Hyperpolarizability tensor factors

We have used that [ "] = E and the fact that Wp M2, and 0)3 are dummy summation indices that run over both positive and negative frequencies. None of Eqs. (59)-(62) is symmetric in the tensor indices y, and 8. As pointed out in connection with Eq. (7), we normally choose our hyperpolarizability tensors to possess intrinsic symmetry, and it is clear that we can accomplish this without changing the polarization of the molecule by taking the average of the six terms generated by permuting pairs of the dummy indices (j3, Wj), (% M2), and (8, 0)3) we denote this operation with the symbol l/6J2 i,2,3> where tlie factor of one sixth is required to maintain the same value of the polarization. The third-order polarization in Eq. (36) can then be written as... 

In the above equations is the component of the molecular first-order hyperpolarizability tensor p along the principal molecular axis, and are angular averages which describe the degree of polar order,/"/ " are the local field correction factors at [Pg.125]

Bold quantities are operators, vectors, matrices or tensors. Plain symbols are scalars. a Polarizability a, P Spin functions a, p Dirac 4x4 spin matrices ap-jS Summation indices for basis functions F Fock operator or Fock matrix Fy, Eajd Fock matrix element in MO and AO basis Y Second hyperpolarizability yk Density matrix of order k gc Electronic g-factor... 

The oriented gas model was first employed by Chemla et al. [4] to extract molecular second-order nonlinear optical (NLO) properties from crystal data and was based on earlier work by Bloembergen [5]. In this model, molecular hyperpolarizabilities are assumed to be additive and the macroscopic crystal susceptibilities are obtained by performing a tensor sum of the microscopic hyperpolarizabilities of the molecules that constitute the unit cell. The effects of the surroundings are approximated by using simple local field factors. The second-order nonlinear response, for example, is given by... 

Notation. The symbols a, f, y are used throughout to denote the electric field polarizability, first and second hyperpolarizabilities respectively, suitably qualified by frequency factors where necessary. The magnetizability is denoted by y and the nuclear screening tensor by a. The numerous but well-known acronyms specifying the computational procedures are used without definition. The possibly rather less well-known acronyms for the principal gauge invariant procedures are given in Table 1. 

These expressions include averaging over all orientations of the chromo-phore with respect to the field and the polarization of the measuring light The factors Uij and bijj in Eqs. (B4.15.2) and (B4.15.3) are elements of the transition polarizability tensor (a) and the transition hyperpolarizability (b), which describe the effects of the external field on the dipole strength of the absorption band. These effects probably are relatively minor in most cases, and are neglected in Eq. (4.64). The transition polarizabifity tensor... 

It is also worth mentioning that the Taylor series expansion in Eq. (8.1) is sometimes replaced by a power series. In this case, the numerical factors l/ are impUcitly included into the 0/y tensor components. These different conventions (referred to as the T convention for the Taylor expansion and B convention for the power series expansion) have often been the causes of confusion in the community when comparing the hyperpolarizability values obtained from different experimental or computational approaches [39, 40]. 

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