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International Journal of Quantum Chemistry
Exact expressions for energy functional in the time‐dependent density functional theory Ágnes Nagy
Published Jan 1, 2003 • 2 citations
Abstract Abstract Exact equations for correlation energy and kinetic energy correction functionals are derived in the time‐dependent density functional theory. Hierarchies of equations are also obtained and approximate functionals are proposed by truncation of the hierarchies at different orders. Explicit relations for the correlation energy and its kinetic component along the adiabatic path are given in terms of powers of the moments of the electron density. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
DOI 10.1002/qua.10509 Related papers New perspectives on the fundamental theorem of density functional theory
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Quantal density functional theory of excited states: Application to an exactly solvable model
Marlina Slamet, Viraht Sahni
International Journal of Quantum Chemistry • Jan 1, 2001• 31 citations
Abstract Exact equations for correlation energy and kinetic energy correction functionals are derived in the time‐dependent density functional theory. Hierarchies of equations are also obtained and approximate functionals are proposed by truncation of the hierarchies at different orders. Explicit relations for the correlation energy and its kinetic component along the adiabatic path are given in terms of powers of the moments of the electron density. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
Abstract Exact time‐dependent density functionals depend on both the entire history of the density and the initial wave function. Recently we have shown that the two effects are entangled by an exact condition that should be useful as a test in building accurate memory‐dependent functionals. Here we consider this condition for small perturbations and derive an exact relation connecting the exchange‐correlation kernel with initial‐state derivatives. We discuss other exact conditions on the initial‐state derivatives which shed some light on the elusive memory‐dependence in time‐dependent density functional theory. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005
Several explicit formulas for the kinetic energy of a many-electron system as a functional of the k-electron density are derived, with emphasis on the electron pair density. The emphasis is on general techniques for deriving approximate kinetic energy functionals and features generalized Weisacker bounds and methods using density-matrix reconstruction. Adapting results from statistical mechanics, a hierarchy of equations is derived that links electron pairs, triplets, quadruplets, etc.; this may be used to derive more accurate approximations. Several methods for defining the exact kinetic energy functional are presented, including the generalizations of the Levy and Lieb formulations of density-functional theory. Together with N-representability constraints on the k-density, this paper provides the basis for “generalized density functional theories” based on the electron pair density. There are also implications for conventional density-functional theory, notably regarding the development of more accurate density functionals for the kinetic energy.
Abstract The fundamental theorem of time‐independent/time‐dependent density functional theory due to Hohenberg–Kohn (HK)/Runge–Gross (RG) proves the bijectivity between the density ρ(r)/ρ(rt) and the Hamiltonian / (t) to within a constant C/function C(t), and wave function Ψ/Ψ (t). The theorems are each proved for scalar external potential energy operators. By a unitary or equivalently a gauge transformation that preserves the density, we generalize the realm of validity of each theorem to Hamiltonians, which additionally include the momentum operator and a curl‐free vector potential energy operator defined in terms of a gauge function α (R)/α (Rt). The original HK/RG theorems then each constitute a special case of this generalization. Thereby, a fourfold hierarchy of such theorems is established. As a consequence of the generalization, the wave function Ψ/Ψ (t) is shown to be a functional of both the density ρ(r)/ρ(rt), which is a gauge‐invariant property, and a gauge function α(R)/α(Rt). The functional dependence on the gauge function ensures that as required by quantum mechanics, the wave function written as a functional is gauge variant. The hierarchy and the dependence of the wave function functional on the gauge function thus enhance the significance of the phase factor in density functional theory in a manner similar to that of quantum mechanics. Various additional perspectives on the theorem are arrived at. These understandings also address past critiques of time‐dependent theory. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008
We consider an analytically solvable model of two interacting electrons that allows for the calculation of the exact exchange-correlation kernel of time-dependent density functional theory. This kernel, as well as the corresponding density response function, is studied in the limit of large repulsive interactions between the electrons and we give analytical results for these quantities as an asymptotic expansion in powers of the square root of the interaction strength. We find that in the strong interaction limit the three leading terms in the expansion of the kernel act instantaneously while memory terms only appear in the next orders. We further derive an alternative expansion for the kernel in the strong interaction limit on the basis of the theory developed previously [Phys. Chem. Chem. Phys. 18, 21092 (2016)] using the formalism of strictly correlated electrons in the adiabatic approximation. We find that the first two leading terms in this series, corresponding to the strictly correlated limit and its zero-point vibration correction, coincide with the two leading terms of the exact expansion. We finally analyze the spatial nonlocality of these terms and show when the adiabatic approximation breaks down. The ability to reproduce the exact kernel in the strong interaction limit indicates that the adiabatic strictly correlated electron formalism is useful for studying the density response and excitation properties of other systems with strong electronic interactions.
Both model results and some exact relations concerning kinetic, exchange, and correlation energy density functionals will be presented. As to single-particle kinetic energy, a model of harmonic confinement, motivated by recent experimental activity on Fermion vapors, of an arbitrary number of Fermions will be used to construct exact nonlocal kinetic energy density functionals in both one and three dimensions. Then, some further exact results concerning nuclear cusp conditions for exchange only, and for exchange plus correlation, will be discussed. Finally, in Appendices, the one-shell harmonic confinement is generalized to include Coulomb repulsion in the two-electron Hookean atom and some results on the Pauli potentials for many-Fermion systems are obtained. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
Abstract The quantal density functional theory (Q‐DFT) of excited states is the description of the physics of the mapping from any bound nondegenerate excited state of Schrödinger theory to that of the s‐system of noninteracting Fermions with equivalent density ρk(r), energy Ek, and ionization potential Ik. The s‐system may either be in an excited state with the same configuration as in Schrödinger theory or in a ground state with a consequently different configuration. The Q‐DFT description of the s‐system is in terms of a conservative field ℱk(r), whose electron‐interaction ℰee(r) and correlation‐kinetic 𝒵 (r) components are separately representative of electron correlations due to the Pauli exclusion principle and Coulomb repulsion, and correlation‐kinetic effects, respectively. The sources of these fields are expectations of Hermitian operators taken with respect to the system wavefunction. The local electron‐interaction potential vee(r) of the s‐system, representative of all the many‐body correlations, is the work done to move an electron in the force of the field ℱk(r). The electron interaction Eee and correlation‐kinetic Tc components of the total energy Ek may be expressed in integral virial form in terms of their respective fields. The difference between the s‐system in its ground or excited state representation is due entirely to correlation‐kinetic effects. The highest occupied eigenvalue of the s‐system differential equation in either case is minus the ionization potential Ik. In this work we demonstrate the transformation of an excited state of Schrödinger theory, as represented by the first excited singlet state of the exactly solvable Hooke's atom, to that of noninteracting Fermions in their ground state with equivalent excited state density, energy, and ionization potential. To further prove the Fermions are in a ground state, we solve the corresponding singlet s‐system differential equation numerically for the vee(r) determined, and obtain the excited state density from the zero node orbitals generated. The resulting total energy is also the same. In addition, the single eigenvalue determined corresponds to minus the ionization potential of the excited state. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001