<—
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Some Recent Developments of
High-Order Response Theory
Y. LUO,1 D. JONSSON,1 P. NORMAN,1 K. RUUD,1 O. VAHTRAS,1
1
˚
B. MINAEV,1 H. AGREN,
A. RIZZO, 2 K. V. MIKKELSEN 3
1
Department of Physics and Measurement Technology, Linkoping
University, S-58183, Linkoping,
¨
¨
Sweden
2
Istituto di Chimica Quantistica ed engergetica Molecolare, Consiglio Nazionale delle Ricerche, Via
Risorgimento 35, I-56126, Pisa, Italy
3
Department of Chemistry, University of Copenhagen, DK-2100 Copenhagen, Denmark
Received 24 October 1997; revised 11 February; revised 26 February 1998
ABSTRACT: We review some recent developments of high-order response theory and
illustrate the utility of this theory for a selection of nonlinear properties. 䊚 1998 John
Wiley & Sons, Inc. Int J Quant Chem 70: 219᎐239, 1998
Introduction
A
lthough response theory by now is a wellestablished branch of quantum chemistry for
predictions of properties and spectra, it still undergoes a vivid development. This holds for new
types of reference states, giving improved accuracy of properties for a given system, for applicability to a larger spectrum of systems with respect
to both size and type, as well as for the order and
the nature of the properties that can be addressed.
This means properties generated by crossing electric and magnetic fields, which can be time-independent or time-dependent, and which can be of
internal, external, or mixed origin. It was the purpose of this article to briefly review a selected set
˚
Correspondence to: H. Agren.
International Journal of Quantum Chemistry, Vol. 70, 219᎐239 (1998)
䊚 1998 John Wiley & Sons, Inc.
of topics falling under the development of highorder response theory and which were presented
at the IXth International Congress of Quantum
Chemistry in Atlanta in 1997. This includes implementations of ‘‘new’’ high-order properties, applications on extended systems, and predictions of
macroscopic quantities. We will, in particular, consider the following topics:
1. The calculation of fourth-order properties using self-consistent-field and multiconfigurational self-consistent-field reference functions. The power of the response theory to
address nonlinear optical properties will be
demonstrated for the highly correlated
molecule acetonitrile, as well as for the study
of nonlinear properties of large molecules
like fullerenes, diphenylpolyenes, and oligothiophenes using direct and parallel implementations of the response theory.
CCC 0020-7608 / 98 / 010219-21
LUO ET AL.
2. We will show how a parameter-free dielectric
continuum model can be constructed and use
this model to calculate nonlinear response
properties, taking proper account of the inertial and optical polarization of the solvent.
This enables accurate predictions of macroscopic quantities using Žmulticonfigurational.
self-consistent reaction field response theory.
3. Excited-state molecular properties can be obtained from an ordinary ground-state reference wave function by taking a double
residue of the response function. We will
show how this allows studies of excited states
of multireference or open-shell character using a restricted Hartree᎐Fock reference function, as well as the calculation of excited-state
properties using integral-direct and parallel
approaches, making it possible to study excited-state properties of large molecules.
4. Molecules with unpaired electrons have a
permanent magnetic dipole moment and thus
often display special electron-spin-related
properties. Such properties require that
proper account is taken of the electron spin
in the evaluation of the response functions.
We will demonstrate how response theory
calculations can be used to study magnetic
phosphorescence and electronic g tensors.
5. Finally, we will consider the application of
response theory to the magnetooptical effect
known as the Cotton᎐Mouton effect ŽCME..
The use of higher-order response theory allows studies of the dispersion of the CME,
and by a suitable choice of perturbation-dependent basis sets, size extensivity may be
ensured in the calculation, enabling studies
of the CME of liquid substances using semicontinuum models.
Response Functions and
Molecular Properties
Response theory is a formulation of time-dependent perturbation theory where the response functions describe how a molecular property responds
to an external perturbation. The Fourier coefficients in the expansion of the expectation value of
a molecular property A in the presence of a timedependent perturbing field V ␻ define response
220
functions of different orders:
²˜
0 Ž t . < A <˜
0 Ž t .:
s ²0 < A <0: q
⬁
⬁
⬁
⬁
⬁
Hy⬁²² A; V
␻1
␻ 1 :: yi ␻ 1
e
d␻1
, V ␻ 2 :: eyiŽ ␻ 1q ␻ 2 . d ␻ 1 d ␻ 2
q 12
Hy⬁Hy⬁²² A; V
q 16
Hy⬁Hy⬁Hy⬁²² A; V
⬁
␻1
, V ␻ 2 V ␻ 3 ::
=eyi Ž ␻ 1q␻ 2q␻ 3 . d ␻ 1 d ␻ 2 d ␻ 3 ,
Ž1.
where, for example, the cubic response function
²² A; V ␻ 1 , V ␻ 2 , V ␻ 3 :: contains all terms of third
order in the perturbation expansion. The time development of the reference state is governed by the
time-dependent Schrodinger
equation
¨
Ž H0 q V t . <˜
0 Ž t .: s i
d
dt
<˜
0 Ž t .: ,
Ž2.
where H0 is the Hamiltonian for the unperturbed
system, and V t , a time-dependent perturbation
operator. As an example, the dipole moment ␮ i n d
induced by an electric field F can be written as
␮ i n d s ␮ q 12 ␣ F q 16 ␤ FF q
1
24
␥ FFF q ⭈⭈⭈
Ž3.
Here, the expansion coefficients define polarizabilities of different orders: ␮ is the permanent dipole
moment, ␣ is the polarizability, ␤ is the Žfirst.
hyperpolarizability, and ␥ is the second hyperpolarizability.
When the perturbations are dipole operators,
the linear and nonlinear optical properties are given
directly by the response functions w compare Eqs.
Ž1. and Ž3.x . These relations are most easily seen in
the spectral representation of the response functions which for exact states are identical to the
sum-over-states expressions obtained from timedependent perturbation theory.
SPECTRAL REPRESENTATION
The linear response function describes secondorder properties such as the polarizability:
␣ A B Ž ␻ 0 ; ␻ 1 . s y²² A; B ::␻ 1
s PÝ
p
²0 < A < p :² p < B <0:
␻p y ␻1
.
Ž4.
VOL. 70, NO. 1
HIGH-ORDER RESPONSE THEORY
The residue at an excitation energy gives the transition moment to the corresponding excited state.
lim Ž ␻ 1 y ␻ e . ␣ A B Ž ␻ 0 ; ␻ 1 . s ²0 < A < e :² e < B <0: .
␻ 1ª ␻ e
Ž5.
The quadratic response function describes thirdorder properties such as the Žfirst. hyper-polarizability:
␤A B C Ž ␻ 0 ; ␻ 1 , ␻ 2 . s y²² A; B, C ::␻ 1 , ␻ 2
sPÝ
²0 < A < p :² p < B < q :² q < C <0:
p, q
Ž ␻ p q ␻ 0 .Ž ␻ q y ␻ 2 .
Ž6.
A double residue determines first-order properties
of an excited state:
lim
␻ 1ªy ␻ e
Ž ␻ 1 q ␻ e . lim Ž ␻ 2 y ␻ e .
␻ 2ª ␻ e
= ␤A B C Ž ␻ 0 ; ␻ 1 , ␻ 2 .
s ²0 < B < e :² e < A < e :² e < C <0:
s ²0 < B < e :Ž² e < A < e : y ²0 < A <0:.² e < C <0: . Ž 7 .
The cubic response function describes fourth-order
properties such as the second hyper-polarizability:
␥A B C D Ž ␻ 0 ; ␻ 1 , ␻ 2 , ␻ 3 .
s y²² A; B, C, D ::␻ 1 , ␻ 2 , ␻ 3
²0 < A < p :² p < B < q :² q < C < r :² r < D <0:
sP
Ý
p, q , r
PÝ
yP
p, q
Ž ␻ p q ␻ 0 .Ž ␻ q y ␻ 2 y ␻ 3 .Ž ␻ r y ␻ 3 .
²0 < A < p :² p < B <0:²0 < C < q :² q < D <0:
Ž ␻ p q ␻ 0 .Ž ␻ p y ␻ 1 .Ž ␻ q y ␻ 2 .
,
Ž8.
and where excited-state second-order properties
can be calculated from a double residue:
lim
␻ 2ªy ␻ e
Ž ␻ 2 q ␻ e . lim Ž ␻ 3 y ␻ e .
␻ 3ª ␻ e
=␥A B C D Ž ␻ 0 ; ␻ 1 , ␻ 2 , ␻ 3 .
s ²0 < C < e :Ž ␣ A0 B Ž ␻ 0 ; ␻ 1 . y ␣ Ae B Ž ␻ 0 ; ␻ 1 ..
=² e < D <0: ,
Ž9.
where P permutes the operators and their corresponding frequencies, y␻ 0 s ␻ 1 q ␻ 2 q ⭈⭈⭈ and
² p < B < q : s ² p < B y ²0 < B <0:< q :.
A major advantage of the response formalism
is that the traditional method of summations over
excited states as indicated in the equations above is
replaced by the solution of linear sets of equations
that can be solved without prior knowledge of the
excited states. This makes it possible to apply the
formalism to problems with large dimensions, enabling both studies of large molecular systems as
well as the use of accurate reference wave functions Žinvolving , 10 9 parameters.. In a sumover-states approach, such large problems would
require the number of excited states included in
the summation to be truncated, but as the contribution from the individual intermediate states vary
considerably in both magnitude and sign, this may
lead to inaccuracies which thus are avoided in the
response formulation. Another advantage of the
response formalism compared to energy-derivative
techniques and finite-field methods is that it allows calculations to be carried out at the frequencies of the experiments, as well as avoiding the
inaccuracies inflicted by numerical differentiation.
COMPUTATIONAL STRATEGIES
Response functions up to third order Žthe cubic
response function. and related residues have been
implemented in the DALTON quantum chemistry
program w 1x for both SCF and MCSCF reference
wave functions, following the general response
function formalism of Olsen and Jørgensen w 2x . The
working formulas for the most recent extension of
the response function formalism to SCF and MCSCF cubic response functions were presented in w 3x
and w 4x , respectively.
Several features of our implementation enables
large linear, quadratic, and cubic response calculations: Ž1. the iterative solutions of the response
equations without explicit construction of the matrices involved; Ž2. the evaluation of response
functions as a contraction over response vectors,
again avoiding explicit construction of the involved matrices; and Ž3. the implementation of the
different terms and contributions to the response
function as expectation values or gradients of multiply one-index transformed operators, which can
be expressed as modified density matrices.
Three additional features enable applications of
response theory to systems of the same size as
those that can be reached by direct SCF: Ž1. The
solution of the response equations and the evaluation of the response function is accomplished us-
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
221
LUO ET AL.
ing Fock matrices that are generated ‘‘on-the-fly’’
in the atomic orbital basis; Ž2. the Fock matrix constructions are parallelized using both the Message
Passing Interface ŽMPI. and the Parallel Virtual
Machine ŽPVM. protocols with more than 95%
slave efficiency in calculations using up to 64 processors; and, finally, Ž3. the use of integral screening
which, as shown in w 5x , can be based on any
general density matrix, enabling integral screening
to be applied in all parts of the calculation of the
response functions.
scribe dynamic correlation effects. In both cases,
the active spaces have been chosen on the basis of
an MP2 natural orbital occupation analysis.
We note that the polarizability displays only
modest correlation effects and that all methods
provide results only slightly smaller than experiment. The SCF dynamic values for the polarizability and the second hyperpolarizability are found to
be in good agreement with the experimental results. The dispersion for ␥ is well described at the
SCF level, the SCF dispersion Ž46%. being close to
the RAS dispersion Ž43%.. We note that even
though there are significant discrepancies among
the correlated results for ␥ , our RAS wave function
gives good agreement with experiment when dispersion is taken into account.
For the dipole moment ␮ and the first hyperpolarizability ␤ , the situation is completely different,
as both properties depend strongly on electron
correlation. The SCF dipole moment value Ž1.67
au. overshoots experiment by 8%, whereas the
correlated results from the CAS and RAS wave
functions are in good agreement with both experiment as well as MP2 Ž1.53 au. and CCSDŽT. Ž1.52
au. calculations w 7x Žsee Table I. The SCF value of
␤ is too small compared to experiment and only a
minor correlation correction is obtained with the
CAS reference state. However, we would expect
dynamic correlation to be important for this
molecule. As shown in Table I, the RAS results for
␤ compare well with both the MP2 and CCSDŽT.
results.
Response Calculations of
Fourth-Order Properties
In this section, we apply cubic SCF and MCSCF
theory to calculate the nonlinear optical properties
of molecules when electron correlation is important, here illustrated by the acetonitrile molecule,
as well as to study the nonlinear properties of
large molecules like C 60 and large oligomers.
ELECTRON CORRELATION: ACETONITRILE
We recently calculated the static and dynamic
electronic properties of gas-phase acetonitrile w 6x
using both SCF and MCSCF wave functions. The
calculated nonlinear properties are related to experiment through the third-order nonlinear susceptibility ␹ Ž3. defined by
␳
␹ Ž3. Ž y2 ␻ ; ␻ , ␻ , 0 . s
4
l␻e sh g ²␥ :ES H G ,
Ž 10.
FULLERENES: C6 0
where
²␥ :ES H G s ␥ Ž y2 ␻ ; ␻ , ␻ , 0 . q
␮ z ␤ z Ž y2 ␻ ; ␻ , ␻ .
3kT
Ž 11.
␥ Ž y2 ␻ ; ␻ , ␻ , 0 . s
␤ z Ž y2 ␻ ; ␻ , ␻ . s
1
15
1
5
Ý Ž2␥ i i j j q ␥ i j ji . ,
Ž 12.
i, j
Ý Ž ␤ z i i q 2 ␤i i z . .
Ž 13.
i
For the polarizability ␣, the average value is defined as one-third of the trace of the polarizability
tensor. In the MCSCF calculations, we used both a
small complete active space ŽCAS. wave function
to investigate static correlation effects and a restricted active space ŽRAS. wave function to de-
222
,
Ab initio calculations of the polarizability and
hyperpolarizability of C 60 in the gas phase were
recently presented in w 9x . For such a large molecule,
the calculation of the cubic response function in
the random-phase approximation ŽCRPA. relies
heavily on the use of an integral-direct and parallel implementation of the response function, as
described previously w 10x .
Our results together with a selection of other
theoretical estimates of the static polarizability and
hyperpolarizability of C 60 are collected in Table II.
We note that the published RPA polarizabilities w 9,
11, 12x vary significantly, clearly showing the importance of using a properly designed basis set in
the calculation of electric properties. The basis set
used in w 9x contains diffuse p and d functions and
has been tailored for the calculation of optical
molecular properties and extensively tested on the
VOL. 70, NO. 1
HIGH-ORDER RESPONSE THEORY
TABLE I
Optical properties for acetonitrile in the gas phase (au) ᎏpermanent dipole moment ␮, polarizability ␣ (y␻; ␻ ),
and hyperpolarizabilities ␤(y2␻; ␻, ␻ ) and ␥ (y2␻; ␻, ␻, 0).a
Static
SCF
␮z
␣
Dispersion
␤z
Dispersion
␥
Dispersion
␮ z ␤ z / 3 kT d
␥ + ␮ z ␤ z / 3 kT d
4 ␹ (3 ) / ␳ d
b
MP2
b
CCSD(T
)b
1.69
28.3
1.53
28.7
1.52
28.9
6.12
27.8
24.2
3012
3870
3.65
6.67
15.0
18.9
4240
13.0
17.2
Dynamic
SCF
CAS
RAS
1.67
28.81
1.53
27.87
1.52
27.93
5.44
8.67
8.67
5.89
8.86
8.86
12.5
15.8
15.8
SCF
b
MP2
2
SCF
CAS
RAS
Exp.c
1.69
1.53
1.67
1.53
1.52
1.54
29.1
29.5
29.62
28.63
28.70
30.4
2.8% 2.8%
2.8%
2.7%
2.8%
9.22
10.89
23.29
10.5
31.1
12.84
13.88
32.13
26.3
72%
12%
39%
27%
38%
3227
2979
3269
4427
4720
4225
4678
4619
47%
46%
42%
43%
6.27
10.7
7.57
12.3
12.3
7.50
11.7
11.7
17.3
22.0
22.0
14.3
18.9
18.9
a
SCF / CAS / RAS data from [6]. Dynamic values at 514.5 nm. Temperature is 298 K.
Quoted from [7].
c
Experimental values for polarizabilities are from [8], and values for dipole moment and hyperpolarizabilities are quoted from [7].
d
Quantities in =10 3 au.
b
hyperpolarizabilities of smaller organic molecules
w 10, 16᎐18x . We believe that the value obtained in
w 9x for ␣, 75.3 = 10y2 4 cm 3, should be close to a
future experimental gas-phase value. The results
for the polarizability obtained from different LDA
calculations w 14, 15x overshoot our ab initio result.
In w 9x , we also presented the first ab initio result
for the hyperpolarizability of C 60 , 5.73 = 10y3 5 esu.
There has been confusion among various experimental observations, partly due to the use of different conventions for the hyperpolarizability, and
the experimental data are quite disparate, see
w 19᎐21x . There are a number of factors that make a
comparison among these results difficult, as for
instance, the type of optical process used, the phase
TABLE II
Calculated values for static polarizability and
hyperpolarizability of C 60 .
␣ (10 y 2 4 cm 3 )
␥ (10 y 3 6 esu)
Methods
75.3
65.4
78.8
77.9
82.7
82.9
57.3
DD᎐CRPA [9]
RPA [11]
RPA [12]
LDA [13]
LDA [14]
LDA᎐RPA [15]
15.9
42.0
29.4
of the sample, and the reference standard w 22x ,
which vary among the different investigations. As
an example, we note that the resonant third-order
nonlinear susceptibility of C 60 recently was measured by degenerate four-wave mixing w 21x , reporting an estimated upper bound to ␥ at about
37 = 10y3 6 esu, which is lower than all other experimental values reported. The situation becomes
even more tricky when comparing experimental
and theoretical values, since experiments are carried out at a finite frequency and since a considerable amount of resonant contributions is incorporated in most experiments.
The hyperpolarizability of conjugated molecules
is closely connected with the delocalized nature of
the ␲ electrons. The optical nonlinearity of C 60 has
often been compared to that of the benzene
molecule, and different values for the ratio between Žhyper-.polarizabilities of C 60 and benzene
have been reported w 21, 23, 24x . We calculated the
polarizability and hyperpolarizability of benzene
at the same level of approximation as for C 60 w 9x .
The average values for these quantities were found
to be about eight times smaller than the corresponding values of C 60 . The expected increase of
nonlinearity going from benzene to C 60 is thus not
as steep as previously suggested w 21, 24, 25x , indicating that pure fullerenes are not likely candidates for nonlinear materials.
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
223
LUO ET AL.
OLIGOMERS: DIPHENYLPOLYENES,
OLIGOTHIOPHENES
In the oligomer approach, one considers a sequence of polymer segments Žoligomers. as models of the full polymer and studies the convergence
of a molecular property or a spectrum as this
sequence increases. Of special interest is the limiting behavior of the properties of the monomer
building blocks as the oligomers approach a size
that may be considered to represent the infinite
Žpolymer. system. In the case of polarizabilities
and hyperpolarizabilities, one tries to reach the
correlation length, which is the number of monomer
units after which each additional unit only gives
an additive contribution to the property. The
oligomer approach allows an easy decomposition
of spectra into local contributions and gives an
interpretation in terms of building blocks. However, the method is, in general, computationally
demanding and requires size-extensive methods
that must be applicable also to larger systems, that
is an appropriate scaling of the computational effort with the system size is required. When using
RPA, this means that the full set of computational
strategies—including the double-direct approach,
the parallelization, and the integral screening techniques briefly commented in the section Computational Strategies—must be employed. Theory can
play an important part in studies of the correlation
length as experimental studies on the saturation
behavior of optical properties of oligomers are rare
because of the difficulties of synthesis.
Large-scale cubic response calculations were recently used to show that the correlation length can
be theoretically obtained for both the polarizability
and the hyperpolarizability w 5x . In the particular
case of the diphenylpolyenes, the polarizability
and the second hyperpolarizability are saturated at
approximately 30 and 40 polyene units, respectively, as seen in Figure 1. These results indicate
that the second hyperpolarizability saturates much
faster if an idealized polymer geometry is used w 5x .
It is thus important to account for the large changes
in the bond-length alternation that are present in
the polymer as this bond-length alternation converges slowly with the number of monomer units
in the oligomer. The theoretically predicted saturation occurs at a significantly smaller number of
repeating units than what has been observed experimentally, possibly due to disorder in the experimental samples and other structural defects in
the chain w 26x .
224
FIGURE 1. The polarizability (xx component) and the
second hyperpolarizability (xxxx component) per polyene
unit as functions of the number of polyene units in the
diphenylpolyenes. X is the number of polyene units plus
three. The two horizontal lines show the saturation region,
defined as the region where the per-unit value of the
property is within 5% of the estimated limiting value of
the value of the per-unit property.
VOL. 70, NO. 1
HIGH-ORDER RESPONSE THEORY
Closely related to the saturation of the polarizability and hyperpolarizability is the saturation of
the optical band gap. The development of efficient
quantum chemistry methods w 18, 26᎐29x such as
the response techniques in the RPA approximation
enables calculations of the saturation of the band
gap. In contrast to linear polyenes, for which the
saturation is observed to be slow w 30x , the saturation is comparatively fast for, for example oligothiophenes w 31x for which measurements have been
made. In these experiments, the saturation region
was found to start at about seven repeat units.
RPA results for the optical band gap of oligothiophenes are compared with experimental data
w 31, 32x in Figure 2. The agreement improves as the
chain length increases. In fact, except for the thiophene monomer, the RPA results are within 0.2 eV
of the experimental data, confirming previous observations for trans- and diphenylpolyenes that the
RPA provides excellent results for the optical band
gap of conjugated oligomers w 17, 33, 34x .
Solvent Modeling and
Macroscopic Properties
An alternative way to obtain ‘‘scalability’’ toward larger systems is to model the interaction
with an environment, for instance, by the solvent
reaction-field model. This model was combined
with response theory by Mikkelsen et al. w 35x who
presented the working equations for linear solvent
response theory using SCF and MCSCF reference
states assuming equilibrium solvation, that is, a
model in which a single dielectric constant ⑀ st is
used to characterize the surrounding dielectric
medium and in which the solvent polarization is
equilibrated with the solute charge distribution.
This approach was later refined w 36x to divide the
solvent polarization into two parts—one equilibrated optical polarization arising from the instant
response of the solvent’s electronic degrees of freedom, and one nonequilibrated inertial polarization
arising from the slow response of the solvent’s
nuclear degrees of freedom. Hence, the nonequilibrium solvent model involves two dielectric constants, ⑀ st and ⑀ o p , for the two respective types of
polarization. In addition, the model requires the
radius of the spherical cavity in which the solute is
placed to be given. If, however, one demands
consistency between the macroscopic and microscopic polarization, only ⑀ st is needed to describe
the dielectric medium together with consistent local-field factors. Hence, a parameter-free cavity
model can be realized, as shown in the following
discussion. We recently extended the nonequilibrium solvation theory to include quadratic and
cubic response properties w 37, 38x at the SCF and
MCSCF levels of theory.
The molecular properties obtained from the reaction-field response theory at any computational
level can, in general, be written as
␮ s R␮ Ž ⑀ , a . ,
␣ s R ␣ Ž ⑀ , a.
␤ s R␤ Ž ⑀ , a . ,
FIGURE 2. Comparison between RPA and
experimental optical band gaps ( E g ) of oligothiophenes.
Two sets of experimental data, Exp. I (filled circle) [31]
and Exp. II (diamond) [32], are shown.
Ž 14.
with parametric dependencies on the dielectric
constant ⑀ and the cavity radius a.
One of the major weaknesses of this model is
the involvement of one or more cavity-size the
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
225
LUO ET AL.
parameters, commonly determined on the grounds
of physical intuition, and different choices for the
cavity radius can therefore be found in the literature. Recently, we presented a method based on
the combined use of the classical continuum approach and modern quantum chemistry reactionfield theory for determining the Onsager spherical
cavity radius of pure liquids w 39x . This method
provides a unique value for the radius of the
cavity for each electronic structure model. It has
been shown that this method not only enables an
accurate determination of the solute’s molecular
properties, but also a calculational procedure to
obtain local-field factors and thereby the macroscopic properties of pure liquids at the same theoretical level w 40, 41x .
Within the framework of a solvent continuum
spherical cavity model, the macroscopic and microscopic properties are connected through the sow 42x relation
called Onsager᎐Bottcher
¨
Ž ⑀ 0 y 1 .Ž 2 ⑀ 0 q 1 .
3⑀
s
0
4␲ N
1 y f 0 Ž a. ␣ 0
␮2
= ␣0 q
,
3kT 1 y f 0 Ž a . ␣ 0
1
ž
/
1 2 Ž ⑀ 0 y 1.
Ž 16.
a3 2 ⑀ 0 q 1
in the dipole approximation, introducing the cavity radius a.
By combining Eqs. Ž14. and Ž15., a deviation
function is introduced:
F 1 Ž ⑀ 0 , a. s
Ž ⑀ 0 y 1 .Ž 2 ⑀ 0 q 1 .
y
3⑀ 0
4␲ N
1 y f 0 Ž a . R ␣ 0Ž ⑀ 0 , a .
ž
1
R␮2 Ž ⑀ 0 , a .
3kT 1 y f 0 Ž a . R ␣ 0Ž ⑀ 0 , a .
/
, Ž 17.
and the cavity radius a 0 can thus be determined as
a zero crossing of the deviation function F 1 Ž ⑀ 0 , a..
The molecular optical properties of the solvated
226
Žn
˜␻2 y 1.Ž2 n˜␻2 q 1.
3ñ␻2
y
4␲ NR ␣Ž ␻ . Ž n
˜␻ , a0 .
1 y f␻ Ž a 0 . R ␣Ž ␻ . Ž n
˜␻ , a0 .
,
Ž 18.
and that the proper refractive index Ž n␻ . is found
at the zero crossing of this function F 2 Ž n
˜␻ , a0 . s 0.
The dynamic polarizabilities of the solute molecule
are then given by R ␣ Ž n␻ , a 0 . and R␤ Ž n␻ , a 0 ..
In a similar fashion, the expression for the DFRI
can be found to be
d n␻2
dd
s
1
n␻2 Ž n␻2 y 1 .Ž 2 n␻2 q 1 .
d 2 n4 q 1 y 2 Ž n2 y 1 . 2rv
␻
␻
3
,
Ž 19.
where v s 43 ␲ Na2 represents the fraction of the
volume occupied by the molecules.
Local-field factors are important for obtaining
nonlinear optical susceptibilities. The general expressions for the static and dynamic local-field
factors in the framework of the continuum model
can be written as
l0 s
⑀0 y 1
4␲ N R ␣ 0Ž ⑀ 0 , a 0 . q R␮2 Ž ⑀ 0 , a 0 . r3kT
l␻ s
= R ␣ 0Ž ⑀ 0 , a .
q
F2Žn
˜␻ , a0 . s
Ž 15.
and the static reaction field factor f 0 is defined as
f 0 Ž a. s
molecule are given by R␮Ž ⑀ 0 , a 0 ., R ␣ Ž ⑀ 0 , a0 ., and
R␤ Ž ⑀ 0 , a 0 . and so on.
A general approach was recently proposed for
calculating dielectric and optical properties such as
the refractive index, the density fluctuation of the
refractive index ŽDFRI., and the nonlinear optical
susceptibility of pure liquids at the ab initio level
w 40, 41x . It was shown that for a given cavity
radius a 0 one can introduce another deviation
function F 2 Ž n
˜␻ , a0 . w 40x :
n␻2 y 1
4␲ NR ␣Ž ␻ . Ž n␻ , a 0 .
.
Ž 20.
Ž 21.
The expressions in Eqs. Ž18. ᎐ Ž21. can be compared
with the similar Lorenz᎐Lorentz expressions which
can be found, for instance, in w 42x .
Liquid benzene and acetonitrile are good model
molecules for nonpolar and strong polar dielectrics, respectively, and were used in w 40x to
construct deviation functions and consistent cavity
radii according to the theory outlined above. The
deviation functions F Ž ⑀ 0 , a. for benzene obtained
at the RPA level are shown in Figure 3, in which
the vibrational contribution to the polarizability
␣ s R ␣ Ž ⑀ , a. also has been included. The values
VOL. 70, NO. 1
HIGH-ORDER RESPONSE THEORY
FIGURE 3. Deviation functions F ( ⑀ , a ) versus cavity radius a. The zero crossing of the functions gives the real cavity
radius. (a) For liquid benzene with different basis sets. The three curves shown correspond to the cases: ( —x— ) using
basis set b0, only electronic contributions; ( ᎐`᎐ ) using basis set b0, total electronic and vibrational contributions;
( ᎐ U ᎐ ) using basis set b1, total electronic and vibrational contribution. The three commonly used cavity radii, (i) ᎐ (iii)
(explained in the text), are indicated by dashed lines. (b) For acetonitrile. The van der Waals-based cavity radius is
6.085 au, and the liquid density (0.786 g cm y3 )-based cavity radius is 5.103 au.
obtained for the cavity radius a 0 are strongly dependent on basis set and to the inclusion of the
pure vibrational contributions. The refractive index, on the other hand Žlisted for different frequencies in Table III., is quite stable with respect to the
basis set and agrees with experimental data to
within 1%. For comparison, the widely used
Lorenz᎐Lorentz ŽL᎐L. equation for the refractive
index is also listed, and we note that there are
significant differences.
We calculated the frequency-dependent polarizabilities using three different cavity radii com-
monly used in the literature: Ži. from the density of
the pure liquid w 47x , Žii. the molecular length plus
the van der Waals’ radius of the outermost atom,
˚ . to the
and Žiii. the addition constant term Ž0.5 A
Ž
.
w
x
value obtained from model ii 48, 49 . For the
lowest frequency Ž ␻ s 0.043 au., the refractive index becomes 1.584, 1.545, and 1.538 for these three
choices of cavity radius, respectively, which are to
be compared with our value of 1.494 and the
experimental value of 1.4825 w 43x . The DFRI calculated at different frequencies obtained with Eq.
Ž19. also shows excellent agreement with the ex-
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
227
LUO ET AL.
TABLE III
Refractive index, density fluctuation of the refractive indices, and local-field factor for liquid benzene and
liquid acetonitrile.
Refractive index
␻
Benzene
0.000
0.0430
0.0770
0.0834
0.0908
0.1050
Acetonitrile
0.0000
0.0770
0.0937
0.1049
[40]
Density fluctuation
a
Exp.
[40]
L᎐L
1.494
1.515
1.4825 b
1.4979 c
1.525
1.537
c
1.5075
1.5196 c
1.655
1.732
1.758
1.776
1.837
1.910
2.007
2.039
2.068
2.146
1.320
1.324
1.327
1.3416 c
1.3456 c
1.3490 c
1.067
1.083
1.095
1.296
1.317
1.335
Local-field factor
Exp.
1.80 d
1.12 e
This work
L᎐La
1.343
1.332
1.352
1.355
1.358
1.366
1.425
1.399
1.415
1.419
1.424
1.436
1.701
1.191
1.195
1.198
1.91
1.267
1.270
1.273
a
Experimental values of refractive index were used.
Experimental value quoted from [43].
c
Experimental values quoted from [44].
d
Experimental values quoted from [45].
e
Experimental value quoted from [46]
b
perimental data, whereas the result from the L᎐Ltype formula overshoots by 13%.
In Table III, we present the consistently obtained refractive indices, DFRI, and local-field factors for liquid acetonitrile and compare them with
the experimental data. The theoretical values for
the refractive index are all within 2%, which is
comparable to what was found for benzene. The
DFRI calculated from the model at ␻ s 0.043 au
underestimates the experimental value by 6%.
However, this is a considerable improvement compared to the L᎐L value, which overshoots experiment by 17% Žcf. Table III.. Finally, we present in
Table III the local-field factors that relate the external Maxwell field and the molecular field. Accurate local-field factors are a prerequisite for a
comparison between hyperpolarizability measurements in the liquid phase and theoretical models,
in particular since the measurement involves the
combination of one static and three optical fields,
thereby yielding a total field factor of l 0 l␻2 l 2 ␻ .
Having prepared a consistent cavity size and
local-field factors, we can now return to the nonlinear optical properties of acetonitrile, analyzed in
the section Response Calculations of Fourth-Order
Properties, and predict the corresponding macroscopic quantities of liquid acetonitrile, in particular, the temperature-dependent third-order nonlin-
228
ear susceptibility ␹ Ž3. w cf. Eq. Ž10.x . Results at the
SCF, CAS, and RAS levels are shown in Table IV.
Compared with the gas-phase results collected in
Table I, dramatic solvent effects on the hyperpolarizabilities can be observed, as ␤ and ␥ increase by
approximately 100 and 20%, respectively. In contrast, the dispersion of the hyperpolarizabilities is
quite similar in the gas and liquid phases—about
25% with the nonequilibrium solvent model and
about 45% with the equilibrium solvent model,
which thereby marks the importance of treating
properly the various degrees of freedom in the
solvent. The experimental value of 1.4 = 10 5 au for
the third-order nonlinear susceptibility, ␹ Ž3., is in
excellent agreement with our ‘‘best’’ estimation of
1.44 = 10 5 au, evaluated at 298 K with the RAS
wave function. We note that dispersion, nonequilibrium polarization, and a consistent cavity radius
and local-field factors are all necessary ingredients
in order to obtain agreement with experiment.
Response Calculations of
Excited-State Properties
In contrast to ground-state properties, which
often can be determined with high accuracy both
by experiment and ab initio calculations, experi-
VOL. 70, NO. 1
HIGH-ORDER RESPONSE THEORY
TABLE IV
Optical properties for acetonitrile in the liquid phase (au) ᎏpermanent dipole moment ␮, polarizability
␣ (y␻; ␻ ), and hyperpolarizabilities ␤(y2␻; ␻, ␻ ) and ␥ (y2␻; ␻, ␻, 0).
Dynamic
Static
␮z
␣
Dispersion
␤z
Dispersion
␥
Dispersion
␮ z ␤ z / 3 kT c
␥ + ␮ z ␤ z / 3 kT c
4 ␹ (3 ) / ␳ c
Equilibrium
SCF
CAS
RAS
1.83
31.11
1.66
30.02
1.66
30.14
31.87
30.75
51.26
4434
20.6
24.9
208
4087
18.0
22.1
185
4562
30.1
34.7
289
SCF
CAS
Nonequilibrium
RAS
SCF
CAS
RAS
Exp.b
1.83
1.66
1.66
1.83
1.66
1.66
32.07
30.90
31.04
30.48
29.43
29.56
3.1%
2.9%
3.0%
y2.0%
y2.0%
y2.0%
46.98
42.85
74.06
40.56
37.62
63.18
47%
39%
44%
27%
22%
23%
6592
5891
6690
5590
4998
5649
49%
44%
47%
26%
22%
24%
30.4
37.0
125
25.1
31.0
105
43.4
50.1
169
26.2
31.8
108
22.1
27.1
91.7
37.0
42.6
144
140
a
Dynamic values at 514.5 nm except the experimental hyperpolarizability value, which is obtained at 532 nm. Temperature is 298
K. Data are from [40].
b
Experimental value is quoted from [50].
c
Quantities in =10 3 au.
mentally determined excited-state properties are
less commonly available and, in general, of lower
accuracy. Most experimental techniques rest on
measurement in solution and depend on models
with a large number of parameters w 51᎐55x . Recent
advances in laser Stark spectroscopy w 56᎐60x have,
however, made it possible to determine excitedstate properties directly, but such measurements
are still difficult to perform. The progress in experiment that now takes place is to some degree
matched by similar developments of ab initio tools
for excited states w 61᎐64x . Response theory provides a viable path in this respect as presented in
the section Response Functions and Molecular
Properties. The formal and computational advantages of the double-residue cubic response method
for excited-state second-order properties can be
summarized in a few items. One needs only to
optimize the ground state, which, with its positive
definite Hessian, in general, is easier to optimize
than are excited states. This also enables property
calculations for excited states that are of an openshell and multideterminant nature by means of a
single-determinant and closed-shell SCF reference
state, allowing the use of large-scale direct AObased and parallel techniques w 10, 34, 33x . The
excited states are reached by the solution of generalized eigenvalue problems, which, apart from
generally being easier to perform than the explicit
optimization, makes it possible to obtain several
excited states in ‘‘one’’ calculation. It is relevant to
point out that response theory ensures spectroscopically pure states and that the double-residue
cubic response becomes equivalent to a separatestate linear response for fully correlated wave
functions Žfull CI.. The spectral representation is,
in this case, also identical to the sum-over-states
expression of time-dependent perturbation theory.
However, we note that the current implementations are based on nondegenerate perturbation theory which will break down for degenerate or
near-degenerate states. Thus, in practice, only the
lowest well-separated excited states are available,
which, however, should be exactly those possible
to determine in an experiment. In this respect, it is
essential that the ground-state linear response calculations give a correct ordering of the main states
for those states contributing to the polarizability of
the excited state in question.
SAMPLE CALCULATIONS
A set of molecules with varying ground-state
configurations and properties were considered in
w 65x : water, ozone, formaldehyde, ethylene, butadiene, cyclobutadiene, pyridine, pyrazine, and tetrazine. The results from the ground-state cubic
response were compared to excited-state experiments when available and with linear response
calculations of the multideterminant optimized ex-
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
229
LUO ET AL.
cited state. Results from such comparisons are
recapitulated in Figures 4᎐6. We note that where
measurements of excited-state polarizabilities exist, for example, formaldehyde, tetrazine, and
naphthalene w 66x , the single-determinant results
compare well both with experiment as well as
with multireference-based calculations. The feasibility of a direct atomic-orbital-based implementa-
tion to address excited-state properties of larger
systems has been demonstrated for the cis-, trans-,
and diphenylpolyenes in w 4, 33x . The sample calculations reported in w 65x indeed support a single-determinant-based approach to second-order excitedstate properties Žsee Fig. 4.. As expected, inherent
multideterminant species, like the ozone and cyclobutadiene molecules, show more problems for a
FIGURE 4. Comparison between single and multideterminant reference-state cubic response excited-state
polarizabilities. The ground-state polarizabilities are calculated by a linear response.
230
VOL. 70, NO. 1
HIGH-ORDER RESPONSE THEORY
FIGURE 5. Ground- and excited-state polarizability for tetrazine. The bars represent (from left to right) SCF
reference-state cubic response double residue, CASSCF reference-state cubic response double residue, CASCSF
finite-field calculation by Schutz
¨ et al. [69], and CASPT2 finite-field calculation by Schutz
¨ et al. [69]. The dots represents
experimental values from [57]. All values in atomic units and ␦1 = 1 / 2( ␣ x x + ␣ y y ) y ␣ z z and ␦ 2 = ␣ z z (11B1u ) y
␣ z z ( X 1A g ).
single-determinant-based approach than do ordinary species.
Formaldehyde, the simplest of carbonyl compounds, has excited states of both ␲␲ U and n␲ U
character, where n denotes the oxygen-localized
lone-pair orbital. The calculated polarizabilities for
the lowest singlet states can be found in Figure 6
together with two experimental values. For the
ground-state polarizability, the effect of correlation
is small, the SCF and CAS results differ less than
4%, and the CAS ␣ is 10% short of the experimental value of w 67x . There are notable differences in
magnitude of the excited-state polarizabilities; the
change for the states of n␲ U valence character is
minor, but for excitation to the more diffuse 3 p
and 3s Rydberg-type orbitals, it is significant, as
might be expected. Causley and Russell w 68x estimated ␣Ž11 B2 . to be 6.1 " 2.7 = 10y2 3 cm 3 or 410
" 180 au from a gas-phase electrochromism experiment. Both the cubic response values ŽSCF and
CAS. fall within the rather large error bars of the
experiment.
Schutz
¨ et al. w 69x employed accurate ab initio
methods, including CASPT2, in order to obtain
equilibrium geometry, excitation energy, and polarizabilities of the first singlet excited 1 B1 u state of
s-tetrazine. Their approach for computing excitedstate polarizabilities by means of numerical differentiation is, in the limit of small fields, equivalent
to the static limit of the separate-state linear response method. Comparison between calculations
and experiment can be found in Figure 5. As for
the ground state, we note a relatively good agreement between theoretical estimations of the excited-state polarizability also for the 1 B1 u state,
although both basis set and level of correlation
vary significantly. The experimental agreement,
however, is poor, for example, the excited- to
ground-state difference in out-of-plane polarizability ␦ 2 is determined experimentally to y17.5 au,
whereas all theoretical estimations fall in the range
from y0.2 ŽCASPT2. to y1.6 ŽCAS cubic response..
The agreement for the measurable ␦ 1 parameter
denoting the difference between in-plane and outof-plane components is better for the 1 B1 u state but
poor for the ground state Žsee Fig. 5.. Considering
the accuracy found for other molecules of similar
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
231
LUO ET AL.
FIGURE 6. Ground- and excited-state polarizabilities for formaldehyde. The bars represent (from left to right) SCF
reference-state cubic response double residue, CASSCF reference-state cubic response double residue, and CASCSF
reference-state linear response. The dots represents experimental values from [67] for the ground state and from [68]
for the excited state. All values in atomic units.
size and electronic distribution, an experimental
reinvestigation on this point would be desirable
with an alternative method.
Spin-Dependent Properties
Parallel to property calculations of singlet states,
there has been a concomitant development concerning triplet linear and quadratic response theory. In this section, we consider two spin-dependent phenomena and use the oxygen molecule to
illustrate what now can be achieved with this
theory. The purpose is two-fold: to study the natural transition rates Žspontaneous emission. from
excited states to the ground state and to study the
interaction of the ground-state molecule with an
external magnetic field. In the first case, we are
232
considering spin-forbidden transitions Žaccording
to normal dipole selection rules. which are characteristic of phosphorescence phenomena, and in the
second case, we are considering the parameters
which form the electronic g-tensor, measured in
electron resonance spectroscopy ŽESR.. The two
phenomena that we describe are apparently very
different in nature, but in both cases, spin᎐orbit
coupling plays a crucial role.
MAGNETIC PHOSPHORESCENCE
By magnetic phosphorescence, we mean a magnetic dipole transition that is forbidden according
to magnetic dipole Žboth spin and orbital. selection
rules, but which is allowed in the presence of
spin᎐orbit coupling. Given the perturbed initial < ˜
i:
and final state < f˜:, the magnetic dipole ŽM. matrix
VOL. 70, NO. 1
HIGH-ORDER RESPONSE THEORY
element is expanded in terms of the intermediary
states < k : which are eigenstates of the unperturbed
Hamiltonian with eigenvalues Ek ,
² f˜<M < ˜
i: s
Ý
k/i
² f <M < k :² Hso < i :
Ei y Ek
q
² f < Hso < k :² k <M < i :
E f y Ek
,
Ž 22.
where Hso is the spin᎐orbit coupling operator,
with one- and two-electron parts. The magnetic
dipole transition rate is given by
⌫i ª f s
␣ 5␻ 3
3
<² f <M < i :<
Ž 23.
in atomic units, where ␣ is the fine-structure constant, and ␻ , the transition frequency Ž Ei y E f .. If
both states involved in the transition have similar equilibrium geometries and if the transition
moments are constant over an adequate bondlength range, the calculation of the transition moment may be carried out at a single point and
the vibrational effects may be represented by a
Franck᎐Condon factor. However, in oxygen, these
conditions are not met; the equilibrium geometries
are similar, but the electronic transition moments
depend strongly on the geometry. The transition
moment in Eq. Ž23. must therefore be calculated
with vibrational averaging for definite vibronic
initial and final states. In addition, there may be
factors related to the degeneracy of the initial and
final states.
The phosphorescence transition amplitudes may
be calculated as a residue of a quadratic response
function,
lim Ž ␻ y ␻ f .²² M , Hso , C :: 0, ␻
␻ª ␻ f
Ž 24.
Ž C being arbitrary.. This approach has been used
successfully for ‘‘ordinary’’ phosphorescence in a
series of articles w 70᎐72x . In magnetic phosphorescence, the electric dipole operator is replaced by
the magnetic dipole operator. For the spin part, the
summation in Eq. Ž22. is reduced to a single term,
which in Eq. Ž24. corresponds to the residue of a
linear instead of a quadratic response function. The
geometry dependence of the transition amplitude
can be understood in terms of this term; a spin᎐
orbit matrix element divided by the transition
frequency. The geometry dependence of the spin᎐
orbit matrix element is modest while the denominator approaches zero for large values of the internuclear separation Žsee Fig. 7, bottom..
We applied the above equation to the three
lowest states of molecular oxygen, which can be
formed from two atomic p 4 configurations; the
triplet ground state, X 3 ⌺y
g ; and two low-lying
singlet excited states, a1⌬ g and b 1 ⌺q
g . The b᎐X
transition was studied in detail in w 70x . The relatively short lifetime of approximately 12 s is explained by a large spin᎐orbit coupling between the
b and X states, which is the matrix element that
defines the spin᎐dipole contribution. The Einstein
coefficients were calculated for several vibronic
transitions. When experimental potential-energy
curves were used, we obtained a Ž0, 0. transition
probability of 0.079 sy1 , approximately 10%
smaller than the most recent of experimental result
of Ritter and Wilkerson of 0.089 sy1 w 73x . However,
our results are in excellent agreement with experiment for all other known vibronic probabilities
Žsee Table V..
The a᎐X transition has no spin contribution; the
orbital magnetic moment gives a transition lifetime for an isolated molecule of more than 1 h.
This is, however, very sensitive to collisions and
its lifetime is reduced by several orders of magnitude in solvents and dense gases. For instance, two
oxygen molecules approaching in the a1⌬ g state
can be conceived to decay to two ground-state
molecules with opposite spin, a process which
obeys normal dipole selection rules. From our calculations, the predicted lifetime is approximately 3
h, which is longer than in most previous observations ŽTable VI..
TABLE V
Calculated and experimental Einstein coefficients
⌫v X , v Y for different vibrational bands of the
b 1 ⌺ g+ ( v X ) ª X 3 ⌺ gy ( v Y ) system.a
vX, vY
[74] (calcd.)
[75]
[76]
[77]
0, 0
0, 1
1, 0
1, 1
2, 0
3, 0
0.0794
0.0047
0.0064
0.0667
0.0003
0.000007
0.084
ᎏ
0.0069
0.067
0.00032
0.0000067
ᎏ
0.0047
ᎏ
ᎏ
ᎏ
ᎏ
0.06
ᎏ
0.0056
ᎏ
0.00027
ᎏ
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
a
From [70].
233
LUO ET AL.
TABLE VI
Calculated and experimental Einstein coefficients
⌫0,0 of the a1⌬ g ( v = 0) ª X 3 ⌺ gy ( v = 0) system.
[74] (cald.)
[78]
[79]
[80]
[81]
0.83
1.47
2.58
1.9
1.5
Here, we consider only the second-order contribution to the g-tensor which can be calculated as
the linear response function
²² L; Hso :: 0 .
FIGURE 7. (Top) Paramagnetic contribution to the
electronic g-shift. (Bottom) Magnetic dipole transition
curves for the a y X and b y X transitions of molecular
oxygen, with orbital (L) and spin (S) contributions.
ELECTRONIC g-TENSOR
The theory for molecular g-tensors is well documented, as in the excellent textbook by Harriman
w 82x and in the presentation given by Hegstrom
w 83x . The g-tensor, which has a number of contributions, is written as
g s g e1 q ⌬ g ,
Due to the similarities between nucear and electron magnetic resonance, it is natural to speak of
paramagnetic and diamagnetic contributions to the
g-tensor Žrather than gauge corrections., as suggested by Schreckenbach and Ziegler w 84x .
The calculated g-shift as a function of internuclear distance w 74x is seen in the upper part of
Figure 7. The increase in the g-factor as a function
of internuclear distance is due to the denominator
where an energy splitting between the ground
state and the lowest 3 ⌸ g state approaches zero in
the separation limit. Since the ground state is a ⌺
state, it is only the perpendicular component of the
g-tensor that can be calculated this way. There are
two conflicting sets of measurements for the electronic g-tensor, one due to Bowers et al. w 85x and
one due to Hendrie and Kusch w 86x . The calculated
paramagnetic contribution gives g-factors that favor Hendrie’s experiment Žsee Table VII., but since
the experiments differ by a few ppm only, all other
contributions to the g-shift need to be taken into
account before an accurate comparison with experiment can be made.
Magnetic Properties: The
Cotton᎐Mouton Effect
Ž 25.
where g e is the free-electron value and ⌬ g is
referred to as the g-shift, the quantity that we wish
to calculate. A complete calculation of the g-shift
to second order must account for three contributions: the mixed second-order energy associated
with the orbital Zeeman interaction and the electronic spin᎐orbit coupling Žone- and two-electron.
which usually is the dominating term, the firstorder gauge-correction terms Žthe magnetic momentum part in the spin᎐orbit coupling., and the
relativistic mass correction to the spin Zeeman
term.
234
Ž 26.
As an illustration that also various nonlinear
magnetic and magnetooptic properties readily can
TABLE VII
Calculated (paramagnetic contribution, with
vibrational averaging) and experimental g-shifts for
molecular oxygen.
[85] (exp.)
[86] (exp.)
[74] (cald.)
⌬ gH
gH
2519
2850
2982
2.004838
2.005169
2.005301
VOL. 70, NO. 1
HIGH-ORDER RESPONSE THEORY
be studied by response theory, we round off this
review by giving a brief account of the Cotton᎐
Mouton effect ŽCME.. This effect, named after the
scientists who first studied it systematically w 87x , is
observed when light is passed through a sample in
a direction perpendicular to an applied magnetic
field and manifests itself as an anisotropy in the
refractive index of the incident light, induced by
the magnetic field.
A theoretical description of the CME of diamagnetic substances was given by Buckingham and
Pople w 88x . They showed that the effect is proportional to the square of the magnetic-field-induced
intensity through the so-called molar Cotton᎐
Mouton constant m C, which can be written as a
sum of a temperature-independent part—related
to the hypermagnetizability anisotropy ⌬␩ —and a
temperature-dependent part Q ŽT . —originating
from molecular reorientation in the sample Ž NA is
Avogadro’s number.:
mC
sm C d i a s
2␲ NA
27
w ⌬␩ q Q Ž T .x .
Ž 27.
The hypermagnetizability anisotropy is defined as
ŽEinstein summation convention used throughout.
⌬␩ s
1
15
Ž 3␩␣ ␤ , ␣ ␤ y ␩␣ ␣ , ␤␤ . ,
Ž 28.
with the first two indices referring to the electric
field, and the last two, to the magnetic field induction. The orientational Žor ‘‘Langevin’’. term involves the electric polarizability Ž ␣ . and the magnetizability Ž ␰ . anisotropies through the equation
Ž k is Boltzmann’s constant.
QŽT . s
1
5kT
Ž 3␣ ␣ ␤ ␰ ␣ ␤ y ␣ ␣ ␣ ␰␤␤ . .
Ž 29.
The case of paramagnetic species, which requires a fully quantum mechanical treatment, was
included in the general derivation made by Kielich
w 89x and Chang w 90x and was discussed in detail by
Kling et al. w 91x . In the high-temperature limit, an
additional Žparamagnetic. term arises in the semiclassical expression of Eq. Ž27.:
mC
mC
p ar a
s
sm C d i a qm C p ar a ,
2␲ NA
1
27
45k 2 T 2
q
2
15kT
Ž 3␣ ␣ ␤ m ␣ m␤ y ␣ ␣ ␣ m␤ m␤ .
Ž 3␨ ␣ ␤ , ␣ m␤ y ␨ ␣ ␣ , ␤ m␤ . ,
where we have introduced the magnetic moment
m and the first hypermagnetizability ␨ . For
homonuclear diatomic molecules, symmetry reduces this expression to
mC
s
2␲ NA
27
ž
⌬␩ q
2
15kT
q
⌬␣ ⌬␰
2
15k 2 T 2
⌬ m2 ⌬ ␣ ,
/
Ž 31.
where ⌬ ␣ s ␣ z z y ␣ x x , ⌬ ␰ s ␰ z z y ␰ x x , and ⌬ m2
s m2z y m2x . We note that the validity of the above
equations can be extended from gases to dense
fluids Žliquids., provided that appropriate localfield corrections are made w 92x .
By fitting experimental data to Eq. Ž31. obtained
at different temperatures and at constant pressure,
one can obtain the hypermagnetizability anisotropy
⌬␩ and the products ⌬ ␣ ⌬ ␰ and ⌬ m2 ⌬ ␣. If one of
the three anisotropies Ž ⌬ ␣, ⌬ ␰ , or ⌬ m2 . is known
from other sources, a CME experiment could furnish the other two. Our interest is focused mainly
on ⌬␩ , extracted from experiment by taking the
T ª ⬁ limit according to Eq. Ž31..
In the language of response theory, we may
obtain the hypermagnetizability tensor as a sum of
a quadratic and a cubic response function:
␩␣ ␤ , ␥␦ Ž y␻ ; ␻ , 0, 0 . s y 14
= ²² r ␣ ; r␤ , L␥ , L␦ ::␻ , 0, 0
y ²² r ␣ ; r␤ , Q␥␦ ::␻ , 0 , Ž 32.
where L is the angular momentum operator around
a gauge origin O and Q is the diamagnetic susceptibility tensor.
Using the standard angular momentum operator to evaluate the hypermagnetizability as indicated in Eq. Ž32. will, however, introduce an artificial dependence on the choice of the gauge origin
in our calculated results. The gauge-origin dependence can be decreased or eliminated by increasing the basis set. However, this is not a viable
approach for anything but small molecules. Furthermore, the dependence on the gauge origin
means that our calculated results will not be sizeextensive.
The problem of gauge-origin dependence and
lack of size-extensivity can very elegantly be solved
by introducing local gauge origins through London atomic orbitals w 93, 94x ,
Ž 30.
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
␻␮ Ž B. s exp Ž yiA ␮ ⭈ r . ␹␮ ,
Ž 33.
235
LUO ET AL.
where the vector potential appearing in the phase
factor moves the global gauge origin to a local
gauge origin situated at the nucleus to which the
conventional spherical Gaussian basis function ␹␮
is attached:
A ␮ s 12 B = R ␮ O .
Ž 34.
In addition to removing the dependence on the
gauge origin, the London atomic orbitals improve
the basis-set convergence, making it possible to
study larger molecular systems. However, London
orbitals have not yet been implemented in the full
quadratic or cubic response functions. An alternative approach to determine the hypermagnetizability anisotropy involves the use of a finite electricfield approach. In the presence of static electric
fields, we express the hypermagnetizability as an
energy derivative:
␩sy
⭸ 4␧
⭸ B2 ⭸ E 2
B , Es0
sy
⭸ 2␣
⭸ B2
Bs0
sy
⭸ 2␰
⭸ E2
.
Es0
Ž 35.
This equation allows evaluation of the hypermagnetizability anisotropy using London orbitals—
thus ensuring size extensivity—by taking finite
differences of analytically calculated magnetizabilities in the presence of a finite electric field.
Several ab initio calculations of the Cotton᎐
Mouton constant of gaseous molecules have been
presented and have been recently reviewed—
together with experimental measurements—by
Rizzo et al. w 95x . In the following, we discuss two
recent examples of our investigations of the CME.
The first one involves a paramagnetic species in
the gas phase ŽO 2 . w 96x ; the second concerns the
determination of the Cotton᎐Mouton constant of
liquid water w 97, 98x .
HYPERMAGNETIZABILITY ANISOTROPY OF
GASEOUS O 2
Experimental determinations of the CME of gaseous O 2 are not rare in the literature w 91, 99, 100,
101x . Huttner
and coworkers w 91x and Ritchie and
¨
w
his group 100x studied the temperature dependence of m C and gave estimates of ⌬␩ at a frequency of 632.8 nm, their results being y737 "
401. and y603." 469. au, respectively. Both groups
obtained the hypermagnetizability anisotropy by
subtracting from the experimental data the same
semiempirical estimate of m C p ar a w Eq. Ž30.x , fitting
the remaining diamagnetic contribution to Eq. Ž27..
236
Equation Ž32. was employed in w 96x to compute
the hypermagnetizability anisotropy of molecular
oxygen using a multiconfigurational SCF quadratic
and cubic response. The effects of electron correlation, basis-set incompleteness, and the choice of
the gauge origin were investigated in detail. Note
that the latter two need particular attention since
London atomic orbitals have not been implemented for the quadratic and cubic response functions. Frequency dispersion, vibrational averaging,
as well as pure vibrational contributions to the
tensor elements of the hypermagnetizability were
studied. The calculated result for ⌬␩ at 632.8 nm is
q2.65 au, more than two orders of magnitude
smaller in absolute value and with the opposite
sign compared to the most probable experimental
estimate. Basis-set incompleteness and residual
electron correlation effects not accounted for were
estimated to lead to an overall incertitude of about
1 au in ⌬␩. Other possible sources of errors—for
instance, gauge-origin dependence, inaccurate account of the effects of vibration or approximations
on the magnetic moment operator Ž m f ␮ B gS with
␮ B indicating the Bohr magneton; g, the electron
g-factor tensor; and S, the electron spin operator.
—were considered uninfluential.
Although we would like to stress that the experimental datum carries wide error bars, it seems
worthwhile to try to investigate the origins of this
apparent discrepancy. The electric polarizability
and magnetizability anisotropies were computed
at the frequency of experimental interest, and vibrational effects were included. This furnished values of the diamagnetic contribution to the CME of
molecular oxygen w Eq. Ž27.x . It can be observed
that the effect of the orientational term in this
system overshadows that arising from the pure
electronic rearrangements due to the external magnetic field, contained in the hypermagnetizability
anisotropy ⌬␩ w QŽ293.15K . s y1969.6 au at 632.8
nmx .
The calculation of the paramagnetic contribution w Eq. Ž30.x to the CME of molecular oxygen
requires an accurate determination of ⌬ m2 . To the
very least, this means knowing with good accuracy the anisotropy of the square of the electron
g-factor, that is, ⌬ g 2 s g z2 y g x2 . It can be easily
shown that experimental estimates of this
anisotropy can lead to very large uncertainties. For
instance, employing Evenson and Mizushima data
w 102x gives ⌬ g 2 s y0.0096 " 0.0036. This carries a
38% error bar, which is transferred to any semiem-
VOL. 70, NO. 1
HIGH-ORDER RESPONSE THEORY
pirical determination of m C p ar a. We believe, based
on this observation and without an accurate estimate of m C p ar a available, that the determination of
⌬␩ through a fit of m C ym C p ar a as a function of
temperature using Eq. Ž27. and an extrapolation to
infinite temperatures is a rather inaccurate procedure.
CME OF LIQUID WATER
The use of London orbitals was crucial in our
recent study of the CME of liquid water w 97, 98x ,
which is the first ab initio investigation of this
nonlinear magnetooptic effect of a liquid sample.
The main results from this investigation are recapitulated in Table VIII.
Several models were used to describe the effects
of the solvent. In the simplest, the water molecule
is surrounded by a dielectric continuum as described in the section Solvent Modeling and Macroscopic Properties. The extra terms arising because of the use of London atomic orbitals were
discussed in w 97x . From Table VIII, it can be seen
that the effects of the dielectric continuum on the
hypermagnetizability and Langevin terms are substantial, far exceeding the changes induced by
electron correlation.
We note that the molar Cotton᎐Mouton constant of water plus dielectric is farther off the
experimental results of Williams and Torbet w 103x
than are the gas-phase results. This result is not
surprising, as it is well known that the dielectric
continuum model is not an adequate approximation to model liquid water, as has been demon-
strated for the second-harmonic generation ŽSHG.
effect w 104x as well as magnetic properties like
nuclear shieldings and magnetizabilities w 105x .
To improve on the description of the solvent
structure, the water molecule was enclosed by its
first solvation shell, which, in turn, was surrounded by the dielectric continuum, thus modeling the liquid water by a semicontinuum. As the
Cotton᎐Mouton constant is an extensive property,
the Cotton᎐Mouton constant of the solvated water
molecule was extracted by the differential shell
approach w 104x in which two calculations are performed:
1
mC
, water solute, first solvation shell, and the
dielectric medium; and
mC
2
, first solvation shell and dielectric medium.
From these calculations, the molar Cotton᎐Mouton
constant of the solute is obtained from the difference m C 1 ym C 2 .
Table VIII shows that the Langevin term as well
as the total Cotton᎐Mouton constant now change
sign. A similar sign change upon solvation has
also previously been observed for the nonlinear
electrooptical SHG effect w 104x . One furthermore
notes that whereas the properties of the gas-phase
molecule were dominated by the hypermagnetizability anisotropy—a property very unique to the
water molecule w 95x —this situation is dramatically
changed upon solvation, in which the contribution
from the hypermagnetizability anisotropy is negligible.
TABLE VIII
Dependence of the Cotton᎐Mouton constant m C = A + B / T of liquid water ( A = ⌬ ␩ in units of
10 y 2 0 G y 2 cm 3 mol y 1 , B = Q( T ) = T in units of 10 y 2 0 G y 2 cm 3 mol y 1 K), and average value
( m C av e, in units of 10 y 2 0 G y 2 cm 3 mol y 1) in the 283.15᎐293.15 K temperature range.
Phase
Wave function a
Gas
Gas
Liquid
Liquid
Liquid
SCF
MCSCF
SCF
MCSCF
SCF
Liquid
Liquid
Experiment
Experiment b
a
b
Solvent
model
None
None
Continuum
Continuum
Semi-continuum
Cavity
radius
A
B
˚
3.98 A
˚
3.98 A
˚
8.73 A
8.87
9.05
14.45
14.94
0.04
230.14
311.5
67.06
64.58
y7826.0
mC
av e
Ref.
9.67
10.13
14.68
15.16
y27.6
[97]
[97]
[97]
[97]
[98]
118(15)
y74(9)
[103]
[103]
Results obtained at the SCF and MCSCF level with various solvent models and the daug-cc-pVTZ basis set.
Employing local-field corrections for the electric field.
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
237
LUO ET AL.
The calculations of the CME of liquid water
could reproduce the experimental sign, a sign induced by close-range interactions in the liquid
water structure. However, quantitative agreement
with experiment is still lacking. It is not clear
whether this disagreement is caused by a lack of
electron correlation in the Langevin term or if the
rigid structure of the solvation shell is too crude
an approximation of the rather unconstrained tumbling of the water molecules in the liquid.
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