Wave equation many-electron

In order to analyze the consequences of basis set expansion of the relativistic wave equation for electrons, it is sufficient to consider the one-electron Dirac equation. The only other fundamental complication we foresee when going to a many-electron model is continuum dissolution (Brown-Ravenhall disease), which we have already dealt with in chapter 5 and so need not consider in this context. 

One of the advantages of this method is that it breaks the many-electron Schrodinger equation into many simpler one-electron equations. Each one-electron equation is solved to yield a single-electron wave function, called an orbital, and an energy, called an orbital energy. The orbital describes the behavior of an electron in the net field of all the other electrons. 

In the VB method, a wave equation is written for each of various possible electronic structures that a molecule may have (each of these is called a canonical form), and the total )/ is obtained by summation of as many of these as seem plausible, each with its weighting factor ... 

As discussed above, it is impossible to solve equation (1-13) by searching through all acceptable N-electron wave functions. We need to define a suitable subset, which offers a physically reasonable approximation to the exact wave function without being unmanageable in practice. In the Hartree-Fock scheme the simplest, yet physically sound approximation to the complicated many-electron wave function is utilized. It consists of approximating the N-electron wave function by an antisymmetrized product4 of N one-electron wave functions (x ). This product is usually referred to as a Slater determinant, OSD ... 

An important consequence of the only approximate treatment of the electron-electron repulsion is that the true wave function of a many electron system is never a single Slater determinant We may ask now if SD is not the exact wave function of N interacting electrons, is there any other (necessarily artificial model) system of which it is the correct wave function The answer is Yes it can easily be shown that a Slater determinant is indeed an eigenfunction of a Hamilton operator defined as the sum of the Fock operators of equation (1-25)... 

We have just explained that the wave equation for the helium atom cannot be solved exacdy because of the term involving l/r12. If the repulsion between two electrons prevents a wave equation from being solved, it should be clear that when there are more than two electrons the situation is worse. If there are three electrons present (as in the lithium atom) there will be repulsion terms involving l/r12, l/r13, and l/r23. Although there are a number of types of calculations that can be performed (particularly the self-consistent field calculations), they will not be described here. Fortunately, for some situations, it is not necessary to have an exact wave function that is obtained from the exact solution of a wave equation. In many cases, an approximate wave function is sufficient. The most commonly used approximate wave functions for one electron are those given by J. C. Slater, and they are known as Slater wave functions or Slater-type orbitals (usually referred to as STO orbitals). 

In his first communication23 on the new wave mechanics, Schrodinger presented and solved his famous Eq. (1.1) for the one-electron hydrogen atom. To this day the H atom is the only atomic or molecular species for which exact solutions of Schrodinger s equation are known. Hence, these hydrogenic solutions strongly guide the search for accurate solutions of many-electron systems. 

In a line of reasoning that many of the younger quantum physicists regarded as reactionary, Schrodinger built his treatment of the electron on the well-understood mathematical techniques of wave equations as partial differential equations involving second derivatives. Schrodinger s equation for stationary electron states, as written in the Annalen der Physik in 1926, took the form... 

In applying this notion to many-electron systems, Pauling reasoned that a wave function might be set up to represent each of the possible classical valence, or electron-pair, bonds in compounds like carbon dioxide or benzene. Each equation corresponds to a combination of ionic and covalent character... 

Clearly, however, electrons exist. And they must exist somewhere. To describe where that somewhere is, scientists used an idea from a branch of mathematics called statistics. Although you cannot talk about electrons in terms of certainties, you can talk about them in terms of probabilities. Schrodinger used a type of equation called a wave equation to define the probability of finding an atom s electrons at a particular point within the atom. There are many solutions to this wave equation, and each solution represents a particular wave function. Each wave function gives information about an electron s energy and location witbin an atom. Chemists call these wave functions orbitals. 

Solutions to the One-Electron Many-Center Wave Equation... 

Molecular orbital an initio calculations. These calcnlations represent a treatment of electron distribution and electron motion which implies that individual electrons are one-electron functions containing a product of spatial functions called molecular orbitals hi(x,y,z), 4/2(3 ,y,z), and so on. In the simplest version of this theory, a single assignment of electrons to orbitals is made. In turn, the orbitals form a many-electron wave function, 4/, which is the simplest molecular orbital approximation to solve Schrodinger s equation. In practice, the molecular orbitals, 4 1, 4/2,- -are taken as a linear combination of N known one-electron functions 4>i(x,y,z), 4>2(3,y,z) ... 

Importantly, the anti-Hermitian CSE may be evaluated through second order of a renormalized perturbation theory even when the cumulant 3-RDM is neglected in the reconstruction. The anti-Hermitian part of the CSE [27, 31, 63] is the stationary condition for two-body unitary transformations of the A-particle wave-function [31, 32], and hence the two-body unitary transformations may easily be evaluated with the anti-Hermitian CSE and RDM reconstruction without the many-electron Schrodinger equation. The contracted Schrodinger equation in conjunction with the concepts of reconstruction and purification provides a new, important approach to computing the 2-RDM directly without the many-electron wavefunction. 

The Hy-CI function used for molecular systems is based on the MO theory, in which molecular orbitals are many-center linear combinations of one-center Cartesian Gaussians. These combinations are the solutions of Hartree-Fock equations. An alternative way is to employ directly in Cl and Hylleraas-CI expansions simple one-center basis functions instead of producing first the molecular orbitals. This is a subject of the valence bond theory (VB). This type of approach, called Hy-CIVB, has been proposed by Cencek et al. (Cencek et.al. 1991). In the full-CI or full-Hy-CI limit (all possible CSF-s generated from the given one-center basis set), MO and VB wave functions become identical each term in a MO-expansion is simply a linear combination of all terms from a VB-expansion. Due to the non-orthogonality of one-center functions the mathematical formalism of the VB theory for many-electron systems is rather cumbersome. However, for two-electron systems this drawback is not important and, moreover, the VB function seems in this case more natural. 

The determinential wave functions shown in equations (42)-(44) have the correct normalization for many-electron Sturmians (i.e. the normalization required by equation (6)). To see this, we can make use of the Slater-Condon rules, which hold for the diagonal matrix elements of... 

The Schrodinger equations for many-electron systems are generahzations of the one-electron problem. The wave function is a function of the space and spin coordinates of all the electrons ... 

In the absence of a magnetic field, the spin does not appear explicitly. We shall see that electron spin constrains the solutions of the many-electron Schro-dinger equation as a condition on the symmetry of the wave function with respect to exchange of electrons. 

Spatial extension, as expressed by the expectation value (r), is roughly comparable for 4 f and 5 f wave functions (Figs. 7 and 8). However, the many-electron wave functions resulting from the solution of the relativistic Dirac equation may also be used to calculate a number of physically interesting quantities, i.e. expectation values of observable... 

Many phenomena such as dislocations, electronic structures of polyacetylenes and other solids, Josephson junctions, spin dynamics and charge density waves in low-dimensional solids, fast ion conduction and phase transitions are being explained by invoking the concept of solitons. Solitons are exact analytical solutions of non-linear wave equations corresponding to bell-shaped or step-like changes in the variable (Ogurtani, 1983). They can move through a material with constant amplitude and velocity or remain stationary when two of them collide they are unmodified. The soliton concept has been employed in solid state chemistry to explain diverse phenomena. 

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