Symmetry particle interchange

Here a given function involves an n-fold product of MOs, to which is applied some projection operator or operators 0. As electrons are fermions, the solutions to Eq. (2) will be antisymmetric to particle interchange, and it is usually convenient to incorporate this into the n-particle basis, in which case the will be Slater determinants. The Hamiltonian given in Eq. (1) is also spin-independent and commutes with all operations in the molecular point group, so that projection operators for particular spin and spatial symmetries could also appear in 0. The O obtained in this way are generally referred to as configuration state functions (CSF s). 

Particle interchange symmetry is of course retained. The Kramers permutation operator is effectively a unit operator when applied to a charge density, for... 

For the case of Bose-Einstein statisties, the eigenfune-tions [ipmaiq ) are required to be symmetrie with respect to particle interchange. Thus, the eigenfunetions (pi iq )] must be symmetric. These symmetrie eigenfunctions can be written... 

This is essentially the quantum-mechanical generalization of Pauli s exclusion principle. The connection is easily made in the case N = 2, where the wavefunction may be written as a product of space and spin factors for if two electrons are put into the same orbital with the same spins (i.e. into the same spin-orbital) the wavefunction can only be symmetric (cf. IPl and above), in violation of the antisymmetry requirement. Two electrons cannot therefore (in an IPM description) occupy the same state or— in Pauli s statement—possess identical sets of quantum numbers. The generality of the principle, which applies for any number of electrons in any kind of system and even when interaction is admitted, will be discussed further in Chapter 3. For more than two electrons the wavefunction has no simple symmetry for interchange of space or spin variables separately, exchange of particles implies exchange of space and spin variables together and (1.2.27) applies to this case only. 

The wave fiinetion for a system of N identical particles is either symmetric or antisymmetric with respect to the interchange of any pair of the N particles. Elementary or eomposite particles with integral spins (s = 0, 1,2,. ..) possess symmetrie wave functions, while those with half-integral spins (s = 1. .)... 

Fig. 4.14 Symmetry of rotational levels of a homonuclear diatomic molecule. The letters s and a refer to the nuclear-interchange symmetry of the wave function with the nuclear-spin factor omitted. The signs + and - refer to the parity of the wave function with respect to inversion of all particles.

However, the symmetry of the situation can be restored if we interchange the words right and "left in the description of the experiment at the same time that we exchange each particle with its antiparticle. In the above experiment, this is equivalent to replacing the word clockwise with counterclockwise. When this is done, the positrons arc emitted in the downward direction, just as the electrons m the original experiment. The laws of nature are thus found to be invariant to the simultaneous application of charge conjugation and mirror inversion. 

Systems containing more than one identical particles are invariant under the interchange of these particles. The permutations form a symmetry group. If these particles have several degrees of freedom, the group theoretical analysis is essential to extract symmetry properties of the permissible physical states. Examples include Bose-Einstein, Fermi-Dirac, Maxwell-Boltzmann statistics, Pauli exclusion principle, etc. 

H is even with respect to the interchange of any particle index (double symmetry group for each pair). As a result of group theory we immediately derive that the corresponding eigenstates are either even or odd ... 

With two electrons and four available spin orbitals, 2 determinantal collective states may be then built. However, Pauli s exclusion principle coupled to the notion of particle indiscernibiUty contributes to reduce this number to 6. Let us label Xs,sz) the collective states X = U (ungerade) or X = G (gerade) refers to the symmetry of the orbital part with respect to the interchange of A) and S) S and describe the total spin configuration. We shall denote V,[Pg.236]

Ptj, the symmetry operation involving interchange of identical particles (nuclei or electrons). All particles axe either Bosons or Fermions, and the total wavefunction must, respectively, be even or odd upon interchange of any pair of identical particles. The total wavefunction of a homonuclear molecule, exclusive of the nuclear spin part, is classified s or a according to whether it is even or odd with respect to nuclear exchange. Since electrons are Fermions, the total molecular wavefunction must be odd with respect to permutation of any two electrons. This requirement is satisfied by the determinantal form of the electronic wavefunction (see Section 3.2.4). 

We can deliberately interchange particles (1) and (2). There is some special symmetry in the mathematical formulation. In the same way as done in Eq. (5.8), we can define the potential resulting from particle (1) at the place of particle (2) as... 

These transformations have the common characteristic of being able to transform the nuclear framework into one which is indistinguishable from the starting model they interchange identical framework particles. Under (a), (d), and (e) the identical nuclei would just interchange, and under (b), (c), and (f) the nuclei would remain unaffected. Transformations of this type are called point symmetry operations of the system. The K>tential is invariant under symmetry operations, but not under arbitrary coordinate transformations such as in Eq. 1.28. 

The requirement for symmetric or antisymmetric wave functions also applies to a system containing two or more identical composite particles. Consider, for example, an molecule. The nucleus has 8 protons and 8 neutrons. Each proton and each neutron has i = j and is a fermion. Therefore, interchange of the two nuclei interchanges 16 fermions and must multiply the molecular wave function by (—1) = 1. Thus the molecular wave function must be symmetric with respect to interchange of the nuclear coordinates. The requirement for symmetry or antisymmetry with respect to interchange of identical nuclei affects the degeneracy of molecular wave functions and leads to the symmetry number in the rotational partition function [see McQuarrie (2000), pp. 104-105]. 

When required, Equation (6.14) should be divided by a symmetry factor to account for the interchange of identical particles in symmetric molecules. While this expression may at first sight seem difficult to evaluate, it can actually be computed quite efficiently. The quantity N E, x) should be precalculated on a large grid of E, x) values which is used to construct an interpolant. The numerical integrations can then be carried out extremely rapidly using numerical quadrature. 

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