Electronic-structure methods

Electronic structure methods use the laws of quantum mechanics rather than classical physics as the basis for their computations. Quantum mechanics states that the energy and other related properties of a molecule may be obtained by solving the Schrodinger equation  

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  

Semi-empirical methods, such as AMI, MINDO/3 and PM3, implemented in programs like MOPAC, AMPAC, HyperChem, and Gaussian, use parameters derived from experimental data to simplify the computation. They solve an approximate form of the Schrodinger equation that depends on having appropriate parameters available for the type of chemical system under investigation. Different semi-emipirical methods are largely characterized by their differing parameter sets. 

4 Ab initio methods, unlike either molecular mechanics or semi-empirical methods, use no experimental parameters in their computations. Instead, their computations are based solely on the laws of quantum mechanics—the first principles referred to in the name ah initio—and on the values of a small number of physical constants  

Exploring Chemistry with Electronic Structure Methods 

In order to introduce the subject of electronic structure methods, we start from the Bom-Oppenheimer (BO) Hamilton operator, which only considers the leading electrostatic interactions (in atomic units) [17,22]  

The terms describe the kinetic energy of the electrons, the electron-nuclear attraction, the electron-electron repulsion, and the nuclear-nuclear repulsion, respectively. In Eq. (2), i, j sum over electrons at positions ti, A, B over nuclei with 

MOs first appear in the framework of the Hartree-Fock (HF) method, which is a mean-field treatment [17,22]. The basic idea is to start from an A-particle wave-function that is appropriate for a system of non-interacting electrons. Having fixed the Ansatz for the A-particle wavefunetion in this way, the variational principle is used in order to obtain the best possible approximation for the fully interacting system. Such independent particle wavefunctions are Slater-determinants, which consist of antisymmetrized products of single-particle wavefunctions (x)J (the antisymmetry brought about by the determinantal form is essential in order to satisfy die Pauh principle). Thus, the Slater-determinant is written as 

The result is an upper bound to the total molecular energy. The minimization leads to die following pseudo-single-particle equations (HF-equations) that must be fulfilled by die orbitals (x)  

Here s. is the orbital energy for the fth MO. The Coulomb and exchange operators Jj and K. are defined by their actions on an orbital y/. (x)  

In this section, we provide a brief account of the different theoretical methods used in the study of electronic structure. This includes two families of methods that arise from the principles of quantum mechanics, the ab initio methods of computation of electronic wave functions and the methods based on DFT. The choice of a particular computational method must contemplate the problem to be solved and in any case is a compromise between accuracy and feasibility. Details of the methods outlined in this section can be found in specialized references, monographs [71], and textbooks [72]. 

Density Functional Theory—Based Methods in Quantum Chemistry 

The Schrodinger equation provides a way to obtain the A -electron wave function of the system and the approximate methods described in the previous section permit reasonable approaches to this wave function. The total energy can be obtained from the approximate wave function as an expectation value and the different density matrices, in particular, the one-particle density matrix, can be obtained in a straightforward way as 

In the Kohn-Sham formalism, one assumes that there is a fictitious system of N noninteracting electrons experiencing the real external potential and this has exactly the same density as the real system. This reference system permits to treat the iV-electron system as the superposition of N one-electron systems and the corresponding iV-electron wave function of the reference system will be a Slater determinant. This is important because in this way DFT permits to handle both discrete and periodic systems. To obtain a trial density one needs to compute the energy of the real system and here it is when a model for the unknown functional is needed. To this purpose, the total energy is written as a combination of terms, all of them depending on the one-electron density only  

5 Density Functional Theory Versus Wave Function Methods Cu on MgO 

If we are interested in describing the electron distribution in detail, there is no substitute for quantum mechanics. Electrons are very light particles, and they cannot be described even qualitatively correctly by classical mechanics. We will in this and subsequent chapters concentrate on solving the time-independent Schrodinger equation, which in short-hand operator fonn is given as 

If solutions are generated without reference to experimental data, the methods are usually called ab initio (latin from the beginning ), in contrast to semi-empirical models, which are described in Section 3.9. 

A word of caution before we start. A rigorous approach to many of the derivations requires keeping track of several different indices and validating why certain transformations are possible. The derivations will be performed less rigorously, trying to illustrate the flow of arguments, rather than focus on mathematical details. 

Critics of first-principles models often point out that this AE corresponds to immobile atoms at zero Kelvin. For most catalytically interesting problems, the quantum mechanical AE is the largest contribution to a finite temperature reaction enthalpy or free energy. Statistical mechanical models can be used to determine and compute finite temperature and entropy contributions when necessary. In practice, the greater challenge is to solve the Schrodinger equation sufficiently accurately to give chemically useful values of AE. 

One way to reduce the computational cost of DFT (or WFT) calculations is to recognize that the core electrons of an atom have only an indirect influence on the atom chemistry. It thus makes sense to look for ways to precompute the atomic cores, essentially factoring them out of the larger electronic structure problem. The simplest way to do this is to freeze the core electrons, or to not allow their density to vary from that of a reference atom. This frozen core approach is generally more computationally efficient. One class of frozen core methods is the pseudopotential (PP) approach. The pseudopotential replaces the core electrons with an effective atom-centered potential that represents their influence on valence electrons and allows relativistic effects important to the core electrons to be incorporated. The advent of ultrasoft pseudopotentials (US-PPs) [18] enabled the explosion in supercell DFT calculations we have seen over the last 15 years. The projector-augmented wave (PAW) [19] is a less empirical and more accurate and transferable approach to partitioning the relativistic core and valence electrons and is also widely used today. Both the PP and PAW approaches require careful parameterizations of each atom type. 

Some insight into the methodological challenges in theoretical studies of f elements can already be gathered from looking at the uranyl double cation UO l which is a closed-shell molecule important in the treatment of nuclear waste. In this molecule the uranium atom has formal oxidation state + W, but from projection analysis [55] at the HF level we find a charge of -1-2.94 and configuration 5/ - 7/ of the uranium atom in the molecule. 

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Many phenomena in solid-state physics can be understood by resort to energy band calculations. Conductivity trends, photoemission spectra, and optical properties can all be understood by examining the quantum states or energy bands of solids. In addition, electronic structure methods can be used to extract a wide variety of properties such as structural energies, mechanical properties and thennodynamic properties. 

PAW is a recent addition to the all-electron electronic structure methods whose accuracy appears to be similar to that of the general potential LAPW approach. The implementation of the molecular dynamics fonnalism enables easy stmcture optimization in this method. 

Goedecker S 1999 Linear scaling electronic structure methods Rev. Mod. Phys. 71 1085... 

Page M, Doubleday C and Mclver J W Jr 1990 Following steepest descent reaction paths. The use of higher energy derivatives with ab initio electronic structure methods J. Chem. Phys. 93 5634 and references therein... 

Eoresman, J. B. Erisch A., 1996. Exploring Chemistry with Electronic Structure Methods 2nd ed. Gaussian Inc. Pittsburgh, PA. 

J. B. Foresman, JE. Frisch, Exploring Chemutry with Electronic Structure Methods Gaussian, Pittsburgh (1996). 

R. A. Albright, Orbital Interactions in Chemistry John Wiley Sons, New York (1998). A. R. Leach, Molecular Modelling Principles and Applications Longman, Essex (1996). J. B. Foresman, JE. Frisch, Exploring Chemistry with Electronic Structure Methods Gaussian, Pittsburgh (1996). 

D. F. Feller, MSRC Ah Initio Methods Benchmark Suite—A Measurement of Hardware and Software Performance in the Area of Electronic Structure Methods WA Battelle Pacific Northwest Labs, Richland (1993). 

Exploring Chemistry with Electronic Structure Methods... 

Gaussian offers the entire range of electronic structure methods. This work provides guidance and examples in using all of the most important of them. 

Recently, a third class of electronic structure methods have come into wide use density functional methods. These DFT methods are similar to ab initio methods in many ways. DFT calculations require about the same amount of computation resources as Hartree-Fock theory, the least expensive ab initio method. 

More accurate methods become correspondingly more expensive computationally. Recommended uses of each level of theory will be discussed throughout the work, and a consideration of the entire range of electronic structure methods is the subject of Chapter 6. 

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