Response theory cubic

Dispersion coefficients for second hyperpolarizabilities using coupled cluster cubic response theory... 

All the linear and nonlinear optical properties introduced above are therefore expressed in terms of linear, quadratic and cubic response functions. They can be computed with high efficiency using analytical response theory [9] with a variety of electronic structure models [8],... 

The contributions, due to the interactions with the solvent, to the cubic response theory are obtained by adding the third-order solvent contribution to the corresponding vacuum equations and we have from Equation (2.308)... 

Here, we provide the theoretical basis for incorporating the PE potential in quantum mechanical response theory, including the derivation of the contributions to the linear, quadratic, and cubic response functions. The derivations follow closely the formulation of linear and quadratic response theory within DFT by Salek et al. [17] and cubic response within DFT by Jansik et al. [18] Furthermore, the derived equations show some similarities to other response-based environmental methods, for example, the polarizable continuum model [19, 20] (PCM) or the spherical cavity dielectric... 

The ab initio calculation of NLO properties has been a topic of research for about three decades. In particular, response theory has been used in combination with a number of electronic structure methods to derive so-called response functions [41 8], The latter describe the response of a molecular system for the specific perturbation operators and associated frequencies that characterize a particular experiment. For example, molecular hyperpolarizabilities can be calculated from the quadratic and cubic response functions using electric dipole operators. From the frequency-dependent response functions one can also determine expressions for various transition properties (e.g. for multi-photon absorption processes) and properties of excited states [42]. 

In the next section we summarize the theoretical background for coupled cluster response theory and discuss certain issues related to their actual implementation. In Sections 3 and 4 we describe the application of quadratic and cubic response in calculations of first and second hyperpolarizabilities. The use of response theory to calculate magneto-optical properties as e.g. the Faraday effect, magnetic circular dichroism, Buckingham effect, Cotton-Mouton effect or Jones birefringence is discussed in Section 5. Finally we give some conclusions and an outlook in Section 6. 

The possibility of simultaneous absorption of two quanta was concluded on purely theoretical basis by Goppert-Mayer in 1931 [64]. First experimental observations of this process in organic systems were reported in early 196O s by Peticolas et al. [96, 97]. Basically, there are two methods of calculation of two-photon absorptivities. The first technique is based on response theory [90]. The two-photon absorption cross section can be determined by the single residue of the cubic... 

Calculations of analytic excited state properties for correlated methods have been reported by several groups [107-118]. Excited state dynamic properties from cubic response theory were first obtained by Norman et al. at the SCF level [55] and by Jonsson et al. at the MCSCF [56] level, and in a subsequent study a polarizable continuum model was applied to account for solvation effects [119]. Hattlg et al. presented a general theory for excited state response functions at the CC level using a quasi-energy formulation [120] which was subsequently implemented and applied at the CCSD level [121, 122]. The first ID DFT calculation of dynamic excited state polarizabilities, which we will shortly review here, was presented in [58] for pyrimidine and -tetrazine utilizing the double residue of the cubic response function derived in Section 2.7.3. 

The expression for the cubic response function is given in Eq. (2.60) of Olsen and Jorgensen (1985). All the propagators that are derived from response theory are retarded polarization propagators. The poles are placed in the lower complex plane. This is specified through the energy variables Ei+itj and 2 + ii . The Pjj operator in Eq. (35) permutes Ei and 2 and it is assumed that the - 0 limit must be taken of the response functions. 

Quadratic response theory in combination with self-consistent field (SCF), MCSCF, and coupled-cluster electronic structure methods have been used for calculation of excitation energies and transition dipole moments between excited electronic states <2000PCP5357>. The excited state polarizabilities for r-tetrazine are given by the double residues of the cubic response functions <1997CPFl(224)201>. 

Using time-dependent density functional cubic response theory, a scheme has been designed to analyze the static and dynamic second hyperpolarizabilities in terms of y densities as well as in terms of contributions from natural bond orbitals (NBOs) and natural localized molecular orbitals (NLMOs). This approach, which has been implemented for both hybrid and nonhybrid TDDFT schemes and which is based on Slater-type basis functions, constitutes an extension of previously proposed schemes limited to the static responses. 

Lan, Y.-Z., Feng, Y.-L., Wen, Y.-H., Teng, B.-T. (2008). Dynamic second-order hyperpolarizabilities of Sis and Si4 clusters using coupled cluster cubic response theory. Chemical Physics Letters, 461(1-3), 118-121. 

The higher-order bulk contribution to the nonlmear response arises, as just mentioned, from a spatially nonlocal response in which the induced nonlinear polarization does not depend solely on the value of the fiindamental electric field at the same point. To leading order, we may represent these non-local tenns as bemg proportional to a nonlinear response incorporating a first spatial derivative of the fiindamental electric field. Such tenns conespond in the microscopic theory to the inclusion of electric-quadnipole and magnetic-dipole contributions. The fonn of these bulk contributions may be derived on the basis of synnnetry considerations. As an example of a frequently encountered situation, we indicate here the non-local polarization for SFIG in a cubic material excited by a plane wave (co) ... 

The incentive and the main goal of this section are to consistently extend the conventional theory on the case of a nonlinear response and by that to confirm its validity. While doing that we propose practical schemes (both exact and approximate) to handle linear and cubic dynamic responses in the framework of classical superparamagnetism. Applying our results to the reported data on the nonlinear susceptibility of Cu-Co precipitates, we demonstrate that a fairly good agreement may be achieved easily. 

B. Jansik, P. Salek, D. Jonsson, O. Vahtras, H. Agren, Cubic response functions in time-dependent density functional theory, J. Chem. Phys. 122 (2005) 054107. 

Buzza and Gates (102) also addressed the question whether disorder or the increased dimensionality from two to three dimensions is responsible for the observed experimental behavior of the shear modulus. In particular, they explored the lack of the sudden jump in G from zero to a finite value at 0 = 0Q that is predicted by the perfectly ordered 2-D model. We have seen above that disorder appears to remove that abrupt jump in two dimensions (90). For drops on a simple cubic lattice, Buzza and Cates analyzed the drop deformation in uniaxial strain close to 0 = 0q, first using the model of truncated spheres . (For reasons given above, we believe this to be a very poor model.) They showed that this model did not eliminate the discontinuous jump in G. An exact model, based on a theory by Morse and Witten (103) for weakly deformed drops, led to G a 1/ In (0 - 0q), which eliminates the discontinuity, but still shows an unrealistically sharp rise at 0 = 0q and is qualitatively very different from the experimentally observed linear dependence of G on (0 - 0q). Similar conclusions were reached by Lacasse and coworkers (49, 104). A simulation of a disordered 3-D model (104) indicated that the droplet coordination number increased from 6 at to 10 at 0 = 0.84, qualitatively similar to what is seen in disordered 2-D systems (90). Combined with a suitable (anharmonic) interdroplet force potential, the results of the simulation were in close agreement with experimental shear modulus and osmotic pressure data. It therefore appears again that disor-... 

---