Independent 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. 

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. 

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 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... 

In an independent particle picture, one has a discrete state, like 3p 3d + interacting with the continnum 3d ef, ep. Configuration interaction mixes the discrete state, which loses its separate existence, in the continnum. The transition moment... 

The interpretation of these autoionizing excited states even from the outset recognized the inadequacy of the independent-particle picture.2 To account for the number of observed series and their intensities, it was necessary to invoke strong mixing of configurations, each with its own individual-particle quantum numbers. To obtain even a minimally satisfactory description of the... 

Several approaches were pursued in the process of finding an interpretation more physically intuitive than the somewhat hollow, ex post facto interpretation of Eq. (1), that is, that the independent-particle picture represented by a single configuration is spoiled by electron-electron correlation, particularly by angular correlation, because the second configuration leaves the radial distribution relatively unaffected but changes the angular distribution. 

In this very special case, the results even for a binary system look like the independent particle picture (cf. Section 5.8.)... 

PE spectroscopy . For the interpretation of the bands one has to go beyond the independent particle picture. Corrections are necessary because electron correlations and relaxation effects must be considered (see Section I). Calculations applied to interpret the PE spectra of N2 and CO are based on the Green s function method or the configuration interaction technique ... 

Regarding the analytical model further comments may apply. The general theoretical framework stands as the DFT however, this venture was developed on its conceptual rather than on its computational virtues. This way, the approximate energetic functional approaches (Nalewajski, 1996 Putz, 2008b) were systematically avoided by considering the independent-particle picture ofthe softness kernel formulation, see Sections 4.6.1-4.6.4. 

Up to this point we have tailored the second-quantization formalism in close connection to the independent-particle picture introduced before. However, the formalism can be generalized in an even more abstract fashion. For this we introduce so-called occupation number vectors, which are state vectors in Fock space. Fock space is a mathematical concept that allows us to treat variable particle numbers (although this is hardly exploited in quantum chemistry see for an exception the Fock-space coupled-cluster approach mentioned in section 8.9). Accordingly, it represents loosely speaking all Hilbert spaces for different but fixed particle numbers and can therefore be formally written as a direct sum of N-electron Hilbert spaces. 

In this section, we have demonstrated that this requirement is satisfied by the solutions of the exact electronic Schrodinger equation and by the solutions of the eigen-problem for some approximate electronic Hamiltonian operator associated with an arbitrary independent particle picture. 

The DF equations written down in Sec. II give an Independent particle picture of atomic structure. One of the Important schemes developed to go beyond the independent picture and to... 

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