Schrodinger equation electronic structure calculations

There are two types of basis functions (also called Atomic Orbitals, AO, although in general they are not solutions to an atomic Schrodinger equation) commonly used in electronic structure calculations Slater Type Orbitals (STO) and Gaussian Type Orbitals (GTO). Slater type orbitals have die functional form... 

Until the last decade, for all but a few theoreticians, electronic structure calculations meant calculations based on approximate solution of the time-independent Schrodinger equation,... 

Separation of the movement of the nuclei and electrons. This is possible because the electrons move much more rapidly (smaller mass) than the nuclei. The position of the nuclei is fixed for the calculation of the electronic Schrodinger equation (in MO calculations the nuclear positions are then parameters, not quantum chemical variables). Born-Oppenheimer surfaces are energy vs. nuclear structure plots which are (n + 1)-dimensional, where n is 3N- 6 with N atoms (see potential energy surface). 

In the above formula, Q is the nuclear coordinate, p, and I/r are the ground state and excited electronic terms. Here Kv is provided through the traditional Rayleigh-Schrodinger perturbation formula and K0 have an electrostatic meaning. This expression will be called traditional approach, which has, in principle, quantum correctness, but requires some amendments when different particular approaches of electronic structure calculation are employed (see the Bersuker s work in this volume). In the traditional formalism the vibronic constants P0 dH/dQ Pr) can be tackled with the electric field integrals at nuclei, while the K0 is ultimately related with electric field gradients. Computationally, these are easy to evaluate but the literally use of equations (1) and (2) definitions does not recover the total curvature computed by the ab initio method at hand. 

The potential energy surface is the central quantity in the discussion and analysis of the dynamics of a reaction. Its determination requires the solution of the many-body electronic Schrodinger equation. While in the early days of theoretical surface science quantum chemical methods had a significant impact, nowadays electronic structure calculations using density functional theory (DFT) [20, 21] are predominantly used. DFT is based on the fact that the exact ground state density and energy can be determined by the minimisation of the energy functional E[n ... 

Two broad classes of technique are available for modeling matter at the atomic level. The first avoids the explicit solution of the Schrodinger equation by using interatomic potentials (IP), which express the energy of the system as a function of nuclear coordinates. Such methods are fast and effective within their domain of applicability and good interatomic potential functions are available for many materials. They are, however, limited as they cannot describe any properties and processes, which depend explicitly on the electronic structme of the material. In contrast, electronic structure calculations solve the Schrodinger equation at some level of approximation allowing direct simulation of, for example, spectroscopic properties and reaction mechanisms. We now present an introduction to interatomic potential-based methods (often referred to as atomistic simulations). 

In a typical electronic-structure calculation, the Bom-Oppenheimer approximation is invoked. This means that for a system of M nuclei and N electrons one first seeks the electronic energy by solving the electronic Schrodinger equation... 

In most electronic-structure calculations, the Born-Oppenheimer approximation is invoked. This leads to flow charts like that of Figure 1, where the electronic properties are determined for a given set of nuclear positions. This is done by solving some kind of time-independent electronic Schrodinger equation,... 

In contrast to force-field calculations in which electrons are not explicitly addressed, molecular orbital calculations, use the methods of quantum mechanics to generate the electronic structure of molecules. Fundamental to the quantum mechanical calculations that are to be performed is the solution of the Schrodinger equation to provide energetic and electronic information on the molecular system. The Schrodinger equation cannot, however, be exactly solved for systems with more than two particles. Since any molecule of interest will have more than one electron, approximations must be used for the solution of the Schrodinger equation. The level of approximation is of critical importance in the quality and time required for the completion of the calculations. Among the most commonly invoked simplifications in molecular orbital theory is the Bom-Oppenheimer [13] approximation, by which the motions of atomic nuclei and electrons can be considered separately, since the former are so much heavier and therefore slower moving. Another of the fundamental assumptions made in the performance of electronic structure calculations is that molecular orbitals are composed of a linear combination of atomic orbitals (LCAO). 

Since the density in turn depends on the solutions of the effective one-electron Schrodinger equation (1.1), we are forced to perform self-consistent electronic-structure calculations. This proves to be a formidable task, and practicable, accurate solutions are possible only on large-scale digital computers. 

The Kohn-Sham-Dirac equation (27) introduced in the last section is the basis of most relativistic electronic structure calculations in solid state theory. There are certain aspects which make the numerical solution of this four-component equation more involved than its non-relativistic coimterpart The Hamiltonian of the Kohn-Sham-Dirac equation is, unlike its Schrodinger equivalent and unlike the field-theoretical Hamiltonian (7) with the properly chosen normal order, not bounded below. In the limit of free, non-interacting particles the solutions of the Kohn-Sham-Dirac equation are plane waves with energies e(k) = cVk -I- c, where positive energies correspond to electrons and states with negative energy can be interpreted as positrons. For numerical procedures, which preferably use variational techniques to find electronic solutions, this property of the Dirac operator causes a severe problem, which can be circumvented by certain techniques like the application of a squared Dirac operator or a projection onto the properly chosen electronic states according to their above definition after Eq. (19). 

Quantum-chemical electronic structure calculations deal with obtaining a numerical solution to the electronic Schrodinger equation. An important... 

In recent years, an alternative approach to implementing the Schrodinger equation for quantitative electronic structure calculations has appeared. Instead of calculating wave-functions of the sort we have described, these methods focus on the electron density (p) across the entire molecule. It has been shown that if p is known precisely, one can in principle determine the total energy (and all other properties of the system) precisely. In addition, p is certainly simpler than the complicated total wavefunction used in the orbital approximation (Eq. 14.13). 

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