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Physical Review A

Differentiation of density functionals that conserves the normalization of the density

Tamás Gál

Published Jan 9, 2001• 23 citations
Publisher

Abstract

The formula for differentiation of functional $A[\ensuremath{\rho}]$ with respect to $\ensuremath{\rho}(\mathbf{r})$ while keeping the normalization $\ensuremath{\int}\ensuremath{\rho}(\mathbf{r})d\mathbf{r}$ of $\ensuremath{\rho}(\mathbf{r})$ fixed is derived, and basic properties arising from it are concerned. The results are then generalized for time-dependent theories. One of the essential consequences of normalization conservation for functional differentiation, namely, that it ruins the symmetry in r of multiple derivatives, is shown to give the resolution of the causality paradox of response functions in time-dependent density-functional theory. The formula for the differentiation of functionals $A[\ensuremath{\rho}]$ that conserves the shape of $\ensuremath{\rho}(\mathbf{r})$ is also presented.

DOI
10.1103/physreva.63.022506

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The formula for differentiation of functional A[\ensuremathρ]A[\ensuremath{\rho}]A[\ensuremathρ] with respect to \ensuremathρ(r)\ensuremath{\rho}(\mathbf{r})\ensuremathρ(r) while keeping the normalization \ensuremath∫\ensuremathρ(r)dr\ensuremath{\int}\ensuremath{\rho}(\mathbf{r})d\mathbf{r}\ensuremath∫\ensuremathρ(r)dr of \ensuremathρ(r)\ensuremath{\rho}(\mathbf{r})\ensuremathρ(r) fixed is derived, and basic properties arising from it are concerned. The results are then generalized for time-dependent theories. One of the essential consequences of normalization conservation for functional differentiation, namely, that it ruins the symmetry in r of multiple derivatives, is shown to give the resolution of the causality paradox of response functions in time-dependent density-functional theory. The formula for the differentiation of functionals A[\ensuremathρ]A[\ensuremath{\rho}]A[\ensuremathρ] that conserves the shape of \ensuremathρ(r)\ensuremath{\rho}(\mathbf{r})\ensuremathρ(r) is also presented.
We resolve an existing paradox regarding the causality and symmetry properties of response functions within time-dependent density-functional theory. We do this by defining a new action functional within the Keldysh formalism. By functional differentiation the new functional leads to response functions which are symmetric in the Keldysh time contour parameter, but which become causal when a transition to physical time is made. The new functional is further used to derive the equations of the time-dependent optimized potential method.
It is pointed out that the kernel of the inverse response function $\ensuremath{\delta}v(\mathbf{r},t)/\ensuremath{\delta}\ensuremath{\rho}({\mathbf{r}}^{\ensuremath{'}}{,t}^{\ensuremath{'}}),$ whose existence is guaranteed by the Runge-Gross theorem, cannot vanish for $t<{t}^{\ensuremath{'}}.$ As such the kernel fxc,{f}_{\mathrm{xc}},fxc​, which is the functional derivative of the time-dependent exchange-correlation potential with respect to the density, can be symmetric with respect to an interchange between t and ${t}^{\ensuremath{'}}$ without violating causality. Thus there is no conflict between the symmetry requirement of this kernel and causality in time-dependent density-functional theory.
I show that the so-called causality paradox of time-dependent density-functional theory arises from an incorrect formulation of the variational principle for the time evolution of the density. The correct formulation not only resolves the paradox in real time, but also leads to an expression for the causal exchange-correlation kernel in terms of Berry curvature. Furthermore, I show that all the results that were previously derived from symmetries of the action functional remain valid in the present formulation. Finally, I develop a model functional theory which explicitly demonstrates the workings of the formulation.
Time-ordered superoperators are used to develop a unified description of nonlinear density response and spontaneous fluctuations of many-electron systems. The pthp\mathrm{th}pth-order density response functions are decomposed into 2p+1{2}^{p+1}2p+1 non-causal Liouville space pathways. Individual pathways are symmetric to the interchange of their space, time, and superoperator indices and can thus be calculated as functional derivatives. Other combinations of these pathways represent spontaneous density fluctuations and the response of such fluctuations to an external field. The resolution of the causality paradox of time-dependent density-functional theory (TDDFT) is shown to be intimately connected with the nonretarded nature of fluctuations.
In this work, we investigate the functional differentiability of the time-dependent many-body wave function and of derived quantities with respect to time-dependent potentials. For properly chosen Banach spaces of potentials and wave functions, Fréchet differentiability is proven. From this follows an estimate for the difference of two solutions to the time-dependent Schrödinger equation that evolve under the influence of different potentials. Such results can be applied directly to the one-particle density and to bounded operators, and present a rigorous formulation of non-equilibrium linear-response theory where the usual Lehmann representation of the linear-response kernel is not valid. Further, the Fréchet differentiability of the wave function provides a new route towards proving basic properties of time-dependent density-functional theory.
The Runge-Gross [E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984)] action functional of time-dependent density-functional theory leads to a well-known causality paradox; that is, a perturbation of the electronic density in the future affects the response of the system in the present. This paradox is known to be caused by an inconsistent application of the Dirac-Frenkel variational principle. In view of the recent solutions to this problem, the action functional employed by Runge and Gross in their formulation of time-dependent density-functional theory is analyzed in the context of the Keldysh contour technique. The time-dependent electronic density and the concept of causality are extended to the contour. We derive a variational equation that obeys causality and relates the exchange-correlation potential with its kernel and the functional derivative of the exchange-correlation action functional with respect to the density. It is shown that the adiabatic local-density approximation is a consistent solution of this equation and that the time-dependent optimized potential method can also be derived from it. The formalism presented here can be used to find new approximation methods for the exchange-correlation potential and to avoid the causality dilemma.