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arXiv:cond-mat/9412100 · Physical Review Letters

Center of Mass and Relative Motion in Time Dependent Density Functional Theory

Giovanni Vignale

Published Apr 17, 1995• 131 citations
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Abstract

It is shown that the theorem asserting separability of the center of mass motion for a system of interacting particles in a harmonic external potential is satisfied in the time dependent density functional theory, provided that the exchange-correlation potential satisfies a simple symmetry under transformation to an accelerated frame of reference. Examples of approximations for the exchange-correlation potential which satisfy or violate this symmetry are given.

DOI
10.1103/physrevlett.74.3233

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It is shown that the theorem asserting separability of the center of mass motion for a system of interacting particles in a harmonic external potential is satisfied in the time dependent density functional theory, provided that the exchange-correlation potential satisfies a simple symmetry under transformation to an accelerated frame of reference. Examples of approximations for the exchange-correlation potential which satisfy or violate this symmetry are given.
We consider interacting particles in an external harmonic potential. We extend the B=0B=0B=0 case of the generalized Kohn theorem, giving a "harmonic-potential theorem" (HPT), demonstrating rigid, arbitrary-amplitude, time-oscillatory Schr\"odinger transport of a many-body eigenfunction. We show analytically that the time-dependent local-density approximation (TDLDA) satisfies the HPT exactly. Other approximations, such as linearized TDLDA with frequency-dependent exchange correlation kernel and certain inhomogeneous hydrodynamic formalisms, do not. A simple modification permits such explicitly frequency-dependent local theories to satisfy the HPT, however.
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.
We formulate equations of time-dependent density functional theory (TDDFT) in the comoving Lagrangian reference frame. The main advantage of the Lagrangian description of many-body dynamics is that in the comoving frame the current density vanishes, while the density of particles becomes independent of time. Therefore a comoving observer will see a picture which is very similar to that seen in the equilibrium system from the laboratory frame. It is shown that the most natural set of basic variables in TDDFT includes the Lagrangian coordinate, $\mathbf{\ensuremath{\xi}}$, a symmetric deformation tensor ${g}_{\ensuremath{\mu}\ensuremath{\nu}}$, and a skew-symmetric vorticity tensor, ${F}_{\ensuremath{\mu}\ensuremath{\nu}}$. These three quantities, respectively, describe the translation, deformation, and the rotation of an infinitesimal fluid element. Reformulation of TDDFT in terms of new basic variables resolves the problem of nonlocality and thus allows us to regularly derive a local nonadiabatic approximation for exchange correlation (xc) potential. Stationarity of the density in the comoving frame makes the derivation to a large extent similar to the derivation of the standard static local density approximation. We present a few explicit examples of nonlinear nonadiabatic xc functionals in a form convenient for practical applications.
Frequency dependent exchange-correlation kernels for time-dependent density functional theory can be used to construct approximate exchange-correlation potentials. The resulting potentials are usually not translationally covariant nor do they obey the so-called zero-force condition. These two basic symmetry requirements are essential for using the potentials in actual applications (even in the linear regime). We provide two pragmatic methods for fully imposing these conditions for both linear and nonlinear regimes. As an example, we take the Gross and Kohn frequency dependent XC functional [Phys. Rev. Lett. 55, 2850 (1985)], correct it, and numerically test it on a sodium metal cluster. Violation of the basic symmetries causes instabilities or spurious low frequency modes.
We propose a memory form of exchange-correlation potential vXC(r,t){v}_{\mathrm{XC}}(\mathbf{r},t)vXC​(r,t) for time-dependent interacting many-particle systems. Unlike previous memory-XC potentials, our vXC{v}_{\mathrm{XC}}vXC​ is not limited to the linear response regime. The proposed form of vXC{v}_{\mathrm{XC}}vXC​ is a generalized local-density approximation chosen so as to satisfy the nonlinear harmonic potential theorem and Newton's third law. For the case of the inhomogeneous electron gas, we give an explicit prescription for vXC{v}_{\mathrm{XC}}vXC​ based solely on an existing parametrization of the linear XC response kernel ${f}_{\mathrm{XC}}^{\mathrm{hom}}(n,\ensuremath{\omega})$ of the uniform gas. Application to quantum wells seems promising.