Local Schrodinger equation method

Because spatially localized functions are the natural choice for isolated molecules, the quantum chemistry methods developed within the chemistry community are dominated by methods based on these functions. Conversely, because physicists have historically been more interested in bulk materials than in individual molecules, numerical methods for solving the Schrodinger equation developed in the physics community are dominated by spatially periodic functions. You should not view one of these approaches as right and the other as wrong as they both have advantages and disadvantages. 

In this paper we present preliminary results of an ab-initio study of quantum diffusion in the crystalline a-AlMnSi phase. The number of atoms in the unit cell (138) is sufficiently small to permit computation with the ab-initio Linearized Muffin Tin Orbitals (LMTO) method and provides us a good starting model. Within the Density Functional Theory (DFT) [15,16], this approach has still limitations due to the Local Density Approximation (LDA) for the exchange-correlation potential treatment of electron correlations and due to the approximation in the solution of the Schrodinger equation as explained in next section. However, we believe that this starting point is much better than simplified parametrized tight-binding like s-band models. 

In 1972 T. L. Allen used Monte Carlo for FSGO method by least squares solution of the Schrodinger equation for many electron atoms and molecules. The least squares solution of the Schrodinger equation was introduced by D. H. Weinstein in 1934, and developed by others. Let us define the local energy, for our system of interest as... 

The one-dimensional potential depicted in Fig. 7(a) provides an illustration of this effect. The Schrodinger equation can be solved with the method used for the square-well case above. Each well gives rise to a nearly independent progression of states. For fi = 2p = 1 and other potential parameters indicated in Fig. 7 one finds that the system has two bound states and a resonance state at 9.46 — i 0.11 localized above the deep outer well. There is also another resonance in the system, E = 9.8 - i 0.002. Its width is very small because this state belongs to the shallow inner well, which is separated from the continuum by a potential barrier. Suppose that we force — by varying a parameter in the Hamiltonian — the narrow state (denoted n) in the shallow well to move across the broader resonance (b) belonging to the deep minimum. The relative positions of the two states can be, for example, controlled by shifting the infinite wall at the... 

The electronic structure of solids and surfaces is usually described in terms of band structure. To this end, a unit cell containing a given number of atoms is periodically repeated in three dimensions to account for the infinite nature of the crystalline solid, and the Schrodinger equation is solved for the atoms in the unit cell subject to periodic boundary conditions [40]. This approach can also be extended to the study of adsorbates on surfaces or of bulk defects by means of the supercell approach in which an artificial periodic structure is created where the adsorbate is translationally reproduced in correspondence to a given superlattice of the host. This procedure allows the use of efficient computer programs designed for the treatment of periodic systems and has indeed been followed by several authors to study defects using either density functional theory (DFT) and plane waves approaches [41 3] or Hartree-Fock-based (HF) methods with localized atomic orbitals [44,45]. 

Because they are so computationally intensive, ab initio and semiempirical studies are limited to models that are about 10 rings or less. In order to study more reahstic carbon structures, approximations in the form of the Hamiltonian (i.e., Schrodinger equation) are necessary. The tight-binding method, in which the many-body wave function is expressed as a product of individual atomic orbitals, localized on the atomic centers, is one such approximation that has been successfully applied to amorphous and porous carbon systems [47]. 

The above demonstrated possibility of obtaining numerical virtual orbitals indicate that the FD HF method can also be used as a solver of the Schrodinger equation for a one-electron diatomic system with an arbitrary potential. Thus, the scheme could be of interest to those who try to construct exchange-correlation potential functions or deal with local-scaling transformations within the functional density theory (32,33). 

To motivate the Kohn-Sham method, we return to molecular Hamiltonian [Eq. (2)] and note that, were it not for the electron-electron repulsion terms coupling the electrons, we could write the Hamiltonian operator as a sum of one-electron operators and solve Schrodinger equation by separation of variables. This motivates the idea of replacing the electron-electron repulsion operator by an average local representation thereof, w(r), which we may term the internal potential. The Hamiltonian operator becomes... 

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