Two-particle operators of physical quantities

Since all the two-electron interaction operators are invariant under rotations in the space of total momentum J of two particles, we shall consider only the scalar two-electron operators 

If the specific tensorial structure (14.58) of a two-electron operator is known, then we can obtain its representation in terms of the product of operators (14.30) acting in the space of states of one shell. In fact, if we substitute into the two-electron matrix element which enters into (13.23), the operator (14.58) in the form 

Let us now put (14.59) into the general formula (13.23) and rearrange the creation and annihilation operators so that when the second-quantization operators are placed side by side the rank projections enter into the same Clebsch-Gordan coefficient. A summation over the projections gives 

This expression is the second-quantization form of an arbitrary two-electron operator with tensorial structure of the kind (14.57). Examination of this formula enables us to work out expressions for operators that correspond to specific physical quantities. Many of these operators possess a complex tensorial structure, and their two-electron submatrix elements have a rather cumbersome form [14]. Therefore, by way of example, we shall consider here only the most important of the two-electron operators - the operator of the energy of electrostatic interaction of electrons (the last term in (1.15)). If we take into account the tensorial structure of that operator and put its submatrix element into (14.61), we arrive at 

Two-electron operators can also be represented in terms of tensorial products of operators (14.40) and (14.42). To this end, we expand the 

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Among all the possible two-particle operators for physical quantities for lN configuration we have only considered in detail the electrostatic interaction operator for electrons here too we shall confine ourselves to the examination of this operator. The explicit form of the two-electron matrix elements of the electrostatic interaction operator for electrons (the... 

Operators corresponding to physical quantities can also be expanded in terms of irreducible tensors in the quasispin space of each individual shell. To this end, it is sufficient to go over to tensors (17.43) and next to provide their direct product in the quasispin space of individual shells. This procedure can conveniently be carried out for a representation of operators such that the orbital and spin ranks of all the one-shell tensors are coupled directly. Here we shall provide the final result for the two-particle operator of general form (14.57)... 

Operators corresponding to physical quantities, in second-quantization representation, are written in a very simple form. In the quantum mechanics of identical particles we normally have to deal with two types of operators symmetric in the coordinates of all particles. The first type includes N-particle operators that are the sum of one-particle operators. An example of such an operator is the Hamiltonian of a system of noninteracting electrons (e.g. the first two terms in (1.15)). The second type are iV-particle operators that are the sum of two-particle operators (e.g. the energy operator for the electrostatic interaction of electrons - the last term in (1.15)). In conventional representations these operators are... 

In Chapter 14 we have shown how an expansion in terms of irreducible tensors in the spaces of orbital and spin angular momenta for one shell can be obtained for the operators corresponding to physical quantities. The tensors introduced above enable the terms of a similar expansion to be also defined in the space of a two-shell configuration. So, for the one-particle operator of the most general tensorial structure (14.51) we find, instead of (14.52),... 

For operators corresponding to physical quantities, we can also obtain an expansion in terms of irreducible tensors in quasispin space. Specifically, for two-particle operators (13.23) that are scalars with respect to the total momentum J... 

The above relationships describe the behaviour of tensors (23.52) and their tensorial products. A further task is to express the operators corresponding to physical quantities in terms of these tensors. Specifically, two-particle operator (23.27) is expanded by going over to the second-quantization representation and coupling the quasispin ranks... 

Similar data for the dN configuration may be found in [91]. Applying such tables we can directly find the matrix elements of one-particle operators corresponding to physical quantities. The matrix elements of two-particle... 

With this brief discussion of the three basic steps (a)-(c) from Subsecfion 4.1, we have arrived at our original destination to represent all physical quantities of interest as sum (of products) of one- and two-parficle amplifudes. In practice, each of these expansions are often lengthy and the complexity of fhese expansions increases rapidly if the number of particle and holes is increased in the valence shells. The latter can be seen easily from the fact that each (valence-shell) particle of hole introduces an additional creation or annihilation operator into the operator strings a)a and (a ), respectively. In contrast to other, e.g. multiconfigurational, expansions of the wave functions, however, the explicit form of the approximate states IV a) cannot be derived so easily in MBPT or the CCA. For this reason also, a straightforward and simply handling of the perturbation expansions decides how successfully the theory can be applied to open-shell atoms and molecules in the future. 

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