Density matrix two-electron

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

J. M. Herbert, Reconstructive Approaches to One- and Two-Electron Density Matrix Theory, Ph.D. thesis, University of Wisconsin, Madison, WI, 2003. 

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... 

Eventually, quantities D can be considered as the Ath element of vector 5, so we can write the element of two-electron density matrix 0 as a scalar product of two vectors ... 

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... 

In the previous section we described existing HFR-based semiempirical methods and demonstrated their hybrid nature in a wide sense. We have also shown that for certain physical situations the semiempirical methods may become invalid due to the necessity to explicitly address nontrivial electronic correlations manifesting themselves either in numerous Slater determinants to be included in the consideration or in nonvanishing matrix elements of the cumulant of two-electron density matrix whose presence must be somehow reproduced in the calculation. 

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 ... 

In the tetrahedral methane molecule (its parameters then correspond to subscript 0 in eqs. (3.132), (3.133)), we notice that the 57X57XE matrix further simplifies as s nm = 1 and, therefore, simple analytical expressions become possible. Also, we notice that the FA approximation is adequate here as, for example, even very large elongation of one C-H bond by 0.1 A leads to changes of the bond geminal amplitudes u, v, and w not exceeding 0.003. The same applies to the expectation values of the pseudospin (f) operators representing the one- and two-electron density matrix elements. 

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. 

Determine all of the nonvanishing one- and two-electron density matrix elements... 

It is interesting to compare the possibilities and errors of different hybrid QM/MM schemes. The careful examination and comparison of link atom and LSCF techniques was performed in Ref. [128] using the CHARMM force field [114] and the AMI method [143] as a quantum chemical procedure. In the case of the link atom procedure two options were used QQ - the link atom does not interact with the MM subsystem and HQ - link atom interacts with all MM atoms. The main conclusion of this consideration is that the LSCF and the link atom schemes are of similar quality. The error in the proton affinity determination induced by these schemes is several kcal/mol. It is noteworthy that all the schemes work rather badly in description of conformational properties of n-butane. The large charge on the MM atoms in the proximity of the QM subsystem (especially on the boundary atom) cause significant errors in the proton affinity estimates for all methods (especially, in the case of the LSCF approach where the error can be of tens of kcal/mol). This is not surprising since the stability and transferability of intrabond one- and two-electron density matrix elements Eq. (19) is broken here. It proves that the simple electrostatic model is not well appropriate for these schemes and that a detailed analysis of the... 

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. 

The only two-electron density-matrix elements now needed are those with all four indices active. Likewise, a full MO transformation of the two-electron integrals is no longer necessary. Since... 

---