Wave function methods for fractional electrons.
Determining accurate chemical potentials is of considerable interest in various chemical and physical contexts: from small molecular charge-transfer complexes to bandgap in bulk materials such as semi-conductors. Chemical potentials are typically evaluated either by density functional theory, or, alternatively, by computationally more intensive Greens function based GW computations. To calculate chemical potentials, the ground state energy needs to be defined for fractional charges. We thus explore an extension of wave function theories to fractional charges, and investigate the ionization potential and electron affinity as the derivatives of the energy with respect to the electron number. The ultimate aim is to access the chemical potential of correlated wave function methods without the need of explicitly changing the numbers of electrons, making the approach readily applicable to bulk materials. We find that even though second order perturbation theory reduces the fractional charge error considerably compared to Hartree-Fock and standard density functionals, higher order perturbation theory is more accurate and coupled-cluster approaches are even more robust, provided the electrons are bound at the Hartree-Fock level. The success of post-HF approaches to improve over HF relies on two equally important aspects: the integer values are more accurate and the Coulomb correlation between the fractionally occupied orbital and all others improves the straight line behavior significantly as identified by a correction to Hartree-Fock. Our description of fractional electrons is also applicable to fractional spins, illustrating the ability of coupled-cluster singles and doubles to deal with two degenerate fractionally occupied orbitals, but its inadequacy for three and more fractional spins, which occur, for instance, for spherical atoms and when dissociating double bonds. Our approach explores the realm of typical wave function methods that are applied mostly in molecular chemistry, but become available to the solid state community and offer the advantage of an integrated approach: fundamental gap, relative energies, and optimal geometries can be obtained at the same level.
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