Localized functions, expansion

Expansion of localized functions in terms of scattering states... 

Fig. 13.3. Left-hand side Contour plots of the modulus square of the lowest four anti-symmetric eigenfunctions of H2O in the electronic ground state, multiplied with the X — A transition dipole function. They have a node on the symmetric stretch line. The assignment rnn ) is based on the local mode expansion (13.5)...

The main issue involved in using DFT and the KS scheme pertains to construction of expressions for the XC functional, Exc[n], containing the many-body aspects of the problems (1.38). The main approaches to this issue are (a) local functionals the Thomas Fermi (TF) and LDA, (b)semilocal or gradient-dependent functionals the gradient-expansion approximation (GEA) and generalized gradient approximation (GGA), and (c) nonlocal functionals hybrids, orbital functionals, and SIC. For detailed discussions the reader is referred to the reviews [257,260-272]. 

The energy equation has been solved numerically [40] and analytically, using general Eigen functions expansion [42], and the details can be found in related references. Using the temperature distribution, the local Nusselt number may be determined as... 

Many approximate functionals have been proposed for the remaining part of the total energy, i.e. the correlation energy. The development of an improved local form of this functional is of special interest for large systems because the complexity of the system will be significantly reduced. Recently, a new form of E(,[p] based on the functional expansion and the adiabatic connection formulation of DF has been advanced which is a sum of integrals of various powers of the density [37] ... 

Liu, S. (1996). Local-density approximation, hierarchy of equations, functional expansion, and adiabatic connection in current-density-functional theory. Phys. Rev. A 54, 1328-1336. 

Next on, one will consider the non-local functionals this can be achieved through the gradient expansion in the case of slowly varying densities - that is assuming the expansion (Murphy, 1981) ... 

We have reported this discussion in some detail because it combines several points treated separately by many authors. These remarks shed some light on the difficulty of getting local multipole expansions which are reliable, easy to handle, and at the same time transferable from molecule to molecule. We have in fact considered, until now, the problem of getting multipole expansions from an already known (r) function, without touching the more important problem of formulating local expansions transferable from molecule to molecule. We shall see later how these problems may be partially solved. 

Orbital-dependent functional represent the next step towards more accurate DFT methods. KS orbitals are ultra-non-local functionals of the density and thus represent an advantageous alternative to the gradient expansion to include non-local contributions in the XC functionals. In addition KS orbitals are readily available in the self-consistent procedure without additional cost... 

In view of the dramatic failure of the local thermodynamic approximations (35), it is natural to attempt to improve them by considering density functional expansions which take into account the nonlocal gradient correction terms. Standard density functional methods yield... 

In HTST, a harmonic expansion of the PES is invoked both in the IS and in the saddle point separating the IS and the FS. The HTST is therefore appUcable under the same general assumptions as mentioned for TST but further demands that the PES is smooth enough for a local harmonic expansion of the PES to be reasonable. This means that it is necessary that the potential is reasonably well represented by its second-order Taylor expansion around these two expansion configurations. The general idea is that the partition functions in Equation (4.20) can be evaluated analytically for the harmonic expansion of the PES around the expansion points. This leads to very simple expressions for the rate constants and gives reasonable rate constants for... 

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