Ellinger Y., Defranceschi M. (eds.) Strategies and applications in quantum chemistry (Kluwer, 200

Since

where the symbol s denotes the set of spin variables, the replacement operators can be further generalised. Thus, in the operator

the sum of annihilators destroy q electrons which are in the state J and the sum of creators create the state I.

Therefore, in this basis one has:

2.2.2. The Hamiltonian Operator and the Energy

The spin free many-body Hamiltonian Operator can be written in compact form by employing the 2-RO

where

In this latter formula, the two electron repulsion integral is written following Mulliken convention and the one electron integrals are grouped in the matrix In this way, the one-electron terms of the Hamiltonian are grouped together with the two electron ones into a two electron matrix. Here, the matrix is used only in order to render a more compact formalism.

An element of the matrix representation of the Hamiltonian in the. N-electron space is therefore:

or equivalently

A particular case of this is the well known expression of the energy of a normalized state

2.2.3. The Holes Replacement Operators

An important operator, complementary to the RO, is the Holes Replacement Operator (HRO). The general form of this operator is:

The HRO’s are holes density operators and operate by first filling orbitals with electrons (i.e. they annihilate holes) and then removing electrons from orbitals (i.e. they create holes). These operators generate the Holes Reduced Density Matrix (HRDM) which in our notation takes the form:

I wish to stress that the meaning of the word Hole here is different and far more general than in Many Body Perturbation Theory. Indeed, no specific reference state is required in this definition and the difference between the RO’s and the HRO’s follows exclusively from the different order of the creator operators with respect to the annihilator operators in E and in respectively.

3.The Contraction Mapping in Matrix Form (MCM)

The MCM is at the basis of the two formalisms which are described in this work. Its general form for is:

which may correspond to simple Slater determinants, to eigenstates of the spin op-

stand for uniquely defined states depending on both space and spin variables.

Relation (13) allows us to contract any q – RDM and what is more, it also allows us to contract any q – HRDM by replacing the number N by the number (2K – N).

The derivation of the MCM is based on the important and well known relation

which must be used q – p times in order to obtain relation (13).

In many cases a simpler form of this mapping may be used. Thus, the RDM by itself, when it is not involved in matrix operations it can be contracted by using

of the electronic integrals (relation 8) one gets after some simple algebra

At first sight, this looks rather complicated, and in fact, evaluating this general relation is not trivial. On the other hand, the results are simple, closed form expressions. The derivation of the algorithms for different values of p have been described in detail previously [25,26,28]. Here we will just sketch the results obtained.

The main point about the SRH matrix is that its only non zero elements are those where the set of the i indices and the set of the j indices are equal or differ at the most in two indices.

The general form of the three different classes of the matrix elements are [28]:

•The set of i indices is equal to the set of j indices (although any ordering of the indices is allowed).

•An index of the set i differs from that of an index of the set j, say (although any ordering of the indices is allowed).

A very convenient feature of this formalism is that the values of the A and B coefficients only depend on three numbers: the number of electrons (N), the number of orbitals of the basis (K), and the Spin quantum number (S).

The different orderings of the matrix element indices ( superscript in relations 19,20,21) give rise to different values of the A and B coefficients. Finally, the value of these coefficients also depends on whether one or more of the matrix element indices are repeated (appears twice).

For p = 1,2,3 the values of these coefficients have been explicitly obtained; and for large p, a set of diagramatic rules have been reported [28] in order to determine the coefficients in each case.

To calculate a p – SRH once the electronic integrals are known, is therefore a very simple and rapid task.

4.2. SOME PROPERTIES OF THE p – SRH MATRICES

Let us now derive from the fundamental relation (17), some properties of the p–SRH matrices. Taking the trace of this matrix we find that:

Since the trace of is an invariant of the system, relation (22) establishes that the trace of the p – SRH matrix is also an invariant of the system.

The eigen – values of the p – SRH matrices have the form:

where the symbol I denotes the corresponding p – SRH eigen-state. This relation shows up the average character of the p – SRH eigen-values. However, until now we have not been able to find the relation linking the values with the energy observables of the N-electron system.

In what follows we will focus our attention on the p = 2 and p = 1 cases which are the most useful ones. The eigen-vectors of the p – SRH for these values of p are geminals and orbitals respectively. In order to simplify the interpretation of the geminals and to reduce the size of the matrices involved in the calculations it is convenient to apply to the 2 – SRH a linear transformation which factorizes this matrix into two blocks according to the representations of the Symmetric Group of Permutations. In this way the eigen-geminals of both blocks have a clear physical meaning since those of the symmetric block describe the space part of singlet pair states and those of the antisymmetric part describe that of triplet states.

Although the physical meaning of the 2- and 1- electron eigen-states of the 2 – SRH has not been established rigorously we interpret them as describing states of two/one electrons which in average can be considered independent. This interpretation was justified [29] through the analysis of the asymptotic form of the 2 – SRH in the coordinate representation for . . In this analysis, Karwowski et al. showed that the eigen-geminals of the asymptotic 2 – SRH described isolated pairs of electrons. Another important feature of the SRH formalism is that it can be generalized by

4.3.APPLICATIONS OF THE SRH THEORY

An outline of the main applications of the SHR theory is presented in this section. In 6.1 the advantage of using the eigen-vectors the 1-SRH as a basis in CI calculations is discussed. The main application until now of this theory is summarized in the following subsection. Then in 6.3 other applications which have been less developed are mentioned.

4.3.1. Performance of the eigen-vectors of the 1-SRH as a basis in CI calculations

A set of calculations [31] was recently carried out in order to compare the performance of the 1 – S R H eigen-orbitals with other known and easy to get basis sets when doing

CI calculations. Some of the results reported in (Ref. 31) for the ground state of the

Water molecule (minimal basis set) are given in Table 1.

These results show that convergence is favored when the basis used are either the

1–SRH or the Absar and Coleman [32,33] Reduced Hamiltonian (RH) eigen-orbitals. The other two basis used were the SCF orbitals and the core Hamiltonian eigen- orbitals (Core H)

4.3.2. The 2 – SRH independent pair model

or the 2 – RDM of a particular eigen-state of the system cannot be filtered out from the contributions of the other eigen-states. To deal with this difficulty and based on the physical interpretation given above of the eigen-geminals and eigen-orbitals of the SRH, the following working hypothesis was proposed [34] in 1985:

“Let us substitute the study of the N-electron system by that of an ensemble of

non interacting pairs of electrons, described by the eigen-geminals of the 2 – SRH” .

should be fulfilled.

Now, when a pair is described by a state I, its Density Matrix ( D M ) is

and by applying this working hypothesis, the 2 – RDM corresponding to the N- electron eigen-state can be approximated by

It should be noted that this relation is formally identical to the spectral resolution of the 2 – RDM. That is, in this model, all happens as if the eigen-vectors of the

This is called [35] Mixed Pair State approximation ( M P S ) . The name, which probably is not the best one, refers to the fact that the barring exceptions, has a value smaller than one, which means that the electron pair is not in the pure state I.

value are thus selected. Now, the eigen-vector whose highest coefficient (in absolute value) is, is allocated the occupation number:

This approximation was denoted initially by the acronym IQG [34] and later on by IP (Independent Pairs) [35]. It gave satisfactory results in the study of the Beryllium atom and of its isoelectronic series as well as in the BeH system. The drawback of this approximation is that when the eigen-vectors are diffuse, i.e. there is more than one dominant two electron configuration per eigen-vector, the determination of the corresponding is ambiguous. In order to avoid this problem the MPS approximation, which does not have this drawback, was proposed.

A detailed discussion of these and other variants was given in (Ref.35). Attention must be called to the fact that these methods are not variational which causes the energies obtained with them to be lower than those obtained with the FCI method. The counterpart to this deffect is that excited states, open-shell systems, and radicals, can be calculated with as much ease as the ground state and closed-shell systems. Also, the size of the calculation is determined solely by the size of the Hilbert subspace chosen and does not depend in principle on the number of electrons since all happens as if only two electrons were considered.

While from the energy point of view, the correlation effects seem to be overestimated, the RDM’s are particularly satisfactory. Thus, when comparing the 2-RDM’s obtained with these approximations for the ground state of the Beryllium atom with the corresponding FCI one, the standard deviations are: 0.00208236 and 0.00208338 for the MPS and IP respectively. For this state, which has a dominant four electron configuration of the type, the more important errors, which nevertheless can be considered small, are given in table 2.

In table 2, the elements which are equal due to symmetry have been omitted.

It can be seen that, even for this case where no ambiguity exists in applying the IP approximation, the results are slightly better with the MPS variant which seems to favor this latter approximation.

The data given in table 2 have been obtained using a double zeta basis [36] and transforming it to the basis which diagonalizes the 1-SRH matrix which as we saw in subsection 5.3 is a basis both good and simple to determine.

Analysis of the different terms of the energy

Another interesting analysis of this method can be carried out by applying a partitioning of the energy [37] which shows up the role played by the 1–RDM, the

1-HRDM and the 2 - H R D M .

Thus, it has been shown that the energy can be partitioned as

where:

The only operation used for obtaining this partitioning is the anticommutation rule of the fermion operators. Note, that by adding the F and G terms one falls into the unitarily invariant Absar and Coleman partitioning [32,33] which was obtained by using a Group theoretical approach.

The interesting point about relation (30) is that each of the terms has a clear physical interpretation. Thus the term involving F is a sum (for N electrons) of generalised Hartree-Fock energy levels and clearly is a one-electron term. The term involving G gives the sum (for N electrons) of the energy of an electron in the average field

In my opinion this partitioning is particularly suitable for analysing electronic correlation effects. To illustrate this point a set of calculations for the three lowest singlet states of the Beryllium atom are reported in table 3 (in all cases

Hartrees).

Let us start the analysis of the results given in table 3 by commenting on the FCI one. It is interesting to note that the one-electron term energy becomes lower as the degree of excitation of the state increases. I find this result rather unexpected, since in principle, the low energy orbitals will become more empty, At any rate the stabilization caused by the term is more than counter-balanced by a large

increase of the positive terms of the energy, in particular by

The most stricking features, when comparing the FCI results with the IP and MPS

•Finally in the third state the two approximations give very similar energy values, both with higher energy than the FCI one. In each approximation, the error of the different terms compensate each other to a certain extent.