Schrodinger equation electron correlation methods

The quantum chemist s traditional way to approximate solutions of the electronic Schrodinger equation is so-called ab initio, wave function-based electron correlation methods. These methods improve upon the HF mean-field approximation by adding many-body corrections in a systematic way [15]. As of the time of this writing, efforts to accelerate ab initio calculations with GPUs are scarce. However, it is expected that this will change in the near future because these methods are of critical importance whenever higher accuracy is required than what can be achieved by DFT or for types of interactions and properties for which DFT breaks down. 

One of the original approximate methods is the wavefunction-theory-based Hartree-Fock (HF) method [40]. The HF method is a single determinant method that does not include any correlation interactions between the electrons, and as such has limited accuracy [41, 42]. Higher level wavefunction-based methods such as coupled cluster [43 5], configuration interaction [40,46,47], and complete active space [48-50] methods include multiple determinants to incorporate some of the electron-electron correlation. Methods based on perturbation theory, such as second order Mpller-Plesset perturbation theory [51], go beyond the HF method by perturbatively adding electron correlation. These correlated wavefunction-based methods have well-defined ways in which they approach the exact solution to the Schrodinger equation and thus have the potential to be extremely accurate, but this accuracy comes at a price [52]. 

The difference between the Hartree-Fock energy and the exact solution of the Schrodinger equation (Figure 60), the so-called correlation energy, can be calculated approximately within the Hartree-Fock theory by the configuration interaction method (Cl) or by a perturbation theoretical approach (Mpller-Plesset perturbation calculation wth order, MPn). Within a Cl calculation the wave function is composed of a linear combination of different Slater determinants. Excited-state Slater determinants are then generated by exciting electrons from the filled SCF orbitals to the virtual ones ... 

In most work reported so far, the solute is treated by the Hartree-Fock method (i.e., Ho is expressed as a Fock operator), in which each electron moves in the self-consistent field (SCF) of the others. The term SCRF, which should refer to the treatment of the reaction field, is used by some workers to refer to a combination of the SCRF nonlinear Schrodinger equation (34) and SCF method to solve it, but in the future, as correlated treatments of the solute becomes more common, it will be necessary to more clearly distinguish the SCRF and SCF approximations. The SCRF method, with or without the additional SCF approximation, was first proposed by Rinaldi and Rivail [87, 88], Yomosa [89, 90], and Tapia and Goscinski [91], A highly recommended review of the foundations of the field was given by Tapia [71],... 

The study of behavior of many-electron systems such as atoms, molecules, and solids under the action of time-dependent (TD) external fields, which includes interaction with radiation, has been an important area of research. In the linear response regime, where one considers the external held to cause a small perturbation to the initial ground state of the system, one can obtain many important physical quantities such as polarizabilities, dielectric functions, excitation energies, photoabsorption spectra, van der Waals coefficients, etc. In many situations, for example, in the case of interaction of many-electron systems with strong laser held, however, it is necessary to go beyond linear response for investigation of the properties. Since a full theoretical description based on accurate solution of TD Schrodinger equation is not yet within the reach of computational capabilities, new methods which can efficiently handle the TD many-electron correlations need to be explored, and time-dependent density functional theory (TDDFT) is one such valuable approach. 

The Hartree-Fock approach derives from the application of a series of well defined approaches to the time independent Schrodinger equation (equation 3), which derives from the postulates of quantum mechanics [27]. The result of these approaches is the iterative resolution of equation 2, presented in the previous subsection, which in this case is solved in an exact way, without the approximations of semiempirical methods. Although this involves a significant increase in computational cost, it has the advantage of not requiring any additional parametrization, and because of this the FIF method can be directly applied to transition metal systems. The lack of electron correlation associated to this method, and its importance in transition metal systems, limits however the validity of the numerical results. 

The method of many-electron Sturmian basis functions is applied to molecnles. The basis potential is chosen to be the attractive Conlomb potential of the nnclei in the molecnle. When such basis functions are used, the kinetic energy term vanishes from the many-electron secular equation, the matrix representation of the nnclear attraction potential is diagonal, the Slater exponents are automatically optimized, convergence is rapid, and a solution to the many-electron Schrodinger eqeuation, including correlation, is obtained directly, without the use ofthe self-consistent field approximation. 

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. 

For anything but the most trivial systems, it is not possible to solve the electronic Schrodinger equation exactly, and approximate techniques must instead be used. There exist a variety of approximate methods, including Hartree-Fock (HF) theory, single- and multireference correlated ab initio methods, semiempirical methods, and density functional theory. We discuss each of these in turn. In Hartree-Fock theory, the many-electron wavefunction vF(r1, r2,..., r ) is approximated as an antisymmetrized product of one-electron wavefunctions, ifijfi) x Pauli principle. This antisymmetrized product is known as a Slater determinant. 

HF is the simplest of the ab initio methods, named after the fact that they provide approximate solutions to the electronic Schrodinger equation without the use of empirical parameters. More accurate, correlated, ab initio methods use an approximate form for the wavefunction that goes beyond the single Slater determinant used in HF theory, in that the wavefunction is approximated instead as a combination or mixture of several Slater determinants corresponding to different occupation patterns (or configurations) of the electrons in the molecular orbitals. When an optimum mixture of all possible configurations of the electrons is used, one obtains an exact solution to the electronic Schrodinger equation. This is, however, not computationally tractable. 

Quantum mechanics (QM) can be further divided into ab initio and semiempiri-cal methods. The ab initio approach uses the Schrodinger equation as the starting point with post-perturbation calculation to solve electron correlation. Various approximations are made that the wave function can be described by some functional form. The functions used most often are a linear combination of Slater-type orbitals (STO), exp (-ax), or Gaussian-type orbitals (GTO), exp (-ax2). In general, ab initio calculations are iterative procedures based on self-consistent field (SCF) methods. Self-consistency is achieved by a procedure in which a set of orbitals is assumed and the electron-electron repulsion is calculated. This energy is then used to calculate a new set of orbitals, and these in turn are used to calculate a new repulsion energy. The process is continued until convergence occurs and self-consistency is achieved. 

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