The pseudopotential is an attempt to replace the complicated effects of the motion of the core 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 the Coulombic potential term for coreelectrons normally found in the Schrödinger equation. 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 pseudo-wavefunctions with significantly fewer nodes. This allows the pseudo-wavefunctions to be described with far fewer Fourier modes, thus making plane-wave basis sets practical to use. In this approach usually only the chemically active valence electrons are dealt with explicitly, while the core electrons are 'frozen', being considered together with the nuclei as rigid non-polarizable ion cores. It is possible to self-consistently update the pseudopotential with the chemical environment that it is embedded in, having the effect of relaxing the frozen core approximation, although this is rarely done. In codes using local basis functions, like Gaussian, often effective core potentials are used that only freeze the core electrons. First-principles pseudopotentials are derived from an atomic reference state, requiring that the pseudo- and all-electron valence eigenstates have the same energies and amplitude outside a chosen core cut-off radius. Pseudopotentials with larger cut-off radius are said to be softer, that is more rapidly convergent, but at the same time less transferable, that is less accurate to reproduce realistic features in different environments. Motivation:
The small-core approximation assumes that there is no significant overlap between core and valence wave-function. Nonlinear core corrections or "semicore" electron inclusion deal with situations where overlap is non-negligible.
Early applications of pseudopotentials to atoms and solids based on attempts to fit atomic spectra achieved only limited success. Solid-state pseudopotentials achieved their present popularity largely because of the successful fits by Walter Harrison to the nearly free electron Fermi surface of aluminum and by James C. Phillips to the covalent energy gaps of silicon and germanium. Phillips and coworkers later extended this work to many other semiconductors, in what they called "semiempirical pseudopotentials".
Norm-conserving pseudopotential
Norm-conserving and ultrasoft are the two most common forms of pseudopotential used in modern plane-waveelectronic structure codes. They allow a basis-set with a significantly lower cut-off to be used to describe the electron wavefunctions and so allow proper numerical convergence with reasonable computing resources. An alternative would be to augment the basis set around nuclei with atomic-like functions, as is done in LAPW. Norm-conserving pseudopotential was first proposed by Hamann, Schlüter, and Chiang in 1979. The original HSC norm-conserving pseudopotential takes the following form: where projects a one-particle wavefunction, such as one Kohn-Sham orbital, to the angular momentum labeled by. is the pseudopotential that acts on the projected component. Different angular momentum states then feel different potentials, thus the HSC norm-conserving pseudopotential is non-local, in contrast to local pseudopotential which acts on all one-particle wave-functions in the same way. Norm-conserving pseudopotentials are constructed to enforce two conditions. 1. Inside the cut-off radius, the norm of each pseudo-wavefunction be identical to its corresponding all-electron wavefunction: 2. All-electron and pseudo wavefunctions are identical outside cut-off radius.
Ultrasoft pseudopotentials
Ultrasoft pseudopotentials relax the norm-conserving constraint to reduce the necessary basis-set size further at the expense of introducing a generalized eigenvalue problem. With a non-zero difference in norms we can now define: and so a normalised eigenstate of the pseudo Hamiltonian now obeys the generalized equation where the operator is defined as where are projectors that form a dual basis with the pseudo reference states inside the cut-off radius, and are zero outside: A related technique is the projector augmented wave method.
: This webpage maintained by the provides a searchable database of pseudopotentials for density functional codes as well as links to pseudopotential generators, converters, and other online databases.
: Website of David Vanderbilt with links to codes that implement ultrasoft pseudopotentials and libraries of generated pseudopotentials.
: This site hosts the GBRV pseudopotential library
: This site collates tested pseudo potentials sorted by type, accuracy, and efficiency, shows information on convergence of various tested properties and provides download options.
