Taylor expansions density functional theory

Recently there has been a great deal of interest in nonlinear phenomena, both from a fundamental point of view, and for the development of new nonlinear optical and optoelectronic devices. Even in the optical case, the nonlinearity is usually engendered by a solid or molecular medium whose properties are typically determined by nonlinear response of an interacting many-electron system. To be able to predict these response properties we need an efficient description of exchange and correlation phenomena in many-electron systems which are not necessarily near to equilibrium. The objective of this chapter is to develop the basic formalism of time-dependent nonlinear response within density functional theory, i.e., the calculation of the higher-order terms of the functional Taylor expansion Eq. (143). In the following this will be done explicitly for the second- and third-order terms... 

As in the theory of functions, a calculus exists for functionals. This calculus provides the tools necessary to develop and implement density functional theory. We begin with the discussion of expansions of functionals, which plays an important role in developing models within DFT and in deriving perturbation expansions. Analogous to the Taylor Series expansion for a function, a functional can be expanded about a reference function. This expansion, called a Volterra expansion, exists provided the functional has functional derivatives to any order and provided the last term in the infinite expansion has limit zero. Assuming these conditions, the Volterra expansion of ft[p] about a reference function, po, is given by... 

In order to start the mathematical development of this theory, several assumptions and conventions are introduced. The yield versus energy dependences are based entirely on density of states considerations assuming that matrix elements do not vary rapidly near threshold. The optical absorption should vary slowly near threshold. Strict energy conservation is always assumed. Energy losses are treated on an all or nothing basis, and this assumption is certainly not always adequate. Taylor expansions to lowest nonvanishing order for functions of the energy bands always are made around... 

Figure 6, which is for the same simulation as Fig. 4, shows sectional averages of the pressure, particle velocity, and density as a function of position with the dashed lines denoting the initial values in the undisturbed crystal. These quantities peak at the front and then relax during the reaction and expansion behind the shock. The shockfront shape is in accord with ZND continuum theory for unsupported planar detonations, which predicts a von Neumann peak near the front followed by a reacting flow and a (Taylor) rarefaction wave. The peak pressure around 1.0 eV/A (Fig. 6, top), which corresponds to an effective pressure of approximately 400 kbar, and maximum particle velocity of 4.8 km/s (Fig. 6, middle) are consistent... 

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