Ellinger Y., Defranceschi M. (eds.) Strategies and applications in quantum chemistry (Kluwer, 200
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M. TADJEDDINE AND J. P. FLAMENT |
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Owing to the very large discrepancies in the data on |
we have made new com- |
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putations with other basis sets |
but with the same process; they converge |
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at less than 4% from the previous ones, giving confidence in our results and in our procedure.
3.4. APPLICATION TO Be AND Ne
The two preceding applications showed that our hydrogenic model fits well with the helium atom and the dihydrogen molecule for the determination of the polarization
functions except that their exponent |
is different from |
which is the exponent of |
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the genuine basis set |
It is obvious that the hydrogenic model will fit less and |
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less as the atom will be described by more and more electrons. |
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Nevertheless our method of |
and |
calculations has been successfully extended to |
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Be and Ne atoms (21). Let us resume the principal results : |
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1. For less |
than 1% error for |
it |
is sufficient |
to |
”polarize” only the valence |
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electrons in Be; the polarization of the 1s orbital leads to an |
value within |
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0.1% of HFL value. Contrary to Be, the polarization of the inner shell is now |
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absolutly negligible for Ne. |
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2. |
The transfer of exponent to the |
set leads to good |
values of |
: for Be : |
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70 (42). |
instead of the HF limit |
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and for Ne : 68 instead of |
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3. |
Contrary to the previous applications |
we observe an increase of the |
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values of |
with the |
polarization function |
because |
contains p func- |
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tions improving the first order wavefunction which, despite its size, was not at |
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the HF limit. |
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4. |
At last, it is possible to still |
improve the results on |
by using two different |
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values of the exponent : |
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by optimization of |
for the STO of |
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by optimization of |
for the STO of |
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This last result is important for the generalization of our procedure to more complicated systems.
4.Conclusion
The computation of polarizabilities requires consideration of two complementary problems: the computational method and the basis set used.
We have first been concerned with the computational point of view. Through the calculation of the dynamic polarizability of CO, we have developed a method based on the conventional SCF–CI method, using the variational– perturbation techniques : the first–order wavefunction includes two parts (i) the traditional one, developed over
the excited states and (ii) additional terms obtained by multiplying the zeroth–order function by a polynomial of first–order in the electronic coordinates. This dipolar
AB INITIO CALCULATIONS OF POLARIZABILITIES IN MOLECULES |
277 |
factor makes an extrapolation procedure possible in critical cases (for example when the low–lying states are of Rydberg character as the states of CO).
This calculation has shown the importance of the basis set and in particular the polarization functions necessary in such computations. We have studied this problem through the calculation of the static polarizability and even hyperpolarizability. The very good results of the hyperpolarizabilities obtained for various systems give proof of the ability of our approach based on suitable polarization functions derived from an hydrogenic model. Field–induced polarization functions have been constructed from the first– and second–order perturbed hydrogenic wavefunctions in which the exponent is determined by optimization with the maximum polarizability criterion.
We have demonstrated the necessity of describing the wavefunction the best we can, so that the polarization functions participate solely in the calculation of polarizabilities or hyperpolarizabilities.
References
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M. TADJEDDINE AND J. P. FLAMENT |
16.K.P. Huber and G. Herzberg, Molecular spectra and molecular structure IV: Constants of diatomiques molecules, Van Nostrand Reinhold, New York, 1979.
17.M. Rérat, M. Mérawa and C. Pouchan, to be published
M.Mérawa, Thèse, Université de Pau et des Pays de 1’Adour (1991)
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19.K.K. Sunil and K. Jordan, Chem. Phys. Letters 145, 377, (1988)
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Coupled Hartree-Fock Approach to Electric Hyperpolarizability Tensors in Benzene
P. LAZZERETTI, M. MALAGOLI and R. ZANASI
Università di Modena, Dipartimento di Chimica, via G. Campi 183, 41100 Modena, Italy
1. Introduction
In the presence of a static, spatially uniform electric field |
the electronic cloud |
of |
atomic and molecular systems gets polarized. The energy, |
W, can be written as |
a |
Taylor series [1–3]
where |
is the unperturbed energy, |
is the permanent electric dipole moment |
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and the coefficients |
etc. |
are known as (static) electric polarizabilities. |
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Non-linear response of the system is rationalized via hyperpolarizabilities
(sum over repeated Greek indices is implied), etc.. The total electric dipole moment of the molecule in the presence of the electric field is [1–3]
According to (1) and (2), the response tensors are defined
279
Y. Ellinger and M. Defranceschi (eds.), Strategies and Applications in Quantum Chemistry, 279–296.
© 1996 Kluwer Academic Publishers. Printed in the Netherlands.
280 |
P. LAZZERETTI ET AL. |
Owing to permutational symmetry of the tensor indices, only
components are distinct for a tensor of rank r appearing in (1). Thus the number of independent values which completely characterize the various tensors in eq. (1) is 3,
6, 10, 15, 21, .. . respectively for |
Molecular point |
symmetry further reduces the number of linearly independent components, see, for instance, Refs. [4], [5]. For any tensor appearing in (1), denoted in general by
let us rearrange its components as a column vector in cartesian space, i.e.,
If the basis set of unit vectors in cartesian 3-space transforms
under an operation T, then the direct product matrix
can be introduced, so that
If T belongs to a group G and brings the physical system into self-coincidence, then the array of components will be stable under G, i.e.,
In addition one can always find a transformation leading to a symmetry adapted basis so that T is brought to the block diagonal form T via the associated similarity
transformation. The matrix can be written as a direct sum
COUPLED HARTREE-FOCK APPROACH |
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281 |
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where the different blocks of |
are classified |
according to the irreducible representa- |
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tion |
with frequency |
of G, and its |
appearence. Accordingly, in the new |
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basis, the symmetry adapted tensor components are [4] |
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for every operation of the group and for each block |
This implies either that |
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or that, being invariant under the operations of G, it carries the one- |
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dimensional totally symmetric representation, |
Thus, if |
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the totally symmetric representation occurs m times in the direct product representation, then the tensor is fully determined by just m numbers. Therefore theoretical procedures for evaluating the higher-rank polarizability tensors appearing in (1) and
(2) should efficiently exploit the symmetry properties of a given molecule to save computer effort. The number of independent parameters can be conveniently eval- uated a priori via simple techniques based on symmetrized Kronecker products [4].
Tables reporting data for a number of groups are available [1].
Besides the elementary properties of index permutational symmetry considered in eq. (7), and intrinsic point group symmetry of a given tensor accounted for in eqs.
(8)-(14), much more powerful group-theoretical tools [6] can be developed to speed up coupled Hartree-Fock (CHF) calculations [7–11] of hyperpolarizabilities, which are nowadays almost routinely performed in a number of studies dealing with non linear response of molecular systems [12–35], in particular at the self-consistent-field (SCF) level of accuracy.
The present paper is aimed at developing an efficient CHF procedure [6–11] for the entire set of electric polarizabilities and hyperpolarizabilities defined in eqs. (l)-(6) up to the 5-th rank. Owing to the 2n + 1 theorem of perturbation theory [36], only 2-nd order perturbed wavefunctions and density matrices need to be calculated. Explicit expressions for the perturbed energy up to the 4-th order are given in Sec. IV.
A computer program for the theoretical determination of electric polarizabilities and hyperpolarizabilitieshas been implemented at the ab initio level using a computa- tional scheme based on CHF perturbation theory [7–11]. Zero-order SCF, and first- and second-order CHF equations are solved to obtain the corresponding perturbed wavefunctions and density matrices, exploiting the entire molecular symmetry to reduce the number of matrix element which are to be stored in, and processed by, computer. Then and tensors are evaluated. This method has been applied to evaluate the second hyperpolarizability of benzene using extended basis sets of Gaussian functions, see Sec. VI.
2. Solution of first-order CHF equation
The Hartree-Fock equations for the i-th element of a set containing occ occupied
molecular orbitals |
in a closed shell system with n = 2occ electrons are [8] |
where the orthonormality conditions are written
282 P. LAZZERETTI ET AL.
All the quantities appearing in (15) are expanded in powers of a formal perturbation parameter which is finally put equal to unity, so that, for instance,
The matrix of Lagrange multipliers is usually chosen diagonal to zero order, so that
and
with
To first order in
with |
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Taking the Hermitian product with |
in eq. (21) one has |
Taking the product in eq. (21) with |
where k labels another occupied orbital in |
a non degenerate problem, |
using (19) and (20), |
Owing to the arbitrary nature of the Lagrange multipliers, one can choose
so that the projection of the first-order i-th orbital on the subspace spanned by occ–1
occupied MO’s |
vanishes. From the orthogonality condition (22) one has, |
for the i-th orbital, |
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For real perturbations, e.g., in the presence of a static electric field, which is the case studied in the present paper, the zeroand first-order orbitals can always be chosen real, so that also
At any rate, the projection of the first-order orbitals on the subspace of occupied is not needed within the McWeeny approach [7], where choice (25) is implicitly
assumed; it is sufficient to calculate the projection |
on the subspace of virtual |
COUPLED HARTREE-FOCK APPROACH |
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283 |
zero-order orbitals. Taking the Hermitian product with an unoccupied |
in eq. (21) |
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one has |
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and left-multiplying by the ket |
and summing over k gives |
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where
is the Hartree-Fock propagator [10], [11] and the projector
is equivalent to the identity operator when acting on the subspace of virtual orbitals [7].
3. Solution of second-order CHF equation
We discuss a method to evaluate the second-order molecular orbitals appearing in eq. (17) consistent with the first-order computational scheme outlined in the previous section. In particular we take advantage of definition (30) to develop a compact approach explicitly oriented to numerical applications.
The second-order CHF equation for the i-th occupied orbital is
where the orbitals satisfy the orthonormality condition to second order,
Taking in (32) the Hermitian product with |
and using eqs. (19), (21), and (22), |
one finds
where the index j = i can be omitted in the sum, in the present case of real pertur- bations, owing to eq. (27). Summing over i occupied, the last term in (34) vanishes.
284 |
P. LAZZERETTI ET AL. |
Left-multiplying (32) by |
occupied in a non degenerate case, using (19), |
(21), and (33) one obtains |
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This equation shows that secondand first-order Lagrangian multipliers are not in-
dependent, so that a specific selection of |
will bias |
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Thus the choice (25) for |
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the first-order Lagrangian multipliers makes the sum over j |
in eq. |
(35) vanish in the |
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case of real perturbations, but, in general, there is no choice of |
in eq. (35) which |
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annihilates the projection of |
on occupied |
and |
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However, using the McWeeny approach [7], it is sufficient to calculate only the pro-
jection on the subspace of virtual zero-order orbitals in order to get the second hyperpolarizability tensor. This projection is evaluated via a procedure similar to the one used in solving the first-order equation (21). Taking in (32) the Hermitian
product with the unoccupied and using (19), one finds
Multiplying on the left by the ket |
and summing over k, one finds |
where
is available from the solutions (29) to the first-order eq. (21).
4. Computational scheme
We will now discuss an iterative scheme based on the CHF approach outlined in Sections II and III, using the McWeeny procedure [7] for resolving matrices into
components, by introducing projection operators |
and |
with respect to the |
subspaces spanned by occupied and virtual molecular orbitals. |
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Expanding the occupied over an orthonormal atomic basis set |
of order m (which |
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is assumed independent of the perturbation), one has |
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COUPLED HARTREE-FOCK APPROACH |
285 |
The projection operators have matrix representations
and
A perturbation expansion analogous to (17) holds for any matrix, for instance, the Fock matrix
and the density matrix
The firstand second-order coefficients |
and |
can also be resolved into projec- |
tions on the subspaces of occupied and virtual molecular orbitals:
The projections |
with k labeling another occupied orbital, vanish according to |
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choice (25), and |
the projection |
vanishes for a real perturbation, see eq. (27). |
Any matrix A can now be resolved into projection components [7] with respect to the occupied and virtual subspaces, that is,
For instance, the first-order density matrix can be written
where
The iterative scheme for the first-order coefficients |
becomes |