Independent-particle approximation

The purpose of this chapter is to provide an introduction to tlie basic framework of quantum mechanics, with an emphasis on aspects that are most relevant for the study of atoms and molecules. After siumnarizing the basic principles of the subject that represent required knowledge for all students of physical chemistry, the independent-particle approximation so important in molecular quantum mechanics is introduced. A significant effort is made to describe this approach in detail and to coimnunicate how it is used as a foundation for qualitative understanding and as a basis for more accurate treatments. Following this, the basic teclmiques used in accurate calculations that go beyond the independent-particle picture (variational method and perturbation theory) are described, with some attention given to how they are actually used in practical calculations. 

Applications of quantum mechanics to chemistry invariably deal with systems (atoms and molecules) that contain more than one particle. Apart from the hydrogen atom, the stationary-state energies caimot be calculated exactly, and compromises must be made in order to estimate them. Perhaps the most useful and widely used approximation in chemistry is the independent-particle approximation, which can take several fomis. Conuiion to all of these is the assumption that the Hamiltonian operator for a system consisting of n particles is approximated by tlie sum... 

It should be mentioned that the single-particle Flamiltonians in general have an infinite number of solutions, so that an uncountable number of wavefiinctions [/ can be generated from them. Very often, interest is focused on the ground state of many-particle systems. Within the independent-particle approximation, this state can be represented by simply assigning each particle to the lowest-lying energy level. If a calculation is... 

Although a separation of electronic and nuclear motion provides an important simplification and appealing qualitative model for chemistry, the electronic Sclirodinger equation is still fomiidable. Efforts to solve it approximately and apply these solutions to the study of spectroscopy, stmcture and chemical reactions fonn the subject of what is usually called electronic structure theory or quantum chemistry. The starting point for most calculations and the foundation of molecular orbital theory is the independent-particle approximation. 

A superb treatment of applied molecular orbital theory and its application to organic, inorganic and solid state chemistry. Perhaps the best source for appreciating the power of the independent-particle approximation and its remarkable ability to account for qualitative behaviour in chemical systems. 

A highly readable account of early efforts to apply the independent-particle approximation to problems of organic chemistry. Although more accurate computational methods have since been developed for treating all of the problems discussed in the text, its discussion of approximate Hartree-Fock (semiempirical) methods and their accuracy is still useful. Moreover, the view supplied about what was understood and what was not understood in physical organic chemistry three decades ago is... 

In the independent particle approximation, the simplifying assumption is made that V i) is an average potential due to a core that consists of the nuclei and all elections other than elechon i... 

The idea in perturbation methods is that the problem at hand only differs slightly from a problem which has already been solved (exactly or approximately). The solution to the given problem should therefore in some sense be close to the solution of the already known system. This is described mathematically by defining a Hamilton operator which consists of two part, a reference (Hq) and a perturbation (H )- The premise of perturbation methods is that the H operator in some sense is small compared to Hq. In quantum mechanics, perturbational methods can be used for adding corrections to solutions which employ an independent particle approximation, and the theoretical framework is then called Many-Body Perturbation Theory (MBPT). 

Although in many cases, particularly in PE spectroscopy, single configurations or Slater determinants 2d> (M+ ) were shown to yield heuristically useful descriptions of the corresponding spectroscopic states 2 f i(M+ ), this is not generally true because the independent particle approximation (which implies that a many-electron wavefunction can be approximated by a single product of one-electron wavefunctions, i.e. MOs 4>, as represented by a Slater determinant 2 j) may break down in some cases. As this becomes particularly evident in polyene radical cations, it seems appropriate to briefly elaborate on methods which allow one to overcome the limitations of single-determinant models. 

On the other hand, accurate calculations beyond the level of the independent particle approximation are very rare. Recently, a detailed study of (HF) 2 was published 82> in which the influence of electron correlation on the properties of the hydrogen bond was investigated. 

We have so far made two implicit assumptions. The first of these is that the gas of electrons is not scattered by the underlying ionic lattice. This can be understood by imagining that the ions are smeared out into a uniform positive background The second assumption is that the electrons move independently of each other, so that each electron feels the average repulsive electrostatic field from all the other electrons. This field would be completely cancelled by the attractive electrostatic potential from the smeared-out ionic background. Thus, we are treating our sp-valent metal as a metallic jelly or jellium within the independent particle approximation. 

Approximations towards the Cooper-Zare model The formulations so far have been rather general, and in the following they will be reduced to the formulations in the independent-particle approximation. First, the Dy are replaced by Dy which leads to the formulation of a and / in the Jij J-coupling case (see [WWa73]). Because only one electron configuration is present in Dy, such a dipole matrix element can be reduced further into the... 

The term "electron correlation energy" is usually defined as the difference between the exact nonrelativistic energy and the energy provided by the simplest MO wave function, the mono-determinantal Hartree-Fock wave function. This latter model is based on the "independent particle" approximation, according to which each electron moves in an average potential provided by the other electrons [14]. Within this definition, it is customary to distinguish between non dynamical and dynamical electron correlation. 

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. 

The previous components of the interaction energy can be derived in the independent particle approximation and so appear within the context of Hartree-Fock level calculations. Nevertheless, inclusion of instantaneous correlation will affect these properties. Taking the electrostatic interaction as an example, the magnimde of this term, when computed at the SCF level, will of course be dependent on the SCF electron distributions. The correlated density will be different in certain respects, accounting for a different correlated electrostatic energy. The difference between the latter two quantities can be denoted by the correlation correction to the electrostatic energy. 

The strong admixture of such states results in a multipeak structure in the spectra and the independent particle approximation MO model may break down completely. A good example for this case is the carbonyl sulfide demonstrated in [35]. In the case of tetrahedral oxyanions, the 0 2s re on of the valence band spectra may also belong to this group. 

The least ambiguous and most appropriate description of the atom after the collision is in terms of the density matrix (Blum, 1981), whose elements are bilinear combinations of scattering amplitudes for different magnetic substates. For the sake of simplicity we restrict ourselves to the most common case, in which the target is initially in an S state and the excitation involves the transfer of one electron from an s orbital to a p orbital in the independent-particle approximation. In atoms with one active electron the transition is — P. If there are two active electrons it is — P. We use the LS-coupling scheme. 

How can we improve this so-called independent-particle approximation such that the motions of the electrons are correlated Often the set of occupied orbitals (i.e., those functions that compose the Slater determinant above) is chosen from a larger set of one-electron functions. These extra functions are frequently referred to as virtual orbitals and may, for example, arise as a byproduct of the SCF procedure." Within the space described by the full set of orbitals, any function of N variables may be written in terms of N-tuple products of the (j)p. For example, a function of two variables may be constructed by using all possible binary products of the set of one-electron functions ... 

The multicentre one-electron space functions (r) describing electron distribution in molecules are called molecular orbitals (MOs). In the independent-particle approximation convenient MOs are constructed by the linear combination of atomic orbitals (LCAO) with coefficients determined by the Ritz method of Chapter 1. 

Nunes and Gonze [153] have recently extended DFPT to static responses of insulating ciystals for any order of perturbation theory by combining the variation perturbation approach with the modern theory of polarization [154]. There are evident similarities between this formalism and (a) the developments of Sipe and collaborators [117,121,123] within the independent particle approximation and (b) the recent work of Bishop, Gu and Kirtman [24, 155,156] at the time-dependent Hartree Fock level for one-dimensional periodic systems. 

The extension of the basis can improve wave functions and energies up to the Hartree-Fock limit, that is, a sufficiently extended basis can circumvent the LCAO approximation and lead to the best molecular orbitals for ground states. However, this is still in the realm of the independent-particle approximation 175>, and the use of single Slater-determinant wave functions in the study of potential surfaces implies the assumption that correlation energy remains approximately constant on that part of the surface where reaction pathways develop. In cases when this assumption cannot be accepted, extensive configuration interaction (Cl) must be included. A detailed comparison of SCF and Cl results is available for the potential energy surface for the reaction F + H2-FH+H 47 ). 

An orbital or independent particle approximation is defined as one in which we can, in a loose way, associate each electron with a given space orbital. The most general wavefunction of this form is 26... 

Here Ao is the Hamilton operator for a single electron in a fixed nuclei field, and j and k are the Fock- and Hartree-operators accounting for Coulomb and exchange interactions between electrons. This simplification is called the independent particle approximation. According to the variational theorem, every... 

There are cases for which more than one solution is found, and it is possible that both may possess physical reality under certain conditions [12] (this will arise again in chapter 11). Furthermore, the Hartree-Fock method can be made multiconfigurational, i.e. several configurations can be mixed or superposed. An electron is then shared between different states, which goes beyond the independent particle approximation. The self-consistent method allows the mixing coefficients to be determined, but the configurations to be included must be specified at the outset, and there is no simple prescription as to which ones should be chosen or left out. 

As a consequence of the breakdown of the independent particle approximation, it then emerged that the quantisation of individual electrons was not completely reliable. This was referred to in the classic texts on the theory of atomic spectra [309] as a breakdown in the I characterisation, and it manifests itself in the appearance of extra lines, which could not be classified within the independent electron scheme. The proper solution would, of course, be to revisit the initial theory and correct its inadequacies by a proper understanding of the dynamics of the many-electron problem, including where necessary new quantum numbers to describe the behaviour of correlated groups of electrons. Unfortunately, this plan of action cannot be followed through it would require a deeper understanding of the many-body problem than exists at present (see, e.g., chapter 10 for some of the difficulties). 

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