New ideas to reduce the computational complexity of non- orthogonal geminal methods for strongly-correlated electronic systems
Résumé
The Lewis electron-pair picture pervades all of chemistry and a substantial
domain of materials science. “Geminals” are electron-pair quantum states which
are used to express approximate solutions of the Schrödinger equation of polyelec-
tronic systems. In quantum chemistry, geminal-based methods are potentially more
effective than the traditional ones based on the orbital picture: electrons occupy
orbitals and experience only the average effect of each other. This is especially true
for strongly correlated-systems. However, the computational cost of using arbitrary
geminal wave functions is non-polynomial in the system size.
It is therefore an important challenge to find additional constraints on geminal
models which preserve the accuracy of the wavefunction description while reducing
the calculations to polynomial cost. The Ayers’ group at Mc Master has developed
one such successful geminal model called AP1roG because the wavefunction is an
antisymmetric product of geminals with one distinct reference orbital occupied in
each geminal [1]. This ansatz relaxes the so-called “strong-orthogonality” constraint
between geminals. The latter makes wavefunction matrix element computations
very easy but appears too drastic from the physical point of view, since the geminalelectron pairs are then distinguishable.
We have followed a different path to relax the strong-orthogonality constraint. We
have defined the geometrical concept of n-orthogonality [2] which can be viewed as
a graded indistinguishability measure for electronic states: 1-orthogonality coincides
with strong orthogonality, the larger n the less distinguishable the n-orthogonal elec-
tronic states will be. We have studied geminal models constrained by n-orthogonality
relations for different values of n, imposed to different combinations of geminal prod-
uct states [3]. However, in this work the scaling of matrix element computations was
still unsatisfactory. We will present new ideas developed since then to remedy this
problem.
References
[1] P. A Limacher, P. W. Ayers, P. A. Johnson, S. De Baerdemacker, D. Van Neck and P. Bultinck, J. of Chem. Theory and Computation 9, 1394-1401 (2013).
[2] P. Cassam-Chenaı̈, Phys. Rev. A77, 032103 (2008).
[3] P. Cassam-Chenaı̈, V. Rassolov, Chem. Phys. Lett. 487, 147-152 (2010).