Irreducible representations electronic wave function

For nonlinear molecules, each electronic wave function is classified according to the irreducible representation (symmetry species) to which it belongs the symmetry properties of i cl follow accordingly. For example, for the equilibrium nuclear configuration of benzene (symmetry ), the... 

In the second-quantization representation the atomic interaction operators are given by relations (13.22) and (13.23), which do not include the operators themselves in coordinate representations, but rather their one-electron and two-electron matrix elements. Therefore, in terms of irreducible tensors in orbital and spin spaces, we must expand the products of creation and annihilation operators that enter (13.22) and (13.23). In this approach, the tensorial properties of one-electron wave functions are translated to second-quantization operators. 

The energy spectrum of atoms and ions with j j coupling can be found using the relativistic Hamiltonian of iV-electron atoms (2.1)-(2.7). Its irreducible tensorial form is presented in Chapter 19. The relativistic one-electron wave functions are four-component spinors (2.15). They are the eigenfunctions of the total angular momentum operator for the electron and are used to determine one-electron and two-electron matrix elements of relativistic interaction operators. These matrix elements, in the representation of occupation numbers, are the parameters that enter into the expansions of the operators corresponding to physical quantities (see general expressions (13.22) and (13.23)). 

Compliance with the octet rule in diamond could be shown simply by using a valence bond approach in which each carbon atom is assumed sp hybridized. However, using the MO method will more clearly establish the connection with band theory. In solids, the extended electron wave functions analogous to MOs ate called COs. Crystal orbitals must belong to an irreducible representation, not of a point group, but of the space group reflecting the translational periodicity of the lattice. 

The many-electron wave function in a crystal forms a basis for some irreducible representation of the space group. This means that the wave function, with a wave vector k, is left invariant under the symmetry elements of the crystal class (e.g. translations, rotations, reflections) or transformed into a new wave function with the same wave vector k. 

The space functions and the spin functions form bases for mutually dual irreducible representations of the symmetric group. The Pauli principle requires that the total electronic wave function is antisymmetric with respect to the simultaneous permututation of space and spin coordinates of the electrons, that is... 

The notation just introduced is rather more than a convenient shorthand for specifying which orbitals are occupied and by how many electrons. It expresses the fact that the MO approximation to the molecular wave function is a product of one-electron wave functions, i.e. orbitals, each taken to a power equal to the number of electrons occupying it. We recall that the irreducible representation of a product of coordinates is the direct product of their irreps extending the same idea to the product of orbitals, we see that the irrep of an electron configuration is simply the direct product of the irreps of its occupied... 

Equation 144 is just the equation for the electronic energy which was discussed in Chapter XI. The electronic wave functions F x, y, z, r) therefore have the symmetry properties of the various irreducible representations of the groups Dooa or Coop according as the nuclei are identical or different. The vibrational wave function R r) depends only on the distance between the two nuclei and therefore belongs to the totally symmetrical representation. The complete wave fimction will thus have the symmetry properties of the product F x, y, z, r)U(6, x)-In order to discuss the nature of the solutions of 14 6 it will be con-... 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

The correct permutational symmetry was implemented into the wave functions by projection onto irreducible representations of the total symmetry group for heteronuclear species and for homonuclear species, where e refers to electron exchange and H refers to nuclear exchange. The irreducible representations chosen were singlets in all cases. 

The importance of symmetry in the study of the electronic structure of atoms and molecules depends on the fact that wave functions must transform according to one of the symmetry species of the symmetry group of the molecule. Stated precisely, the eigenfunctions of a Hamiltonian form bases for irreducible representations of the symmetry group of the Hamiltoirian. This principle allows wave functions to be classified according to symmetry species it assists... 

Various schemes exist to try to reduce the number of CSFs in the expansion in a rational way. Symmetry can reduce the scope of the problem enormously. In die TMM problem, many of die CSFs having partially occupied orbitals correspond to an electronic state symmetiy other than that of the totally symmetric irreducible representation, and dius make no contribution to the closed-shell singlet wave function (if symmetry is not used before the fact, die calculation itself will determine the coefdcients of non-contributing CSFs to be zero, but no advantage in efdciency will have been gained). Since this application of group dieoiy involves no approximations, it is one of the best ways to speed up a CAS calculation. 

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