Operator of averaged electrostatic interaction

It is to be stressed that, although the two-electron submatrix elements in (14.63) and (14.65) are defined relative to non-antisymmetric wave functions, some constraints on the possible values of orbital and spin momenta of the two particles are imposed in an implicit form by second-quantization operators. Really, tensorial products (14.40) and (14.42), when the sum of ranks is odd, are zero. Thus, the appropriate terms in (14.63) and (14.65) then also vanish. 

The operator of the energy of electrostatic interaction of electrons in (14.65) is represented as a sum of second-quantization operators, and the appropriate submatrix element of each term is proportional to the energy of electrostatic interaction of a pair of equivalent electrons with orbital Lu and spin S12 angular momenta. The values of these submatrix elements are different for different pairing states, since, as follows from (14.66), the two-electron submatrix elements concerned are explicitly dependent on L12, and, hence, implicitly - on S12 (sum L12 + S12 is even). It is in this way that, in the second-quantization representation for the lN configuration, the dependence of the energy of electrostatic interaction on the angles between the particles shows up. This dependence violates the central field approximation. 

The second-quantization counterpart of this approach is the replacement (for the lN configuration) of operator (14.65) by some effective operator, whose two-particle submatrix elements are independent of characteristics L12, S12 of the pairing state of electrons. To this end, we introduce the submatrix element averaged over the number of various antisymmetric pairing states in shell, equal to (4/ + 2)(4/ + l)/2  

The first factor under the summation sign takes care of the antisymmetry of the two-electron states of the shell. The submatrix elements are summed in accordance with the statistical weights of these states. Using (14.66), (6.25), (6.26) and (6.18), we sum the right side of (14.67) in the explicit form 

Since the right side of this expression is independent of the values of parameters Ln, S12, the summation over these parameters in the effective operator 

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Reasoning along the same lines, we also find the averaged operator of exchange electrostatic interaction between the shells... 

We can also introduce another effective operator, one that explicitly includes the exchange interaction of electrons. For this purpose, we shall have, in averaging, to take into account the dependence of the electrostatic interaction of two particles on their total spin. For a pair of equivalent electrons the number of various singlet states (S12 = 0) is given by... 

These operators can be averaged in the same manner as in Chapter 14 where we have introduced the average operator of electrostatic interaction of electrons in a shell. The main departure of the case at hand is that the Pauli exclusion principle, owing to the fact that electrons from different shells are not equivalent, imposes constraints neither on the pertinent two-particle matrix elements nor on the number of possible pairing states, which equals (4/i + 2)(4/2 + 2). The averaged submatrix element of direct interaction between the shells will then be... 

Let us emphasize that in single-configurational approach the terms of the Hamiltonian describing kinetic and potential energies of the electrons as well as one-electron relativistic corrections, contribute only to average energy and, therefore, are not contained in, which in the non-relativistic approximation consists only of the operators of electrostatic interaction e and the one-electron part of the spin-orbit interaction so, i.e. 

The terms included in equation 2, are respectively the kinetic operator, the attraction of a single electron with all nuclei centered in all cells, the averaged electrostatic potential of all electrons and the averaged exchange interaction. 

To complete the averaging operation in Equation 5-11, the radial dependence of c must be known. In the most general case, this expression should account for long-range (e.g., electrostatic) and short-range (e.g., steric) interactions between the pore walls and the particle [9]. Here, as in previous analyses [12], we assume that the average concentration in the pore, (c), is only a function of z, so that Equation 5-11 reduces to... 

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