Hyperpolarizability tensors, first

The next terms in the series, denoted. .. in equation 17.1 above, are called the dipole hyperpolarizabilities. The first one is and this also is a tensor. It has three indices, and the corresponding formula for the induced dipole, equation 17.3, becomes... 

Trans-polyenes H-(-HC=CH-),, -H, trans-polyenynes H-(HC=CH-C=C) -H, cumulenes H2C=(C=C) =CH2 and polyynes H-(C=C) -H have been studied (M=N-1). For eentrosymmetrie molecules, the first order hyperpolarizability p is equal to zero so that non linear effects are of second order nature. Furthermore, (the x axis goes through the middle of the C-C bonds of the polyenes, or is the intemuclear axis in the case of linear molecules) is the most important component of the second order y hyperpolarizability tensor, the other components being negligible. Both y and the mean hyperpolarizability... 

Heterogeneous dielectric media models have included the developments of Jprgensen et al. [7-9] (reviewed here) and Corni and Tomasi [52,53], Generally, the number of methods for determining frequency-dependent molecular electronic properties, such as the polarizability or first- and second hyperpolarizability tensors of heterogeneously solvated molecules, is very limited. 

Our present focus is on correlated electronic structure methods for describing molecular systems interacting with a structured environment where the electronic wavefunction for the molecule is given by a multiconfigurational self-consistent field wavefunction. Using the MCSCF structured environment response method it is possible to determine molecular properties such as (i) frequency-dependent polarizabilities, (ii) excitation and deexcitation energies, (iii) transition moments, (iv) two-photon matrix elements, (v) frequency-dependent first hyperpolarizability tensors, (vi) frequency-dependent polarizabilities of excited states, (vii) frequency-dependent second hyperpolarizabilities (y), (viii) three-photon absorptions, and (ix) two-photon absorption between excited states. 

As will be explained in Section 8.3.3, the reduction spectrum of rank-three tensors contains, in the general case, one (pseudo)scalar, three vectors, two (pseudo)devia-tors, and one septor [13]. Here, we only have to deal with the vectorial parts. They transform as a vector. Hence, the scalar product of the dipole moment vector with a vector part of the first hyperpolarizability tensor can be written as the simple product of two scalars, when we designate 9 as the angle between the molecular axis (z-axis) and the dipole moment axis (Eq. (14)). 

P Ptjk Pzzz P first hyperpolarizability tensor, tensor components... 

In the present contribution we will discuss the direction of the changes of the NLO response and the solvatochromic behavior as a function of solvent polarity of the D-tt-A chromophores. The best starting point for these considerations seems to be the simple two-state model for the first-order hyperpolarizability (/ ) [8]. To avoid the extreme complexity of the sum-over-states (SOS) expression [101], Oudar and Chemla proposed the relation between the dominant component of 13 along the molecular axis (let it be the x-axis) and the spectroscopic parameters of the low-lying CT transition [8]. The use of the two-level approximation in the static case (ru = 0.0) has lead to the following expression for the static component of the first-order hyperpolarizability tensor ... 

In formula (34), /u, (f) = (0 /Ai(f) 0 ) is the ground state, permanent, electric dipole moment of species f, while ) is the molecular first hyperpolarizability tensor defined by... 

Butadiene. - The ab initio calculation of the linear polarizability and second hyperpolarizability tensors for f/ww-butadiene has attracted a great deal of attention. The molecule is the smallest stable structure showing -electron delocalisation and is the first member of the fully conjugated polyene series. Extended conjugated structures are associated with low excitation energies and large response functions. 

However, using the McWeeny approach [7], it is sufficient to calculate only the projection P >v/K on the subspace of virtual zero-order orbitals in order to get the second hyperpolarizability tensor. This projection is evaluated via a procedure similar to the one used in solving the first-order equation (21). Taking in (32) the Hermitian product with the unoccupied and using (19), one finds... 

Bancewicz et al have discussed colKsion-induced HRS spectra of gaz-eous H2-Ar mixture on the basis of ab initio calculations of the collision-induced first hyperpolarizability tensor, Ap. The independent spherical components of the Ap values, evaluated at the MP2 level using the [%s6p5dAf and [6sAp2d[ basis sets for Ar and H atoms, have been fitted to analytical formulas expressing their dependence on the intermolecular distance and then employed as input data to analyze the HRS data. The good agreement between the quantum and semiclassical approaches can be treated as a benchmark proof of both the method s feasibility and accuracy. 

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