Polarizability first-hyper

First hyper-polarizability, 35 Forces on the nuclei, 14 Free-energy functional, 5, 6, 10, 11, 47, 49 Frequency dependent dielectric permittivity, 25, 53... 

Solvent reaction potential, 1-3, 6, 8, 10 State specific excitation energies, 37, 56 Static first hyper-polarizability, 14 SVEP method, viii... 

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. 

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. 

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.)... 

An introduction to the phenomena of NLO will be given first (Section 2), followed by the evaluation of molecular second-order polarizabilities by theoretical models that both allow their rationalization and the design of promising molecular structures (Section 3). It will be necessary to develop different models for molecular symmetries, but the approach will remain the same. NLO effects and experiments used for the determination of molecular (hyper)polarizabilities will be dealt with in Section 4. Finally, experimental investigations will be dealt with in Section 5, followed by some concluding remarks. 

In the Bishop and Kirtman (BK) perturbation treatment [17-19] two basic additional assumptions are made. First, when K> is an intermediate excited electronic state it is assumed that, under ordinary non-resonant conditions, one may ignore the optical frequency term zr j. /2- -w in the corresponding energy denominator as compared to the electronic excitation energy. Then, after summing over all intermediate states other than K = 0, one is left with die pure vibrational (hyper)polarizability, P". The latter may be expressed compactly in terms of so-called square bracket quantities. Thus,... 

The P" and P contributions to the static property value may be obtained from this power series in the following manner. First, we impose the minimum condition (i.e. 5V(Q, F)/5g = 0 Vi) on the potential of Eq. (8). This leads to analytical expressions for the field-dependent equilibrium geometry in terms of field-free normal coordinate displacements. Then, substitution of these displacements back into Eq. (8) yields a power series in the static electric field and that gives directly the nuclear relaxation contribution, P" ", to the static (hyper)polarizability [24], For instance, < /0 0)and (0 0,0) are obtained in this manner as ... 

Marder. S.R.. Beratan, D.N.. Cheng. L.-T. Approaches for optimizing the first electronic hyper-polarizability of conjugated organic molecules. Science 252, 103-106 (1991)... 

The PCM calculation are performed by using the CPHF formalism [51] for the static case, and to the TD-CPHF formalism for the frequency dependent case [52]. There are also calculations at higher levels of the QM theory which have not been fully analyzed. The formulas are quite complex, and we refer the interested readers to the two source papers. What is worth remarking here is that (hyper)polarizability values are quite sensitive to the cavity errors. In passing from 7I ) (i.e. a, the polarizability tensor) to 7 1 (i.e. /9, the first hyperpoljirizability) and to 7 (i.e. 7, the second hyperpolarizability) the problem of cavity errors become worse and worse. 

This description of quantum mechanical methods for computing (hyper)polarizabilities demonstrates why, nowada, the determination of hyperpolarizabilities of systems containing hundreds of atoms can, at best, be achieved by adopting, for obvious computational reasons, semi-empirical schemes. In this study, the evaluation of the static and dynamic polarizabilities and first hyperpolaiizabilities was carried out at die Time-Dependent Hartree-Fock (TDOT) [39] level with the AMI [50] Hamiltonian. The dipole moments were also evaluated using the AMI scheme. The reliability of the semi-empirical AMI calculations was addressed in two ways. For small and medium-size push-pull polyenes, the TDHF/AMl approach was compared to Hartree-Fock and post Hartree-Fock [51] calculations of die static and dynamic longitudinal first hyperpolarizability. Except near resonance, the TDHF/AMl scheme was shown to perform appreciably better than the ab initio TDHF scheme. Then, the static electronic first hyperpolaiizabilities of the MNA molecule and dimer have been calculated [15] with various ab initio schemes and compared to the AMI results. In particular, the inclusion of electron correlation at the MP2 level leads to an increase of Paaa by about 50% with respect to the CPHF approach, similar to the effect calculated by Sim et al. [52] for the longitudinal p tensor component of p-nitroaniline. The use of AMI Hamiltonian predicts a p aa value that is smaller than the correlated MP2/6-31G result but larger than any of the CPHF ones, which results fi-om the implicit treatment of correlation effects, characteristic of die semi-empirical methods. This comparison confirms that a part of die electron... 

The correlation, vibrational and relativistic effects to L NLO properties have been studied by selecting as model systems the Group Ilb sulfides ZnS, CdS and HgS [15]. These weakly bound systems are expected to have quite large vibrational (hyper) polarizabilities. To the best of our knowledge this was the first study which included the computation of all three contributions to the (hyper) polarizabilities. 

The obtained electronic (hyper)polarizabilities were compared with values published previously by Campbell et al. [70, 71] which were computed using an uncoupled approximation to coupled-perturbed HF theory, with the 6-3IG basis set. Their values were very different from those obtained in Ref. [60], e.g. for the first hyperpolarizabilities they reported Pyyy —7,000 au, Pyyy 15,000 au, about an order of magnitude larger than in Ref. [61]. For the second hyperpolarizability Yxxxx 320 X 10 au, Vyyyy 540 X 10 au, yzzzz —320 x 10 au. These large differences may be interpreted as an additional indication that HF theory is unreliable for the hyperpolarizabiUties of Li C6o-... 

The nuclear relaxation (NR) contributions were computed using a finite field approach [73,74]. In this approach one first optimizes the geometry in the presence of a static electric field, maintaining the Eckart conditions. The difference in the static electric properties induced by the field can then be expanded as a power series in the field. Each coefficient in this series is the sum of a static electronic (hyper) polarizability at the equilibrium geometry and a nuclear relaxation term. The terms evaluated in Ref. [61] were the change of the dipole moment up to the third power of the field, and that of the linear polarizability up to the first power ... 

In [170] the authors obtain a test set of ten molecules of specific atmospheric interest in order to evaluate the performance of various Density Functional Theory (DFT) methods in (hyper)polarizability calculations as well as established ab initio methods. The authors make their choice for these molecules based on the profound change in the physics between isomeric systems, the relation between isomeric forms and the effect of the substitution. In the evaluation analysis the authors use arguments chosen from the information theory, the graph theory and the pattern recognition fields of Mathematics. The authors mentioned the remarkable good performance of the double hybrid functionals (namely B2PLYP and mPW2PLYP) which are for the first time used in calculations of electric response properties. 

The (monochromatic) electric fields are characterized by Cartesian directions indicated by the Greek letters and by circular optical frequencies, coi, a>2, and The induced dipole moment oscillates at a> = EjO,. and are such that the jS and y 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, o(0 0) and a(—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, (0 0,0), a>,a>)... 

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 (hyperjpolarizability 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 ... 

The MP2/aug-cc-pVTZ method has been applied within the finite field approach to calculate the polarizability and first hyperpolarizability of the Li2F and LigF systems. These systems present an alkalide character, i.e. some of its alkali metal atoms bear a negative charge, which is loosely bound and is therefore at the origin of large (hyper)polarizabilities. Using unrestricted MP2 calculations, the first hyperpolarizability has been shown... 

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