The two-electron density matrix

We now turn our attention to the two-electron density matrix. We begin by noting that the two-electron density-matrix elements (1.7.6) are not all independent because of the anticommutation 

The following elements are therefore zero in accordance with the Pauli principle  

To avoid these redundancies in our representation, we introduce the two-electron density matrix T with elements given by 

There are M(M — l)/2 rows and columns in this matrix with composite indices PQ such that P Q. The elements of T constitute a subset of the two-electron density elements (1.7.6) and differ from these by a reordering of the middle indices  

The reason for introducing this reordering is that it allows us to examine the two-electron density matrix by analogy with the discussion of the one-electron density in Section 1.7.1. Thus, as in the one-electron case, we note that the two-electron density matrix T is Hermitian 

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Integrating the A -electron density matrix over coordinates 3 to N generates the two-electron density matrix (2-RDM) ... 

The two-electron density matrix elements are given in similar fashion ... 

This is the well-known full-CI scheme. The two-electron density matrix is defined by the formula... 

The expressions Eqs. (2), (4) are completely general. To address the aspects important for the TMCs modelling, i.e. the energies of the corresponding electronic states, we notice that the statement that the motion of electrons is correlated can be given an exact sense only with use of the two-electron density matrix Eq. (4). Generally, it looks like [35] (with subscripts and variables notations w omitted for brevity) ... 

The attempts to construct an acceptable parameterization for TMCs are almost exclusively undertaken within the framework of the HFR MO LCAO paradigm. It is easy to understand that the nature of failures which accompany this direction of research as long as it exists lays precisely in the inadequate treatment of the cumulant of the two-electron density matrix by the HFR MO LCAO. 

A capacity of a theoretical method to reproduce such characteristics is intimately related to the grammatically) correct treatment of the cumulant of the two-electron density matrix. Let us assume that we want to fit some experimental data to the model... 

It turned out, however, that for the d-shells the recipe [92] of constructing the two-electron density matrix does not work for a major part of the atomic electronic terms of the transition metal ions [93]. Further studies revealed that constructs similar to [92] are, nevertheless, possible also for some other terms for which the name of the non-Roothaan terms [94,95] was coined not very conveniently (the point is that the Roothaan and non-Roothaan terms together do not exhaust the entire set of terms). The Roothaan and non-Roothaan terms together are those where it is possible... 

In the context of the EHCF construct described in the previous Section, the problem of semiempirical modelling of TMCs electronic structure is seen in a perspective somewhat different from that of the standard HFR MO LCAO-based setting. The EHCF provides a framework which implicitly contains the crucial element of the theory the block of the two-electron density matrix cumulant related to the d-shell. Instead of hardly systematic attempts to extend a parameterization to the transition metals it is now... 

We have already mentioned that the HFR lacks the cumulant of the two-electron density matrix. As we have shown above, it is indispensable for describing the mul-tiplet structure of central transition metal ion. The specific form of the wave function allowing for it will be used in the semiempirical context for constructing a method targeted at the transition metal complexes (TMCs). It will be described in Section 2.4.2. 

Intrageminal elements of the two-electron density matrix easily write through the amplitudes of the corresponding geminal ... 

The nuclear-electron terms in the operator on the right-hand side refer to the electronic coordinate Ti, which is excluded from the integration in eqn (6.78). The subscript T is not indicated on the coordinate appearing in p. The two-electron terms involve the coordinates Tj and f2 and integration of >j/ j/ over the coordinates of the remaining electronic coordinates and multiplication by the factor N N - l)/2, as indicated, yields the two-electron density matrix r (i i.< 2) (eqn (EL3)). Integration of the density in eqn (6.78) for the final electronic coordinate = r over the basin of the atom yields the basin virial T" (12) as indicated in eqn (6.79)... 

It is a fundamental fact of quantum mechanics, that a spin-independent Hamiltonian will have pure spin eigenstates. For approximate wave functions that do not fulfill this criterion, e.g. those obtained with various unrestricted methods, the expectation value of the square of the total spin angular momentum operator, (5 ), has been used as a measure of the degree of spin contamination. is obviously a two-electron operator and the evaluation of its expectation value thus requires knowledge of the two-electron density matrix. 

The alpha part of the density matrix is obtained in the same manner. The operation count for calculating the one-electron density matrix (11.8.72) is identical to the operation count of one direct Cl iteration for a one-electron operator. Likewise, a well-designed algorithm for the construction of the two-electron density matrix will have an operation count identical to the count of a direct Cl iteration for a two-electron operator. 

Also, the two-electron density matrix T is positive semidefinite since its elements are either trivially equal to zero or inner products of states in F(A7, N — 2). 

We recall that the diagonal elements of D correspond to expectation values of ON operators (1.7.9) and are interpreted as the occupation numbers of the spin orbitals. We now examine the diagonal elements of the two-electron density matrix ... 

For a state containing a single ON vector, the two-electron density matrix has a particularly simple diagonal structure with the following elements... 

For a general electronic state, containing more than one ON vector and providing a correlated treatment of the electronic system, the two-electron density matrix is in general not diagonal and cannot be generated directly from the one-electron density elements. As in the one-electron case (1.7.19), we may invoke the Schwarz inequality to establish an upper bound to the magnitude of... 

From Section 5.2, we recall that it is impossible to tell from its occupation number alone whether the orbital decreases or increases the probability of having the two electrons close to the same nucleus. Indeed, this information can be obtained only from an inspection of the two-electron density matrix or the wave function itself. The one-electron occupation numbers are nevertheless useful for analysing the electronic state, providing information about the importance of the individual otbitals for the one-electron density. 

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