Single electron function

Although the one-dimensional model bears little resemblance to any real molecular system, many of its features carry over to cases of practical interest. Suppose we consider three-dimensional motion in a central field as in atoms. The orbitals or single-electron functions now become atomic orbitals and can be classified in the usual manner as Is, 2a,. . . 2p, 3p,. . . 3d,. Suppose we are dealing with an atom in which there are two electrons of the same spin (a, say) occupying the 2a and 2p orbitals (inner shells being ignored for the present). Then the antisymmetric product function is... 

For larger molecules it is assumed that a molecular wave function, , is an anti-symmetric product of atomic wave functions, made up by linear combination of single-electron functions, called orbitals. The Hamiltonian operator, H which depends on the known molecular geometry, is readily derived and although eqn. (3.37) is too complicated, even for numerical solution, it is in principle possible to simulate the operation of H on d>. After variational minimization the calculated eigenvalues should correspond to one-electron orbital energies. However, in practice there are simply too many electrons, even in moderately-sized molecules, for this to be a viable procedure. 

Note that all of the above expressions are written in terms of single electron functions and no reference is made to many-electron functions. This is a fundamental characteristic of the many-body perturbation theoretic approach to the correlation problem. 

When molecular orbitals are expressed in terms of a basis set of K single electronic functions ( ) (often atomic orbitals), the electron density is expressed as ... 

The bond-pair wave functions in Equations 6.27, 6.28, and 6.29 were specially constructed to describe two electrons localized between two atoms as a single chemical bond between the atoms. These wave functions should not be called MOs, because they are not single-electron functions and they are not de-localized over the entire molecule. The corresponding single bonds (see Figs. 6.37, 6.38, and 6.39) are called cr bonds, because their electron density is cylindrically symmetric about the bond axis. There is no simple correlation between this symmetry... 

To set up the total time-independent wave function of the many-electron system the independent-particle model is used in general, resulting in an antisymmetrized Hartree product of four-component orthonormal one-electron functions. Independent of the system (atom, molecule or solid) these four-component single electron functions, the spinors, may be written as... 

Equations 32-9 and 32-12 are identical, so that by using the variation method with a product-type variation function we obtain the same single-electron functions as by applying the criterion of the self-consistent field. 

Electronic structure calculations - The numerical approach We will focus in this subsection on electronic structure calculations (non-relativistic and relativistic) using numerical techniques, i.e., we do not use an expansion of single-electron functions (orbitals or spinors) in terms of analytic basis functions. Nowadays such numerical calculations are routinely feasible only for atoms and diatomics. Within the non-relativistic approach we have to determine rad.ial functions Pj(r) (z = nl) as parts of single-centre functions... 

The corresponding functions X , Xy then define what are known as the normal coordinates of vibration, and the Hamiltonian can be written in terms of these in precisely the form given by equation (A 1.1.691. with the caveat that each term refers not to the coordinates of a single particle, but rather to independent coordinates that involve the collective motion of many particles. An additional distinction is that treatment of the vibrational problem does not involve the complications of antisymmetry associated with identical fermions and the Pauli exclusion principle. Products of the normal coordinate functions nevertheless describe all vibrational states of the molecule (both ground and excited) in very much the same way that the product states of single-electron functions describe the electronic states, although it must be emphasized that one model is based on independent motion and the other on collective motion, which are qualitatively very different. Neither model faithfully represents reality, but each serves as an extremely useful conceptual model and a basis for more accurate calculations. 

If the density is determined by a set of single-electron functions then what we require in order to compute the density is an equation which determines these functions. Generally speaking, such an equation and its boundary conditions will be generated from a variation principle, and the Kohn-Hohenberg proof is based on the ordinary Schrodinger variation principle so the prospects are optimistic. 

There are numerous alternative methods that introduce electron correlation in the molecular calculations at a more precise level that can be profitably used. We mention here the MC-SCF approach (the acronym means that this is a variant of HF (or SCF) procedure starting fi om the optimization no more of a single antisymmetric orbital product, but of many different products, or configurations), the Coupled-Cluster theory, etc., all methods based on a MO description of single-electron functions. 

The self-consistent field approximation, which was briefly introduced earlier, is used to reduce the A -electron problem into the solution of n-single-electron systems. It reduces a 3n variable problem into n single electron functions that depend on three variables each. The individual electron-electron repulsive interactions shown in Eq. (A4) are replaced by the the repulsive interactions between individual electrons and an electronic field described by the spatially dependent electron density, p(r). This avoids trying to solve the difficult multicenter integrals that describe electron-electron interactions. The only trouble is that the electron density depends upon how each electron interacts with it. At the same time, the electron interaction with the field depends upon the density. A solution to this dilemma is to iterate upon the density until it convergences. The electron density that is used as the input to calculate the electron-field interactions must be equivalent, to within some tolerance, to that which results from the convergence of the electronic structure calculation. This is termed the self-consistent field (SCF). 

Electron density from 2c-spinors. Most of the time, the V-electron ground state wave functions are approximated by an antisymmetrized product of N orthonormal single-electron functions (spin-orbitals) and are expressed in terms of a Slater determinant y/>. The electron density is then the expectation value of the one-electron density operator ... 

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