Small Core Approximation Sample Clauses

Small Core Approximation. ‌ The electrons in DFT can be separated into two broad groups. Firstly, there are localized electrons confined to a relatively small inner-core region whose one-electron eigenfunctions ψα = ϕc are approximately equal to those of an isolated atom. These electrons move rigidly with the nuclei. Secondly, there are valence electrons whose eigenfunctions ψα = ψk are concentrated in the interstitial regions. Under the small core approximation [93], we treat these two classes of electrons separately. This sepa- ration is not always easy or possible. For the simple metals of primary interest in this thesis, this distinction is sharp and the small-core approximation gives rise to negligible errors. The simple metal core is defined to be the last complete noble gas configura- tion. The valence electrons are the remaining Z electrons which can be read off from their periodic table group. On the other hand, the small-core approximation yields non-negligible errors for the noble or transition metals. The filled or partially-filled d bands overlap significantly with neighbouring cores and cannot be simply incorporated into the valence. The core eigenfunctions obey the following Schrödinger equation Σ − 2m∇ + VKS (r)Σ ϕc (r) = Ecϕc (r) (2.39) where Ec is the eigenvalue of the core state c. Whilst the core eigenfunctions ϕc resemble those of the free atom very closely, the eigenvalues Ec are shifted. The valence eigenstates ψk obey the following Schrödinger equation Σ ℏ2 2 − 2m∇ + VKS (r)Σ ψk (r) = Ekψk (r) (2.40) where k is a wavevector and {ψk} are orthogonal to the core electrons {ϕc}. The valence orbitals ψk are Xxxxx wavefunctions and thus can be written as ψk (r) = eik·ruk (r) (2.41)