Many-electron functions

The variational problem may again be formulated as a secular equation, where the coordinate axes are many-electron functions (Slater determinants), <, which are orthogonal (Section 4.2). 

The problem of evaluating the effect of the perturbation created by the ligands thus reduces to the solution of the secular determinant with matrix elements of the type rp[ lICT (pk, where rpj) and cpk) identify the eigenfunctions of the free ion. Since cpt) and cpk) are spherically symmetric, and can be expressed in terms of spherical harmonics, the potential is expanded in terms of spherical harmonics to fully exploit the symmetry of the system in evaluating these matrix elements. In detail, two different formalisms have been developed in the past to deal with the calculation of matrix elements of Equation 1.13 [2, 3]. Since t/CF is the sum of one-electron operators, while cpi) and cpk) are many-electron functions, both the formalisms require decomposition of free ion terms in linear combinations of monoelectronic functions. 

We have used for the row vectors of the respective entities, while we denote by ( ) and O the orbitals and many-electron functions, and by O and T(0) the two corresponding linear transformations, respectively. Various types of many-electron space for which such transformations may be carried out have been described by Malmqvist [34], In general, O may be non-unitary, possibly with subsidiary conditions imposed for ensuring that the corresponding transformation of the V-electron space exists e.g. a block-diagonal form according to orbital subsets or irreducible representations). 

In the notation of Eq. (1), O signifies the many-electron functions defined in terms of the VB-orbital basis, while is a set of many-electron functions defined in... 

In the notation ofEq. (1), signifies the many-electron functions defined in terms... 

Here A is the antisymmetrizer for all particles and any given factor for sub stem R, describes the state of a group of Nr electrons. This ansatz is a generalized product function [1] - anaJogous to a Slater determinant, but with each spin-orbital replaced by a many-electron function. 

We begin by reviewing perhaps the most fundamental selection rule in quantum chemistry. Let the functions f Vi form a basis of partner functions for irrep a, and similarly ipj for irrep /3. Let O denote an operator that commutes with all elements of the group Q O is a totally symmetric operator in the terminology of Sec. 1.4. At this stage, it should be noted, our basis functions can be one- or many-electron functions. Consider now the matrix element... 

The application of quantum-mechanical methods to the prediction of electronic structure has yielded much detailed information about atomic and molecular properties.13 Particularly in the past few years, the availability of high-speed computers with large storage capacities has made it possible to examine both atomic and molecular systems using an ab initio variational approach wherein no empirical parameters are employed.14 Variational calculations for molecules employ a Hamiltonian based on the nonrelativistic electrostatic nuclei-electron interaction and a wave function formed by antisymmetrizing a suitable many-electron function of spatial and spin coordinates. For most applications it is also necessary that the wave function represent a particular spin eigenstate and that it have appropriate geometric symmetry. 

Point (3) above requires some amplification. At the quantitative level, the ultimate aim of either a VB or an MO calculation is to obtain the total molecular wave function. Such a function will lead to an electron density map for the molecule which should yield information about its bonding and insights into its reactivity. The function may also be manipulated in order to calculate various molecular constants whose theoretical values can be compared with experimental ones, if available. The kind of function we are talking about is a many-electron function it contains the coordinates of all the electrons in the molecule, and is usually expressed as a product of one-electron functions (i.e. orbitals). In MO theory, these are the MOs. The constraints of symmetry and orthogonality ensure that these MOs are amenable in themselves to quantum-mechanical manipulations. In VB theory, however, the one-electron functions are localised bond orbitals which are not quite respectable and are not immediately amenable to manipulation. The total molecular wave function obtained from a VB calculation is not necessarily inferior to its MO counterpart however, its factorisation into one-electron functions is designed to preserve the useful and successful notion of the localised electron-pair bond. This has the disadvantage that the one-electron functions are less useful for quantum-mechanical purposes. 

Analytic, exact solutions cannot be obtained except for the simplest systems, i.e. hydrogen-like atoms with just one electron and one nucleus. Good approximate solutions can be found by means of the self-consistent field (SCF) method, the details of which need not concern us. If all the electrons have been explicitly considered in the Hamiltonian, the wave functions V, will be many-electron functions V, will contain the coordinates of all the electrons, and a complete electron density map can be obtained by plotting Vf. The associated energies E, are the energy states of the molecule (see Section 2.6) the lowest will be the ground state , and the calculated energy differences En — El should match the spectroscopic transitions in the electronic spectrum. 

However, to obtain results with good accuracy, it is also necessary to include a pseudostate (or localized) channel, which accounts for those closed channels that have not been included, as well as for the multiple ionization channels, since the set of single-ionization PWCs alone would be incomplete. The localized channel may comprise a large number of normalized many-electron functions /, built from localized orbitals... 

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. 

The unprojected UHF equations can be obtained again46 by calculating the expectation value of hamiltonian (3) with the DODS many-electron function and performing the variation of the coefficients in the Bloch functions... 

We now specialize the discussion to the ligand field theory situation and define the orthonormal set of spin-orbitals we shall use in the determinantal expansion of the many-electron functions Vyy for the groups M and L. First we suppose that we have a set of k orbitals describing the one-electron states in the metal atom these will be orthonormal solutions of a Schrodinger equation for a spherically symmetric potential, V<,(r), which may be thought of as the average potential about the metal atom which an electron experiences ... 

Electrons interchanged. When the many-electron function of a molecule is written in the form of a determinant, the fundamental antisymmetry principle (the Pauli exclusion principle) of quantum mechanics is satisfied. According to that principle an A-electron function must be antisymmetric, i.e. it must change sign whenever spatial and spin variables of any two electrons are interchanged ... 

Of what valne is it to have a semicondnctor that can condnct an electric cnrrent by the flow of electrons if it is n-type or by the flow of holes if it is p-type Many electronic functions can be fnlfilled by semiconductors that possess these properties, but the simplest is rectification—the conversion of alternating current into... 

This spectral form has the advantage that a single diagonalization of the Hamiltonian allows one to construct easily the R-matrix at all energies. In order to accomplish this it is necessary to introduce a basis set of many-electron functions and solve equation... 

Suppose states D > and A > are one-determinant many-electron functions, which are written in terms of (real) molecular orbitals and where a is the spin index, a a, p. These are the optimized canonical orbitals obtained from Hartree-Fock calculations of states D and A. Using the standard rules of matrix element evaluations[18], one can obtain an appropriate expression for Eq. (1) in terms of MO s of the system. 

McWeeny extended the use of group functions, Eq. (146), to intermolecular forces by mixing to the second order a set of such many-electron functions Af, A on each atom A and B, respectively. The functions of one atom A were assumed orthogonal in the sense of Eq. (1) not only to each other but also to those of B. There is no way of obtaining these functions. Also, the assumed Eq. (1) is too restrictive between A and B it is equivalent to the no overlap assumptions of the London theory. These limitations were observed on an application to He He. 

When the single determinant many-electron functions are constructed from canonical Hartree-Fock orbitals, the excited functions, and , are doubly excited with respect to the reference function . The second term in the third order energy expression cancels diagonal components for which p = v in the first term. The principal term in the fourth order energy expression has the form... 

This is deliberately vague it means continuous, square integrable, smooth and differentiable at least twice almost everywhere and having all the desirable properties which a many-electron function should have and which the spin-orbitals themselves are required to have . 

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