Electron irreducible representation

As usual the numerical factors denote how many irreducible representations of a given kind are present. It is well known from general theory on vibronic coupling (see, for instance, the Bersuker and Polinger book [8]) that only coordinates transforming as the symmetric part of the electronic irreducible representation r can be coupled. Forp-like functions T = Ei, for d-like functions r = E2 and so on. Thus for p-like functions the only vibrational combinations that can be coupled are of type E2 and are coming from [Eif, [Eif,..., namely... 

The rotation-vibration-electronic energy levels of the PH3 molecule (neglecting nuclear spin) can be labelled with the irreducible representation labels of the group The character table of this group is given in table Al.4.10. 

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 remaining combinations vanish for symmetry reasons [the operator Lz transforms according to // (A") irreducible representation]. The nonvanishing of the off-diagonal matrix element B+ is responsible for the coupling of the adiabatic electronic states. 

Step 1. Norbornadiene C7H8 of symmetry C2V contains 8 CH single, 8 CC single and 2 jr-bonds, occupied by 36 electrons. (We disregard the inner carbon ls-orbitals). Accordingly, a SCF treatment yields 18 bonding canonical molecular orbitals (CMOs) irreducible representation Ai, 2 to A2, 4 to B and 5 to 82- We collect these 18 CMOs in a column vector... 

Ti, T Irreducible representation, denoted in the text and in the tables by small letters if referring to M.O.s of the parent molecule and by capital letters if referring to many-electron states of the + 1 ions. 

There exists no uniformity as regards the relations between localized orbitals and molecular symmetry. Consider for example an atomic system consisting of two electrons in an (s) orbital and two electrons in a (2px) orbital, both of which are self-consistent-field orbitals. Since they belong to irreducible representations of the atomic symmetry group, they are in fact the canonical orbitals of this system. Let these two self-consistent-field orbitals be denoted by Cs) and (2p), and let (ft+) and (ft ) denote the two digonal hybrid orbitals defined by... 

Labeling these electronic energy levels in a proper way (irreducible representations) and determine their degeneracy. 

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. 

Formaldehyde belongs to the symmetry group Czv and the electronic states can be classified into the four irreducible representations Ai, A 2, B, and B -... 

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