Independent particle methods

We will look at the convergence as a function of basis set and amount of electron correlation (Figure 4.3). For independent-particle methods (FIF and DFT) we will use the polarization consistent basis sets (pc-n), while for correlated methods (MP2 and CCSD(T)) we will use the correlation consistent basis sets (cc-pVXZ, X = D,T, Q, 5, 6). Table 11.1 shows how the geometry changes as a function of basis set at the HF level of theory, where the quality of the basis sets is indicated by the maximum angular momentum function included in the basis set. 

The independent-particle methods have the decisive advantage that they appeal to our inclination to picture electronic processes as promoting one electron at a time and, using a MO interpretation, to assign individual molecular orbitals, or a simple combination of them, to each line in the electronic spectrum. The independent-particle MO analysis of Auger spectra starts out from the simple expression for double electron ionization energies ... 

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. 

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 summary, we have made three assumptions 1) the Bom-Oppenheimer approximation, 2) the independent particle assumption governing molecular orbitals, and 3) the assumption of n-molecular orbital theory, but the third is unique to the Huckel molecular orbital method. 

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). 

The OVGF function method provides a quantitative account of ionisation phenomena when the independent-particle picture of ionisation holds and as such is most applicable in the treatment of outer-valence orbitals. It provides an average absolute error for vertical ionisation energies below 20 eV of 0.25 eV for closed shell molecules. The TDA and ADC(3) methods allow for the breakdown of one particle picture of ionisation and so enable the calculation of the shake up spectra. The ADC(3) is correct up to 3rd order, is size consistent and includes correlation effects in both the initial and final states. 

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. 

The most widely used qualitative model for the explanation of the shapes of molecules is the Valence Shell Electron Pair Repulsion (VSEPR) model of Gillespie and Nyholm (25). The orbital correlation diagrams of Walsh (26) are also used for simple systems for which the qualitative form of the MOs may be deduced from symmetry considerations. Attempts have been made to prove that these two approaches are equivalent (27). But this is impossible since Walsh s Rules refer explicitly to (and only have meaning within) the MO model while the VSEPR method does not refer to (is not confined by) any explicitly-stated model of molecular electronic structure. Thus, any proof that the two approaches are equivalent can only prove, at best, that the two are equivalent at the MO level i.e. that Walsh s Rules are contained in the VSEPR model. Of course, the transformation to localised orbitals of an MO determinant provides a convenient picture of VSEPR rules but the VSEPR method itself depends not on the independent-particle model but on the possibility of separating the total electronic structure of a molecule into more or less autonomous electron pairs which interact as separate entities (28). The localised MO description is merely the simplest such separation the general case is our Eq. (6)... 

During the last decade MO-theory became by far the most well developed quantum mechanical method for numerical calculations on molecules. Small molecules, mainly diatomics, or highly symmetric structures were treated most accurately. Now applicability and limitations of the independent particle, or Hartree-Fock (H. F.), approximation in calculations of molecular properties are well understood. An impressive number of molecular calculations including electron correlation is available today. Around the equilibrium geometries of molecules, electron-pair theories were found to be the most economical for actual calculations of correlation effects ). Unfortunately, accurate calculations as mentioned above are beyond the present computational possibilities for larger molecular structures. Therefore approximations have to be introduced in the investigation of problems of chemical interest. Consequently the reliability of calculated results has to be checked carefully for every kind of application. Three types of approximations are of interest in connection with this article. 

The various methods used in quantum chemistry make it possible to compute equilibrium intermolecular distances, to describe intermolecular forces and chemical reactions too. The usual way to calculate these properties is based on the independent particle model this is the Hartree-Fock method. The expansion of one-electron wave-functions (molecular orbitals) in practice requires technical work on computers. It was believed for years and years that ab initio computations will become a routine task even for large molecules. In spite of the enormous increase and development in computer technique, however, this expectation has not been fulfilled. The treatment of large, extended molecular systems still needs special theoretical background. In other words, some approximations should be used in the methods which describe the properties of molecules of large size and/or interacting systems. The further approximations are to be chosen carefully this caution is especially important when going beyond the HF level. The inclusion of the electron correlation in the calculations in a convenient way is still one of the most significant tasks of quantum chemistry. 

In section 2, we provide a description of the methods employed in the present study the generation of Gaussian-type basis sets, the independent particle model and the treatment of electron correlation effects, and, the computational details. Results are presented and discussed in section 3. Section 4 contains our conclusions. 

Since his appointment at the University of Waterloo, Paldus has fully devoted himself to theoretical and methodological aspects of atomic and molecular electronic structure, while keeping in close contact with actual applications of these methods in computational quantum chemistry. His contributions include the examination of stability conditions and symmetry breaking in the independent particle models,109 many-body perturbation theory and Green s function approaches to the many-electron correlation problem,110 the development of graphical methods for the time-independent many-fermion problem,111 and the development of various algebraic approaches and an exploration of convergence properties of perturbative methods. His most important... 

There are many ways to improve this independent-particle model by incorporating electron correlation in the spatial part Hylleraas function [Hyl29] and the method of configuration interaction (Cl) will be used as illustrations. 

The other approach most frequently used to describe a correlated wavefunction beyond the independent-particle model is based on configuration interaction (Cl). (If the expansion is made on grounds of other basis sets, the approach is often called superposition of configurations, SOC, in order to distinguish it from the Cl method.) According to the general principles of quantum mechanics, the exact wavefunction which is a solution of the full Hamiltonian H can be obtained as an expansion in any complete set of basis functions which have the same symmetry properties ... 

The variational method of quantum chemistry for the determination of the energy has a direct analog in QMC. This is a consequence of the capability of the MC method to perform integration, and should not be confused with MC integration of the integrals that arise in basis set expansion methods. An important branch of QMC is the development of compact and accurate wave functions characterized by explicit dependence on interparticle distances electron-electron and electron-nucleus that are typically written as a product of an independent particle function and a correlation function. Such wave functions lead one immediately to the VMC method for evaluation [3-5]. Wave functions constructed following VMC can also serve as importance functions for the more accurate DMC variant of QMC. 

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