Functional derivative of the universal density functional in Fock space
- Department of Chemistry, University of British Columbia, 2036 Main Mall, Vancouver, British Columbia, V6T 1Z1 (Canada)
Within the framework of zero-temperature Fock-space density-functional theory (DFT), we prove that the Gateaux functional derivative of the universal density functional, {delta}F{sup {lambda}}[{rho}]/{delta}{rho}(r)|{sub {rho}={rho}{sub 0}}, at ground-state densities with arbitrary normalizations (<{rho}{sub 0}(r)>=n is a member of R{sub +}) and an electron-electron interaction strength {lambda}, is uniquely defined, but is discontinuous when the number of electrons n becomes an integer, thus providing a mathematically rigorous confirmation for the 'derivative discontinuity' initially discovered by Perdew et al. [Phys. Rev. Lett. 49, 1691 (1982)]. However, the functional derivative of the exchange-correlation functional is continuous with respect to the number of electrons in Fock space; i.e., there is no 'derivative discontinuity' for the exchange-correlation functional at an integer electron number. For a ground-state density {rho}{sub 0,n}{sup v,{lambda}}(r) of an external potential v(r), we show that {delta}F{sup {lambda}}[{rho}]/{delta}{rho}(r)|{sub {rho}={rho}{sub 0,n}{sup v{lambda}}}== {mu}{sub SM}{sup n}-v(r), where the constant {mu}{sub SM}{sup n} is given by the following chain of dependences: {rho}{sub 0,n}{sup v,{lambda}}(r)map[v]mapE{sub 0}{sup v,{lambda}}(n)map{mu}{sub SM}{sup n}={partial_derivative}E{sub 0}{sup v,{lambda}}(k)/{partial_derivative}kl{sub k=n}. Here [v] is the class of the external potential v(r) up to a real constant, and {mu}{sub SM}{sup n} is the chemical potential defined according to statistical mechanics. At an integer electron number N, we find that there is no freedom of adding an arbitrary constant to the value of the chemical potential {mu}{sub SM}{sup N}, whose exact value is generally not the popular preference of the negative of Mulliken's electronegativity, -(1/2)(I+A), where I and A are the first ionization potential and the first electron affinity, respectively. In addition, for any external potential converging to the same constant at infinity in all directions, we resolve that {mu}{sub SM}{sup N}=-I. Finally, the equality {mu}{sub DFT}={mu}{sub SM}{sup n} is rigorously derived via an alternative route, where {mu}{sub DFT} is the Lagrangian multiplier used to constrain the normalization of the density in the traditional DFT approach.
- OSTI ID:
- 20646355
- Journal Information:
- Physical Review. A, Vol. 70, Issue 4; Other Information: DOI: 10.1103/PhysRevA.70.042503; (c) 2004 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA); ISSN 1050-2947
- Country of Publication:
- United States
- Language:
- English
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