Symmetry of wave function

Mermin s "generalised crystallography" works primarily with reciprocal space notions centered around the density and its Fourier transform. Behind the density there is however a wave function which can be represented in position or momentum space. The wave functions needed for quasicrystals of different kinds have symmetry properties - so far to a large extent unknown. Mermin s reformulation of crystallography makes it attractive to attempt to characterise the symmetry of wave functions for such systems primarily in momentum space. 

In parallel there exist some attempts trying to introduce a field theory (FT) starting from the standard description in terms of phase space [4—6], Of course, the best way to derive a FT for classical systems should consist in taking the classical limit of a QFT in the same way as the so called classical statistical mechanics is in fact the classical limit of a quantum approach. This limit is not so trivial and the Planck constant as well as the symmetry of wave functions survive in the classical domain (see for instance [7]). Here, we adopt a more pragmatic approach, assuming the existence of a FT we work in the spirit of QFT. 

So much for considerations depending upon particle identities and symmetry of wave functions. An entirely different set of considerations arise from the fact that as the two atoms approach, the potential energy assumes a more complicated form, since in addition to nucleus-electron interactions there are repulsions between the two nuclei and the two electrons respectively and attractions between each nucleus and the electron of what was originally the other atom. 

The Symmetry Properties of Wave Functions of Li3 Electronically Ground State in [Pg.582]

Another aspect of wave function instability concerns symmetry breaking, i.e. the wave function has a lower symmetry than the nuclear framework. It occurs for example for the allyl radical with an ROHF type wave function. The nuclear geometry has C21, symmetry, but the Cay symmetric wave function corresponds to a (first-order) saddle point. The lowest energy ROHF solution has only Cj symmetry, and corresponds to a localized double bond and a localized electron (radical). Relaxing the double occupancy constraint, and allowing the wave function to become UHF, re-establish the correct Cay symmetry. Such symmetry breaking phenomena usually indicate that the type of wave function used is not flexible enough for even a qualitatively correct description. 

In the other approach one looks at the VB content of the symmetry adapted) wave function, for instance for N2... 

It is important to distinguish between mmetiy properties of wave functions on one hand and those of density matrices and densities on the other. The symmetry properties of wave functions are derived from those of the Hamiltonian. The "normal" situation is that the Hamiltonian commutes with a set of symmetry operations which form a group. The eigenfunctions of that Hamiltonian must then transform according to the irreducible representations of the group. Approximate wave functions with the same symmetry properties can be constructed, and they make it possible to simplify the calculations. 

Based on the same two step proeedure as presented above for C2H4 (MCSCF ealeulations followed by Schmidt orthogonalization of Rydberg functions), a systematic search was conducted by progressively incorporating groups of orbitals in the active space. Two types of wave functions proved well adapted to the problem, one for in-plane excitations, the other for out-of-plane excitations from the carbene orbital. The case of the Ai states will serve as an illustration of the general approach done for all symmetries and wave functions. 

The s-states have spherical symmetry. The wave functions (probability amplitudes) associated with them depend only on the distance, r from the origin (center of the nucleus). They have no angular dependence. Functionally, they consist of a normalization coefficient, Nj times a radial distribution function. The normalization coefficient ensures that the integral of the probability amplitude from 0 to °° equals unity so the probability that the electron of interest is somewhere in the vicinity of the nucleus is unity. 

The major difference between classical and quantum mechanical ensembles arises from the symmetry properties of wave functions which is not an issue in classical systems. 

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