LCAO ansatz

Introduction of a Basis The LCAO Ansatz in the Kohn-Sham Equations... 

Here, Aa denotes the atom at which the atomic orbital [Pg.308]

It is a beautiful mathematical consequence of the above LCAO approach that, without having to specify the Hamiltonian any further, it is possible to arrive at a general quantum-chemical solution for the hydrogen molecule H2 (or any other two-nuclei one-electron molecule). Starting with an LCAO ansatz for the molecular orbital as in... 

For a given electronic Hamiltonian (that is, a proper description of the electron-nucleus and electron-electron interactions), the LCAO ansatz may deliver the molecular orbitals ipi and the many-electron wave function Y provided that there is a set of useful basis functions (p, for example, atomic orbitals. At this point, it is probably the right time to review briefly what type of atomic orbitals are mostly used within molecular quantum chemistry. By... 

Before showing more complex DOS, we will first concentrate on how to characterize the electron density distribution in molecules and solids because that is where we started this is also needed to further decompose the density-of-states and characterize the electronic structures in terms of bonding properties. Recalling the LCAO ansatz for the molecular orbital (MO) of the H2 molecule as given in Equation (2.11), we may square-integrate the MO over all space and yield... 

The DFTB method is based on an LCAO Ansatz for the Kohn-Sham molecular orbitals of basis functions (j (Eq. 1). 

In the Bra-Ket notation, the LCAO ansatz (Eq. (5.1)) for the crystal wave function reads... 

Insertion of the LCAO ansatz Eq. (5.50) into the Schrodinger equation (H— ) I h) and multiplication with l from the left yields the following set of algebraic equations... 

If the terms with the factor 4 C2 were zero we could fulfil Eq. (15). This term thus contains the kind of electron correlation introduced by our ansatz. In order to get a pictorial idea of this t) of correlation, we introduce as further simplification the LCAO approximation. 

We reiterate that, just as in LCAO-MO theory, the originally unknown function tp k,r) is expressed by a set of known functions, but the latter are entirely delocalized and not localized as atomic orbitals. While this ansatz ultimately leads us away from the notion of a solid-state material being composed of atoms with their associated atomic functions (this ansatz is a very... 

One should note that a four-component relativistic AO always contains only two different radial-dependent parts (large and small components). The four different components are then formed with the help of different combinations of these two radial functions with the well-known angular-dependent parts of the wave functions [see, for instance, equations (54), (99), (105), and (107) in the book by Mott and Sneddon ]. Already in the 1 = 0, j = s = case (l5 electron) four different components are obtained in this way for the relativistic AO. Therefore, if one wishes to perform an LCAO procedure, the basis functions must possess four different components which have to be varied independently. This is the reason for ansatz (1.108). 

The ansatz Eq. (2.41) is the Linear Combination of Atomic Orbitals (LCAO) approach. The coefficients a are the parameters of variation when minimizing the functional E[ ]. The set of functions belonging to a single atom 0 I j 1,..., rik) is called atomic basis set. 

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