## Calculating lower bound to the FCI energy using Löwdin's bracketing function

__Zsuzsanna TÖTH__^{1}, and Ágnes Szabados^{2}

^{1}MATHERIALS research group, Ecole des Ponts ParisTech, Champs-sur-Marne, France

^{2}Laboratory of Theoretical Chemistry, Eötvös University, Budapest, Hongrie

Mercredi 26 Février 2020, 11h00

bibliothèque LCT, tour 12 - 13, 4^{ème} étage

Common electron correlation methods either give upper bound to the FCI energy or it is not
known from which side they approximate. It is simple to obtain upper bound: any expectation
value gives upper bound to the ground state energy. Opposite to this, currently there is no widely
used computational method to calculate lower bound.

Lödin's bracketing function[1], *f*_{br}(ε) gives a lower bound if the argument *ε* is an appropriate upper bound. In general the argument and the value of the function „brackets” at least one eigenvalue. Taking a normalised reference function *φ*, the bracketing function is expressed as:

Calculation of the bracketing function is computionally demanding due to the operator inverse.
To explore practical approaches we used several numerical approximations. The approximate
lower bounds are evaluated at each step of a Davidson FCI iteration sequence on the example of
simple molecules[2].

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References :

[1] P. O. Löwdin, *Phys. Rev.*, 139 (1965) A357.

[2] Zs. Tóth, Á. Szabados, *J. Chem. Phys.*, 143 (2015) 084112.