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 exploitation of the (2n + 1) rule to be extended to non-variational wavefunctions. The selection of either a variational or a non-variational method still has an important bearing on the procedures subsequently employed for the computation of the polarizabilities. In a variational method the energy must be optimized, in the presence of the perturbing field, with respect to any of the parameters that are allowed to vary in the wavefunction. Inmost calculations of response functions the relevant parameters are the molecular orbital coefficients and the state vector coefficients in a configuration interaction (CI) procedure. In a few cases the orbital exponents may also be allowed to vary. It follows that the variational procedures are those where a full Hartree-Fock minimization is carried out in the presence of the perturbation (Coupled Hartree-Fock methods, CHF and CPHF) and, if CI is included, methods where both the CI coefficients and the orbital coefficients are optimized (Multiple Configuration Hartree-Fock, MCSCF). Perturbation methods (Moeller-Plesset (MP2, MP4 etc.) are clearly non-variational. The CI method [other than full CI (FCI)], without redetermination of the molecular orbital coefficients in the presence of the perturbation, is also non-variational as is the coupled-cluster method (CC), where blocks of configurations are added without optimization of the individual components within each block. In the non-variational cases the dipole moment and polarizabilities calculated from equation (6) will differ from those obtained via equation (8). As approximate solutions are improved to approach the exact function these differences become less significant, even for the non-variational methods, since the Hellman-Feynman theorem holds for the exact wavefunction.

3 Frequency-Dependent Polarizabilities: General Theory

The general theory of time-dependent response functions has been described in many publications. The response is non-local in time and the Fourier transforms of the general time-dependent functions lead to the definitions of the frequency-dependent response functions which are the quantities most easily related to experimental measurements and potential applications. The notation can be established by writing the formulae for the first three molecular response functions explicitly

[FORMULA NOT REPRODUCIBLE IN ASCII] (9)

The time-dependent field components are here taken to be the complex forms

F(t) = Fexp(ωpt), p= 1,2,3 ... (10)

which combine in the nonlinear interaction to give an induced dipole com ponent at the sum frequency, Σωp. The negative sign indicates an 'output' frequency. For real fields combinations of real and imaginary parts give additional numerical factors when two or more of the input frequencies are degenerate. [The case of second harmonic generation, below, illustrates this point.]

3.1 Time-Dependent Perturbation Theory: The Sum over States Method. - The most direct approach to the calculation of the time-dependent polarizabilities is to apply standard time-dependent perturbation theory to the evolution of the wavefunction in the time-dependent Schrodinger equation. The wavefunction is formally expanded in terms of the complete set of molecular eigenfunctions (ground and excited states) and the solutions are obtained in terms of matrix elements of the perturbation between these states and the corresponding eigenvalues. As an example we write the formula for the second harmonic generation (SHG) hyperpolarizability in schematic form:

[FORMULA NOT REPRODUCIBLE IN ASCII] (11)

where the notation is intended to indicate that a number of combinations of products involving permutations of the excited state indices, n, m with the terms containing the energy denominators are to be taken. The omission of the ground state, indicated by the primed summation, implies that the matrix elements are interpreted as

[FORMULA NOT REPRODUCIBLE IN ASCII] (12)

The subtraction of what is essentially the ground state dipole from the diagonal terms is the consequence of a canonical transformation that ensures that the ground state can be omitted from the perturbation sums, thereby removing a term that becomes infinite at zero frequency. This modification of the matrix elements for polar molecules has important consequences for the qualitative interpretation of second order nonlinearities. Explicit formulae up to third order are given in the paper by Ward and Orr. An earlier paper by Ward was well known as a standard reference for perturbation formulae for nonlinear optical phenomena. Hameka and Svendsen pointed out that Ward's formulae could lead to infinite terms of several different kinds. Their general criticism of the sum over states approach as a method of calculation is debatable. More specific points were resolved in the Orr and Ward paper by the introduction of an implicit rearrangement of the third order terms, which removed some apparently infinite contributions. That such a rearrangement can be made was also noted by Butcher and Cotter and has been discussed a little more explicitly by Morley et al. It follows from the elementary perturbation theory. The earlier part of the Orr and Ward paper addresses the question of line widths using the method of averages of Bogoliubov and Mitropolsky but this technique is not necessary to derive the above formulae. Another extensive set of time-dependent perturbation theory formulae for many kinds of bulk and molecular response function can be found in the article by Flytzanis, which uses a density matrix approach. Line widths are discussed, among other aspects, by Butcher and McLean and by Wherrett. The sum over states (SOS) method, which is simply the numerical implementation of the time-dependent perturbation theory, will be discussed as a computational method in the next section. Here we use it to establish some general features of the time-dependent polarizabilities.

First, it is apparent from the nature of the formulae that, under the right circumstances, near resonance with one of the excitation energies may lead to one term dominating the response. Pre-resonant enhancement can be studied through SOS formulae provided all the photon energies are such that no combination produces a transition energy lying within the line width.

3.1.1 Second Order Effects.-Second order effects occur only when the molecule lacks a centre of symmetry. The diagonal quantities, rnn. vanish for all states of centrosymmetric molecules and equation (12) indicates that excited states that have dipole moments very different from the ground state will be particularly effective in producing high quadratic susceptibility. The excitation of such a state from the ground state will involve charge transfer across the molecule and high hyperpolarizabilities in organic molecules have been associated with the presence of such charge transfer states. Early in the history of the nonlinear optics of organic materials a two-state model was introduced, in which only the ground state and one excited state were retained. This model has been widely used, apparently successfully, to identify molecules with large β-hyperpolarizabilities. An excited state with large charge transfer from the ground state (large rnn) and high oscillator strength (essentially r2on) is required. For preresonant enhancement the excitation energy of n should be a little above the doubled photon energy, thereby leaving both fundamental and doubled beams in a transparent region but substantially reducing one of the energy denominators. Given that the favoured wavelengths for fibre optic communications are in the 1 to 2 micron region, donor/acceptor substituted aromatic molecules with one or two rings are an obvious choice. The archetypical molecule is 4-nitroaniline. Other second order effects, especially the linear electrooptic (Pockels) effect, are treated in much the same way although the absence of the pre-resonant effect at the doubled frequency would be expected to reduce the plausibility of the two state model.

3.1.2 Third Order Effects.- Third order response, like first order response, is present in all molecules and so the associated effects (Kerr effect, frequency tripling etc.) will occur in centrosymmetric and noncentrosymmetric materials. Third order effects are much weaker than second order effects, which is the reason why so much effort has been expended in the investigation of noncentrosymmetric materials likely to have large second order responses. In the third order the extra link in the chain of matrix elements means that excited states not directly accessible from the ground state via the dipole operator now come into play. Amongst these is the ground state itself and it is found that the states with ground state intermediate make a predominantly negative contribution to the susceptibility. The overall susceptibilities (calculated and measured) are usually found to be positive In noncentrosymmetric molecules diagonal terms involving charge transfer type states make a positive contribution and should therefore enhance the overall response. There is some evidence that noncentrosymmetric molecules have larger third order polarizabilities than centrosymmetric molecules.

3.2 Measurement of the Dynamic Hyperpolarizabilities.- The most commonly reported measurement of a second order molecular property is the electric field induced second harmonic generation (EFISH) measurement on a solution where the active molecule is present as solute. In comparing experiment with theory it is pertinent to note that the measured quantity is the vector part of the second order SHG tensor projected onto the direction of the molecular dipole (x). Since the βSHG tensor has up to 18 components the experimental information for testing the details of a theory is usually by no means complete. Other methods of estimating βSHG are through hyper-Rayleigh scattering and through measurements of solvatochromic shifts in solution. The latter method depends on the rather arbitrary choice of a cavity radius which does not drop out of the final answer. Frequency tripling and EFISH (from centrosymmetric molecules with no β)are the usual sources of experimental data on the γ-hyperpolarizabilities.