Particle picture

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. 

Now that the wave and particle pictures were reconciled it became clear why the electron in the hydrogen atom may be only in particular orbits with angular momentum given by Equation (1.8). In the wave picture the circumference 2nr of an orbit of radius r must contain an integral number of wavelengths... 

The ionization process can be described as a one-particle event, when only one term dominates in summation (3), yielding a pole strength close unity. In this case, 1 - Fu gives an estimate of the fraction of photoemission intensity dispersed in many-body effects. On the contrary, small pole strengths are indicative (18-20) of a breakdown of the one-particle picture of ionization. 

As noticed from this expression, the CP calculation has to be basically carried out on the quasi-particle picture. Formally, quasi-particle energies and wave functions have to be evaluated by solving... 

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. 

In a series of calculations on ethylene, butadiene and hexatriene, Deleuze and co-workers [105] showed that the ADC(3) method can provide a very accurate picture of the electronic processes associated with ionisation in the valence region. Poly(acetylene) has a large feature above 21 eV, which was previously assigned to shake up. The theoretical work showed conclusively that in fact even the band at around 17eV, which had previously been assigned to a C 2s excitation could not be explained by a single particle picture but was due to satellite excitations. 

Darwin, 1929 Mott, 1930). The incident particle has momentum HKg before any interaction its momentum after exciting atoms 1 and 2 respectively into the nth and mth states is represented by hKnm. Mott showed that the entire process has negligible cross section unless the angular divergences are comparable to or less than (K a)-1, where a denotes the atomic size. As Darwin (1929) correctly conjectured, the wavefunction of the system before any interaction is the uncoupled product of the wavefunctions of the atom and of the incident particle. After the first interaction, these wavefunctions get inextricably mixed and each subsequent interaction makes it worse. Also, according to the Ehrenfest principle, the wavefunction of the incident particle is localized to atomic dimensions after the first interaction therefore, the subsequent process is adequately described in the particle picture. 

Whereas Si and s2 are true one-electron spin operators, Ky is the exchange integral of electrons and in one-electron states i and j (independent particle picture of Hartree-Fock theory assumed). It should be stressed here that in the original work by Van Vleck (80) in 1932 the integral was denoted as Jy but as it is an exchange integral we write it as Ky in order to be in accordance with the notation in quantum chemistry, where Jy denotes a Coulomb integral. 

In elementary particle physics the need to eliminate virtual processes is emphasized in many excellent texts.5,17 For quite different reasons we come to a conclusion rather near to that derived from the 5-matrix theory. There is a consistent particle picture. But at this point we have lost mechanics in the usual sense. We no longer deal with forces, correlations, and virtual particles, but with scattering cross sections and lifetimes. 

AEAt ft) we have here well-defined excited levels in the quasi-particle picture. There is no contradiction as the superposition of both processes, Eqs. (55) and (56), gives to the total cross section a finite line width which is in agreement with the uncertainty principle. 

Apart from the demands of the Pauli principle, the motion of electrons described by the wavefunction P° attached to the Hamiltonian H° is independent. This situation is called the independent particle or single-particle picture. Examples of single-particle wavefunctions are the hydrogenic functions (pfr,ms) introduced above, and also wavefunctions from a Hartree-Fock (HF) approach (see Section 7.3). HF wavefunctions follow from a self-consistent procedure, i.e., they are derived from an ab initio calculation without any adjustable parameters. Therefore, they represent the best wavefunctions within the independent particle model. As mentioned above, the description of the Z-electron system by independent particle functions then leads to the shell model. However, if the Coulomb interaction between the electrons is taken more accurately into account (not by a mean-field approach), this simplified picture changes and the electrons are subject to a correlated motion which is not described by the shell model. This correlated motion will be explained for the simplest correlated system, the ground state of helium. 

In the independent particle picture, the ground state of helium is given by Is2 xSo. For this two-electron system it is always possible to write the Slater determinantal wavefunction as a product of space- and spin-functions with certain symmetries. In the present case of a singlet state, the spin function has to be... 

The n-particle Picture and the Calculation of the Electronic Structure of Atoms, Molecules,... 

The first viewpoint defines the N-particle picture and leads in principle to an exact solution of Eq.(3). Leaving aside the question of whether or not one could extract useful information from the complicated form that such an exact solution for large values of N would be bound to take, the computational hindrance associated with this approach as N increases has mitigated against widespread use of such a direct procedure. At the same time, where it has been implemented[29], such as in connection with the He atom, this approach has yielded results which agree to great precision with those of exact numerical procedures. 

Unlike the AT-particle picture, which in principle leads naturally to the exact solution of the many-particle Schrodinger equation, the single-particle picture has led to the development of a number of different approximation schemes designed to address particular issues in the physics of interacting quantum systems. A particularly pointed example of this state of affairs is the strict dichotomy that has set in between so-called single-particle theories and canonical many-body theory[31, 32]. Each of the two methodologies can claim a number of successful applications, which tends to reinforce the perceived formal gap between them. 

One of the applications of the n-particle picture made below involves the generalization of the Hatree-Fock approximation to n-particle states. If one attempted to determine n-particle states directly from the equations, coupled with an antisymmetrization procedure with respect to particle indices, one would encounter precisely the same difficulty as noted above. Instead, the problem must be approached from a different perspective. 

In this approach, we consider the evolution of a system of particles described by means of the generalized HF equations as the interparticle interaction is turned on, starting from a single Slater determinant. The determi-nantal state corresponding to the zero-interaction limit provides an initial condition for solving the generalized HF equations within the n-particle picture. The states which evolve out of this procedure are known to satisfy the Pauli principle in the zero-interaction limit, and the generalized HF procedure to be described below maintains the correct symmetry as well as the requirements of the exclusion principle. 

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