Electrons Hartree approximation

The Hartree approximation is usefid as an illustrative tool, but it is not a very accurate approximation. A significant deficiency of the Hartree wavefiinction is that it does not reflect the anti-synnnetric nature of the electrons as required by the Pauli principle [7], Moreover, the Hartree equation is difficult to solve. The Hamiltonian is orbitally dependent because the siumnation in equation Al.3.11 does not include the th orbital. This means that if there are M electrons, then M Hamiltonians must be considered and equation A1.3.11 solved for each orbital. 

The response was calculated in the Hartree approximation, and only a linear response was considered. Suppose an external potential U(r) leads to a change in the electron density 8p. This then... 

The third term of Eq (54) is the electronic Hartree potential, whereas the fourth one represents the exchange-correlation potential. This last term is usually obtained from a model exchange-correlation energy functional xc[pl To a first order approximation, the effective KS potential compatible with the electron density p f) given in Eq (51) may be written as ... 

Now, if the many-body (electron) problem can be arranged in such a way that the many-body, nonseparable wave function is expressed in terms of a separable wave function, which depends on N single-particle wave functions (Hartree approximation), i.e.,... 

Messina et al. consider a system with two electronic states g) and e). The system is partitioned into a subset of degrees of freedom that are to be controlled, labeled Z, and a background subset of degrees of freedom, labeled x the dynamics of the Z subset, which is to be controlled, is treated exactly, whereas the dynamics of the x subset is described with the time-dependent Hartree approximation. The formulation of the calculation is similar to the weak-response optimal control theory analysis of Wilson et al. described in Section IV [28-32], The solution of the time-dependent Schrodinger equation for this system can be represented in the form... 

The HF equations are approximate mainly because they treat electron-electron repulsion approximately (other approximations are mentioned in the answer suggested for Chapter 5, Harder Question 1). This repulsion is approximated as resulting from interaction between two charge clouds rather than correctly, as the force between each pair of point-charge electrons. The equations become more exact as one increases the number of determinants representing the wavefunctions (as well as the size of the basis set), but this takes us into post-Hartree-Fock equations. Solutions to the HF equations are exact because the mathematics of the solution method is rigorous successive iterations (the SCF method) approach an exact solution (within the limits of the finite basis set) to the equations, i.e. an exact value of the (approximate ) wavefunction l m.. 

Application of ab initio MO theory usually begins at the monoconfigurational level, with the Hartree-Fock-Roothaan or LCAO-SCF methodology [4,5]. In this scheme the wave function for a closed-shell molecule containing N electrons is approximated as an antisymmetrized product (determinant) of spin-orbitals, ... 

Hartree-Fock with Proper Dissociation Internally Consistent Self Consistent Orbitals Independent Electron Pair Approximation Intermediate Neglect of Differential Overlap Intermediate Retention of Differential Overlap Iterative Natural Orbital Ionization Potential... 

This chapter begins a series of chapters devoted to electronic structure and transport properties. In the present chapter, the foundation for understanding band structures of crystalline solids is laid. The presumption is, of course, that said electronic structures are more appropriately described from the standpoint of an MO (or Bloch)-type approach, rather than the Heitler-London valence-bond approach. This chapter will start with the many-body Schrodinger equation and the independent-electron (Hartree-Fock) approximation. This is followed with Bloch s theorem for wave functions in a periodic potential and an introduction to reciprocal space. Two general approaches are then described for solving the extended electronic structure problem, the free-electron model and the LCAO method, both of which rely on the independent-electron approximation. Finally, the consequences of the independent-electron approximation are examined. Chapter 5 studies the tight-binding method in detail. Chapter 6 focuses on electron and atomic dynamics (i.e. transport properties), and the metal-nonmetal transition is discussed in Chapter 7. 

The simplest approach to approximating a solution to Eq. 4.3 is to assume that all the electrons move independently of one another. That is, imagine they mumally interact only via an averaged potential energy. This is known as the Hartree approximation. It enables us to write the Hamiltonian for the A -electron system as a sum of N one-electron Hamiltonians, and the many-body wave function as a product of N one-electron wave... 

The details of the modified electron-gas (MEG) ionic model method have been fully described by Gordon and Kim (1972). The fundamental assumptions of the method are (1) the total electron density at each point is simply the sum of the free-ion densities, with no rearangements or distortion taking place (2) ion-ion interactions are calculated using Coulomb s law, and the free-electron gas approximation is employed to evaluate the electronic kinetic, exchange, and correlation energies (3) the free ions are described by wave functions of Hartree-Fock accuracy. Note that this method does not iterate to a self-consistent electron density. 

The many-body ground and excited states of a many-electron system are unknown hence, the exact linear and quadratic density-response functions are difficult to calculate. In the framework of time-dependent density functional theory (TDDFT) [46], the exact density-response functions are obtained from the knowledge of their noninteracting counterparts and the exchange-correlation (xc) kernel /xcCf, which equals the second functional derivative of the unknown xc energy functional ExcL i]- In the so-called time-dependent Hartree approximation or RPA, the xc kernel is simply taken to be zero. 

If the indirect part of the electronic Coulomb repulsion is neglected, we do not get the Hartree approximation as might be expected. Instead, we get a less accurate method, which will be called the neglect of indirect Coulomb repulsion (NICR) method. If only the direct part of the Coulomb repulsion is included, the electron-electron repulsion is... 

Perdew and Zunger (1981), in the Xa-like equivalent of the Hartree approximation, advocate subtracting the total self-interaction of each electron in Xa-like models. This proposal would remove the m dependence of hydrogenic systems. Since the self-interaction of each electron (orthonormal orbital), as well as their sum, is not invariant under a unitary transformation among the orbitals, in contrast to the first-order density matrix and thus Xa-like models, Perdew and Zunger propose picking out a unitary transformation... 

The AIMP method in its present form starts from a quasirelativistic all-electron Hartree-Fock calculation for the atom under consideration in a suitable electronic state and approximates the operators on the left-hand side of Equation (3.10) for an atomic core X as described in the following. 

The quantum-mechanical SCF method for obtaining the vibrational energy levels is a direct adaptation of the Hartree approximation for electronic struc-true calculations, which dates back to the early stages of quantum theory. The introduction of the method for vibrational modes is, however, rather recent and is due to Bowman and co-workers,6,7 Carney et al.,8 and Cohen et al.9 The semiclassical version of the SCF, the SC-SCF method, proposed by Gerber and Ratner,10 relies on the characteristically short de Broglie wavelengths typical of vibrational motions (as opposed to electronic ones) to gain some further simplification, but is otherwise based on the same physical considerations as the quantum-mechanical approximation. A brief review of the SCF and SC-SCF methods can be found in Ref. (11). 

Degeneracies of the SCF states are an obvious cause for breakdown of the approximation in the form discussed in the previous sections. We discuss now an extension of the method that applies to such cases, that is, to resonances and near-resonances between SCF modes. Just as the vibrational SCF method is an adaptation of the Hartree approximation from electronic structure calculations, so is the generalization discussed here an application of the configuration interaction (Cl) method, which uses for the wavefunctions a linear combination of the strongly interacting SCF states. Quantum Cl for polyatomic vibrations was introduced by Bowman and co-workers,7-21 the semi-classical version is due to Ratner et al.33... 

As a result of the mean-field approximation, those pairs of Slater determinants that differ by more than one spin-orbital no longer contribute. This approximation is based on an idea similar to the conversion of the ordinary full electronic Hamiltonian into the one-electron Hartree-Fock operator and can be interpreted as describing electronic motion in an averaged field of the other electrons. 

When solved self-consistently, the electron densities obtained from Eq. [6] can be used in Eq. [5] to calculate the total electronic energy. This is equivalent to the relationship between Eqs. [3] and [4] for the Hartree approach. Unlike the Hartree approximation, however, this expression takes into account exchange and correlation interactions between electrons directly, and requires no other approximations other than the form of the density functional. 

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