Response function quadratic

The second contribution in (3.14) contains the quadratic response function, (rur2) ... 

We obtain the terms for the solvent modifications of the quadratic response functions, denoted wj J, by collecting all terms lor n = 2 in Equation (2.308)... 

Also in response theory the summation over excited states is effectively replaced by solving a system of linear equations. Spin-orbit matrix elements are obtained from linear response functions, whereas quadratic response functions can most elegantly be utilized to compute spin-forbidden radiative transition probabilities. We refrain from going into details here, because an excellent review on this subject has been published by Agren et al.118 While these authors focus on response theory and its application in the framework of Cl and multiconfiguration self-consistent field (MCSCF) procedures, an analogous scheme using coupled-cluster electronic structure methods was presented lately by Christiansen et al.124... 

In the earlier sections of this chapter we reviewed the many-electron formulation of the symmetry-adapted perturbation theory of two-body interactions. As we saw, all physically important contributions to the potential could be identified and computed separately. We follow the same program for the three-body forces and discuss a triple perturbation theory for interactions in trimers. We show how the pure three-body effects can be separated out and give working equations for the components in terms of molecular integrals and linear and quadratic response functions. These formulas have a clear, partly classical, partly quantum mechanical interpretation. The exchange terms are also classified for the explicit orbital formulas we refer to Ref. (302). 

The expressions in Eqs. (13-39) and (13-40) define the linear and quadratic response functions implicitly. 

Table 13-1. Matrices and vectors for linear and quadratic response functions...

The explicit PE contributions to the quadratic response function enter the E matrix (Eq. (88)) and the vector (Eq. (89)). Contributions that appear... 

The linear response function in Eq. (11) has the same structure as the second-order energy expression in Eq. (2) and we note that for A = V and wj = 0 they are identical, except for a factor of two. Similarly, Eq. (10) defines the quadratic response function... 

From the discussion above it is clear that for the evaluation of quadratic response functions it is desirable to contract two indices of E with two vectors simultaneously, giving... 

Henne Hettema, Hans Jorgen Aa. Jensen, Poul Jorgensen, and Jeppe Olsen (1992). Quadratic response functions for a multi-configurational self-consistent-field wave-function. J. Chem. Phys. 97, 1174-1190. 

For the quadratic response function, we have that the energy difference between the excited state q) and the ground state 0) is co o = q-Eo and for the external electric field we have the frequencies co and > . We are able to write the spectral representation of the quadratic response function as... 

Semi-empirical ZINDO SOS (sum over states) and ab initio quadratic response function (DDRPA) calculations on a series of D-A-substituted 7t-conjugated chromophores based on styryl benzothiazoles were used to aid in the design of dyes with high nonlinear optical properties <2004PCP495>. 

A ((P Q)) propagator is called a linear response function, since it measures the response of P to a perturbation Q. The ((r r)) propagator thus determines the polarizability, which is a second-order property. The concept may be generalized to higher orders, i.e. the quadratic response function, given as a ((P Q,R)) propagator. 

In the previous section we discussed pure electric-dipole hyperpolarizabilities, in particular second harmonic generation. Another important class of NLO processes includes birefringences and dichroisms which can be rationalized (at least to lowest orders in perturbation theory) in terms of response functions involving, besides the electric-dipole, also magnetic-dipole and electric-quadrupole operators. Prominent examples related to quadratic response functions are ... 

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