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Symmetry Constraints in the Ionization Potentials and on the Formulation of the Hohenberg-Kohn-Sham Theory
Abstract Density functional-theory was given a formal structure with the Hohenberg-Kohn1 (HK) theorems and a succesful practical procedure with the Kohn-Sham2 (KS) scheme to construct the density function as linear combinations of squares of auxiliary functions ψi of symmetry i with weight ci. The functions ψi obey a Schrödinger like wave equation with eigenvalue εi. The overall symmetry of the system has to be consistent with the “occupation” of the i-th “states”. In principle the set {i} should be complete and the total energy E[ρ;{ci,N}] has to be minimized with respect to the parameters {ci,N}, then the HK-KS equations should really be written δE[ρ;{ci,N}] = 0 and $$ \int {\rho d\tau =} \int {\sum\nolimits_i {{c_i}{{\left({{\psi_i}} \right)}^2}}} $$ dτ = N, allowing for fractional ci. Also the ionization potential μi, (with δci (i) ≈ 1) is given by $$- \mu _i = \sum\nolimits_{\rm{i}} {\frac{{\partial E}}{{\partial c_i }}} {\rm{ }}\partial c_i^{\left( i \right)} + {\rm{ }}\sum\nolimits_{{\rm{n > 1}}} {\frac{1}{{{\rm{n!}}}}} \sum\nolimits_{j,k, \cdots } {\left( {\partial ^n E/\left( {\partial c_i \partial c_k \cdots } \right)} \right)\left( {\partial c_i^{\left( i \right)} \partial c_k^{\left( i \right)} \cdots } \right)}$$ with $$\sum\nolimits_j {\partial c_j^{(i)} = 1}$$ . In this way μi = − εi + Δμi and there will be a μk ≤ μi≠k, corresponding to the least energy removal of one electron. Examples and results are discussed and shown. The internal symmetry of the system is an important (hidden or explicit) part of the theory and when properly considered, the results have always compared reasonable well with experiment (see Keller, Amador and de Teresa3, also Trickey4).
Symmetry Constraints in the Ionization Potentials and on the Formulation of the Hohenberg-Kohn-Sham Theory
Abstract Density functional-theory was given a formal structure with the Hohenberg-Kohn1 (HK) theorems and a succesful practical procedure with the Kohn-Sham2 (KS) scheme to construct the density function as linear combinations of squares of auxiliary functions ψi of symmetry i with weight ci. The functions ψi obey a Schrödinger like wave equation with eigenvalue εi. The overall symmetry of the system has to be consistent with the “occupation” of the i-th “states”. In principle the set {i} should be complete and the total energy E[ρ;{ci,N}] has to be minimized with respect to the parameters {ci,N}, then the HK-KS equations should really be written δE[ρ;{ci,N}] = 0 and $$ \int {\rho d\tau =} \int {\sum\nolimits_i {{c_i}{{\left({{\psi_i}} \right)}^2}}} $$ dτ = N, allowing for fractional ci. Also the ionization potential μi, (with δci (i) ≈ 1) is given by $$- \mu _i = \sum\nolimits_{\rm{i}} {\frac{{\partial E}}{{\partial c_i }}} {\rm{ }}\partial c_i^{\left( i \right)} + {\rm{ }}\sum\nolimits_{{\rm{n > 1}}} {\frac{1}{{{\rm{n!}}}}} \sum\nolimits_{j,k, \cdots } {\left( {\partial ^n E/\left( {\partial c_i \partial c_k \cdots } \right)} \right)\left( {\partial c_i^{\left( i \right)} \partial c_k^{\left( i \right)} \cdots } \right)}$$ with $$\sum\nolimits_j {\partial c_j^{(i)} = 1}$$ . In this way μi = − εi + Δμi and there will be a μk ≤ μi≠k, corresponding to the least energy removal of one electron. Examples and results are discussed and shown. The internal symmetry of the system is an important (hidden or explicit) part of the theory and when properly considered, the results have always compared reasonable well with experiment (see Keller, Amador and de Teresa3, also Trickey4).
Symmetry Constraints in the Ionization Potentials and on the Formulation of the Hohenberg-Kohn-Sham Theory
Keller, Jaime (author) / Amador, Carlos (author) / Teresa, Carmen (author) / Flores, Jose A. (author)
Condensed Matter Theories ; 131-140
1987-01-01
10 pages
Article/Chapter (Book)
Electronic Resource
English
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