij
~9!
where the kinetic momentum of the ith electron is given as
p i 52i“ i 1A~ ri !
~10!
and the dependence on the magnetic fields is collected in the
vector potential
A~ r! 5
1
B3rO 1 a 2
2
(
K
mK 3rK
r 3K
,
~11!
where a is the fine structure constant and O denotes the
~arbitrary! gauge origin. The summation runs over all the
atoms in the molecule.
If a perturbation is applied to a molecular system, the
system responds. It is advantageous to try to mimic this
physical response in our basis functions. For external magnetic fields this can be achieved by introducing the London
atomic orbitals5
e
~5!
~6!
v m ~ rM ;AM ! 5e ~ 2iAM –r! x m ~ rM ! ,
~12!
AeM
where
represents the potential at the position of the
nucleus M
AeM 5 12 B3RM O
~13!
and xm is a conventional Gaussian basis function positioned
on that nucleus. The London orbitals respond correctly to the
external magnetic field in the sense that they are correct
through first order in the external magnetic field for a oneelectron, one-center system.11
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Rizzo et al.: Electric field dependence of magnetic properties
As shown by Helgaker and Jo” rgensen,8 the Hamiltonian
integrals that occur for London atomic orbitals can be written
in the following form:
1
2
^ v m ~ AeM ! u p 2 2 (
K
e
5 ^ x m u e iAM N –r
S
ZK
u v n ~ AeN ! &
rK
1
@ 2i“1A~ r! 2AeN # 2 2
2
(
K
D
ZK
u x n& ,
rK
~14!
g mnrs ~ A! 5 ^ x m x n u
exp~ iAeM N –r1 1iAeRS –r2 !
r 12
u x rx s& ,
~15!
where the new vector potentials are defined as
AeM N 5AeM 2AeN 5 21 B3BM N ,
1
A~ r! 2AeN 5 B3rN 1 a 2
2
(
K
~16!
mK 3rK
r 3K
.
H int52E–de ,
~17!
~18!
where de is defined as
( ri .
~19!
i
As the operator in Eq. ~18! commutes with the exponential
operator in the London atomic orbital, the one-electron integrals we need can be written as
e
h mn ~ A! 5 ^ x m u e ~ iAM N –r!
2
(
K
S
1
@ 2i“1A~ r! 2AeN # 2
2
D
ZK
2E–de u x n & ,
rK
We now follow closely the exposition in a recent paper
on MCSCF magnetizabilities using London atomic
orbitals.27 Only the major steps are given, with emphasis on
the differences arising when the electric field effects are investigated.
The wave function perturbed by the external magnetic
field and the nuclear magnetic moments can be expressed in
terms of a unitary rotation of orbital and configuration parameters in the reference wave function optimized at zero
field and zero nuclear magnetic moments
u WF~ B,m! & 5exp~ i k ! exp~ iS ! u RWF& .
~21!
Our reference wave function is optimized at a given electric
field E and constructed as a linear combination of Slater
determinants
u RWF& 5
( C bu b & .
~22!
b
We note that there is no dependence on the gauge origin in
these integrals.
Whereas this Hamiltonian is adequate for the nuclear
shielding constants and magnetizabilities, an additional term
must be introduced in the one-electron integrals when we
want to study the electric field derivatives of these properties
by a finite field approach.
The expressions for the magnetizabilities and nuclear
shieldings are obtained by explicit ~analytic! differentiation
of our modified Hamiltonian with respect to nuclear magnetic moments and the magnetic field. The third and fourth
derivatives are then determined by finite difference of the
results obtained at different electric field strengths. As we
shall see, the use of London atomic orbitals ensures gauge
origin independent third- and fourth-order mixed electric and
magnetic field derivatives.
The effect of an external electric field can be represented
by the interaction between the molecular dipole operator de
and the external electric field as
de 52
8955
The orbital and configuration operators k and S in Eq. ~21!
describe magnetic perturbations and are therefore given by
k 5 ( k Rrs ~ E rs 1E sr ! ,
~23!
r.s
S5
(
S RK ~ u K &^ RWFu1uRWF&^ K u ! .
~24!
KÞ0
Only nonredundant rotations are included in the summation
of Eq. ~23!. The one-electron excitation operators in Eq. ~23!
are defined as
(
E mn 5
s 561/2
1
am
sa ns .
~25!
Our reference wave function is optimized at zero magnetic
field and nuclear magnetic moments, but at a finite electric
field. In this case the Hamiltonian integrals in Eqs. ~15! and
~20! reduce to the ordinary field-free integrals with an extra
interaction operator describing the finite electric field. Thus
our reference function has an implicit dependence on the
external electric field.
The Slater determinants in Eq. ~22! are written as ordered products of creation operators
ub&5
) a b1 u vac&
i
~26!
i
creating electrons in a space of orthonormalized molecular
orbitals ~OMOs!
f OMO
~ B! 5 ( f UMO
~ B! T nm ~ B! .
m
n
~27!
n
~20!
where we have added the electric field interaction integral to
Eq. ~14!. No modification is needed in the expression for the
two-electron integrals.
The unmodified molecular orbitals ~UMOs! f UMO
(B) corren
spond to the optimized orbitals at B50 and m50 and are
linear combinations of London atomic orbitals
f UMO
~ B! 5 ( C n m v m ~ B! .
n
~28!
m
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Rizzo et al.: Electric field dependence of magnetic properties
8956
We note that there is no field dependence in the MO
coefficients. All dependence on the external magnetic field in
our wave function is collected in the London orbitals and the
connection matrix T mn . As discussed by Olsen et al.,18 it is
advantageous to choose this connection as
T5W 21 ~ WS 21 W 1 ! 1/2 ,
~29!
where S is the UMOs overlap matrix
S mn ~ B! 5
( C m m C n n ^ v m ~ B! u v n ~ B! &
~30!
mn
( C n n C m m ^ x n u v m ~ B! & .
~31!
mn
This is the so-called natural connection which ensures that
the OMOs change as little as possible when the magnetic
field is turned on. This is what we expect our orbitals to do,
and it means that the natural connection properly describes
the physical response of the London orbitals to the external
magnetic field. The use of the natural connection leads to a
numerically stable algorithm, in contrast to the symmetric
connection.18,19
It is not obvious from Eqs. ~29! to ~31! that the natural
connection gives gauge origin independent results, as it involves the use of integrals W mn ~B! that depend on the gauge
origin. This gives an origin dependence in our Hamiltonian
that parallels that of the exact Hamiltonian. And, as shown
by Olsen et al.,18 all observable properties calculated from
the Hamiltonian in the natural connection are strictly gauge
origin independent.
Our first attempt at calculating shielding polarizabilities
and hypermagnetizabilities was by the use of the symmetric
connection. However, as this approach is numerically
unstable,18,19 it turned out to be impossible to perform the
numerical differentiation from the shielding constants and
magnetizabilities evaluated at the different electric field
strengths. Thus the evaluation of higher order properties like
shielding polarizabilities and hypermagnetizabilities provides a striking example of the need for a stable algorithm
for the calculation of second-order properties.
Combining the molecular orbitals in Eq. ~27!, with the
Hamiltonian integrals in Eqs. ~15! and ~20!, our Hamiltonian,
in accordance with Helgaker and Jo” rgensen,8 can be written
as
H ~ B! 5
1
( h̃ mn~ B! E mn 1 2 (
mn
g̃ mnpq ~ B! e mn pq ,
~32!
mnpq
where the molecular integrals and two-electron excitation
operator are defined as
h̃ mn ~ B! 5
(
m8n8
* ~ B! h m 8 n 8 ~ B! T n 8 n ~ B! ,
T mm
8
m8n8r8s8
* ~ B! T *nn ~ B! g m 8 n 8 r 8 s 8 ~ B!
T mm
8
8
3T r 8 r ~ B! T s 8 s ~ B! ,
e mn pq 5E mn E pq 2 d pn E mq .
~34!
~35!
As described in Refs. 13 and 27, the nuclear shieldings
and magnetizabilities in an external electric field can now be
calculated by straightforward analytic differentiation of the
energy functional
3exp~ i k ! u RWF& .
( C n n C m m ^ v n~ B0 ! u v m~ B! &
mn
5
(
e ~ B,m! 5 ^ RWFu exp~ 2i k ! exp~ 2iS ! H exp~ iS !
and the nonsymmetric matrix W is given by
W mn ~ B! 5
g̃ mnrs ~ B! 5
~33!
~36!
The nuclear shieldings are obtained by differentiating once
with respect to nuclear magnetic fields and once with respect
to external magnetic field, whereas the magnetizability is obtained by differentiating twice with respect to the external
magnetic field. In calculations without London orbitals this
would result in the ordinary perturbation expression with
MO coefficients modified due to the presence of the electric
field. In calculations using London orbitals we obtain, in
addition to the modified integrals presented elsewhere,12,27
an extra derivative contribution from the electric field interaction operator. Thus the first and second derivatives of the
one-electron Hamiltonian look like
U
] h mn
]B
1
5 ^ x m u LN u x n & 1iQ M N ^ x m u rh u x n &
2
B50
2iEQ M N ^ x m u rdu x n & ,
U
] 2 h mn
] B2
~37!
5 ^ x m u r 2N I2rN rTN u x n & 1iQ M N ^ x m u rLTN u x n &
B50
1Q M N ^ x m u rrT h u x n & Q M N
2EQ M N ^ x m u rrT du x n & Q M N ,
~38!
where h represents the ordinary one-electron Hamiltonian
without contributions from external magnetic and electric
fields, the overbar means that the integral is to be symmetrized, LN is the operator for angular momentum around
nucleus N, LN 52irN 3“, and the elements of the matrix
Q M N are the components of the vector from nucleus N to
nucleus M , see Ref. 8.
The last terms in Eqs. ~37! and ~38! are new to our
implementation, and vanish in the absence of an external
electric field.
We notice that both derivative integrals are independent
of the gauge origin, and that there is no change in the twoelectron integrals, as the electric field interaction operator is
a one-electron operator. The new integrals that appear due to
the presence of the electric field can be quite easily evaluated
following the outline in Ref. 12.
Thus a numerically stable algorithm for gauge origin independent magnetizabilities and shielding constants has been
presented. The shielding polarizabilities and hypermagnetizabilities can be obtained by calculations of the nuclear
J. Chem. Phys., Vol. 102, No. 22, 8 June 1995
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Rizzo et al.: Electric field dependence of magnetic properties
8957
shieldings and magnetizabilities at different electric field
strengths, followed by differentiation with respect to the
electric field strength.
C2 m2 J21 ~SI, MKSA!51.481 847310225 ~4pe0! cm3
~emu!;
—1 a.u. of x 5(e 2 a 20 )/(m e )57.891 04310229 J T22 or
C. Relationship to experiment
7.891 04310230 cm3;
—1 a.u. of j 5(e 3 a 30 )/(m e E h )51.534 562310240
Before concluding this section, we give some definitions
needed to relate the quantities we compute to experiment, in
particular the hypermagnetizability h to the Cotton–Mouton
effect. The first theoretical treatment of the Cotton–Mouton
effect appeared in 1910.28 A fundamental contribution to the
understanding of the topic was given by Buckingham and
Pople in 1956.29 A classical statistical mechanics approach30
shows that for rigid diamagnetic molecules the so-called
Cotton–Mouton constant m C, proportional to the mean electric polarizability in a strong magnetic field, can be written as
m C5
2pN
27
1
H S
S
1
1
h ab , ab 2 h aa , bb
5
3
1
1
a ab x ab 2 a aa x bb
5kT
3
D
DJ
,
~39!
where N is Avogadro’s number, k the Boltzmann constant,
and T the temperature. The Einstein summation ~here and
below! is assumed. For axial molecules the above equation
simplifies to
m C5
F
G
2pN
2
DaDx ,
Dh1
27
15kT
1
3
C2 m2 J21 T2252.682 108310244 ~4pe0! cm3 G22;
—1 a.u. of s 8 5 p pm(a 20 /e)51.944 69310218
mV2155.830 03310214 cm statV21 ~esu!;
—1 a.u. of s 9 5 p pm(a 20 /e) 2 53.781 82310230
m2 V2253.398 92310221 cm2 statV22.
In Eq. ~40! above the factor relating m C to the quantity
in square brackets ~computed in a.u.! is 3.758 738 10221
cm3 G22 mol21 ~4pe0! ~CGSM! or 5.935 561 10231
m5 A22 mol21 ~SI!.
The relationship between the experimental Cotton–
Mouton constant21 C CM and the m C introduced by Buckingham and Pople,29 at a temperature T, 1 atm and in the CGSM
unit system, is
C CM~cm21 G22 )5
27
1
•
C ~ cm3 G22 mol21 !
2V m l ~ cm! m
5
0.164 518
C ~ cm3 G22 mol21 ! ,
T ~ K!•l~cm! m
~40!
where Da5ai2a' , Dx5xi2x' , and the hypermagnetizability anisotropy is defined as
D h 5 51 ~ h ab , ab 2
C m T2251.379 196310226 ~4pe0! cm4 s21 G21;
—1 a.u. of h 5(e 4 a 40 )/(m e E 2h )52.984 25310252
h aa , bb !
~41!
thus differing by a factor of five from that used by Hüttner
and collaborators.31 Notice also that some authors ~e.g., Ref.
31! employ König’s32 definition of m C, a factor of 9 larger
than Eq. ~39! above. Our results for m C and Dh are given
according to Eqs. ~39! to ~41! above.
The experimentally measured frequency dependent hypermagnetizability can be written in terms of a combination
of quadratic ~diamagnetic contribution! and cubic ~paramagnetic contribution! terms ~r position, L angular momentum,
and Q quadrupole moment operators!33
p
d
h ab , gd ~ 2 v ; v ,0,0 ! 5 h ab
, gd 1 h ab , gd
52
1
2
^^ r a ;r b ,L g ,L d && 2 v ; v ,0,0
2
1
4
^^ r a ;r b ,Q gd && 2 v ; v ,0 .
~42!
By taking advantage of symmetry and thus reducing the expression for the paramagnetic contribution to a Cauchy moment expansion, we were able to study the frequency dependence of the hypermagnetizability anisotropy in the Ne and
Ar atoms using the multiconfigurational response
method.33,34 The same approach cannot straightforwardly be
employed for molecules.
Atomic units are used throughout the paper ~unless otherwise stated!. Conversion factors to some of the units used
by other authors are given in the following:
—1 a.u. of a 5(e 2 a 20 )/(E h )51.648 78310241
where l is the radiation wavelength and V m is the molar
volume at the given temperature of the ~ideal! gas.
III. COMPUTATIONAL DETAILS
A. Computational aspects
There are many practical problems to deal with in the
calculations, in particular the selection of the finite fields, the
choice of basis set and active space for CASSCF and the
dependence of the results on the molecular geometry. The
accuracy of the numerical differentiation depends crucially
on the strength of the applied finite field. Often the field
suitable for shielding derivatives ~where first derivatives may
be nonzero! is too small to compute reliably the magnetizability derivatives ~where the first derivative may vanish because of higher symmetry!. In such cases, different fields
were applied for different properties. Usually field strengths
of 0.01 a.u. were suitable for the magnetizability derivatives
and 0.001 a.u. for the shielding derivatives. For many tensor
components we found intermediate fields suitable for both
calculations. We tried to minimize the number of finite fields,
since the calculations are time consuming. For example, for
s~H! in H2O at least eight separate combinations are required
to obtain all tensor components. A complete calculation for
each of these combinations took a few CPU hours on a Convex 3840 computer.
The convergence of the properties with respect to extension of the basis set and increase of the active space is not
systematic. For the individual components the convergence
differs, some of them being apparently much easier to compute than others. We shall, therefore, describe in the text the
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8958
Rizzo et al.: Electric field dependence of magnetic properties
main conclusions of this computational analysis for all four
molecules. However, as a rule we shall present only the numbers for the best SCF and best correlated calculation. To
illustrate the basis set dependence we quote two sets of values for N2 . The correlation and geometry dependence will be
discussed for H2O. These are typical examples, and at the
same time there are some literature data which we use for
comparison.
All calculations have been performed using a suite of
programs which includes HERMIT35 for the integrals, SIRIUS36
for the wave function and ABACUS37 for the magnetic properties.
It is not reasonable to describe all the aspects of the
calculation in detail. To begin with, there are too many tensor
components for a detailed analysis. An extreme case is the s9
tensor for the H atom in H2O, which has 41 ~28 different!
nonvanishing components.38 Following Grayson and
Raynes,39,40 we shall discuss for HCN and H2O only those
components that do not vanish after rotational averaging. For
N2 and C2H2 we include all nonzero first derivatives.
B. Basis sets
The dependence of the properties on the basis set was
studied for each molecule. We began the construction of the
basis set with the GTO sets referred to as H IV in Ref. 12.
They consist of [11s7 p3d1 f ]/ ^ 8s7p3d1 f & contractions for
C, N, and O atoms and a [6s3p1d]/ ^ 5s3p1d & contraction
for the hydrogen atom. These sets are well suited for the
calculation of magnetizabilities and shielding constants, but
they do not include diffuse functions needed for the electricfield derivatives of these properties. In the larger basis sets
we add diffuse s, p, d, and f ~s, p, and d for H! functions
using a geometric progression for the exponents. As in the
calculation of Verdet constants,41 we observe that the first set
of diffuse functions, giving rise to the sets labeled IVa here
as in Ref. 41, changes the computed properties significantly.
The changes upon adding a second set are smaller. These
sets, called IVb,41 finally include [13s9 p5d2 f ]/
^ 10s9p5d2 f & functions for C, N, and O and [8s5p2d]/
^ 7s5p2d & functions for H. The second f function on C, N,
or O and d function on H were included already in set IVa
for some molecules and only in set IVb for the others.
The basis set convergence was studied for many properties not only at the SCF level, but also at the CASSCF level.
We shall illustrate the basis set dependence for N2 . For C2H2
and HCN we were unable to use basis IVb at the correlated
level. For C2H2 the differences between the SCF h components computed using basis sets IVa and IVb are smaller than
5% ~we exclude in these comparisons the very small tensor
components!, for s8 they do not exceed 1%, and for s9 10%.
Also for HCN the differences between the SCF IVa and IVb
results are smaller than for N2 . For water the effect of the
basis set is larger: about 10% for h and s9 at the SCF level,
and significantly larger for s9 at the CAS A ~see below! level
~on the average, about 30%!. To establish convergence with
the basis set at the CASSCF level we would therefore need
to use a basis set larger than IVb, which is not presently
possible.
C. Geometries
We have used the following geometries:
—for N2 , R~N–N!51.097 513 Å;42
—for C2H2 , R~C–H!51.0606 Å and R~C–C!51.2032
Å;43
—for HCN, R~H–C!51.064 Å and R~C–N!51.156 Å,44
—for H2O, R~O–H!50.972 Å and ,HOH5104.5.12
The linear molecules are placed along the z axis, N2 and
C2H2 symmetrically and HCN with the positive z direction
from H to N. The water molecule has z axis as C2 axis and
lies in the xz plane, with positive z direction from O to the H
atoms. We discuss the shielding derivatives for the atoms
located at (0,0,2z) in N2 and C2H2 and for the (1x,0,1z)
H atom in H2O.
D. MCSCF configuration spaces
The active spaces are labeled here by the number of
active orbitals in the different irreducible representations of
the molecule, using only D 2h and its subgroups. Thus the
notation (n 1 n 2 ...n 8 ) in D 2h indicates the number of active
orbitals in symmetries ( s g p ux p uy d g s u p gx p gy d u ), respectively. In C ` v , the symmetries are ordered as ( sp x p y d ) and
in C 2 v as (a 1 b 2 b 1 a 2 ).
The core orbitals are inactive in all calculations. Since
we are using a finite field technique, we could not exploit the
full point group symmetry. Thus the CAS labels define the
wave function for the unperturbed molecule. Symmetry reduction also means a large increase in the number of determinants: The largest CI expansion used includes over
800 000 determinants. The active spaces were selected based
on the MP2 natural orbital occupation numbers.
The first function used for N2 is a full valence CASSCF,
~21102110!. The second wave function includes five more
orbitals in the active space, giving ~42203110!. For C2H2 the
same CAS choices are made.
For HCN we use only one CASSCF function, ~5220!.
Because of lower symmetry, a wave function corresponding
to the larger CAS of N2 would include more than 4.5 million
determinants and the calculation was not attempted. For H2O
the first function is a ~4220! CAS, the second a ~6331! CAS.
When needed, we shall use the labels CAS A and CAS B
to denote the wave functions with the smaller and larger
active spaces, respectively.
It has been observed that by using small complete active
spaces one may overestimate the correlation corrections for
the shielding.13 We have noticed a similar effect for the
shielding polarizabilities and this is the reason we attempted
to use, whenever possible, at least two different active
spaces.
IV. MAGNETIZABILITY POLARIZABILITIES
The results obtained for the magnetizabilities and hypermagnetizabilities of the four molecules N2 , C2H2 , HCN, and
H2O are summarized in Tables I to IV. Table V, where we
have gathered results for all four molecules, displays the
J. Chem. Phys., Vol. 102, No. 22, 8 June 1995
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Rizzo et al.: Electric field dependence of magnetic properties
8959
TABLE I. N2 magnetizabilities x ab , hypermagnetizabilities h ab,cd and anisotropy Dh ~a.u.!. Here and in the
following tables the sequence for the indices ab,cd in the hypermagnetizability tensor h is: a,b: electric field;
c,d: magnetic field. See the text for details.
SCF
x xx
x zz
h xx,xx
h xx,y y
h xx,zz
h zz,xx
h zz,zz
h xz,xz
Dha
Correlated
Basis IVa
Basis IVb
Ref. 55
CAS B
Basis IVa
CAS B
Basis IVb
Ref. 55 ~MP2!
21.94
23.86
21.94
23.86
21.95
23.86
22.11
23.82
22.11
23.82
22.12
23.81
212.09
263.58
234.53
22.66
221.98
13.33
25.76
263.28
234.14
21.94
221.48
13.04
29.91
267.52
234.07
22.11
221.00
13.10
233.10
275.04
239.78
221.68
227.08
13.46
227.62
276.46
239.90
221.26
226.82
13.46
231.91
274.32
236.47
217.80
223.82
13.14
28.24
30.78
30.38
24.92
27.94
24.46
Experiment: Dh596.9674.6 a.u. ~Ref. 56!.
a
temperature dependence of the Cotton–Mouton constant,
comparing with experiment and other reference calculated
values.
The calculation of molecular magnetizabilities has directly or indirectly been the subject of several studies. Recent
SCF and correlated ~MP2! results for xab in N2 , HCN, and
H2O with inclusion of vibrational corrections have been presented by Cybulski and Bishop.45 Their approach is not
gauge origin independent, but the use of extended basis sets
leads to a good overall reliability of the final results. The
MC-IGLO method, an extension of the SCF-IGLO method
used for studies on H2O,46 HCN, and C2H2 ,47 has been applied by van Wüllen and Kutzelnigg48 to N2 and H2O. The
approach is multiconfigurational and gauge origin independent, in principle less computationally intensive than ours
although somewhat dependent on the choice of the localization scheme for the molecular orbitals and on the completeness of the basis set. Sauer et al.44 have calculated magnetizabilities of N2 and HCN both at RPA ~resorting to
Geertsen’s gauge independent approach49! and correlated
~SOPPA! levels of approximation. We also mention here the
work of Jaszunski et al.42 with RPA and MCRPA results for
N2 and Geertsen’s50 data for H2O. LAOs were employed
within our group to compute magnetizabilities at the SCF
level for a series of diamagnetic molecules including H2O.12
MCSCF magnetizabilities were also computed in an independent work.27 Notice finally that the only literature data we
are aware of for C2H2 was obtained in our group using a
gauge origin dependent approach.41
Very few studies of magnetizability polarizabilities have
been published. Calculations are easier for atoms,33,34,51 due
to the higher symmetry. H2 and D2 were studied by Fowler
and Buckingham,52 benzene by Augspurger and Dykstra.53
Most of the work in this area has been carried out by Bishop
and co-workers, who published correlated results for H2
~D2!54 and very recently accurate SCF and MP2 results for
the hypermagnetizabilities of H2 , N2 , HF, and CO.55 To our
knowledge, there are no literature data for C2H2 , HCN and
H2O.
From the experimental point of view, the quantity most
directly related to the hypermagnetizability is the Cotton–
Mouton constant m C, which depends on the hypermagnetizability anisotropy according to Eqs. ~39! and ~40!. N2 and
C2H2 are the only two molecules studied here for which experimental investigations aimed at measuring the hypermagnetizability anisotropy have been conducted.56,57 For these
molecules the temperature dependent part, unrelated to the
hypermagnetizability anisotropy, gives by far the most important contribution to m C. This is confirmed by our results
and also by the calculations of Cybulski and Bishop55 for N2 .
The experiment itself is extremely difficult, which adds to
the uncertainty in the estimates of Dh.
A. Magnetizability polarizabilities of N2
Table I shows the results for the N2 molecule. The two
sets IVa and IVb give nearly the same results for both the
parallel and the perpendicular components of the magnetizability tensor. Our numbers practically coincide with those of
Ref. 45, both at SCF and correlated levels of approximation.
Apparently, the basis set is saturated at the IVb level as far as
the magnetizability is concerned.
Our best average value for x ~22.68 a.u.! slightly improves our previous MCRPA estimate42 ~22.66 a.u.!. The
MC-IGLO results for the tensor components are 23.76 a.u.
( x zz ) and 22.10 a.u. ( x xx ). All these studies show that correlation increases ~by about 5% in our case! the average
magnetizability ~in absolute value!, essentially increasing the
perpendicular component. This behavior is not reproduced
by SOPPA.44
Convergence with respect to the basis set is not yet satisfactory for the magnetizability polarizabilities, at least for
some of the components. In particular the perpendicular
( h xx,xx ) component is significantly modified going from IVa
to IVb, both at the SCF and MCSCF levels. The SCF results
are in good agreement with those of Ref. 45, with the exception of h xx,xx , where the result of Cybulski and Bishop
~29.91 a.u.! falls between our IVa and IVb results. The authors in Ref. 45 remark that the sum rules45,55 they use to
verify the degree of gauge origin independence are ‘‘less
well satisfied’’ for h ~especially for h xx,xx ! in N2 than in the
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Rizzo et al.: Electric field dependence of magnetic properties
8960
TABLE II. C2H2 magnetizabilities x ab , hypermagnetizabilities h ab,cd and
anisotropy Dh ~a.u.!. See the text for details.
SCF
Basis IVb
x xx
x zz
h xx,xx
h xx,y y
h xx,zz
h zz,xx
h zz,zz
h xz,xz
Dh c
Correlated
Basis IVa-CAS B
24.71a
25.21a
24.65b
25.06b
277.42
2205.34
2193.08
284.24
285.58
38.87
275.16
2202.72
2182.24
289.68
286.36
37.01
88.98
86.85
Reference values: 24.748 ( x xx ), 25.223 ( x zz ), Ref. 47 ~IGLO!.
Reference values: 25.339 ( x xx ), 25.074 ( x zz ), Ref. 41 ~MCSCF, gauge
origin in the center of mass!.
c
Experiment: 20654 ~Ref. 57. Notice the use of König’s definition for Dh,
see the text! and 455634 ~Ref. 31!.
a
b
TABLE III. HCN magnetizabilities x ab , hypermagnetizabilities j ab,c and
h ab,cd and anisotropy Dh ~a.u.!. The sequence of the indices ab – c in j ab,c
both here and in the next table is a,b: magnetic field; c: electric field. See
the text for details.
SCF
Basis IVb
Correlated
Basis IVa
x xx
x zz
23.10a
24.45a
23.20b
24.35b
j xx,z
j zz,z
j xz,x
5.08
0.06
0.90
4.88
0.38
1.08
229.90
298.82
279.40
22.38
242.38
22.86
233.44
2100.37
268.75
219.00
241.67
21.75
42.53
41.40
h xx,xx
h xx,y y
h xx,zz
h zz,xx
h zz,zz
h xz,xz
Dh
a
Reference values: 23.118 ( x xx ), 24.447 ( x zz ), Ref. 45, CHF.
Reference values: 23.118 ( x xx ), 24.370 ( x zz ), Ref. 45, MP2.
b
other molecules, confirming that it is quite difficult to
achieve basis set convergence for the h xx,xx component. The
correlation effect is large, in particular for the electric field
derivatives of x xx ~h xx,xx and h zz,xx !. As already pointed out
in Ref. 45, the hypermagnetizability anisotropy Dh is less
sensitive to the effect of correlation than these individual
components. It decreases by about 10% in our CAS B IVb
calculation ~27.94 a.u.! compared to the corresponding SCF
value ~30.78 a.u.!. The best correlated value of Ref. 45
~24.46 a.u.! is still lower by about 12%. The experiment
gives a value of 96.9674.6 a.u.,56 leaving both us and Ref.
45 within the error bars!
B. Magnetizability polarizabilities of C2H2
The magnetizability and magnetizability polarizabilities
of C2H2 are reported in Table II. For this molecule, as for
HCN and H2O, there are no theoretical results for h available
for comparison. Moreover, as mentioned above, the only correlated numbers for x in the literature were obtained in our
group as a byproduct of a calculation of the Verdet constant,
with a [12s7p4d u 6s3p]/ ^ 6s5p4d u 4s3p & basis set, the
CAS B wave function and using a gauge dependent
approach.41 Our SCF magnetizability tensor elements and
those obtained in the SCF-IGLO approach by Schindler and
Kutzelnigg47 are in good agreement, as well as those by
Ruud et al.58
We mentioned above that the differences between the
components of h at the SCF level using the sets IVa and IVb
are smaller than 5%, and that we cannot use IVb in the
correlated calculations. Correlation effects are smaller in
C2H2 than in N2 ~less than 4% on the average for the hypermagnetizabilities!. The anisotropy decreases by about
2%–3% ~86.85 a.u.! when correlation is introduced compared to SCF ~88.98 a.u.!. There are two very different experimental estimates for Dh of C2H2 : a recent 20654 a.u.
value by Coonan and Ritchie57 and an older 455634 a.u.
value by Kling et al.31 Our result disagrees with both experiments. It falls in between them, but it is much closer to the
first.
C. Magnetizability polarizabilities of HCN
In HCN the first derivatives jab,g are nonzero, see Table
III. For these derivatives, as well as for the second derivatives and Dh, we have no theoretical or experimental numbers to compare with. Our SCF magnetizability is in good
agreement with that computed by Cybulski and Bishop.45
The average value ~23.55 a.u.! practically coincides with
that of Ref. 45 and is close to 23.54 a.u. obtained by Sauer
et al.44 The correlated average value ~23.58 a.u.! may be
compared with 23.52 a.u. MP2 in Ref. 45 and the SOPPA
value of 23.38 a.u. in Ref. 44. Apparently, Cybulski and
Bishop45 find a slightly smaller effect of correlation ~none for
x xx ! than we do, and in the opposite direction for the average
value. All in all correlation plays a minor role for the magnetizability of HCN. Notice that we could not perform correlated calculations with basis set IVb and CAS B.
Correlation plays a greater role for the magnetizability
polarizabilities. In one case ( h zz,xx ) the effect is quite dramatic but, as for the two previous molecules, the influence
on the anisotropy Dh is quite small ~less than 3%!. In axial
molecules one obtains from Eq. ~41!
Dh5
1
15
~ 7 h xx,xx 25 h xx,y y 12 h zz,zz 22 h xx,zz
22 h zz,xx 112 h xz,xz !
~43!
which shows that the contribution of h zz,xx ~219.00 a.u.! to
the anisotropy is, for instance, about an order of magnitude
smaller than that of h xx,y y ~2100.37 a.u.!, which is only
slightly influenced by correlation.
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Rizzo et al.: Electric field dependence of magnetic properties
TABLE IV. H2O magnetizabilities x ab , hypermagnetizabilities j ab,c and
h ab,cd and anisotropy Dh ~a.u.!. See the text for details.
SCF
Correlated
Basis IVb
Basis IVb-CAS A
Basis IVb-CAS B
x xx
xyy
x zz
22.91
22.97a
22.96a
22.99
23.02
23.03
22.97b
23.01b
23.02b
j xx,z
j y y,z
j zz,z
j xz,x
j yz,y
20.33
0.45
0.08
20.22
20.22
20.20
0.69
0.31
20.43
20.36
20.22
0.67
0.31
20.40
20.33
h xx,xx
h xx,y y
h xx,zz
h y y,xx
h y y,y y
h y y,zz
h zz,xx
h zz,y y
h zz,zz
h xy,xy
h xz,xz
h zy,zy
211.28
212.70
221.60
251.42
223.66
251.70
229.50
241.32
214.61
8.02
4.25
13.82
218.87
214.87
226.75
274.72
234.08
268.48
244.25
250.50
218.12
12.35
7.50
18.40
218.12
218.75
229.63
271.60
234.48
267.68
240.87
252.06
220.13
11.50
7.38
38.75
17.71
24.46
32.06
a
Dh
8961
ing from set IVa to set IVb. It appears that we need a set
larger than IVb to be able to claim basis set convergence.
Therefore, we list in the table only IVb results. Both our SCF
and CASSCF magnetizabilities are in good agreement with
the results of Ref. 45. The average SCF magnetizability
~22.95 a.u.! compares well with other results: 22.93 a.u.
~Ref. 45!, 22.95 ~Ref. 48, SCF-IGLO!, and 22.88 ~Ref. 50,
RPA!. Correlation gives xav.523.00 a.u., a value very close
to the MP2 23.02 a.u. result of Ref. 45, the 22.925 a.u.
value of Ref. 48 ~MC-IGLO! and the 22.963 a.u. CCPPA
result of Geertsen.50
Correlation effects are quite important for the magnetizability polarizabilities. The correlated anisotropy ~32.06 a.u.!
is almost twice the SCF value ~17.71 a.u.!. The effects on the
individual tensor components, although never as dramatic as
in some previous cases, are quite strong. As far as convergence with respect to the configuration space is concerned,
the anisotropy increases by 25% going from CAS A to CAS
B, a change closely related to the change in the h zy,zy component, which goes from 18.40 to 38.75 a.u. Apparently we
are still not converged with respect to the correlation treatment.
E. Temperature dependence of the Cotton–Mouton
constant
Reference values: 22.897 ( x xx ), 22.945 ( x y y ), 22.949 ( x zz ), Ref. 45,
CHF.
b
Reference values: 22.992 ( x xx ), 23.041 ( x y y ), 23.033 ( x zz ), Ref. 45,
MP2.
a
It is interesting to discuss the temperature dependence of
the Cotton–Mouton constant m C and to compare with the
literature, see Table V. Our references are here Eqs. ~39! to
~41!. Accordingly, we report in Table V our ‘‘best’’ results for
Dh, Da, and Dx, comparing them with both experiment and
other theoretical estimates. The quantity Q(T) is defined as
D. Magnetizability polarizabilities of H2O
Q~ T !5
Table IV summarizes the results for H2O. We have already mentioned that for this molecule changes of about 10%
in the SCF hypermagnetizabilities were observed when go-
1
@~ a xx 2 a y y !~ x xx 2 x y y ! 1 ~ a y y 2 a zz !
15kT
3 ~ x y y 2 x zz ! 1 ~ a xx 2 a zz !~ x xx 2 x zz !# ,
TABLE V. Temperature dependence of m C. Atomic units used unless explicitly specified. Q(T)51(2)/(15kT)( a i 2 a' )•( x i 2 x' ) for N2 , C2H2 , and HCN.
i, j D a
Q(T) 5 (1)/(15kT) @ ( a xx 2 a y y )( x xx 2 x y y ) 1 ( a y y 2 a zz )( x y y 2 x zz ) 1 ( a xx 2 a zz )( x xx 2 x zz ) # 5 (1)/(15kT) ( x,y,z
ii, j j • D x ii, j j for H2O.
N2
Our
estimate
Basis IVb
CAS B
Dh
27.94
Daa
4.37
Dxb
21.71
Q(273.15 K!
m C(273.15
3 22
~cm G
K!
mol21!31018
C2H2
Our
estimate
Basis IVa
Ref. CAS B
Others
24.46
96.9674.6
4.92
4.75160.088
55
56
61
56
86.85
21.69
21.85060.076
55
56
20.41
21151.83
24.22 24.4
23.92c 24.3c
23.86d 24.160.2d
11.29
HCN
Others
Ref.
20654
455634
12.5760.13
11.58
57
31
62
63
41.40
20.5060.03
20.7960.02
57
31
21.15
2713.49
59
56
60
22.36
22.17c
Our
estimate
Basis IVa
CAS A
8.14
21442.89
23.6160.2c
23.5260.15c
31
57
25.27
H2O
Our
estimate
Basis IVb
Others Ref.
CAS B
Others
Ref.
32.06
9.04
21.25
64
45
0.44
0.79
0.36
0.049
0.042
20.007
3.98
0.41
0.68
0.27
0.049
0.041
20.008
65
45
0.14
ai2a' for the axial molecules. D a xx,y y , D a xx,zz and D a y y,zz for H2O.
xi2x' for the axial molecules. D x xx,y y , D x xx,zz and D x y y,zz for H2O.
c
At 293.15 K.
d
At 298.15 K.
a
b
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Rizzo et al.: Electric field dependence of magnetic properties
8962
TABLE VI. N2 . Shielding constants ~ppm! and shielding polarizabilities ~ppm a.u.!. Here and in the following
8 and s ab,cd
9
is a: nuclear moment; b: magnetic field; c,d:
tables the sequence of the indices ab-c(d) in s ab,c
electric field. See the text for details.
SCF
5 1
15kT
(
i, j
Correlated
Basis IVa
Basis IVb
Ref. 73
Basis IVa
CAS B
Basis IVb
CAS B
Refs. 72 & 73
~MP2!
sav.
Az
B xx
B zz
2109.4
21047.0
2297.0
1375.2
2109.4
21047.3
2297.8
1379.0
2111.38
21051.7
2294.7
1378.1
251.8
2837.6
2897.6
716.2
251.8
2837.9
2890.0
711.6
239.70
2777.0
21029.6
796.8
s xx
s zz
8
s xx,z
8
s zz,z
8
s xz,x
8
s zx,x
9
s xx,xx
9 y
s xx,y
9
s xx,zz
9
s zz,xx
9
s zz,zz
9
s xz,xz
9
s zx,xz
2333.5
338.6
1547.1
46.9
85.4
21402.8
21947.6
4564.2
24070.3
2834.4
2110.5
21130.0
2859.4
2333.4
338.6
1547.5
46.9
85.4
21403.0
21984.5
4602.3
24081.5
2831.0
2111.0
21127.5
2845.2
2336.4
338.6
1554.
46.92
85.81
21409.
22021.
4585.
24086.
2796.3
2111.0
21136.
2834.2
2247.4
339.5
1236.6
39.4
69.4
2653.6
1555.8
4813.0
22089.5
21002.1
2118.4
2898.4
2492.4
2247.4
339.5
1237.1
39.4
69.4
2652.4
1474.1
4865.0
22075.3
2999.2
2118.9
2896.4
2481.5
2229.5
339.8
1147.
36.86
72.35
235.28
2785.
4248.
22332.
2856.2
2119.4
2815.9
2269.8
D a ii, j j •D x ii, j j ,
~44!
x,y,z
and it includes all the temperature dependence of m C, according to Eq. ~39!, for the molecules studied in this work. A
comparison of Q(T) with the temperature independent part,
Dh, shows that for the three axial molecules the hypermagnetizability anisotropy gives only a minor contribution to
m C, compared to the contribution from the T-dependent factor: about 2.5% for N2 , 14% for C2H2 and 3% for HCN at
273.15 K. In all three cases Dh is of opposite sign with
respect to Q(T). For C2H2 , the most recent experiment predicts 22%66% relative contribution of the T-independent
term to m C. 57 The data furnished by Kling and Hüttner56 and
reported in Table V can be combined to predict approximately a 7%65.5% contribution for N2 .
H2O shows a completely different behavior. Our results
give Q(T)53.98 a.u. at 273.15 K, while our best value for
the hypermagnetizability anisotropy is 32.06 a.u. In this case
the temperature dependent term contributes a little more than
4% to m C. The water molecule thus shows a ‘‘quasiatomic’’
response to the electromagnetic perturbation as far as the
Cotton–Mouton effect is concerned: the predicted value of
219
cm3 G22 mol21 at 273.15 K, is quite close to
m C, 1.4310
that computed for argon33 ~9.1310220 cm3 G22 mol21, independent of T! and about an order of magnitude larger than
that obtained for isoelectronic neon atom34 ~1.0310220
cm3 G22 mol21, again independent of T!.
In Table V we also compare the calculated values of
C
with
experiment for N2 and C2H2 . N2 is a molecule of
m
special importance in experiments, since it has been used as
a reference system for determinations of Cotton–Mouton
constants in atomic and molecular systems59 and for calibration of the experimental apparatus.60 Our values of m C of N2
at different temperatures agree quite well with experiment.
There is some disagreement, however, between our values
and experiment for m C in C2H2 . We have already discussed
our limitations and the factors we believe are the sources of
uncertainty in our calculation. For atoms the calculated values of m C are accurate to a few percent.33,34,51 Although the
accuracy of ab-initio calculations in molecules like C2H2 is
lower, we do not believe that the errors in the computed
electric polarizability and magnetizability anisotropies can
explain the discrepancy ~up to 40%! with experiment, and in
view of the relatively minor contribution given by the temperature independent factor to m C, the overall disagreement
can hardly arise from our uncertainty in Dh.
V. SHIELDING POLARIZABILITIES
There have been numerous calculations of the shielding
tensors for all the nuclei in N2 , C2H2 , HCN, and H2O. For a
recent review of various methods used in shielding tensor
calculations and references to earlier results, see Ref. 66. The
same methods which were used for magnetizabilities have
also been applied to compute the shielding constants in these
four molecules. There are IGLO47 and MC-IGLO,48 RPA,
and SOPPA,44,67 SCF-GIAO,68 and MCSCF-GIAO13,69 results for most of the nuclei in these molecules. In addition,
there are MCSCF42 and MP270 values, and recent MBPT15
and coupled cluster ~CCSD!71 calculations using GIAO basis
sets.
We shall not discuss these results in such detail as the
corresponding magnetizability values. There are few other
results for magnetizability derivatives. In contrast, there are
literature data for all the shielding polarizability tensors we
compute and we can compare our results directly to these
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Rizzo et al.: Electric field dependence of magnetic properties
8963
TABLE VII. C2H2 . Shielding constants ~ppm! and shielding polarizabilities ~ppm a.u.!. See the text for details.
C atom
H atom
SCF
115.7
B xx
B zz
146.6
556.6
s xx
s zz
8
s xx,z
8
s zz,z
8
s xz,x
8
s zx,x
9
s xx,xx
9 y
s xx,y
9
s xx,zz
9
s zz,xx
9
s zz,zz
9
s xz,xz
9
s zx,xz
a
119.772
119.8
2750.7
2733.9
116.4
583.1
553.45
40.246
278.825
1097.8
56.5
2756.2
Az
34.3
278.5
1106.0
56.5
242.0
2212.1
2652.1
2689.7
21546.5
23144.0
2246.8
473.7
21129.0
SCF
Basis IVa
CAS B
Othersa
Basis IVb
sav.
Correlated
2542.0
2659.0
21619.8
22815.5
2258.8
30.4
2639.6
269.1
146.9
675.6
146.6
9.0
54.1
279.1
934.9
49.1
232.7
262.3
2677.6
1517.9
21906.7
22694.7
2240.0
455.6
21336.0
25.1
40.9
73.7
60.0
226.8
232.7
2135.0
2507.6
20.6
2236.9
252.8
2163.5
2177.3
Basis IVa
CAS B
Othersb
Basis IVb
129.1
Correlated
30.76
33.7
270.1
267.2
71.3
2191.0
6.8
25.73
40.82
76.3
57.8
30.5
265.6
146.9
25.4
25.3
40.9
70.7
55.4
226.3
220.2
2154.4
2491.7
238.6
2235.5
275.0
2159.1
2147.5
226.5
2428.5
552.0
2226.0
101.8
First number: Ref. 77. Second ~if present!: Ref. 75.
First number: Ref. 39. Second ~if present!: Ref. 75.
b
reference values. Therefore, we discuss primarily the shielding constants obtained in the studies that included at the
same time shielding polarizabilities, and these results are
shown in the tables. We refer to Refs. 16 and 75 for more
data on accurate ab initio and experimental results for s.
For all the nuclei in all four molecules our and other
recent SCF calculations yield similar results. The differences
at the SCF level are usually related to differences in assumed
geometries.
The correlation corrections for s~N! in N2 are large and
fairly well described by our MCSCF wave function. For
HCN, the CAS A function overestimates the correlation effects, while for C2H2 and H2O they are not so large and the
comparison does not really help us estimate the accuracy of
TABLE VIII. HCN. Shielding constants ~ppm! and shielding polarizabilities ~ppm a.u.!. See the text for details.
Atom H
SCF
Basis IVb
sav.
Az
B xx
B zz
s xx
s zz
8
s xx,z
8
s zz,z
9
s xx,xx
9 y
s xx,y
9
s xx,zz
9
s zz,xx
9
s zz,zz
a
29.2
255.9
69.1
80.4
24.3
39.0
57.9
51.9
246.7
2267.8
2227.2
299.8
227.9
Atom C
Correlated
a
Others
31.97
32.7
254.9
254.1
26.9
66.2
86.6
25.09
39.12
56.4
51.8
157.0
2217.0
2184.3
2101.5
228.8
Basis IVa
SCF
Basis IVb
29.2
71.1
255.2
2440.2
48.5
76.3
2283.3
2743.5
24.2
39.2
57.6
50.6
2.5
2190.9
2221.5
2102.6
215.1
232.1
277.5
637.8
45.1
21039.0
3885.9
2293.9
21147.2
2126.8
Atom N
Correlated
b
Others
75.746
76.3
2428.6
2422.6
2290.8
2744.8
2751.45
225.287
277.812
620.3
45.3
2998.5
3855.5
2299.3
21112.5
2129.8
SCF
Basis IVa
Basis IVb
93.8
250.4
2418.7
1949.1
142.1
2513.5
687.6
5680.6
1.5
278.3
608.6
38.9
21137.3
1756.7
1616.1
21472.2
2151.0
2245.2
339.3
22895.1
257.2
2420.4
21037.7
216943.8
22667.6
2195.9
Correlated
c
Others
245.03
245.0
1927.0
1910.1
651.2
5667.0
5668.05
2237.29
339.48
22861.9
257.3
2408.0
21162.0
216901.2
22337.0
2199.5
Basis IVa
17.5
1434.9
2581.4
3551.0
2143.7
340.1
22128.9
246.8
1576.5
3629.0
210587.1
21716.8
2132.1
First number: Ref. 39. Second ~if present!: Ref. 75.
First number: Ref. 77; Second ~if present!: Ref. 75.
First number: Ref. 40; Second ~if present!: Ref. 75.
b
c
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Rizzo et al.: Electric field dependence of magnetic properties
8964
TABLE IX. H2O. Shielding constants ~ppm! and shielding polarizabilities ~ppm a.u.!. See text for details.
O atoma
H atom
SCF
sav.
Ax
Az
B xx
Byy
B zz
B xz
s xx
syy
s zz
8
s xx,x
8
s xx,z
s 8y y,x
s 8y y,z
8
s zz,x
8
s zz,z
9
s xx,xx
9 y
s xx,y
9
s xx,zz
s 9y y,xx
s 9y y,y y
s 9y y,zz
9
s zz,xx
9 y
s zz,y
9
s zz,zz
9
s xx,xz
s 9y y,xz
9
s zz,xz
Correlated
b
c
Our geometry
Geo. Ref. 40
Others
320.5
0.0
2437.1
3582.3
2653.7
2017.5
0.0
358.9
297.7
304.9
0.0
605.0
0.0
320.8
0.0
385.3
25212.6
2716.0
211375.5
28285.0
834.6
2145.1
27996.3
3803.5
2584.6
0.0
0.0
0.0
328.4
0.0
2378.2
3002.8
2547.8
1767.0
0.0
366.4
305.5
313.3
0.0
513.0
0.0
280.9
0.0
340.7
24282.0
2908.8
210068.2
27122.5
791.4
2142.3
26612.3
3404.1
2391.8
0.0
0.0
0.0
329.06,325.3
0.0
2381.6,2401.1
2946.1
2363.3
2074.0,2169.65
0.0
366.78
306.15
314.24
0.0
518.5
0.0
284.3
0.0
342.1
24182.0
21232.5
211045.0
27000.5
458.5
2388.1
26494.0
2954.0
21010.8
0.0
0.0
0.0
SCF
CAS A
CAS B
Our geometry
Geo. Ref. 40
338.3
0.0
2306.4
1845.9
2806.7
1253.5
0.0
372.1
315.0
327.9
0.0
455.5
0.0
220.0
0.0
243.8
23558.9
21064.0
29288.7
25247.6
1099.3
715.6
22268.9
4805.1
1051.8
0.0
0.0
0.0
335.3
0.0
2336.7
2125.2
2800.0
1367.6
0.0
367.7
312.6
325.6
0.0
521.2
0.0
235.0
0.0
254.0
23748.7
2950.5
29866.3
25723.3
1104.0
768.6
23278.9
4646.5
891.8
0.0
0.0
0.0
30.1
79.6
43.5
262.6
96.0
24.8
2273.7
37.9
22.8
29.7
256.1
250.3
287.4
253.6
295.1
226.6
23.2
2190.7
2.7
229.0
2131.6
271.1
149.6
2253.5
280.2
301.9
250.1
269.2
30.8
76.6
42.7
255.9
96.4
24.7
2254.5
38.7
23.2
30.3
253.2
249.7
285.1
252.5
291.5
225.8
24.8
2195.1
2.9
207.2
2129.4
270.1
132.8
2254.1
281.1
280.6
238.8
244.1
Correlated
b
Others
d
32.34,36.7
78.5
47.2,47.3
243.9
97.2
6.1,40.2
2262.4
39.52
25.22
34.59
CAS A
CAS B
30.2
73.3
42.0
229.3
128.3
42.5
2212.7
38.0
22.9
29.7
251.3
250.6
283.3
250.8
285.1
224.7
223.3
2277.8
212.5
137.9
2151.7
2117.0
61.5
2340.1
2125.7
258.4
213.6
166.1
30.2
74.9
42.2
231.0
126.9
45.5
2223.9
38.0
22.9
29.7
252.5
250.5
284.6
251.4
287.4
224.8
229.1
2271.2
223.2
144.5
2154.0
2124.3
70.8
2336.5
2125.6
271.5
220.8
179.5
a
0.0 values exact by symmetry.
Bond lengths and angle as in Ref. 40.
c
First number: Ref. 40. Second ~if present!: Ref. 75.
d
First number: Ref. 74 ~recomputed rotating the axes!. Second ~if present! Ref. 75.
b
the computed shielding derivatives. The strong geometry dependence of the shielding constants makes this comparison
difficult, in particular when various data do not differ significantly.
For the shielding polarizabilities, there are few experimental results, none directly related to the values calculated
here. We refer to the works of Grayson and Raynes39,40,74 for
a discussion of experimental data.
In the tables we use the following definitions for the
‘‘mean’’ shielding polarizability tensors ~Einstein summation
assumed!:
A z 52
1
3
8 ,z ,
s aa
B zz 52
1
6
9 ,zz ,
s aa
~45!
and similarly for the other components. These definitions are
the same as those used by Bishop and co-workers ~see, for
instance, Refs. 72 and 73! and by Raynes and collaborators
~see, e.g., Refs. 39 or 74!, but differ for B from Ref. 75 ~the
quoted values are hence scaled by a factor of 20.5 compared
to the values of Ref. 75!.
A. Shielding polarizabilities of N2
Table VI summarizes the results for the nuclear magnetic
shielding and shielding polarizabilities for N2 . The SCF results for all components are in fairly good agreement with
the data of Bishop and Cybulski.73 The largest difference ~for
9 ! is less than 5%, for the other components the differs zz,xx
ences are of the order of 1%–2% or less. The correlation
corrections computed with the CAS B wave function are in
general in good agreement with the MP2 values of Refs. 72
and 73. Only the final results for the x field derivatives differ
significantly from those of Bishop and Cybulski.72,73 However, the correlation correction has the same sign for all derivatives. For these components, the CAS correlation corrections are about 60% of the MP2 values, e.g., for s 9xx,xx both
SCF values are about 22000, our final result is 11494 and
Bishop and Cybulski obtain 12785. It is not clear at this
stage whether CAS underestimates or MP2 overestimates the
correlation effect. The value of s~N! itself computed in the
MCSCF approximation using large active space, agrees better than MP2 with experiment, which gives 261.660.2 ppm
for s~N! in N2 .16,70,76
B. Shielding polarizabilities of C2H2
Table VII presents the results for C2H2 . Our SCF values
for s~C! and its derivatives agree well with the literature ~see
also Refs. 47 and 68!. The correlation corrections to s8 and
s9, although significant, are much smaller than for N2 . Also
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Rizzo et al.: Electric field dependence of magnetic properties
for the A and B coefficients the SCF approximation yields a
reasonable value. This reflects the fact that for C2H2 , the
SCF values for s~C! and s~H! are, in contrast to s~N! in N2 ,
close to the correlated ~and experimental! results.
C. Shielding polarizabilities of HCN
The HCN results, summarized in Table VIII, are similar
to those for N2 and C2H2 . The correlation corrections are
most important for the N atom, starting with the change in
s~N!. As mentioned above, due to lower symmetry of HCN,
we were not able to use a CAS B active space. The accuracy
of the correlated results for s thus is lower. CAS A overestimates the correlation effects for s~C! and s~N!.71 Presumably, the same applies to the correlation corrections to the
shielding derivatives. Our SCF results differ significantly
from literature SCF results39,75 for some components of
s9~H! in C2H2 and HCN. Calculations placing the gauge origin at the H atom are very demanding and this is probably
the reason we observe for the H atom larger differences than
for other atoms. As we use LAOs and have employed larger
basis sets than those of Refs. 39 and 75, we believe that our
SCF results are more reliable.
D. Shielding polarizabilities of H2O
In Table IX, reporting the results for H2O, we have included only the nonzero derivatives of the diagonal tensor
components of s. We do not discuss the off-diagonal components and their derivatives. Basis IVa and IVb give the
same results for SCF. In Table IX we have shown the results
for CAS A and CAS B wave functions. As for other molecules, CAS A overestimates the correlation corrections.71
For the oxygen shielding derivatives, where the correlation
corrections are most significant, the CAS A results provide a
fair approximation to CAS B, but the differences are noticeable.
We have not analyzed systematically the dependence of
the properties on the geometry. For N2 , we refer to Bishop
and Cybulski72 for a careful study of the geometry dependence and rovibrational effects. We have noticed, however,
that our SCF results for shielding polarizabilities in H2O differ significantly from those of Grayson and Raynes.40 Therefore, we have repeated these SCF calculations using the geometry of Ref. 40 and as shown in Table IX the differences
become much smaller. This indicates that the geometry dependence of the shielding polarizabilities is quite strong,
which is not surprising considering the large changes in s~O!
with molecular geometry.
VI. CONCLUSIONS
For many of the properties studied in this work, our
calculations represent the first analysis going beyond the
SCF approximation. The calculations confirm the strong dependence of these properties on the basis set and correlation
corrections. It appears that for small molecules at the SCF
level it is possible to saturate the basis set. It is much more
difficult to reach convergence in the treatment of correlation
effects.
8965
A mixed analytic-numerical differentiation technique
was used to compute the hypermagnetizabilities and shielding polarizabilities. The calculations, although time consuming, are feasible.
When the first derivatives ~third-order properties! are
non zero, the calculations appear to be rather accurate. For
second derivatives the accuracy is lower, and the allperpendicular tensor components for linear molecules are
particularly difficult to describe. For these components, correlation effects may change both the sign and the order of
magnitude of the SCF results. Considering the size of the
correlation corrections to the second-order magnetic properties in the molecules we have studied, this is not very surprising. However, it shows that an accurate and reliable calculation of magnetizability and shielding derivatives requires
the use of very large basis sets, a method which ensures
gauge origin invariance and, at the same time, a wavefunction that describes all the important correlation effects.
ACKNOWLEDGMENTS
The authors wish to express their thanks to W. T. Raynes
for kindly providing preprints of his recent work and for his
comments on this manuscript. A major part of this work has
been done during a visit of A. B., M. J., and A. R. to Aarhus.
This work has been partly supported by the Danish National
Research Council ~Grant No. 11-0924!.
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J. Chem. Phys., Vol. 102, No. 22, 8 June 1995
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