Schrodinger equation one-electron

For a periodic lattice, it can be shown (Bloch theorem) that the solutions to the one-electron Schrodinger equation are of the... 

Slater then proposed returning to one-electron Schrodinger equations (and therefore one-electron wave functions i and corresponding eigenvalues ,) but now using not the Hartree-Fock (non-local) potential in these equations but the simplified potential... 

But the noninteracting GS problem may be solved alternatively by varying in Eq. (35) the N spin orbitals normalization condition on i(x) this leads obviously to one-electron Schrodinger equations... 

The differential virial theorem (169) for noninteracting systems can alternatively be obtained [31], [32] by summing (with the weights fj ) similar relations obtained for separate eigenfunctions 4>ja(r) of the one-electron Schrodinger equation (40) [in particular the KS equation (50)]. Just in that way one can obtain, from the one-electron HF equations (33), the differential virial theorem for the HF (approximate) solution of the GS problem, as is shown in Appendix B, Eq. (302), in a form ... 

The Hartree-Fock self-consistent field (SCF) method is the primary tool used in this chapter. It is rooted in the time-independent one-electron Schrodinger equation (in atomic units) ... 

As a result of these assumptions, qualitative molecular orbital models can be developed in which one assumes that each mo (f>i obeys a one-electron Schrodinger equation... 

The one-electron Schrodinger equation for an isolated free atom takes the form... 

The concepts which we need for understanding the structural trends within covalently bonded solids are most easily introduced by first considering the much simpler system of diatomic molecules. They are well described within the molecular orbital (MO) framework that is based on the overlapping of atomic wave functions. This picture, therefore, makes direct contact with the properties of the individual free atoms which we discussed in the previous chapter, in particular the atomic energy levels and angular character of the valence orbitals. We will see that ubiquitous quantum mechanical concepts such as the covalent bond, overlap repulsion, hybrid orbitals, and the relative degree of covalency versus ionicity all arise naturally from solutions of the one-electron Schrodinger equation for diatomic molecules such as H2, N2, and LiH. 

Let us consider what happens as two s-valent atoms A and are brought together from infinity to form the AB diatomic molecule as illustrated schematically in Fig. 3.1. The more deeply bound energy level EA could represent, for example, the hydrogenic Is orbital (EA = —13.6 eV), whereas the less deeply bound energy level EB could represent lithium s 2s orbital (EB = — 5.5 eV. cf Fig. 2.16). Each free atomic orbital satisfies its own effective one-electron Schrodinger equation (cf eqn (2.59)), namely... 

We wish to solve the one electron Schrodinger equation for the AB dimer,... 

Eigenfunctions of the one-electron Hamiltonian defined by Eq. (4.33) must satisfy the corresponding one-electron Schrodinger equation... 

Thus, formula (2.18) represents a new form of the non-relativistic wave function of an atomic electron (to be more precise, its new angular part in jj coupling). It is an eigenfunction of the operators I2, s2, j2 and jz, and it satisfies the one-electron Schrodinger equation, written in j-representation. Only its phase multiplier depends on the orbital quantum number to ensure selection rules with respect to parity. 

Hunter s unique expertise and experience enabled him to realize that the square root of the electron density satisfies a one-electron Schrodinger equation and that the effective potential in this Schrodinger equation is, in principle, an exact representation of the motion of a single electron within a many-electron system.232 He subsequently showed that there is a surface enclosing a molecule outside of which the electron s kinetic energy would be negative. He has proposed that this molecular envelope provides a nonarbitrary definition of molecular size and shape that is suitable for implementation in computer graphics. 

These formulas are valid for the nonrelativistic one-electron Schrodinger equation. The Lagrangian density is... 

Zhang, X.-G. and Butler, W.H. (1992). Simple cellular method for the exact solution of the one-electron Schrodinger equation, Phys. Rev. Lett. 68, 3753-3756. 

The procedure for calculating energy level diagrams by the self-consistent field Xa scattered wave (SCF-Xa-SW) method is as follows (Sherman, 1984, 1991). An octahedral cluster such as [FeO6]10 is partitioned into a set of (overlapping) spheres centred about divalent iron and each oxygen atom, and these are surrounded by an outer sphere. Within each atomic sphere the one-electron Schrodinger equation... 

Price and Halley (PH) [136] and Halley, Johnson, Price and Schwalm (HJPS) [137] have described a different theory of electron overspill into the layer between the solvent and metal-ion cores at a metal-electrolyte interface in the absence of specific adsorption of ions. Previous authors avoided the use of Schrodinger s equation altogether by introducing trial functions for the electron density function n(x). In contrast Halley and co-workers (HQ [138-141] used the Kohn-Sham version [122] of the variational principle of Hohenberg and Kohn [121] in which n(x) was described in terms of wave functions obeying Hartree-like equations. An effective one-electron Schrodinger equation is solved... 

As remarked above, considerable progress has resulted from use of the one-body potential of the density description in a one-electron Schrodinger equation approach. In the language of the density description, this is tantamount to treating the single-particle kinetic energy density exactly, as suggested by Kohn... 

In order to define orbitals in a many-electron system, two approaches are possible, which we may refer to as constructive and analytic . The first approach is more common one makes the ad hoc postulate that every electron can be associated with one orbital and the total wave function can be constructed from these orbitals. Then, one is led to an effective one-electron Schrodinger equation from one electron in the field of the other electrons. The underlying model is the independent particle model (IPM). When following the constructive way, one does not know a priori whether the model is a good approximation to the actual physical situation one only knows that it cannot be rigorously correct. The merit of this approach is its relative simplicity from both the mathematical and physical points of view. 

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