Local density functional Hamiltonian

A common type of local density functional Hamiltonian is the SVWN. The local density functional theory represents a severe approximation for molecular systems since it assumes a uniform total electron density throughout the molecular system. Other approaches have been developed that account for variation in total density (non-local density functional theory). This is done by having the functions depend explicitly on the gradient of the density in addition to the density itself. An example of a density functional Hamiltonian that takes this density gradient into account... 

A deeper argument is that local density functional derivatives appear to be implied by functional analysis [2,21,22]. The KS density function has an orbital structure, p = Y.i niPi = X fa- For a density functional Fs, strictly defined only for normalized ground states, functional analysis implies the existence of functional derivatives of the form SFj/ Sp, = e, — v(r), where the constants e, are undetermined. On extending the strict ground-state theory to an OFT in which OEL equations can be derived, these constants are determined and are just the eigenvalues of the one-electron effective Hamiltonian. Since they differ for each different orbital energy level, the implied functional derivative depends on a direction in the function-space of densities. Such a Gateaux derivative [1,2] is equivalent in the DFT context to a linear operator that acts on orbital functions [23]. 

Atomic units will be used throughout. The explicit density functionals representing the different contributions to the energy from the different terms of the hamiltonian are found performing expectation values taking Slater determinants of local plane waves as in the standard Fermi gas model. Those representing the first relativistic corrections are calculated in the Appendix. 

In the case of delocalized basis states tpa(r), the main matrix elements are those with 0 = 7 and f3 = 6, because the wave functions of two different states with the same spin are orthogonal in real space and their contribution is small. It is also true for the systems with localized wave functions tpa(r), when the overlap between two different states is weak. In these cases it is enough to replace the interacting part by the Anderson-Hubbard Hamiltonian, describing only density-density interaction... 

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]. 

Like in molecular quantum chemistry, the localized-delocalized antagonism is omnipresent in the theoretical literature on itinerant magnetism. On the one hand, the Hubbard model [292] and related theories for strongly correlated systems have been employed to study rare-earth and also transition metals. Since the latter do not have flat bands, extensions to the Hubbard theory are required [293-295] also, to make the model Hamiltonians (almost) exactly solvable, simplifications are introduced. On the other hand, density-functional theory is able to extract Stoner s parameters [296,297] for a self-consistent description of itinerant magnetism [298]. As has been illustrated before, the theoretical limits of the LDA became apparent from Fe phase stability problems (see Section 2.12.1) and were solved by using gradient corrections. The present status of DFT in the treatment of cooperative magnetism has also been reviewed [299]. 

During the past three deeades, three main versions of the MCP method have been developed [1,53]. Version I is based on the local approximation. The core-valence Coulomb repulsion is a local interaction and can be satisfactorily approximated by a local potential function. For convenience of the integral evaluation, such a local potential function is chosen to be a linear combination of Gaussian type functions. The core-valence exchange operator is not a local operator. However, in Version I, this non-local interaction is also approximated by the local potential function of Gaussian type. This non-local to local approximation for the exchange operator shares the same concept with Slater s Xa density functional model [69]. Under such an approximation, the one-electron hamiltonian for the valence space in an atom (Eq. 8.5) is rewritten as... 

Several approaches are available in the literature to generate and evaluate Hamiltonian matrix elements with wavefunctions of charge-localized, diabatic states. They differ in the level of theory used in the calculation and in the way localized electronic structures are created [15, 25, 26, 29-31]. When wavefunction-based quantum-chemical methods are employed, the framework of the generalized Mulliken-Hush method (GMH) [29, 32-34], is particularly successful. So far, it has been used in conjunction with accurate electronic structure methods for small and medium sized systems [35-37]. As an alternative to GMH and other derived methods [38, 39], additional methods have been explored for their applicability in larger systems such as constrained density functional method (CDFT) [25, 37, 40, 41], and fragmentation approaches [42-47], which also include the frozen density embedding (FDE) method [48, 49]. 

The present authors have used the dielectric screening method for all their ab-initio computations of the macroscopic dielectric constant and phonon frequencies of Si and Ge. For the calculation of the electron energies and wave functions needed in the expression of the electron density response matrix they apply the local density approximation in the Hamiltonian. The advantage and shortcomings of this approximation are treated at length in the papers by J.T. Devreese, R. Martin, K. Kune, S. Louie, A. Baldereschi and R. Resta in these proceedings. 

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