Quadratic response equations

In order to determine the contributions to the quadratic response equations, one has to expand the electronic wave function O > and the operator T1v to second order. The next step concerns the collection of the appropriate terms for the quadratic response equations and as for the linear response equations it is convenient to define the following effective operators ... 

We determine the contributions to the quadratic response equations by collecting terms that involve the quadratic terms in K t) and S t) and as before we consider the orbital and the configurational parts separately and we have... 

When we perform investigations of nonlinear time-dependent properties of quantum mechanical systems coupled to a structured environment given by a classical system we need to consider the modifications to the response equations. Therefore, we will focus on the modifications related to the term Wqm/cm])-Once more we expand 0 > and Tl and then collect to second order the appropriate terms for the quadratic response equations. It is convenient for us to define the following effective operators... 

We use these effective one-electron operators for expressing the QM/CM contributions to the quadratic response equations... 

In short, for a given property A and two periodic perturbations B and C with associated frequencies and respectively, the evaluation of the quadratic response equation ((A B, is carried out by first solving the three linear... 

This subsection presents the modification of the response equations when considering the quadratic response equations for calculating third order time-dependent molecular properties. 

Having these definitions we are able to write the QM/MM contributions to the quadratic response equations as... 

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)... 

We obtain the contributions to the linear, quadratic and cubic response equations by expanding 0 ) and T1 1 followed by collecting terms for a given order of the expansion ... 

The simplest model that adequately describes the concentration-response relationship should be used, e.g., a linear model is simpler than a quadratic model. At the completion of the validation, evaluation of different regression models must be performed. Justification for using a quadratic regression equation must be documented. 

A consistent study of the linear and lowest nonlinear (quadratic) susceptibilities of a superparamagnetic system subjected to a constant (bias) field is presented. The particles forming the assembly are assumed to be uniaxial and identical. The method of study is mainly the numerical solution (which may be carried out with any given accuracy) of the Fokker-Planck equation for the orientational distribution function of the particle magnetic moment. Besides that, a simple heuristic expression for the quadratic response based on the effective relaxation... 

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). 

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... 

Similarly as for the quadratic case an alternative approach is to solve the adjoint linear response equation for A at frequency Wj + W2 + as done in our implementation. 

The present contribution concerns an outline of the response tlieory for the multiconfigurational self-consistent field electronic structure method coupled to molecular mechanics force fields and it gives an overview of the theoretical developments presented in the work by Poulsen et al. [7, 8, 9], The multiconfigurational self-consistent field molecular mechanics (MCSCF/MM) response method has been developed to include third order molecular properties [7, 8, 9], This contribution contains a section that describes the establisment of the energy functional for the situation where a multiconfigurational self-consistent field electronic structure method is coupled to a classical molecular mechanics field. The second section provides the necessary background for forming the fundamental equations within response theory. The third and fourth sections present the linear and quadratic, respectively, response equations for the MCSCF/MM response method. The fifth 283... 

The equations of motion for the linear and quadratic response functions are now obtained by taking the time derivatives and using the Heisenberg equation of motion... 

As the exact response equations appearing in the above expression are identically zero one may conclude that the property has errors quadratic in the errors of the response, thus higher precision. 

Various mathematical models or algebraic equations can be used to estimate fertilizer response curves such as those shown in Figures 2.10 and 2.11, Mathematical models selected for this purpose are equations that properly represent agricultural production-fertilizer relationships that can be estimated using statistical methods. A quadratic function model is commonly used to estimate fertilizer response functions that properly represent agricultural output increasing at a decreasing rate as a result of increased fertilizer application rates. A quadratic response function model may be written as ... 

The second, third and foiuth terms inside the first summation in equation (3) are the perturbations introduced into the hamiltonian by the effects of the external fields. The fourth term, describing the electric field perturbation, is linear in the external potential or electric field. The second and third terms give rise to linear and quadratic responses ro a constant, uniform magnetic field. Smaller terms, arising from the Dirac equation, which represent spin-orbit coupling etc. have been omitted. 

It is often preferable to use y instead of the TDKS orbitals in practice. is unique (up to a gauge transformation), while the orbitals can be mixed arbitrarily by unitary transformations. Both and are linear in y, while they are quadratic in the orbitals also, the TDKS response equations are inhomogeneous in the orbitals (i.e., the response of each orbital is coupled to the response of all other orbitals) due to the density dependence of Ha, while they are homogeneous in y. A response theory based on the TDKS density matrix is therefore considerably simpler than one based on orbitals. Finally, the use of y is computationally more efficient than using orbitals. 

Continuum solvation models (QM/CSM), viii COSMO method, viii Coupled eluster Bruckner double (BD), 10 Coupled cluster linear response equations, 30 Coupled cluster linear response functions, 24 Coupled cluster method (CC), 6, 8, 20, 30, 31 Coupled cluster quadratic response functions, 31... 

The computational route to the calculation of two-photon circular dichroism spectra was opened only recently [115], when it was recognized that the molecular parameters entering the TPCD rotatory strength equation 2.83 can be obtained, as for the two-photon transition amplitude in Eq. 2.78, as single residues of appropriate quadratic response functions [82] ... 

The formulation of approximate response theory based on an exponential parame-trization of the time-dependent wave function, Eq. (11.36), and the Ehrenfest theorem, Eq. (11.40), can also be used to derive SOPPA and higher-order Mpller-Plesset perturbation theory polarization propagator approximations (Olsen et al., 2005). Contrary to the approach employed in Chapter 10, which is based on the superoperator formalism from Section 3.12 and that could not yet be extended to higher than linear response functions, the Ehrenfest-theorem-based approach can be used to derive expressions also for quadratic and higher-order response functions. In the following, it will briefly be shown how the SOPPA linear response equations, Eq. (10.29), can be derived with this approach. 

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