One-electron wavefunction

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

Once electron repulsion is taken into account, this separation of a many-electron wavefunction into a product of one-electron wavefunctions (orbitals) is no longer possible. This is not a failing of quanmm mechanics scientists and engineers reach similar conclusions whenever they have to deal with problems involving three or more mutually interacting particles. We speak of the three-body problem. 

Don t confuse the state wavefunction with a molecular orbital we might well want to build the state wavefunction, which describes all the 16 electrons, from molecular orbitals each of which describe a single electron. But the two are not the same. We would have to find some suitable one-electron wavefunctions and then combine them into a slater determinant in order to take account of the Pauli principle. 

Where might these one-electron wavefunctions come from I explained the basic ideas of HF and HF-LCAO theory in Chapter 6 we could find the molecular orbitals as linear combinations of appropriate atomic orbitals by solving the HF eigenvalue problem... 

The subsets of d orbitals in Fig. 3-4 may also be labelled according to their symmetry properties. The d ildxi y2 pair are labelled and the d yldxMyz trio as t2g. These are group-theoretical symbols describing how these functions transform under various symmetry operations. For our purposes, it is sufficient merely to recognize that the letters a ox b describe orbitally i.e. spatially) singly degenerate species, e refers to an orbital doublet and t to an orbital triplet. Lower case letters are used for one-electron wavefunctions (i.e. orbitals). The g subscript refers to the behaviour of... 

The left superscript indicates that the arrangements are all spin triplets. The letter T refers to the three-fold degeneracy just discussed and it is in upper case because the symbol pertains to a many-electron (here two) wavefunction (we use lower-case letters for one-electron wavefunctions or orbitals, remember). The subscript g means the wavefunctions are even under inversion through the centre of symmetry possessed by the octahedron (since each d orbital is of g symmetry, so also is any product of them), and the right subscript 1 describes other symmetry properties we need not discuss here. More will be said about such term symbols in the next two sections. 

Solutions to the Schrodinger equation (3.5) are called one-electron wavefunctions or orbitals and take the form in Eq. (3.6)... 

Term wavefunctions describe the behaviour of several electrons in a free ion coupled together by the electrostatic Coulomb interactions. The angular parts of term wavefunctions are determined by the theory of angular momentum as are the angular parts of one-electron wavefunctions. In particular, the angular distributions of the electron densities of many-electron wavefunctions are intimately related to those for orbitals with the same orbital angular momentum quantum number that is. 

Unfortunately, the many-electron wavefunction itself does not necessarily provide insight into the chemistry of complex molecules as it describes the electronic distribution over the whole system. It is therefore assumed that the true many-electron wavefunction. can be represented as the product of a series of independent, one-electron wavefunctions ... 

The simplest solution to this problem is to construct an antisymmetric wavefunction using a linear combination of one-electron wavefunctions. For two electrons, this takes the following form ... 

Although in many cases, particularly in PE spectroscopy, single configurations or Slater determinants 2d> (M+ ) were shown to yield heuristically useful descriptions of the corresponding spectroscopic states 2 f i(M+ ), this is not generally true because the independent particle approximation (which implies that a many-electron wavefunction can be approximated by a single product of one-electron wavefunctions, i.e. MOs 4>, as represented by a Slater determinant 2 j) may break down in some cases. As this becomes particularly evident in polyene radical cations, it seems appropriate to briefly elaborate on methods which allow one to overcome the limitations of single-determinant models. 

A further advantage of using Lagrangian dynamics is that we can easily impose boundary conditions and constraints by applying the method of Lagrangian multipliers. This is particularly important for the dynamics of the electronic degrees of freedom, as we will have to impose that the one-electron wavefunctions remain orthonormal during their time evolution. The Lex of our extended system can then be written as ... 

The most usual starting point for approximate solutions to the electronic Schrodinger equation is to make the orbital approximation. In Hartree-Fock (HF) theory the many-electron wavefunction is taken to be the antisymmetrized product of one-electron wavefunctions (spin-orbitals) ... 

Kohn and Sham later introduced the idea of an auxiliary non-interacting system with the same electron density as the real system. They were able to express the electron density of the interacting system in terms of the one-electron wavefunctions of the non-interacting system ... 

Inglesfield used a Green s function technique to write the one-electron wavefunctions as... 

The first factor will dictate, from the properties of the one-electron wavefunctions, the characteristic dependence of the cross-sections on the initial orbital character of the electron and on the exciting photon energy. 

Evaluation of the Oni s requires detailed atomic calculations, with a good knowledge of the one-electron wavefunctions (the best results, especially for high 1 orbitals, are obtained by relativistic calculations). They are usually expressed as a function of the exciting photon energy hv. 

I now want to generalize the concept of the charge density, and in particular treat spin explicitly rather than averaging it out. I told you earlier that, for a one-electron wavefunction l (r, s), T 2(r, s) dr ds gives the chance of finding the electron in the spatial volume element dr with spin coordinate between s and s + ds. In Cartesian coordinates, dr = dx dy dz. Some authors write x for the space and spin variables and I am going to follow this notation where appropriate. 

In Pauli s model, the sea of electrons, known as the conduction electrons are taken to be non-interacting and so the total wavefunction is just a product of individual one-electron wavefunctions. The Pauli model takes account of the exclusion principle each conduction electron has spin and so each available spatial quantum state can accommodate a pair of electrons, one of either spin. 

Under the Hartree-Fock (i.e., HF) approximation, the function of in variables for the solutions of the electronic Hamiltonian is reduced to n functions, which are referenced as molecular orbitals (MOs), each dependent on only three variables. Each MO describes the probability distribution of a single electron moving in the average field of all other electrons. Because of the requirements of the Pauli principle or antisymmetry with respect to the interchange of any two electrons, and indistinguishability of electrons, the HF theory is to approximate the many-electron wavefunction by an antisymmetrized product of one-electron wavefunctions and to determine these wavefunctions by a variational condition applied to the expected value of the Hamiltonian in the resulting one-electron equations,... 

The symmetry species of the complete electronic wavefunction corresponding to the configuration (21) is given by the direct product of the species of one-electron wavefunction species, taken as many... 

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