Hyperpolarizabilities defined

The present paper is aimed at developing an efficient CHF procedure [6-11] for the entire set of electric polarizabilities and hyperpolarizabilities defined in eqs. (l)-(6) up to the 5-th rank. Owing to the 2n+ theorem of perturbation theoiy [36], only 2-nd order perturbed wavefunctions and density matrices need to be calculated. Explicit expressions for the perturbed energy up to the 4-th order are given in Sec. IV. 

In Equation 9 the terms represent the orientationally averaged second hyperpolarizabilities defined as... 

Last, in Fig. 19 we show the evolution of the DPT values for the interaction-induced hyperpolarizability for the monomer (M) and the per-monomer (PM) value. In fact the PM = y(H20)2/2and M = y(H20) values determine the differ-ential-per-monomer hyperpolarizability, defined as... 

The susceptibility tensors give the correct relationship for the macroscopic material. For individual molecules, the polarizability a, hyperpolarizability P, and second hyperpolarizability y, can be defined they are also tensor quantities. The susceptibility tensors are weighted averages of the molecular values, where the weight accounts for molecular orientation. The obvious correspondence is correct, meaning that is a linear combination of a values, is a linear combination of P values, and so on. 

The matrices F, G, F-, H, A", B, and C which appear in the expression for the second hyperpolarizability in Eq. (30) are defined as partial derivatives of the quasienergy Lagrangian taken at zero field strengths and hence are frequency-independent. 

An alternative compact expansion with coefficients which are independent of the optical process can be derived for the isotropic parallel average of the second hyperpolarizability 7 defined as [13]... 

The tensor of the static first hyperpolarizabilities P is defined as the third derivative of the energy with respect to the electric field components and hence involves one additional field differentiation compared to polarizabilities. Implementations employing analytic derivatives in the Kohn-Sham framework have been described by Colwell et al., 1993, and Lee and Colwell, 1994, for LDA and GGA functionals, respectively. If no analytic derivatives are available, some finite field approximation is used. In these cases the P tensor is preferably computed by numerically differentiating the analytically obtained polarizabilities. In this way only one non-analytical step, susceptible to numerical noise, is involved. Just as for polarizabilities, the individual tensor components are not regularly reported, but rather... 

We have considered scalar, vector, and matrix molecular properties. A scalar is a zero-dimensional array a vector is a one-dimensional array a matrix is a two-dimensional array. In general, an 5-dimensional array is called a tensor of rank (or order) s a tensor of order s has ns components, where n is the number of dimensions of the coordinate system (usually 3). Thus the dipole moment is a first-order tensor with 31 = 3 components the polarizability is a second-order tensor with 32 = 9 components. The molecular first hyperpolarizability (which we will not define) is a third-order tensor. 

It should be noted that polarizabilities of various orders can be defined in an alternative way in the SI system of units to that discussed previously. A quantity having the dimension of volume a = a/47re0 can be considered to be an SI analogue of the cgs polarizability. Analogously, y = -y/47re0 (or y = y/eo) can be used as the third-order hyperpolarizability in the SI system, with y having the units of m5 V 3. The presence or absence of the factor of An in the definition of the hyperpolarizability is, unfortunately, not always obvious in literature data. 

The square of the molecular wave function 2, defines the molecular charge distribution. The wave function of state i, ( ), can be calculated in the presence or absence of an electric field. Details of the zero field occupied and unoccupied states determine the size of the hyperpolarizability. Summing the expectation value of the electronic position over occupied states (1 to M) gives the polarization [11]... 

Among the molecular properties introduced above are the permanent electric dipole moment /xa and traceless electric quadrupole moment a(8, the electric dipole polarizability aajg(—w to) [aiso(to) = aaa(—or, o>)], the magnetizability a(8, the dc Kerr first electric dipole hyperpolarizability jBapy(—(o a>, 0) and the dc Kerr second electric-dipole hyperpolarizability yapys(— ( >, 0,0). The more exotic mixed hypersusceptibilities are defined, with the formalism of modern response theory [9]... 

Second, the effect of applied electric field changing the mixing of neutral and charge-separated forms of the charge-transfer chromophores will define the magnitude of electric field required to achieve a given phase shift. Molecular first hyperpolarizability (3 can be understood within the framework of the parameters of this two-state charge-transfer process, as shown in Eq. (5) (see also the Appendix) ... 

In Eqs (4) and (5) the nonlinearities are defined as the derivatives of the polarization in the electric field. Whether the derivative is performed first for Ej and then for Ek or vice versa is of no importance. Therefore, the second-order hyperpolarizabilities y, (-6)4,ct)3,ct)2>fi>i) have a permutation symmetry in the spatial... 

In the same way, frequency-dependent hyperpolarizabilities can be defined as complex quantities by considering the relations between the nonlinear (quadratic and cubic) components of the induced dipole moment oscilla-... 

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