Permutation of electrons

Let us notice that antisymmetrization involves summations over permutations of electron coordinates, whereas the coupling of momenta and their projections involves summations over functions with different values of the magnetic quantum numbers. Since these are independent finite summations, the order in which they are performed is interchangeable we can, in principle, either antisymmetrize, first, and couple, second, or vice versa. In practice, the Pauli principle restricts the possible values of the projections and as a result of these complications it proves to be more convenient to couple first, and antisymmetrize second, with the antisymmetrization not always being done by means of coordinate permutations. 

A major difficulty is the problem of electron indistinguishability. The natural choice of the unperturbed Hamiltonian is the sum of the Hamiltonians for the separated molecules, but this is not symmetric with respect to permutations of electrons on one molecule with electrons on the other. The order of a term in the perturbation expansion then becomes undefined,129 and although this difficulty can be overcome,130 the application to large systems is probably not in sight. 

The three-body exchange nonadditivity, e ch, is conveniently categorized as related to two general mechanisms depicted in Fig.8. The first, where permutation of electrons encompasses all three monomers and the second, where only two are involved. They are referred to as... 

The (iV ) factor included in these expressions ensures that the determinants are normalized when the orbitals are normalized. Eq. (41) gives an explicit representation of the antisymmetrizer. This summation is over the N permutations of electron coordinates for a fixed orbital order, or equivalently, over the permutations of spin-orbital labels for a fixed order of electrons. The exponent Pp is the number of interchanges required to bring a particular permuted order of electron coordinates, or of spin-orbital labels, back to the original order. Different expansion terms are generated when different spin orbitals are employed in the determinant. For convenience, we will choose this spin-orbital basis to be the direct product of the set of n spatial orbitals and the set of spin factors a, / . A particular spin orbital of this form may be written as where r (= 1 to n) labels the spatial orbital and spin factor, or simply as (j), where the combined index r (= 1 to 2n) labels both the spatial and spin components. The notation used will be clear from the context. 

To put any such search in context it is necessary to consider the behaviour of both the internal coordinates and the Euler angles under the permutation of of identical nuclei. The permutation of electrons yields the standard representation, which need not be further considered. 

Cases of three or more electrons were very difficult to treat by the above methods. For instance, for three-electron systems, it is required to have six terms in the expansion of each basis function in order to comply with the antisymmetry criterion, and each term must have factors containing ri2, ri3, r23, etc., if we want to accelerate the convergence. There is, indeed, a real problem with the size of each trial wave function. A symmetrical wavefunction requires that the trial basis set for helium contain two terms to guarantee the permutation of electrons. For an N-electron system, this number grows as N . For a ten-electron system like water, it would be required that each basis set member have more than 3 million terms, and this is in addition to the dependence on 3N variables of each of the terms. These conditions make the Schrodinger equation intractable for systems of even a few electrons. Just the bookkeeping of these terms is practically impossible. 

It is essential to note that the labels on the variables do not refer to the electrons, but rather to points in space, which are chosen by an observer and are physically distinguishable. The Pauli principle has been respected from the start by insisting that I I p be invariant against any permutation of electron labels . 

The probability distribution of the electrons cannot possibly depend on the way that we have labelled them of course the same must be true for any possible permutation of electronic coordinates. In fact, if P is any one of the n permutations of the n electronic coordinates then... 

This follows since all the MOs are normalized. All matrix elements involving a permutation operator gives zero. Consider for example the permutation of electrons 1 and 2. 

Here P stands for all the sequences of permutations of electron labels that lead to different products (i.e., P is a permutation operator), and p is the number of pairwise permutations in a given sequence. For 1/74 there are 4 sequences P, the simplest being no permutations (hence, p = 0) which produces the leading term. Then there are single permutations, such as Pi,2 (with p = 1), which produces the term —(/>i(2) i(l)(/ 2(3) 2(4). There are also double permutations, etc. According to Eq. (A7-3), any term differing from the leading term by an odd number of... 

Terms also appear in corresponding to more than a single permutation of electron indices. 

The clamped nuclei Hamiltonian is invariant under the permutation of electrons so that its eigenfunctions must be basis functions for irreducible representations (irreps) of the symmetric group of degree N. However, because of the Pauli principle, not all irreducible representations of this group are allowed. 

Perhaps the most commonly used approximation in quantum chemistry is the Haitree-Fock method in which the wavefunction consists of an anti-symmetric product of one-electron functions, to take account of the permutation of electrons as dictated by the Pauli exclusion principle. It is assumed that each electron moves in the average field due to the nucleus and all the other electrons in the system. For the helium atom the required wavefunction takes the form of the determinant of a 2 x 2 matrix,... 

Since, according to (7.4.10), permutations of electron labels are equivalent to inverse permutations of orbitals, it is not surprising that there is a one-to-one correspondence between the functions we need and the standard Young tableaux. If we adopt a similar convention, with orbital indices increasing along rows and down columns, we obtain standard Weyl tableaux... 

The number of rows fs is given by (4.2.2), while the number of columns D m, N, S) is given by (7.6.18). Under a unitary transformation (induced by an orbital basis change) the functions in each row are mixed among themselves and carry an irrep of U(m) while under a permutation of electrons (or orbital indices) the functions in each column are mixed among themselves and carry an irrep Dj of S. The number of functions in the array may be enormous but the classification is simple. 

Unlike the full Hamiltonian H, its zero-order part Ho, as well as the perturbation V, does not possess full symmetry with respect to the permutations of electrons. As a consequence, the zero-order wave function po is not completely antisymmetric - some electrons are assigned to one monomer, the... 

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