Pseudopotential

In quantum mechanics, the pseudopotential approximation is an attempt to replace the complicated effects of the motion of the core (i.e. non-valence) electrons of an atom and its nucleus with an effective potential, or pseudopotential, so that the Schrödinger equation contains a modified effective potential term instead of e.g. the Coulombic potential term for core electrons normally found in the Schrödinger equation. The pseudopotential approximation was first introduced by Hans Hellmann in the 1930s. By construction of this pseudopotential, the valence wavefunction generated is also guaranteed to be orthogonal to all the core states.

The pseudopotential is an effective potential constructed to replace the atomic all-electron potential such that core states are eliminated and the valence electrons are described by nodeless pseudo-wavefunctions. In this approach only the chemically active valence electrons are dealt with explicitly, while the core electrons are 'frozen', being considered togheter with the nuclei as rigid non-polarizable ion cores.Norm-conserving pseudopotentials are derived from an atomic reference state, requiring that the pseudo- and all-electron valence eigenstates have the same energies and amplitude (and thus density) outside a chosen core cutoff radius $r^{c}$.Pseudopotentials with larger cutoff radius are said to be extit{softer}, that is more rapidly convergent, but at the same time less extit{transferable}, that is less accurate to reproduce realistic features in different environments.

Motivation

# "Reduction of basis set size"
# "Reduction of number of electrons"
# "Inclusion of relativistic and other effects"

# "one-electron picture"
# "small-core appr." assumes that there is no significant overlap between core and valence WF. So the exchange correlation potential( see DFT) is: $E\_\{xc\}(n\_\{core\}+n\_\{valence\})=E\_\{xc\}(n\_\{core\})+E\_\{xc\}(n\_\{valence\})$; If this is not true, the "non-linear core corrections" technique (Louie et al., 1982) is used.

Types

Two kinds of PPs are used, Norm-conserving PP and Ultrasoft PP.

Norm-conserving PP

Norm-conserving PP: Outside of a cutoff-radius, the pseudo-wavefunctions are identical to the real all-electron WF.
e.g. BHS-PP, Bachelet,Hamann,Schlüter (Hamann et al., 1982)

Literature

Hans Hellmann in the Journal of Chemical Physics 3 61 (1935) and Journal of Chemical Physics 4 (1936), Harrison (1966), Brust (1968), Stoneham (1975), Heine (1970), Pickett (1989),..

see also: Density functional theory