Ellinger Y., Defranceschi M. (eds.) Strategies and applications in quantum chemistry (Kluwer, 200
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C. VALDEMORO |
These results show that at least for the ground state, the correlation effects described
by the term |
are overestimated in the IP and MPS calculations. |
4.3.3. Other applications
Before concluding this section, it must be pointed out that there are other fields of
application of the SRH formalism. Thus, of the statistical theory of spectra [30,38].
Karwowski et al. have used it in the study Also, the techniques used in developing the
p-SRH algorithms have proven to be very useful in other areas such as the nuclear shell theory [39,40].
We think that the physical meaning of the SRH matrices is not yet fully understood and can therefore be considered an open field of research. It is to be expected that a more complete understanding of the SRH matrices will lead to new applications of this formalism.
5. The Contracted Schrödinge r Equation
In this section I will outline a new line of research recently initiated by our group. It must be emphasized that the only points in common with the SRH formalism previously described are that no call is made upon the N-electron WF of the electronic system and that its basic formal tools are also the MCM and the RDM's.
5.1.THE NAKATSUJI AND COHEN-FRISHBERG EQUATION
The integration with respect to N – q electron variables of the Schrödinger equation was reported simultaneously by H. Nakatsuji [41] and by L. Cohen and C. Frishberg [42] in 1976. The form of this equation (NCF) for q = 2 is:
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67 |
where
is the 2-RDM written in first quantization language. The symbols |
denote |
|||
the Hamiltonians of two and three electrons respectively and |
|
is the two electron |
||
repulsion operator. |
|
|
|
|
Since this integro-differential equation depends not only on |
but also, through the |
|||
two integral terms, on |
and |
it is indeterminate [43]. |
|
|
An important property of the NCF equation is that in it the variational principle is taken implicitly into account [42,44].
5.2. ORBITAL REPRESENTATION OF THE CONTRACTED SCHRÖDINGER
EQUATION (CSchE)
The matrix form in a spin-geminal representation (CSchE) of equation (34) was obtained [18] in 1985 by applying the MCM.
The interest of contracting the matrix form of the Schrödinger equation by employing the MCM, is that the resulting equation is easy to handle since only matrix operations are involved in it. Thus, when the MCM is employed up to the two electron space, the geminal representation of the CSchE has the form [35]:
where the symbols have the same meaning as in the preceding sections. It must be pointed out, that the contraction can also be carried out, up to the first order and the result is:
5.3.ITERATIVE SOLUTION OF THE CSchE
It was suggested [35,45] that the indeterminacy of the CSchE could be removed by replacing in it the 3- and the 4-RDM’s by their corresponding approximations evaluated within the SRH formalism. After this replacement is performed, the matrix equation can be solved with the help of relation (10) and
as auxiliary conditions.
Recently, a more powerful approach has been initiated. The different steps involved in the procedure just proposed for solving the CSchE are:
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C. VALDEMORO |
•From an initial 2-RDM the corresponding 3- and 4- order RDM’s are approximated by using a method which will be described in the following section.
•Then, all the approximated RDM’s are replaced in the r.h.s. of equation (36) so
that its three terms are added into a matrix, say |
and relation (36) becomes: |
• By taking the trace of both sides of equation (39) one obtains since
• The following step is to divide by which gives a new 2-RDM from which the procedure can start again.
All these steps are built into an iterative procedure whose success pivots on the approximation of the higher order RDM’s in terms of the 2-RDM. This important part of the method will be addressed in the next section.
5.4.APPROXIMATING AN RDM IN TERMS OF THE LOWER ORDER ONES
As has been mentioned, the iterative procedure for solving the 2-CSchE will only work if sufficiently precise approximations of the 3- and 4-order RDM’s in terms of the 2-RDM can be obtained. Since the method is based on the N-representabili- ty relations, the subsection 8.1 is dedicated to discuss these fundamental equations. Then in 8.2 the method will be outlined and some examples will be given.
5.4.1. The N-representability conditions
The basic relations for studying the properties of the RDM’s are the anticommutation/commutation relations of groups of fermion operators since their expectation values give a set of N-representability conditions of the RDM’s. Thus,
• The first order condition
From the fundamental rule of anticommutation of an annihilator with a creator operator it follows, in our orbital representation, that:
Since both the RDM’s and the HRDM’s are positive matrices, this relation says that the eigen-value of the 1 - RDM, must be which is the well known ensemble N-representability condition for the 1-RDM [10] represented in an orbital basis (in a spin-orbital representation the upper bound would be 1 instead of 2).
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69 |
• The second order condition
From the commutation of two annihilator with two creator operators follows the also well known Q-condition [9] for the 2 -RDM. In our notation, this condition takes the form:
and replacing the Krönecker deltas by their value according to relation (41) one finds [46]:
Note, that in this last relation, the part involving HRDM’s and that involving RDM’s have the same structure.
• General N-order condition
The aim of the following discussion under this heading is not to describs the formalism but merely to outline the ideas on which the method for approximating a p-RDM from the q-RDM’s with q < p is based. Nevertheless, in order to avoid using vague or imprecise arguments the essential theoretical background supporting the leading ideas must also be included here. The reader interested in going beyond this sketchy discussion is referred to a recent paper [47] where all the details are reported.
The result of commuting/anticommuting (for N even/odd) N annihilator operators with N creator operators is:
where the symbols are N-electron configurations. This relation is very elegant and compact but the following, in the orbital representation (obtained by inference [47]), is more practical for our purpose:
where:
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– CN represents the classes of the Symmetric Group of Permutations S N
– are the permutations (of the indices of the annihilator operators) belonging to a particular class
–represents the parity of the permutations belonging to class C
– The symbol is given by:
– The symbol describes a sum of terms. Each of these terms is a
product of (N - i) Krönecker deltas with a i-RDM element. Now, the terms whose addition is represented by a G symbol are those where the
indices are ordered according to the |
permutation of class |
For instance, for N = 3 and |
the G symbols are: |
Relations (44,45) describe the general form of the N-order condition; However, some terms must be eliminated from relation (45) because they do not occur when the anticommutation/commutation operations are carried out explicitly.
We call these terms spin – forbidden because in all of them the spin correspondence which should exist between the creator and the annihilators forming the p-RO (which generates the p-RDM) is not maintained. These spin-forbidden terms are those having a transposition of at least two indices in their p-RDM. For instance:
which is the third term of |
is spin-forbidden and must be eliminated. |
An equivalent N-order equation having the same structure for the particle part and for the holes part (in a similar way as in (43)), may also be inferred. This equation has the form:
where all the symbols have the same meaning as in relation (45) except for
. This new symbol, like also describes a sum of terms. Each of these
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71 |
terms is a product of (N – i) 1-RDM elements with an i-RDM element, (i.e. the 1-RDM plays a similar role to the Krönecker deltas). Thus, for instance:
is spin-forbidden and must be eliminated.
The inference process leading to equations (45) and (49) was carried out with the help of a set of graphs specially suited for operating with RO’s. This graphical method has been described in several recent publications [26,27,47] and would excessively lengthen this paper; therefore, I would like to mention its usefulness without elaborating.
The interest of relation (49) lies in that the holes and the particle parts of the equation, have the same structure.
Equations (45) and (49) stress the direct connexion existing between the elements and classes of the Symmetric Group of Permutations and the terms derived by commuting/anticommuting groups of fermion operators after summing with respect to the spin variables.
Two important facts concerning the set of relations given above are that all the
N-representability relations known to us, can be derived from (45) (or (44) in a spin- space representation) by varying the value of N and relation (49) condenses them all.
It is interesting to note that relation (45) guaranties that the N-electron state of reference (whose superindex has been omitted) is antisymmetric since the RO's involved on the l.h.s of these equations operate on N-electron states. Now, by contracting this equation to a p-electron space an N-representable equation is obtained (by construction). In view of this, I hoped that a relation obtained in such a way would be a sufficient N-representability condition or at least more stringent than the (45) equation for N = p. Now the contraction of equation (45) gives exactly the same equation where N has been replaced by p. On the other hand the contraction of equation (49) gives a very complicated equation where partial traces of RDM’s of orders (p + l),(p + 2),....(N – 1) appear. This equation although difficult to analyse may prove to be useful and it is being studied at the moment.
5.4.2. Approximation proposed
The method for approximating an RDM in terms of the lower order ones is based on equation (49). The working hypothesis which has been put forward [46] is:
”Let us assume that Holes and Particles are totally different objects. If this assumption were true, equation (49) could be exactly decoupled into two equations, one involving RDM’s of different orders and the other, of similar structure, linking HRDM’s of different orders”.
This hypothesis, given that Holes and Particles are related through the N-repre- sentability conditions, is not true. On the other hand, by taking into account several
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C. VALDEMORO |
auxiliary conditions, very good approximations have recently been obtained by using this method.
In order to see an example of how these suplementary conditions are imposed let us consider the approximation of a 3-RDM in terms of the 2 - RDM . Since an RDM cannot have any negative diagonal element when such an element occurs it is put equal to zero. Until now the negative diagonal elements found were of the type
where M represents the approximated 3-RDM. By comparing the approximated
matrix with the exact one it was apparent that the deffect in |
was compensated |
|
very closely by an excess in the element |
therefore this element was corrected |
|
in accordance. After this correction was performed the new elements (M´) had the value:
It has been shown [48] that the related off-diagonal elements obey definite symmetry relations which must be maintained after the corrections indicated above have been applied.
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73 |
These symmetry relations impose that the following corrections should also be introduced:
• |
must also be made equal to zero. |
• The value |
should be subtracted from |
Under these conditions, the 3-RDM of the three lower states of the Beryllium atom and the two lower ones of the Water molecule were determined [48] by taking as initial data the 2-RDM obtained in a Full Configuration Interaction. In Table 4 some of these results are given and as can be seen they are very satisfactory.
The results for approximating the 4-RDM in terms of the lower order RDM’s are slightly inferior but still very good. In consequence, I expect that the iterative procedure proposed in the previous section may prove to be a realistic one.
In spite of the good results obtained we continue our search for simple auxiliary conditions directed at ensuring that the approximated matrix is positive and that its trace has the correct value. This search is mainly focused at improving the quality of the 2-RDM obtained in terms of the 1-RDM,which at the moment is the less precise procedure [46]. When this latter aim is fulfilled we expect that the iterative solution of the 1-order CSchE will also be successful although in this CSchE the information carried by the Hamiltonian only influences the result in an average way which probably will retard the convergence.
6.Conclusion
The two previous sections outline the main formal and applicative results obtained in our search for a theoretical framework where the number of variables which are explicitly taken into account would be as small as the observables allow. This framework should permit the use of different levels of approximation for the Hamiltonian operator and its orbital representation. That is, the size of the basis set and the kind of approximation used for the integrals should not be predetermined by the formalism.
Both lines of research are far from being closed and we are confident that their development will contribute useful results. However, without considering future performances I think that it can already be said that it is a good strategy to project the future Quantum Chemical methodology in such a way that the WF is by-passed and the 2-RDM or (better still but more difficult) the 1-RDM are directly determined.
Acknowledgements
Investigación Científíca y Técnica delMinisterio de Educación y Ciencia under project PB90-0092.
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