The Independent-Particle Model

Solving the electronic Schrodinger equation means that we need to determine a very complicated function, = Yg/( rj ), whose variables are all electronic coordinates. This implies that the solution will be the more complicated the more electrons are involved in our first-quantized description. In order to simplify the problem and to find a suitable starting point, we introduce a product ansatz. 

The Slater determinant represents the simplest approximation to an electronic ground state, and its orbitals may be optimized to approximate the ground state as closely as possible within the independent particle model. We shall see later in this chapter how the optimization of the orbitals can be done within (Dirac-)Hartree-Fock theory by relying on the variational principle. 

Read has shown that the energies of such autoionization states can be described to a good approximation by a modified Rydberg formula, in which electron correlation is reflected in a phenomenological way. In this approach the energies are given by 

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In Section II.C we gave a general discussion of the Coulomb correlation, and we will now define the correlation error in the independent-particle model in greater detail. It is convenient to study the first- and second-order density matrices and, according to the definitions (Eq. II.9) applied to the symmetryless case, we obtain... 

The general idea of using different orbitals for different spins" seems thus to render an important extension of the entire framework of the independent-particle model. There seem to be essential physical reasons for a comparatively large orbital splitting depending on correlation, since electrons with opposite spins try to avoid each other because of their mutual Coulomb repulsion, and, in systems with unbalanced spins, there may further exist an extra exchange polarization of the type emphasized by Slater. 

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)... 

Fig. 2. Second-order contributions to intermolecular perturbation energies (schematic description by orbital excitations within the framework of the independent particle model AEpol and zI-Echt are represented by single excitations, AEms by correlated double excitations)...

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. 

J. L. Stuber and J. Paldus, Symmetry Breaking in the Independent Particle Model. In E. J. Brandas and E. S. Kryachko (Eds.) Fundamental World of Quantum Chemistry, A Tribute Volume to the Memory of Per-Olov Lowdin, Vol. 1. (Kluwer, Dordrecht, 2003), pp. 67-139. 

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... 

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. 

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 basis functions (r) and vP2(r) are solutions within the independent-particle model (operator H°, equ. (1.3)) ... 

Note that n — N/2 corresponds to the independent particle model analogous to the celebrated Hartree-Fock equations in atomic and molecular physics. We also observe that the fundamental interaction mentioned above is unitarily connected with the electromagnetic interactions between the particle m0 and the antiparticle —m0. Since we do not make any distinctions between the Klein-Gordon and the Dirac equation, we are not able here to integrate the electro-weak theory although in principle this should be possible. 

Fig. 10.1. Energy spectrum predicted by the independent particle model. (From Bliimel and Reinhardt (1992).)...

One of the most basic features of the helium spectrum is its organization into an infinite sequence of ionization thresholds. This feature is not the result of intricate computations. It is already apparent on the level of the independent particle model of the helium atom (see Section 10.1). All predictions on the quantum manifestations of chaos have to... 

This shows that the resonance velocities contain a collective component. The data peak at = 2.15 x 10. This corresponds to a collective drift whose origin and magnitude can be understood qualitatively. Using the independent particle model for a rough estimate of the energy levels we have... 

Detailed study of the chart of nuclides makes evident that for certain values of P and N a relatively large number of stable nuchdes exist. These numbers are 2, 8, 20, 28, 50, 82 (126, only for N). The preference of these magic numbers is explained by the shell structure of the atomic nuclei (shell model). It is assumed that in the nuclei the energy levels of protons and of neutrons are arranged into shells, similar to the energy levels of electrons in the atoms. Magic proton numbers correspond to filled proton shells and magic neutron numbers to filled neutron shells. Because in the shell model each nucleon is considered to be an independent particle, this model is often called the independent particle model. 

The are eigenvalues of Hhf- We interpret them as the one-electron energies of the orbitals ) in the independent-particle model. 

A configuration in the independent-particle model may be of either the closed-shell or open-shell type. In the former the N electrons occupy all the orbitals of the lowest-energy sets with the same symmetry and principal quantum number , called shells. In the latter some orbitals with particular values of the projection quantum numbers are unoccupied. The... 

The dipole moment operator is a one-electron operator, and within the independent particle model, with explicit treatment of electron repulsion as well as without, the transition moment becomes... 

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