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Physical Review A

Wave function and external potential from constrained search in density-functional theory

Ji Luo

Published Jul 16, 2008• 0 citations
Publisher

Abstract

It is proven that in Levy's constrained search or Lieb's ensemble constrained search, if the minimum is achieved by wave functions that satisfy the steadiness condition for the current density kernel and have continuous second derivatives, the minimizing wave functions are eigenstates of some external potential. If the excited states of this potential can only achieve stationarity and cannot achieve a minimum in the constrained search, the minimizing wave functions are ground states and the corresponding electron density is $\mathit{v}$-representable or ensemble $\mathit{v}$-representable. This is also true for the ensemble system with a varying electron number. In time-dependent density-functional theory, if the stationarity of the action integral is achieved in the constrained search by a wave function that satisfies the continuity equation for the current density kernel and has a continuous first derivative with respect to the time coordinate and continuous second derivatives with respect to the space coordinates, the stationary wave function satisfies the time-dependent Schr\"odinger equation of some time-dependent external potential and the corresponding electron density is time-dependent $\mathit{v}$-representable.

DOI
10.1103/physreva.78.012505

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It is proven that in Levy's constrained search or Lieb's ensemble constrained search, if the minimum is achieved by wave functions that satisfy the steadiness condition for the current density kernel and have continuous second derivatives, the minimizing wave functions are eigenstates of some external potential. If the excited states of this potential can only achieve stationarity and cannot achieve a minimum in the constrained search, the minimizing wave functions are ground states and the corresponding electron density is v\mathit{v}v-representable or ensemble v\mathit{v}v-representable. This is also true for the ensemble system with a varying electron number. In time-dependent density-functional theory, if the stationarity of the action integral is achieved in the constrained search by a wave function that satisfies the continuity equation for the current density kernel and has a continuous first derivative with respect to the time coordinate and continuous second derivatives with respect to the space coordinates, the stationary wave function satisfies the time-dependent Schr"odinger equation of some time-dependent external potential and the corresponding electron density is time-dependent v\mathit{v}v-representable.
To construct an NNN-representable time-dependent density-functional theory, a generalization to the time domain of the Levy-Lieb (LL) constrained-search algorithm is required. That the action is only stationary in the Dirac-Frenkel variational principle eliminates the possibility of basing the search on the action itself. Instead, we use the norm of the partial functional derivative of the action in the Hilbert space of the wave functions in place of the energy of the LL search. The electron densities entering the formalism are NNN-representable, and the resulting universal action functional has a unique stationary point in the density at that corresponding to the solution of the Schr\"odinger equation. The original Runge-Gross (RG) formulation is subsumed within the current formalism. Concerns in the literature about the meaning of the functional derivatives and the internal consistency of the RG formulation are allayed by clarifying the nature of the functional derivatives entering the formalism.
This thesis is devoted to the formulation and implications of the time-dependent density functional theory (TDDFT). The work is divided into two main parts. In the first part we develop rigorous theorems for the density-potential mapping in quantum many-body systems on a lattice. We prove the uniqueness of the TDDFT map and demonstrate that a given density is v-representable if the initial many-body state and the density satisfy certain well defined conditions. In particular, we show that for a system evolving from its ground state any density with a continuous second time derivative is v-representable. Then the lattice TDDFT formulation is extended to cover system of interacting lattice electrons strongly coupled to cavity photons.We prove that under some mathematical conditions the electron-photon wave function is a unique functional of the electronic density and the expectation value of the photonic coordinate. Then we further generalize the ground state v representability theorem to include the ground state of a general lattice electron-photon Hamiltonian.
A density-functional formalism comparable to the Hohenberg-Kohn-Sham theory of the ground state is developed for arbitrary time-dependent systems. It is proven that the single-particle potential $v(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}t)$ leading to a given vvv-representable density $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}t)$ is uniquely determined so that the corresponding map $v\ensuremath{\rightarrow}n$ is invertible. On the basis of this theorem, three schemes are derived to calculate the density: a set of hydrodynamical equations, a stationary action principle, and an effective single-particle Schr\"odinger equation.
The premise of density-functional theory is that knowledge of the ground-state density uniquely determines the Hamiltonian, and thereby, via solution of the corresponding time-independent Schr\"odinger equation, all the properties of the system. The density therefore constitutes a basic variable of quantum mechanics. There are at present two paths from the density to the Hamiltonian: the Hohenberg and Kohn proof of the bijectivity between the external potential and the basic variable, and the Percus, Levy, and Lieb constrained-search proof. We argue the Hohenberg- and Kohn-type proof to be the more fundamental, and that this is the case in general when both external electrostatic and magnetostatic fields are present, and the basic variables are the ground-state density and physical current density.
The logical structure and basic theorems of time-dependent current density functional theory (TDCDFT) are analyzed and reconsidered from the point of view of recently proposed time-dependent deformation functional theory (TDDefFT). It is shown that the formalism of TDDefFT allows to avoid a traditional external potential-to-density/current mapping. Instead the theory is formulated in a form similar to the constrained search procedure in the ground state DFT. Within this formulation of TDCDFT all basic functionals appear from the solution of a constrained universal many-body problem in a comoving reference frame, which is equivalent to finding a conditional extremum of a certain universal action functional. As a result the physical origin of the universal functionals entering the theory, as well as their proper causal structure become obvious. In particular, this leaves no room for any doubt concerning the predictive power of the theory.
By utilizing the knowledge that a Hamiltonian is a unique functional of its ground-state density, the following fundamental connections between densities and Hamiltonians are revealed: Given that ${\ensuremath{\rho}}_{\ensuremath{\alpha}}, {\ensuremath{\rho}}_{\ensuremath{\beta}},\dots{},{\ensuremath{\rho}}_{\ensuremath{\omega}}$ are ground-level densities for interacting or noninteracting Hamiltonians H1,H2,…,HM{H}_{1}, {H}_{2},\dots{},{H}_{M}H1​,H2​,…,HM​ (MMM arbitrarily large) with local potentials v1{v}_{1}v1​,v2{v}_{2}v2​,…\dots{}…,vM{v}_{M}vM​, but given that we do not know which $\ensuremath{\rho}$ belongs with which HHH, the correct mapping is possible and is obtained by minimizing $\ensuremath{\int}d\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}} [{v}_{1}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}){\ensuremath{\rho}}_{\ensuremath{\alpha}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})+{v}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}){\ensuremath{\rho}}_{\ensuremath{\beta}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})+\ensuremath{\cdots}{v}_{M}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}){\ensuremath{\rho}}_{\ensuremath{\omega}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})]$ with respect to optimum permutations of the $\ensuremath{\rho}$'s among the vvv's. A tight rigorous bound connects a density to its interacting ground-state energy via the one-body potential of the interacting system and the Kohn-Sham effective one-body potential of the auxiliary noninteracting system. A modified Kohn-Sham effective potential is defined such that its sum of lowest orbital energies equals the true interacting ground-state energy. Moreover, of all those effective potentials which differ by additive constants and which yield the true interacting ground-state density, this modified effective potential is the most invariant with respect to changes in the one-body potential of the true Hamiltonian. With the exception of the occurrence of certain linear dependencies, aaa density will not generally be associated with any ground-state wave function (is not wave function vvv representable) if that density can be generated by a special linear combination of three or more densities that arise from a common set of degenerate ground-state wave functions. Applicability of the "constrained search" approach to density-functional theory is emphasized for non-vvv-representable as well as for vvv-representable densities. In fact, a particular constrained ensemble search is revealed which provides a general sufficient condition for non-vvv representability by a wave function. The possible appearance of noninteger occupation numbers is discussed in connection with the existence of non-vvv representability for some Kohn-Sham noninteracting systems.