CHAPTER 1Chapter 1 An Introduction to the State of the Art in Quantum Chemistry
Introduction
The fundamental building blocks in quantum chemistry are nuclei and electrons. The small electronic mass necessitates the use of quantum mechanics for describing the electron distribution, but the nuclear masses are sufficiently heavy that their motion to a good approximation can be described by classical mechanics. The large difference in mass is the basis for the Born–Oppenheimer approximation, where the coupling between the nuclear and electronic motions is neglected. From the electron point of view, the nuclei are thus stationary, and the electronic Schrödinger equation can be solved with the nuclear positions as parameters. A (large) set of such solutions forms a 3N−6 dimensional potential energy surface (PES) upon which the nuclear motions can be solved subsequently. The multi-dimensional nature of the surface prevents a complete mapping for systems with more than four nuclei, and for larger systems, the effort must therefore be focused on the chemically important low-energy region. Traditionally such investigations have been done by a static approach, by locating minima and first-order saddle points on the PES [1]. Minima describe stable molecules, while first-order saddle points relate to the chemical transformation of one species to another via transition state theory. More recently, the PES has also been explored by direct dynamics, where Newton's equations for the nuclear motions are solved using energies and derivatives generated on-the-fly, as required by the dynamics [2]. In the present context, we will only be concerned with methods for solving the electronic Schrödinger equation and not with methods for exploring the resulting PES.
Solving the electronic Schrödinger equation is difficult for two main reasons:
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The electrons are indistinguishable and the differential equation couples all the electronic coordinates.
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The interaction between electrons is only a factor of Z (nuclear charge) less than the interaction between the nucleus and the electrons.
The standard approach for solving multi-variable differential equations is to find a set of coordinates where the variables can be separated and solve them one at a time. This is not possible for the electronic Schrödinger equation with more than one electron and the relatively large electron–electron interaction compared to the nucleus–electron interaction prevents a central-field approximation. Neglect of the electron–electron interaction leads to a wave function composed of hydrogen-like orbitals, but this is too poor a model to be useful. A qualitatively correct description can be obtained by a mean-field approximation, where the average electron–electron interaction is included, and within the wave function approach, this is known as the Hartree–Fock (HF) method. In order to improve the computational efficiency, various approximations to the HF equations can be made, with the reduction in fundamental accuracy being (partly) made up for by parameterization against experimental data. Such methods can collectively be called semi-empirical methods. Alternatively, the inherent deficiencies due to the mean-field approximation can be reduced by adding many-body corrections and these are called electron correlation methods [3].
Density functional theory (DFT) may be considered as an alternative formulation of quantum mechanics, where the electron density is the fundamental variable, rather than the electron coordinates [4]. DFT can also be considered as an improvement of the HF model, where the many-body correlation is modeled as a function of the electron density. DFT is analogous to the HF method a pseudo one-particle model, leading to a computationally efficient way of determining the electronic structure for large systems. While DFT has been a widely used tool for several decades in solid-state physics, it was only after the introduction of so-called gradient-corrected functionals in the early 1990s that the accuracy improved sufficiently to become a useful tool in computational chemistry.
An integrated element in practical calculations is the expansion of the orbitals (one-particle wave functions) in a set of known functions, the basis set. Only Gaussian-type basis functions will be considered in the present case, as these are used almost universally for application purposes. An ideal basis set should give a good accuracy for a small number of functions, be computationally efficient and allow a systematic way of extrapolating to the basis set limit. Unfortunately, different methods have different basis set requirements and it is not possible to find a single basis set optimum for all purposes.
In the following, we will briefly review the theoretical background for methods aimed at solving the electronic Schrödinger equation and present some highlight of the recent research within each area.
Section snippets
Hartree–Fock
The electronic Schrödinger equation in abbreviated form can be written aswhere the Hamilton operator H contains four terms corresponding to the electron kinetic energy, the nuclear–electron attraction, the electron–electron repulsion and the nuclear–nuclear repulsion. The latter is an additive constant within the Born–Oppenheimer approximation.
The variational principle states that an approximate wave function will always have an energy higher than the exact wave function
Electron correlation methods
The HF model only accounts for the average electron–electron interaction and thus neglects the correlation between electrons. Since HF is the energetically best single determinant wave function, correlated methods must necessarily involve more than one Slater determinant. This also means that the mental picture of each electron residing in a separate orbital must be abandoned. Rather, one must accept a picture with a range of orbitals having a fractional number of electrons. The HF model has N
Density functional theory
DFT rests on the Hohenberg–Kohn theorem, which states that there is a unique one-to-one correspondence between the ground state electron density and the energy of a system [17]. In the Kohn–Sham version of DFT, the density is written as an antisymmetric product of orbitals, analogous to the HF model [18]. The one-electron term and the Coulomb interaction between electrons are identical to those in the HF model, but the exchange and correlation contributions are incorporated as functionals of
Semi-empirical methods
The major computational effort in the HF model is calculating two-electron integrals over basis functions. In order to improve the computational efficiency, and thus allowing treatment of larger systems, it is necessary both to limit the size of the basis set and neglect certain classes of integrals. Semi-empirical methods are characterized by using only a minimum valence basis set, i.e., core electrons are accounted for by reducing the effective nuclear charge. All integrals extending over
Basis sets
The use of nuclear centered basis functions allows a formal way of approaching the basis set limit, by including more and more functions, and of increasingly higher angular momentum. Since a complete (infinite) basis set is computationally infeasible, error cancellation is the key to achieving good results with modest-sized basis sets. A good basis set is characterized by having a balanced composition, i.e., a proper number of functions and angular momenta. Unfortunately, different methods have
Summary
While computer hardware continues to closely follow Moore's law (doubling the performance–price ratio every 18 months), the introduction of new algorithms over the years has given at least the same amount of improvements. For HF and DFT methods, the scaling with system size appears to have been solved, and these methods are well suited for running in parallel on inexpensive cluster-type computers. There is little doubt that systems containing up to thousands of atoms will be attempted in the
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