Electron density matrix elements

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

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 electronic density matrix elements phcld connecting the two orbitals c and d in cells h and /, respectively, are calculated by summation over all occupied states (k,n) in the first Brillouin zone (BZ) ... 

This equation represents the general solution of the non-orthogonality problem , since the construction of the matrix G " (A) does not involve N operations. All the other density matrices needed in the evaluation of the matrix of the Hamiltonian (36) may be derived similarly. Thus, for example, the required one-electron density matrix elements are given by... 

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... 

Hyperconjugation of alkyl substituents in the allyl system leads to charge stabilization, to a reduction of the r-eiectron densities of the trigonal carbons attached to the alkyl groups, and to a reduction of 7r-electron density matrix elements. Therefore, hyperconjugation results in decreased 1,3 r-overlap effects. Table 4 lists MMP2 allyl 1,3 r-overlap energies. 

Let us examine the one-electron density-matrix elements of the CID wave function ... 

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... 

But since, by (5.4.20), the expectation values on the right-hand side are simply 1- and 2-electron density-matrix elements, we obtain at once (treating in the same way)... 

Clearly, all information that is required about the wave function (1.7.2) for the evaluation of expectation values (1.7.4) is embodied in the quantities (1.7.5) and (1.7.6) called the one- and two-electron density-matrix elements, respectively. Overbars are used for the spin-orbital densities to distinguish these from those that will be introduced in Chapter 2 for the orbital basis. Since the density elements play such an important role in electronic-structure theory, it is appropriate here to examine their properties within the framework of second quantization. 

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... 

In the state (5E.2.3), there is only one occupied orbital. The nonzero one-electron density-matrix elements must therefore have p — q=l ... 

The evaluation of the elements of the Fock matrix is now trivial since, for a closed-shell state, the one-electron density-matrix elements are given by... 

When the generalized Fock matrix is evaluated from (10.8.27), (10.8.30) and (10.8.32), only two-electron density-matrix elements with all four indices active are used. If we had used (10.8.23) directly, we would instead have had to reference density elements with indices labelling the full set of occupied MOs. From the definition of the inactive and active Fock matrices (10.8.28) and (10.8.29), we see that these matrices are symmetric - unlike the generalized Fock matrix (10.8.23), which is symmetric only at stationary points. In Section 10.8.5, we shall consider the cost of the evaluation of the Fock matrices in more detail. 

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