Interchange operator

When the system is made up of identical particles (e.g. electrons in a molecule) the Hamiltonian must be symmetrical with respect to any interchange of the space and spin coordinates of the particles. Thus an interchange operator P that permutes the variables qi and (denoting space and spin coordinates) of particles i and j commutes with the Hamiltonian, [.Pij, H] = 0. Since two successive interchanges of and qj return the particles to the initial configuration, it follows that P = /, and the eigenvalues of are e = 1. The wave functions corresponding to e = 1 are such that... 

It is important to note that except for N = 2 the Nl permutations P do not commute among themselves. The reason is that the interchange operators Pij and Ptk (k j) do not commute. The eigenfunctions ip(qx,... < jv) are therefore not in general eigenfunctions of all the N permutation operators P. However, there are two exceptional states which are eigenstates of H and the N permutation operators The totally symmetric state ifisiqi, <7/v) satisfies, for all P, the operations... 

We must now investigate the effect of the binary interchange operators, P/y on the (j)i functions. We suppress the spin-label superscript for these considerations. It is straightforward to determine... 

Second, suitable solutions to the Schrodinger equation for electrons must have appropriate permutational symmetry. That is, an interchange of the space-spin coordinates Xj and Xj must not alter the probability density For this to be true, the interchange operator has to have the effect, = P. For... 

The tliree protons in PH are identical aud indistinguishable. Therefore the molecular Hamiltonian will conmuite with any operation that pemuites them, where such a pemiutation interchanges the space and spin coordinates of the protons. Although this is a rather obvious syimnetry, and a proof is hardly necessary, it can be proved by fomial algebra as done in chapter 6 of [1]. 

How many distinct ways of pemiuting the tliree protons are there For example, we can interchange protons 1 and 2. The corresponding syimnetry operation is denoted (12) (pronounced one-two ) and it is to be understood quite literally protons 1 and 2 mterchange their positions in space. There are obviously... 

This procedure is then repeated after each time step. Comparison with Eq. (2) shows that the result is the velocity Verlet integrator and we have thus derived it from a split-operator technique which is not the way that it was originally derived. A simple interchange of the Ly and L2 operators yields an entirely equivalent integrator. 

Redundant, isomorphic structures have to be eliminated by the computer before it produces a result. The determination of whether structures are isomorphic or not stems from a mathematical operation called permutation the structures are isomorphic if they can be interconverted by permutation (Eq. (6) see Section 2.8.7). The permutation P3 is identical to P2 if a mathematical operation (P ) is applied. This procedure is described in the example using atom 4 of P3 (compare Figure 2-40, third line). In permutation P3 atom 4 takes the place of atom 5 of the reference structure but place 5 in P2. To replace atom 4 in P2 at position 5, both have to be interchanged, which is expressed by writing the number 4 at the position of 5 in P. Applying this to all the other substituents, the result is a new permutation P which is identical to P]. 

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