Polarizabilities and hyperpolarizabilities

Benoít Champagne

DOI: 10.1039/b812904p

This chapter presents and discusses recent achievements towards determining and interpreting polarizabilities and hyperpolarizabilities of atoms, molecules, clusters, supramolecular assemblies, polymers, and aggregates. It evidences both the strong activities in the field and their dual character. Indeed, on the one hand, the polarizabilities and even more the hyperpolarizabilities are difficult quantities to predict, so that they are ideal targets when elaborating new calculation methods. This comes from the fact that many aspects need to be addressed in their evaluation: vibrational versus electronic contributions, frequency dispersion including resonance, electron correlation and relativistic effects, impact of the surroundings. On the other hand, linear and nonlinear responses are evaluated in a strategy of designing new systems with remarkable electric, magnetic, and optical properties. In this second motivation for calculating the polarizabilities and hyperpolarizabilities, the emphasis is also put on the interpretation and the deduction of structure-property relationships while these investigations are parts of multidisciplinary approaches including synthetic and experimental characterizations.

1. Introduction

This Chapter reports on theoretical developments and applications carried out from June 2007 to May 2008 for estimating and interpreting the polarizabilities and hyperpolarizabilities of atoms, molecules, polymers, clusters, and molecular solids. It follows the three Chapters written by D. Pugh in 2000, 2002, and 2006 for the same series of reviews.

After a brief introduction to the polarizabilities and hyperpolarizabilities, this chapter is divided into six sections. The first one (Section 2) deals with methodological developments and implementations with particular emphasis on vibrational contributions as well as on electron correlation effects and the subsequent challenge of using density functional theory approaches. Applications are then classified into four categories. The last section (Section 6) is analyzing the responses of molecular aggregates and of solid phases while the three first ones (Sections 3–5) are concerned with linear and nonlinear responses of molecules, or rather of species of finite size. Indeed, these three sections are not only dealing with molecules but also with clusters, polymers, and supramolecular aggregates. These three sections are further classified into (non-resonant) electric field responses (Section 3), mixed electric dipole, electric quadrupole, and magnetic dipole responses (Section 4), and resonant responses (Section 5). Finally Section 7 presents some challenges. This classification is certainly not unique but appears to the author as a suitable way for presenting the works performed during the dedicated period.

1.1 Theoretical frame and definitions of polarizabilities and hyperpolarizabilities

Investigations of the linear and nonlinear optical properties of molecules, polymers, and clusters often adopt the semi-classical approach. In this approach, the particles are treated quantum mechanically while a classical treatment is applied to the radiation so that the Hamiltonian is written as the sum of two types of terms, one representing the isolated system (H0) and one being the radiation-molecule interaction term (H1). For sufficiently large wavelengths with respect to the system dimensions, H1 can be expressed under the form of a multipole expansion:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where one distinguishes successively the electric dipole, magnetic dipole, and electric quadrupole terms associated with the electric field, magnetic field, and electric field gradient, respectively. In turn, each of these moments can be expressed into Taylor series expansions of the different external perturbations ([??], [??], and [??][??]). In the case of the responses of the dipole moment to external electric fields, the Taylor series expansion for any Cartesian component reads:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

The (monochromatic) electric fields are characterized by Cartesian directions indicated by the Greek letters and by circular optical frequencies, ω1, ω2, and ω3. The induced dipole moment oscillates at ωσ = Σiωi. K(2) and K(3) are such that the β and γ values associated with different NLO processes converge towards the same static value. The "0" superscript indicates that the properties are evaluated at zero electric fields. Eqn (2) is not the unique phenomenological expression defining the (hyper)polarizabilities. Another widely-applied expression is the analogous power series expansion where the 1/2 and 1/6 factors in front of the second- and third-order terms are absent. The static and dynamic linear responses, α(0;0) and α(-ω;ω), correspond to the so-called static and dynamic polarizabilities, respectively. At second order in the fields, the responses are named first hyperpolarizabilities whereas second hyperpolarizabilities correspond to the third-order responses. Different phenomena can be distinguished as a function of the combination of optical frequencies. So, β(0;0,0), β(-ω;ω,0), β(0;ω,-ω), and β(-2ω;ω,ω) are associated with static, dc-Pockels (dc-P), optical rectification (OR), and second harmonic generation (SHG) processes whereas γ(0;0,0,0), γ(-ω;ω,0,0), γ(2ω;ω,ω,0), γ(-ω;ω,-ω,ω), and γ(-3ω;ω,ω,ω,) describe static, dc-Kerr, electricfield-induced second harmonic generation (EFISHG), degenerate four-wave mixing (DFWM), and third harmonic generation (THG) phenomena, respectively. The polarizabilities, first, and second hyperpolarizabilities are second-, third-, and fouth-rank tensors and contain therefore many quantities. For instance, the β tensor contains 27 elements. Nevertheless, in the static limit, only 10 are independent whereas they are 18 independent terms for dc-P and SHG. Moreover, experimental characterizations, which enable to deduce some invariants of the linear and nonlinear responses, are generally not able to address the full set of independent tensor components in absence of symmetry considerations or in absence of approximations concerning the relative amplitude of these elements. The illustrations and applications treat here the case of hyper- Rayleigh scattering (HRS) and EFISHG. For HRS, in the case of plane-polarized incident light and observation made perpendicular to the propagation plane without polarization analysis of the scattered beam, the second-order NLO response that can be extracted from HRS data reads:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

<β2zzz> and <β2xzz> correspond to orientational averages of the β tensor, which, without assuming Kleinman's conditions read:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

In HRS, an interesting quantity is the depolarization ratio, which reads:

DR = <β2zzz>/<β2xzz> (6)

It gives information on the geometry of the chromophore, the part of the molecule responsible for the NLO response (for an ideal donor/acceptor one-dimensional system DR = 5, for an octupolar molecule, DR = 1.5 whereas for a Λ-shape molecule, the amplitude of DR depends on the angle between the chromophore as well as on the D/A groups). On the other hand, the EFISHG measurements give information on the projection of the vector part of β on the dipole moment vector:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where ||[??]|| is the norm of the dipole moment and μi and βi the components of the μ and β vectors. EFISHG measurements can also be used to determine the second hyperpolarizability. In that case, the γ quantity that can be deduced from experiment is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Different averaging expressions hold for the dc-Kerr effect and other phenomena.

Equations similar to eqn (2) can be written for the other perturbations as well as for mixed perturbations and responses. Some of them are introduced in section 4. More information, definitions, and experiment-related issues can be found in other reviews and in books.

1.2 Brief overview of the methods for predicting and interpreting polarizabilities and hyperpolarizabilities

There are many approaches to compute the polarizabilities and hyperpolarizabilities and also different ways to classify them. One convenient division is between perturbation theory approaches, which express the (hyper)polarizability using Summation-Over-States (SOS) expressions and those techniques, which are based on the evaluation of derivatives of the energy (or another property). SOS approaches consist in evaluating energies and transition dipoles that appear in the (hyper)polarizability expressions. For instance, in the case of the frequency-dependent electric-dipole electronic first hyperpolarizability, the SOS expression reads:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where the sums run over all excited states |n> and |m> of energy En and Em. |0> is the ground state wavefunction of energy E0 and ΔEn0 = En - E0. The quantity [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the ζ-component of the transition (electric) dipole moment between states |0> and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and ΣP-σ,1,2 is the sum over the six permutations of the pairs (-ωσ, ζ), (ω1, η) and (ω2, χ). The quality of the computed properties depend therefore on the level of approximation that is used to determine the ground and excited state wavefunctions and energies. It is interesting to recall that, within these schemes, field-perturbed energies and wavefunctions are never explicitly determined but that the (hyper)polarizabilities are evaluated from the knowledge of the unperturbed wavefunctions and energies. One of the big advantages of these approaches lies in the simplicity of the expressions, easily amenable to interpretation. Still nowadays, two- and three-state models continue to receive lot's of success to interpret β and γ values. Propagators or response function approaches, where the SOS expressions are recast under the form of superoperator resolvent and then approximated, also belong the the class of SOS approaches, though the formalism is different.

The other class of methods relies on the numerical or analytical evaluation of the field derivatives of the energy to evaluate the polarizabilities and hyperpolarizabilities. The finite field (FF) approach is the most straight-forward and probably the most employed of these methods to evaluate static (hyper)polarizabilities. It consists in adding the perturbation to the Hamiltonian and in solving the wavefunction/energy equations for different magnitudes and orientations of the electric field, and then in differentiating the energy numerically. This approach is very general because it can be applied to any level of theory for which field-dependent energies can be evaluated. It can also be generalized to other field-dependent quantities, as illustrated by the following equivalence relationships:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where the Taylor series expansion (eqn (2)) is used and where E is the energy. These equivalence relationships are guaranteed for exact wave- functions as well as for methods that satisfy the Hellmann-Feynman theorem. Besides numerical derivative procedures, the (hyper)polarizabilities can be evaluated using analytical methods, which are no more restricted to differentiations with respect to static fields. One of these methods is referred to the time-dependent Hartree-Fock (TDHF) scheme and consists in solving at each order the Hartree-Fock perturbation equations by using an iterative self-consistent procedure. Its static analog is called the coupled-perturbed Hartree-Fock (CPHF) method and it provides similar results to a FF approach based on Hartree-Fock field-dependent energies. Although the above examples are based on the determination of successive responses to the electric field, similar approaches are employed to get mixed responses. Similarly, analogs of the CPHF and TDHF methods have been worked out to calculate these properties by including electron correlation effects. At the DFT levels, the corresponding approaches are known as the coupled-perturbed Kohn-Sham (CPKS) and time-dependent DFT (TDDFT) schemes.

It is however not the scope of this survey to describe in more details these methods. The interested reader is invited to look at refs. 2–6 for more information on these methods and on their implementations.

2. Methodological developments and new implementations

2.1 Ab initio methods

Wang et al. have proposed a density matrix based TDDFT method in both real time and frequency domains to calculate the dynamic hyperpolarizabilities. Illustrations carried out using local density approximation on small compounds (CO, HF, HCl, and LiF) have shown the good agreement between the two methods. In addition, in the real time domain scheme, it has been found that the external field should be turned on slowly to prevent nonadiabatic effects.

The explicit time-dependent configuration interaction (TD-CI) method has been applied to calculate the linear and nonlinear electric responses of H2 and H2O molecules to external time-dependent perturbations. Three variants have been employed to solve the time-dependent Schrödinger equation, namely, the TD-CIS (inclusion of single excitations only), TD-CISD (inclusion of single and double excitations), and TD-CIS(D) (single excitations and a perturbative treatment of the double excitations) methods. The authors have stressed that one of the biggest advantages of this approach is its ability to tackle molecular responses to pulses with arbitrary time-dependence, even beyond perturbation theory.

Eshuis et al. have implemented fully propagated time-dependent Hartree-Fock theory to calculate the real time electronic dynamics of closed- and open-shell molecules in strong oscillating electric fields. This method has been illustrated on the determination of the frequency-dependent polarizability of ethylene and is shown to converge, in the weak field limit, to the same results as the linearized TDHF method.

Gauss et al. have reported the first implementation of the gauge-including atomic orbital (GIAO) or London atomic orbital (LAO) analytical second-derivative approach for the calculation of the magnetizabilities — and rotational g tensors — using arbitrary coupled cluster methods. In this way, hierarchies of approximate schemes [HF, CCSD, CCSD(T), CCSDT, ...] for calculating the magnetizabilities have been benchmarked and the role of electron correlation assessed.

A scheme to calculate frequency-dependent first hyperpolarizabilities for general CC wavefunctions (CCSD, CC3, CCSDT, and CCSDTQ) has been presented by O'Neill et al. This analytical third derivative scheme exploits the similarities between response theory and analytic derivative theory. Illustrations have first confirmed that the inclusion of higher-than-double excitations is essential for a quantitative description of the first hyperpolarizabilities. Moreover, the CC3 approximation has been seen to provide good results for singly-bonded systems, with little multireference character, but that full triples contribution using CCSDT are required for benchmark quality results on other systems. Representative results of ref. 18 are given in Table 1.