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...