Geometrical constraints to reduce complexity in quantum molecular systems
Résumé
The energy of a molecule is not the sum of the energies of its atomic components. Our aim is to compute an accurate approximation of this quantity based on an electron pair model, that is to say by using an antisymmetric product of two-electron wave functions, called “geminals”. In this model, the total wave function can also be described in a more practical way, by a set of matrices, one for each geminal. However, without further restrictions, such a model has a factorial computational complexity with the number of electrons and its applicability is therefore limited to small systems.
We have introduced generalized orthogonality constraints between geminals to reduce the computational effort to a polynomial cost, without sacrificing the indistinguishability of the electrons.
In other words, the electron pairs corresponding to the geminals are not fully distinguishable, so that our model remains complex, in the sense that geminal products have still to be antisymmetrized according to the Pauli principle to form bona fide electronic wave functions.
Our geometrical constraints translate, in terms of geminal matrices, into simple equations involving the traces of products of these matrices. In the simplest non-trivial model, a set of solutions is given by block-diagonal matrices where each block is of size 2x2 and consists of a Pauli matrix or the Identity, multiplied by a complex parameter to be optimized. With this simplified ansatz for geminals, the number of terms in the calculation of the matrix elements of quantum observables is considerably reduced.
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