Electronic structure methods approximation

For any but the smallest systems, however, exact solutions to the Schrodinger equation are not computationally practical. Electronic structure methods are characterized by their various mathematical approximations to its solution. There are two major classes of electronic structure methods ... 

Ab initio molecular orbital theory is concerned with predicting the properties of atomic and molecular systems. It is based upon the fundamental laws of quantum mechanics and uses a variety of mathematical transformation and approximation techniques to solve the fundamental equations. This appendix provides an introductory overview of the theory underlying ab initio electronic structure methods. The final section provides a similar overview of the theory underlying Density Functional Theory methods. 

Making approximations in the Hamiltonian describing the system, e.g. semi-empirical electronic structure methods. 

Ffom a theoretical point of view, stacking fault energies in metals have been reliably calculated from first-principles with different electronic structure methods [4, 5, 6]. For random alloys, the Layer Korringa Kohn Rostoker method in combination with the coherent potential approximation [7] (LKKR-CPA), was shown to be reliable in the prediction of SFE in fcc-based solid solution [8, 9]. 

In combined QM/MM potentials, the system is divided into a QM region and an MM region. The QM region typically includes atoms that are directly involved in the chemical step and they are treated explicitly by a quantum mechanical electronic structure method. The MM region consists of the rest of the system and is approximated by an MM force field. The QM/MM potential is given by ... 

Equation (2) is an example of a sum-over-states (SOS) expression of a molecular response property. It suggests an easy way of computing / , but in practice the SOS approach is rarely taken because of its very slow convergence, i.e., because of the need to compute many excited states wavefunctions. The summation goes over all excited states and also needs to include, in principle, the continuum of unbound states. As it will be shown below, there are more economic ways of computing [1 within approximate first-principles electronic structure methods. 

With a given basis set and electronic structure method, if all possible excitations (corresponding to physically meaningful transitions, or generated as artifacts from using an incomplete basis and an approximate functional) are included in the SOS for... 

The electronic structure methods are based primarily on two basic approximations (1) Born-Oppenheimer approximation that separates the nuclear motion from the electronic motion, and (2) Independent Particle approximation that allows one to describe the total electronic wavefunction in the form of one electron wavefunc-tions i.e. a Slater determinant [26], Together with electron spin, this is known as the Hartree-Fock (HF) approximation. The HF method can be of three types restricted Hartree-Fock (RHF), unrestricted Hartree-Fock (UHF) and restricted open Hartree-Fock (ROHF). In the RHF method, which is used for the singlet spin system, the same orbital spatial function is used for both electronic spins (a and (3). In the UHF method, electrons with a and (3 spins have different orbital spatial functions. However, this kind of wavefunction treatment yields an error known as spin contamination. In the case of ROHF method, for an open shell system paired electron spins have the same orbital spatial function. One of the shortcomings of the HF method is neglect of explicit electron correlation. Electron correlation is mainly caused by the instantaneous interaction between electrons which is not treated in an explicit way in the HF method. Therefore, several physical phenomena can not be explained using the HF method, for example, the dissociation of molecules. The deficiency of the HF method (RHF) at the dissociation limit of molecules can be partly overcome in the UHF method. However, for a satisfactory result, a method with electron correlation is necessary. 

Stable. In this case the electronic structure ordinarily approximates that of a free-elcctron gas and may be analyzed with methods appropriate to free-electron gases. Again, the crystal structure is the determining feature for the classification. When tin has a tetrahedral structure it is a covalent solid when it has a close-packed white-tin structure, it is a metal. Even silicon and germanium, when melted, become close-packed and liquid metals. 

It was, therefore, clear in 1974 that electronic-structure methods were not sufficiently advanced to reproduce experimental data accurately for even a simple ionic oxide such as MgO. The emphasis at the time was on the determination of the effects of different approximations upon the calculated results. Comparison was usually made between one calculation and another rather than between calculation and experiment. The theoretical papers reported the quantities arising directly from the calculations, such as orbital eigenvalues and atomic-orbital charge decompositions, and spectral properties were interpreted primarily in terms of orbital energies. No attempt was made to evaluate equilibrium structural or energetic properties. 

The basic question to which we require an answer is how small may a metal particle become before it loses its metallic properties Or conversely, in the growth of a metallic cluster, at what point does the electronic structure closely approximate to that of the bulk metal Though the questions appear simple the answers are not. This is partly because the many properties that may be used to characterize the metallic state are differently related to cluster size, so that one may be achieved at a much earlier stage in cluster growth than another. Moreover, if one property is arbitrarily selected as a criterion, the various available theoretical methods for the calculation of that property may differ in their predictions as to when, during cluster growth, the metallic state is achieved. 

Figure 1 In a QM/MM calculation, a small region is treated by a quantum mechanical (QM) electronic structure method, and the surroundings treated by simpler, empirical, molecular mechanics. In treating an enzyme-catalysed reaction, the QM region includes the reactive groups, with the bulk of the protein and solvent environment included by molecular mechanics. Here, the approximate transition state for the Claisen rearrangement of chorismate to prephenate (catalysed by the enzyme chorismate mutase) is shown. This was calculated at the RHF(6-31G(d)-CHARMM QM-MM level. The QM region here (the substrate only) is shown by thick tubes, with some important active site residues (treated by MM) also shown. The whole model was based on a 25 A sphere around the active site, and contained 4211 protein atoms, 24 atoms of the substrate and 947 water molecules (including 144 water molecules observed by X-ray crystallography), a total of 7076 atoms. The results showed specific transition state stabilization by the enzyme. Comparison with the same reaction in solution showed that transition state stabilization is important in catalysis by chorismate mutase78.

This variational principle is the basis for most of the currently used electronic-structure methods, i.e., by systematically improving the approximate... 

The use of the SOS expressions in conjunction with approximate states (vibrational or electronic) provide us with methods to use in practical calculations. The electronic wave functions and transition moments can be determined with our standard electronic structure methods (SCF, CI, MCSCF, etc.) and the vibrational wave functions and transition moments (if considered) are determined using the potential energy surfaces. The SOS formulas may be formally separated into electronic and vibrational contributions to the properties (see Section 4), and this fact makes the SOS expressions pertinent in all calculations regardless of the choice of electronic structure method. Criticism of the SOS approach mainly concerns calculations of the electronic contributions to the properties as the SOS technique is often hampered by slow convergence with respect to Ute number of states that need to be included in the summations. It has been used wltlt success only for very small systems, most notably by Bishop and co-workers in a- and y-calculations of calibratlonal quality on helium [12] and Hj [13]. 

One somewhat displeasing detail in the approximate polarization propagator methods discussed in the previous section is the fact that concern needs to be made as to which formulation of wave mechanics that is used. This point has been elegantly resolved by Christiansen et al. in their quasi-energy formulation of response theory [23], in which a general and unified theory is presented for the evaluation of response functions for variational as well as nonvariational electronic structure methods. 

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