Hyper-polarizability

This paper is, thus, a double tribute to Professor Berthier. On one side, G. Berthier has provided excellent analysis of quantum mechanical instabilities [4], while additionally being at the origin of the interest of the Namur group for studies of (hyper)polarizabilities in organic molecules and chains. 

In order to overcome the optimization process of the (hyper) polarizabilities calculations, we have been led to deeply study the perturbational and variational methods and in particular the variation-perturbation treatment introduced by Hylleras (20) since 1930. We will not develop here the theoretical framework of the recent study of N. El Bakali Kassimi (21). We propose criteria for generating adequate sets of polarization functions necessary to calculate (hyper) polarizabilities. 

Where a, P and y are the linear polarizability, the first- and second-hyper polarizabilities, respectively, and are represented by second, third and fourth rank tensors, respectively, and is a static polarizability. 

Cyvin, S. J., Rauch, J. E. and Decius, J. C. (1965) Theory of hyper-Raman effects (nonlinear inelastic light scattering) selection rules and depolarization ratios for the second-order polarizability. 

Spectroscopic applications usually require us to go beyond single-point electronic energy calculations or structure optimizations. Scans of the potential energy hypersurface or at least Taylor expansions around stationary points are needed to extract nuclear dynamics information. If spectral intensity information is required, dipole moment or polarizability hypersurfaces [202] have to be developed as well. If multiple relevant minima exist on the potential energy hyper surface, efficient methods to explore them are needed [203, 204],... 

Hyper)polarizabilities are defined as the coefficients in the Taylor series expansion of the dipole moment - or the energy - in the presence of static and/or oscillating electric fields ... 

This paper concerns the use of ab initio Cl methods to calculate the (hyper)polariz-abilities within the SOS scheme. This work starts by adopting the simplest Cl scheme. 

Just as a is the linear polarizability, the higher order terms p and y (equation 19) are the first and second hvperpolarizabilities. respectively. If the valence electrons are localized and can be assigned to specific bonds, the second-order coefficient, 6, is referred to as the bond (hyper) polarizability. If the valence electron distribution is delocalized, as in organic aromatic or acetylenic molecules, 6 can be described in terms of molecular (hyper)polarizability. Equation 19 describes polarization at the atomic or molecular level where first-order (a), second-order (6), etc., coefficients are defined in terms of atom, bond, or molecular polarizabilities, p is then the net bond or molecular polarization. 

Still within a continuum solvation approach [22,41], a unified treatment of the local field problem has recently been formulated within PCM for (hyper)polarizabilities [47] and extended to several optical and spectroscopic properties, including IR, Raman, VCD and VROA spectra [8,9,11,12],... 

The approach just sketched in terms of effective properties has also been applied to other vibrational spectroscopies, such as Raman [9], IR linear dichroism [10], VCD [11] and VROA [12], as well as to (hyper)polarizabilities [47 19] and birefringences of systems in a condensed phase (see refs. [50,51] and the contribution by Rizzo in this... 

An analytical structure-(hyper)polarizability relationship based on a two-state description has also been derived [49]. In this model a parameter MIX is introduced that describes the mixture between the neutral and charge-separated resonance forms of donor-acceptor substituted conjugated molecules. This parameter can be directly related to BLA and can explain solvent effects on the molecular hyperpolarizabilities. NMR studies in solution (e.g. in CDCl3) can give an estimate of the BLA and therefore allow a direct correlation with the nonlinear optical experiments. A similar model introducing a resonance parameter c that can be related to the MIX parameter was also introduced to classify nonlinear optical molecular systems [50,51]. 

Here, E is the strength of the applied electric field (laser beam), a the polarizability and / and y the first and second hyper-polarizabilities, respectively. In the case of conventional Raman spectroscopy with CW lasers (E, 104 V cm-1), the contributions of the / and y terms to P are insignificant since a fi y. Their contributions become significant, however, when the sample is irradiated with extremely strong laser pulses ( 109 V cm-1) created by Q-switched ruby or Nd-YAG lasers (10-100 MW peak power). These giant pulses lead to novel spectroscopic phenomena such as the hyper-Raman effect, stimulated Raman effect, inverse Raman effect, coherent anti-Stokes Raman scattering (CARS), and photoacoustic Raman spectroscopy (PARS). Figure 3-40 shows transition schemes involved in each type of nonlinear Raman spectroscopy. (See Refs. 104-110.)... 

Although the spherical form of the multipole expansion is definitely superior if the orientational dependence of the electrostatic, induction, or dispersion energies is of interest, the Cartesian form171-174 may be useful. Mutual transformations between the spherical and Cartesian forms of the multipole moment and (hyper)polarizability tensors have been derived by Gray and Lo175. The symmetry-adaptation of the Cartesian tensors of quadrupole, octupole, and hexadecapole moments to all 51 point groups can be found in Ref. (176) while the symmetry-adaptation of the Cartesian tensors of multipole (hyper)polarizabilities to simple point groups has been considered in Refs. (172-175). 

In this section we present summaries of the DRF approach in various fields of computational chemistry, ranging from spectra and (hyper-)polarizabilities to chemical reactions in solution. 

We discussed DRF in perspective with other methods, gave the theoretical background and addressed the implementation. In a short section on the validation of DRF we showed that we can treat a system with QM, MM or QM/MM without significant loss of accuracy. A set of examples of its application ranges from simple solvation energies, spectra to (hyper)polarizabilities and processes of excited states of molecules in solution. These examples employ DRF in combination with—ab initio or semi-empirical—conventional wave function and DFT techniques. 

Duijnen, P.Th. van, Swart M. and Grozema F., QM/MM calculation of (hyper)polarizabilities with the DRFapproach., in Hybrid Quantum Mechanical and Molecular Mechanics Methods, J.Gao and M.A. Thompson, Editors. 1999, ACS Books Washington, DC. p. 220-232. 

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