Particle permutation

Here, 6 is the Dirac delta function, U is the potential energy function, and q represents the 3N coordinates. In this expression, the integral is performed over the entire configuration space - each coordinate runs over the volume of the simulation box, and the delta function selects only those configurations of energy S. The N term factors out the identical configurations which differ only by particle permutation. It is worth noting that the density of states is an implicit function of N and V,... 

For the degenerate ground state, the wave function (10) is relevant only for bosons (deuterons), for the two particles in the same states are indistinguishable. For fermions (protons), spins are correlated in such a way that the total wave function is antisymmetrical with respect to particle permutations, according to the Pauli principle. The spatial wave function can be rewritten as... 

It would appear that identical particle permutation groups are not of help in providing distinguishing symmetry labels on molecular energy levels as are the other groups we have considered. However, they do provide very useful restrictions on the way we can build up the complete molecular wavefimction from basis functions. Molecular wavefunctions are usually built up from basis functions that are products of electronic and nuclear parts. Each of these parts is further built up from products of separate uncoupled coordinate (or orbital) and spin basis functions. When we combine these separate functions, the final overall product states must conform to the permutation symmetry mles that we stated above. This leads to restrictions in the way that we can combine the uncoupled basis functions. 

The integrals described by Eq. (6) are invariant under particle permutations, which include not only permutations of the indices 1, 2, 3 but also changes in the coordinate origin, which correspond to permutations of the type (w2 U2, W3 M3, 2 3)- The symmetry... 

In the above, is an A-particle state vector satisfying the proper symmetry requirements with respect to particle permutation and Ej is the energy of state... 

The T1 transformation does not affect the particle rank of the Hamiltonian. Indeed, the only complication that occurs upon the transformation (13.7.20) is a loss of symmetry in the one- and two-electron integrals, with only the particle-permutation symmetry of the two-electron integrals retained as discussed in Section 13.7.4. Therefore, if we are prepared to work with integrals of reduced symmetry, the T1 transformation of the Hamiltonian (13.7.20) will simplify the subsequent manipulation of the coupled-cluster equations considerably, effectively reducing the complexity of the CCSD equations to that of the CCD equations. 

Let us consider now in more detail the behaviour of the eigenfunctions imder identical particle permutations. Let a particular irreducible representation of the symmetric group of the electronic coordinates be denoted as [A] and let the conjugate representation be denoted as [A]. For electrons (or any spin 1/2 particles) the representation of the symmetric group carried by the spin-eigenfimctions s,Ms,k must be one described by a no more than two-rowed Young diagram, that is [A] = [Ai,A2] where... 

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