Electric Multipoles, Polarizabilities and Hyperpolarizabilities

BY DAVID PUGH

1 Introduction

The study of the response of molecules to external perturbations provides one of the principal sources of information on molecular behaviour. The theory of the response functions is therefore of fundamental importance in molecular science. Over the last two decades the development of optoelectronics has given an additional impetus to work on the nonlinear response of molecules to electro-magnetic fields. The nonlinearities provide the means for amplification, modulation and changing the frequency of optical signals, in the same way that the nonlinear characteristics of valves and transistors facilitate these operations in conventional electronics. Current activity in theoretical modelling of the response functions to some extent reflects this dual motivation. At the more fundamental end of the range, ab initio calculations on small molecules using highly sophisticated theoretical methods are being applied with considerable success. On the other hand, semi-empirical methods, with a good deal of reparametrisation for specific types of molecule and type of calculation, grounding the calculations on experimentally determined spectroscopic and dipole parameters, have been applied to a vast range of compounds with the aim of identifying those with large hyperpolarizabilities of the kind that might lead to applications of the material in optoelectronics.

In this article only the response to the electric field will be treated. Magnetic effects were also included in an earlier SPR 1 but the great expansion of the field in recent years has necessitated a sub-division of the material. Specific applications in optoelectronics and reviews of molecules currently of direct interest in that field can be found in a number of books and edited volumes. Effects interpretable only through quantum electrodynamics are treated in the books by Loudon and by Craig and Thirunamachandran.

The plan of this review is as follows: Sections 2 and 3 attempt to give an overview of the general theory for static and dynamic effects respectively. Section 4 describes specific methods of calculation, with some emphasis on new developments in the period from about 1970 to the present. Section 5 is a literature review of the period 1998-May 1999.

2 Perturbation of Molecules by Static Electric Fields: General Theory

If the electrostatic potential created at the point, r, by the external field is V(r), the perturbed hamiltonian operator is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where the sum is over all the constituent particles of the molecule, H0 is the unperturbed hamiltonian and qα is the charge on the particle at rα. Expanding the potential in a Taylor series in the displacement from an origin in the molecule (often conveniently taken as the electronic charge centroid) leads to the multipole expansion

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

for a neutral molecule, where the field and its derivatives, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], are now to be taken at the co-ordinate origin and the repeated index summation convention is being used. The expressions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

are respectively the components of the dipole, quadrupole and octupole operators.

Traceless forms of the higher multipole operators have been given by Buckingham. Much of the work reviewed will refer to the case of a spatially uniform applied field, when the perturbed hamiltonian reduces to

H = Ho - µiFi (4)

The foregoing formula refer to changes in the hamiltonian of the system. The polarizability and hyperpolarizability terms arise from the changes in the wavefunction induced by the perturbed hamiltonian.

Denoting the unperturbed and perturbed normalised wavefunctions (usually for the ground state) by ψ0 and ψ respectively, we have, for the uniform field,

[FORMULA NOT REPRODUCIBLE IN ASCII] (5)

where the perturbed wavefunction has been expanded as a power series in the field. Formulae for the expectation values of the multipole components can be written directly in terms of the perturbed wavefunction as {ψ|μiψ}, {ψ|Qij|ψ} etc. In particular the expression for the dipole expectation value can be used to define the permanent dipole moment and the polarizability and hyperpolarizabilities:

[FORMULA NOT REPRODUCIBLE IN ASCII] (6)

Here, µ(0) is the permanent dipole moment and the tensors α, β and γ are respectively the linear polarizability and the first (quadratic or second order polarizability)) and second (third order or cubic) hyperpolarizabilities.

2.1 Analytic Derivatives of the Energy. - If the the nth order perturbation to the wavefunction has been calculated equation (6) gives a general prescription for calculating the nth order polarizability tensor. In cases where the perturbed wavefunction is calculated by a variational method, the calculation of the polarizabilities can be simplified by making use of relationships that have been established between the analytical derivatives of the energy. If equation (5) is differentiated with respect to the field the result (assuming for simplicity that the wavefunction is real) is

[FORMULA NOT REPRODUCIBLE IN ASCII] (7)

For variationally determined wavefunctions the second term can be shown to vanish, giving the Hellmann-Feynman theorem,

[FORMULA NOT REPRODUCIBLE IN ASCII] (8)

so that, on comparing equations (8) and (6), the dipole moment and polarizabilities can be identified with the derivatives of the energy at vanishing field strength. In many methods these derivatives can be evaluated analytically, in the sense that they can be obtained as functions of the integrals that have already been computed in the the determination of the ground state. In order to calculate any of the polarizabilities it is necessary to obtain some of the correction functions to the unperturbed wavefunction defined in equation (6), but extensions of the type of argument used in the Hellman-Feynman theorem have shown that, again for variationally determined wavefunctions, a knowledge of the nth order correction to the wavefunction enables the polarizability of order (2n + l) to be calculated. The introduction of Lagrangian methods has also allowed some of the computational advantage obtained by...