Schrodinger space model

Most semi-empirical models are based on the fundamental equations of Hartree-Fock theory. In the following section, we develop these equations for a molecular system composed of A nuclei and N electrons in the stationary state. Assuming that the atomic nuclei are fixed in space (the Born-Oppenheimer approximation), the electronic wavefunction obeys the time-independent Schrodinger equation ... 

Once this equation is solved for all relevant regions of the nuclear configuration space, in the BO framework, the nuclear motion can be treated either via a classical mechanical analysis with the help of computer simulations [6], or it can be treated quantum mechanically for simple models [54], In the latter scheme, the nuclear Schrodinger equation must be solved ... 

Equations (3.23) and (3.24) are valid also for a model space containing several unperturbed energies, e.g. several atomic configurations. These equations will form the basis for our many-body treatment. The generalized Bloch equation is exact and completely equivalent to the Schrodinger equation for the states considered. 

The quantum mechanical model proposed in 1926 by Erwin Schrodinger describes an atom by a mathematical equation similar to that used to describe wave motion. The behavior of each electron in an atom is characterized by a wave function, or orbital, the square of which defines the probability of finding the electron in a given volume of space. Each wave function has a set of three variables, called quantum numbers. The principal quantum number n defines the size of the orbital the angular-momentum quantum number l defines the shape of the orbital and the magnetic quantum number mj defines the spatial orientation of the orbital. In a hydrogen atom, which contains only one electron, the... 

I. Lindgren, The Rayleigh-Schrodinger perturbation and the linked-diagram theorem for a multi configurational model space, J. Phys. B At. Mol. Opt. Phys. 7 (1974) 2441. 

Normal fermionic particles have n = 1 and for photons n — 2. However, these velocities are not defined in four dimensions, but in 5D space, made up of a four-dimensional Euclidean space S = x, y, z, u] and absolute time t. The first three coordinates of S are familiar in 3D space, with u orthogonal to these in 4D and not observable in 3D. Each (moving or stationary) particle has its own inertial system S wherein it moves with velocity c along u. The model therefore contains the surprising results (4.3.2) calculated by Schrodinger [67] and Winterberg [68] that electrons and photons have intrinsic velocities of c and pic respectively. 

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. 

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