Response function approximate expression

More information can be extracted from the linear response function, however. This is most easily seen by studying the exact linear response function. Unlike the approximate theories, the exact linear response function is expressed in terms of the ground and excited states satisfying the time-independent Schrodinger equation... 

The model of a reacting molecular crystal proposed by Luty and Eckhardt [315] is centered on the description of the collective response of the crystal to a local strain expressed by means of an elastic stress tensor. The local strain of mechanical origin is, for our purposes, produced by the pressure or by the chemical transformation of a molecule at site n. The mechanical perturbation field couples to the internal and external (translational and rotational) coordinates Q n) generating a non local response. The dynamical variable Q can include any set of coordinates of interest for the process under consideration. In the model the system Hamiltonian includes a single molecule term, the coupling between the molecular variables at different sites through a force constants matrix W, and a third term that takes into account the coupling to the dynamical variables of the operator of the local stress. In the linear approximation, the response of the system is expressed by a response function X to a local field that can be approximated by a mean field V ... 

From this equation it follows that dg,A Pa is diagonal in the spin indices. We will therefore in the following put density variation 5p (r) determines the potential variation 5vs,(r) only up to a constant (see also [66] ). To find an explicit expression for the above functional derivative we must find an expression for the inverse density response function i A. In order to do this we make the following approximation to the Greens function (see Sharp and Horton [39], Krieger et al. [21]) ... 

The sum-over-states expressions that we have presented in Eqs. (69), (70), and (71) are only true for exact wave functions and they are rather cumbersome methods for calculating time-dependent electromagnetic properties of a quantum mechanical subsystem within a structured environment. The advantage of the sum-overstates expressions is that they illustrate the type of information that is obtainable from response functions. We have utilized modem versions of response theory where the summation over states is eliminated when performing actual calculations, that involve approximative wave functions [21,24,45-47,80-83]. 

In the theory of classical liquids [79],/x<, plus the particle-particle interaction is known as Ornstein-Zernike function. In practice, of course, this quantity is only approximately known. Suitable approximations of will be discussed in section 6. In order to construct such approximate functionals, it is useful to express / c in terms of the full response function x- An exact representation of fxc is readily obtained by solving Eq. (144) for Vi and inserting the result in Eq. (155). Equation (154) then yields... 

When the external electric field is time-dependent, there is no well-defined energy of the molecular system in accordance with Eq. (100), and the wave function response can thus not be retrieved from a variational condition on the energy as in the analytic derivative approach described above. Instead the response parameters have to be determined from the time-dependent Schrodinger equation, a procedure which was illustrated in Section 3 for the exact state case. In approximate state theories, however, our wave function space only partially spans the 7V-electron Hilbert space, and the response functions that correspond to an approximate state wave function will clearly be separate from those of the exact state wave function. This fact is disregarded in the sum-over-states approach, and, apart from the computational aspect of slowly converging SOS expressions, it is of little concern when highly accurate wave function models are used. But for less flexible wave function models, the correct response functions should be used in the calculation of nonlinear optical properties. 

However, the equivalence of the response functions to the property derivatives is in approximate methods not always strict, as, for example, CC response functions as defined in Section 2 do not involve contributions due to orbital relaxation while property derivatives usually do. The incorporation of orbital relaxation effects in the property derivatives is mandatory when perturbation-dependent basis functions such as GIAOs/LAOs are used. Applying the above reformulation to the expressions for a(-u), w) and obtained from the CC response functions takes only relaxation with respect to the (static) external magnetic field into account [70, 71]. The frequency-dependent electric fields are treated in an unrelaxed manner, which avoids spurious poles due to orbital relaxation (see Section 2.2). 

Analytic response theory, which represents a particular formulation of time-dependent perturbation theory, has constituted a core technology in much of the this development. Response functions provide a universal representation of the response of a system to perturbations, and are applicable to all computational models, density-functional as well as wave-function models, and to all kinds of perturbations, dynamic as well as static, internal as well as external perturbations. The analytical character of the theory with properties evaluated from analytically derived expressions at finite frequencies, makes it applicable for a large range of experimental conditions. The theory is also model transferable in that, once the computational model has been defined, all properties are obtained on an equal footing, without further approximations. 

To compute the interacting RPA density-response function of equation (32), we follow the method described in Ref. [66]. We first assume that n(z) vanishes at a distance Zq from either jellium edge [67], and expand the wave functions (<) in a Fourier sine series. We then introduce a double-cosine Fourier representation for the density-response function, and find explicit expressions for the stopping power of equation (36) in terms of the Fourier coefficients of the density-response function [57]. We take the wave functions <)),(7) to be the eigenfunctions of a one-dimensional local-density approximation (LDA) Hamiltonian with use of the Perdew-Zunger parametrization [68] of the Quantum Monte Carlo xc energy of a uniform FEG [69]. 

In the previous sections, we derived general correlation function expressions for the nonlinear response function that allow us to calculate any 4WM process. The final results were recast as a product of Liouville space operators [Eqs. (49) and (53)], or in terms of the four-time correlation function of the dipole operator [Eq. (57)]. We then developed the factorization approximation [Eqs. (60) and (63)], which simplifies these expressions considerably. In this section, we shall consider the problem of spontaneous Raman and fluorescence spectroscopy. General formal expressions analogous to those obtained for 4WM will be derived. This will enable us to treat both experiments in a similar fashion and compare their information content. We shall start with the ordinary absorption lineshape. Consider our system interacting with a stationary monochromatic electromagnetic field with frequency w. The total initial density matrix is given by... 

To an uninitiated user of quantum chemistry programs, mathematical expressions of density functionals may appear esoteric. The analytic form of many functionals is indeed complicated and noninmitive, but it often conceals beautifully simple ideas. Lack of familiarity with these ideas and a black-box attitude toward the alphabet soup of density functional approximations are in part responsible for the wide-spread sentiment that DFT is effectively an empirical method with no prescription for systematic convergence to the right answer. We hope to convince the reader here that this view is unfair and that the development of density functionals can be, in its own way, a rigorous procedure. We will do so by systematizing and explaining the principal ideas behind modern density functional approximations. Because the most important developments in DFT relevant... 

In London theory the vector potential is taken to be space-independent (the rigid approximation). In Pippaid theory (4)> it is taken to be space-dependent with the following Fourier decomposition A(r) > XqA(q)exp(i ). The qii component of the current-density operator is the sum of two terms J(q) = Ji(q) + JL(London). Here reproduces the London theory. Fbr a weak field the current density is linear in A(q) and can be expressed in terms of a linear response function as follows ... 

The Golden Rule formula (9.5) for the mean rate constant assumes the Unear response regime of solvent polarization and is completely equivalent in this sense to the result predicted by the spin-boson model, where a two-state electronic system is coupled to a thermal bath of harmonic oscillators with the spectral density of relaxation J(o)) [38,71]. One should keep in mind that the actual coordinates of the solvent are not necessarily harmonic, but if the collective solvent polarization foUows the Unear response, the system can be effectively represented by a set of harmonic oscillators with the spectral density derived from the linear response function [39,182]. Another important point we would like to mention is that the Golden Rule expression is in fact equivalent [183] to the so-called noninteracting blip approximation [71] often used in the context of the spin-boson model. The perturbation theory can be readily applied to... 

Any evaluation of the polarizability demands the knowledge of the linear response function. The general expression for this function in DFT can be formulated as in Eq. (4.197) with in terms of softness kernel, the local and the global softness, respectively, all related with the introduced local response function L r). However, for atomic systems a very sensitive approximation consists in neglecting all non-local contributions in the linear response function (Putz et al., 2003, 2012b,c) ... 

In the difference term one recognizes a fluctuation term, f(r) the deviation of the Fukui function from the average electron density per electron, multiplied by the functional derivative of E with respect to a(r) at constant number of electrons. This quantity (5E/5o(i))ig is a response function of the type (5E/5v(r))j. mentioned in the Introduction and measures the sensitivity of the system s energy to variations in the shape factor. Its evaluation seems to be far from trivial but it is possible to get already an idea of what might be factors of importance in this response function. Adopting an orbital formalism and using a Koopmans type approximation one arrives at an approximate expression... 

Needless to say, equation (21.8) is a bit cumbersome and its original derivation is rather lengthy. However, many subsequent treatments of the macroscopic theory are now available which provide both a more readily understandable approach and many useful approximate expressions. In fact, by using the method of Parsegian and Ninham to determine the dielectric response function from absorption data and reflectance measurements, it is now quite straightforward to calculate dispersion forces from Lifshitz s theory. 

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