yair kurzweil and roi baer- generic galilean-invariant exchange-correlation functionals with quantum...

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Generic Galilean-invariant exchange-correlation functionals with quantum memory

Yair Kurzweil and Roi Baer*Department of Physical Chemistry and the Lise Meitner Center for Quantum Chemistry, the Hebrew University of Jerusalem, Jerusalem

91904, Israel

Received 25 October 2004; revised manuscript received 21 March 2005; published 6 July 2005

Today, most applications of time-dependent density-functional theory use adiabatic exchange-correlationXC potentials that do not take into account nonlocal temporal effects. Incorporating such memory termsinto XC potentials is complicated by the constraint that the derived force and torque densities must integrate to

zero at every instance. This requirement can be met by deriving the potentials from an XC action that is both

rotationally and translationally invariant, i.e., Galilean invariant GI. We develop a class of simple but flexibleforms for an action that respect these constraints. The basic idea is to formulate the action in terms of the

Eularian-Lagrangian transformation metric tensor, which is itself GI. The general form of the XC potentials in

this class is then derived, and the linear response limit is derived as well.

DOI: 10.1103/PhysRevB.72.035106 PACS numbers: 71.45.Gm, 71.10.Li, 71.15.Mb

I. INTRODUCTION

Time-dependent density-functional theory1 TDDFT isroutinely used in many calculations of electronic processesin molecular systems. Almost all applications use adiabaticpotentials describing an immediate response of the Kohn-Sham potential to the temporal variations of the electron den-sity. The shortcomings of these potentials were studied byseveral authors.25 Some of the problems are associated withself interaction, an ailment inherited from ground-statedensity-functional theory.6 Other deficiencies are known orsuspected to be associated with the adiabatic assumption.The first attempt to include nonadiabatic effects7 was basedon a simple form of the exchange-correlation XC potentialin the linear response limit. Studying an exactly solvablesystem, this form was shown to lead to spurious time-

dependent evolution.8

The failure was traced back to viola-tion of a general rule: the XC force density, derived from thepotential, should integrate to zero.9 Convincing argumentswere then presented,10 demonstrating that nonadiabatic ef-fects cannot be easily described within TDDFT and instead acurrent-density-based theory must be used. Vignale andKohn10 gave an expression for the XC potentials applicablefor linear response and long wavelengths.

That the total XC force is zero is a valid fact not only inTDDFT but also in time-dependent current density-functional theory TDCDFT. It stems from the basic re-quirement that the total force on the noninteracting particlesmust be equal to the total force on the interacting particles.This is so that a different total acceleration results and thetwo densities or current densities will be at variance. In theinteracting system the total Ehrenfest force can only resultfrom an external potential because of Newtons third law theelectrons cannot exert a net force on themselves. InTDCDFT the total force equals the sum of the external force,Hartree force, and XC force. Since the Hartree force inte-grates to zero Newtons third law again the total XC forcedoes so as well. A similar general argument can be applied tothe total torque, showing that the net XC torque must bezero. These requirements then have to be imposed on theapproximate XC potentials.9

The question we deal with in this paper is the how toconstruct approximations to the XC potentials that mani-

festly obey the zero XC force and torque condition. For thispurpose, we develop the concept of an XC action, obtain-ing the potentials as functional derivatives of such an action.This is similar to the concepts of DFT, where the XC poten-tial is functionally derived from an energy functional. Ouraction is a functional Su of the electron fluid velocity fieldur , t =jr , t/nr , t where nr , t and jr , t are the particleand current densities. It also depends on the initial electrondensity n0r. The reader may wonder why our action de-pends only on u and not on the current density j, which is thenatural variable in TDCDFT.10 We have found that workingwith the velocity field is easier to construct GI functionals.Furthermore, this is no violation of the Runge-Gross theorem

since knowledge of n0r and ur , t is sufficient to uniquelydetermine jr , t and nr , t for t0, by solving the follow-ing pair of equations the second of which is the continuityequation, which must hold in TDCDFT:

jr,t = nr, tur,t,

nr,t + jr,t = 0, nr,0 = n0r . 1.1

The critical point is that by ensuring that the action isGalilean invariant GI, the derived potentials are assuredto obey the zero force and torque conditions.9,11 But what dowe mean by a Galilean invariant action? It means that anytwo observers of the same physical system each using hisown coordinate system of reference will report the samevalue for the action. We define GI in detail in Sec. II. Gal-ilean frames can be translationally or rotationally accelerat-ing. Invariance in the first case is called translational invari-ance TI and in the second case, rotational invariance RI.We discuss this in more detail in Sec. II. Kurzweil and Baer 11

have recently developed a TI XC action and derived poten-tials that obey the zero force condition. Their XC action was,however, not RI and so did not enforce the zero torque con-dition. It is the purpose of this paper to further develop thetheory along similar lines, to achieve zero XC torque as well.

PHYSICAL REVIEW B 72, 035106 2005

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Nr,t = nRr,t,t = nRr,t,t = Nr,t. 3.4

Equations 3.3 and 3.4 show that Nr , t is indeed GI.Thus, a simple generic form for the action functional can beimmediately written as

S1u = s1Nu . 3.5

Note that what this equation tells us is that the action, being

a functional of the velocity field u, must depend on it onlythrough the functional dependence of N on u. This latterdependence is explicit, given by Eqs. 3.1 and 3.2

n0,u Rr, tnr,t

Nr,t. 3.6The functional s1 in Eq. 3.5 is any functional of its argu-ment. Once Eq. 3.5 is adopted, the question shifts to pro-ducing an appropriate form for s1. This form is chosen so asto produce specific known physical properties of a generalelectronic system. Since we will generalize Eq. 3.5 in Sec.III B, we will not dwell on the form of s1. We will discuss

the form of the more general case.

B. GI action depending on the ELT metric tensor

Looking for a more general form, we now consider theJacobian matrix of the ELT

Iij = jRir,t. 3.7

Here and henceforth we use the notation i /ri, i=1 , 2, 3. This matrix is TI, as can be straightforwardlyverified.11 However, I is not RI. Indeed, the following trans-formation, derived from the definition of the rotation, R=MtR, must hold

Ir,t = MtIr, t. 3.8

Although I is not RI, its determinant is, since det I=det M det I and det M= 1 because it is a proper rotationmatrix. One can then suggest a generic functional of theform

S2u = s2det Iu . 3.9

Comparing to S1 though, we find S2 contains nothing new.This is because the function Nr , t is directly related to theJacobian determinant. Indeed, the number of particles in afluid element must be constant so n(Rr , t , t)d3R= nr , 0d3r, and thus the ratio of volume elements, which is

the Jacobian is given by the ratio of densities

Jr,t1 detIr, t1 = Nr,t/n0r, 3.10

where n0r = nr , 0. Thus, the functional s1 in Eq. 3.5 canalso be thought of as a functional of detI.

Our first attempt to introduce an action in terms of theJacobian I yielded nothing new beyond Eq. 3.5. So, let usreturn to Eq. 3.8, which describes how the Jacobianchanges under rotations and search for another quantity thatis rotationally invariant. This leads us to consider the 33symmetric positive-definite ELT metric tensor

gr,t = Ir,tTIr,t. 3.11

It is immediately obvious from Eq. 3.8 and the orthogonal-ity of Mt that gr , t = gr , t. Thus g is RI. And since I isTI,11 it is also TI. We conclude that the metric tensor g is GI.

One can also see that g must be GI because of its geomet-ric content: the tensor g essentially tells us how to computethe distance dS between two infinitesimally adjacent fluid

elements, which originally started at r and r + dr, respec-tively,

dS2 = Rr + dr,t Rr,t2 = drT g dr . 3.12

Any two observers rotated or accelerated will agree onsuch a distance between any two electron-fluid parcels. Thisis because the observers, while changing position of theircoordinate axes origins and directions, still preserve the dis-tance scale. Thus we conclude again that gr , t must be GI.

The metric tensor is a natural quantity on which a genericaction can be defined. Thus, we consider the following classof metric-tensor actions:

Sn0

,u = sn0

,gu . 3.13

Here, n0 is the initial ground-state density assuming the sys-tem starts from its ground state. It is comforting to note thatin view of Eq. 3.10 and the fact that det I =det g, thegeneric action in Eq. 3.13 includes S1 in Eq. 3.5 as aspecial case.

Equation 3.13 tells us that the action depends on u onlythrough the dependence of g on u. This is an explicit depen-dence that we designate by

uR I g . 3.14

Writing Eq. 3.13 still tells us nothing about what the formof sn0 , g is. Before we discuss this issue, we first discuss

the method by which the GI potentials can be derived fromthe generic action in Eq. 3.13.

IV. XC VECTOR POTENTIAL

In this section we set out to derive the XC potential that isobtained from the generic metric tensor action by functionalderivation. It is important to realize, following van Leeu-wens work,14 that the action and vector potentials derivedfrom it must be defined using a Keldysh contour. TheKeldysh contour allows us to avoid causality problems inher-ent in any usual action formulated in terms of the densityor current density.

The Keldysh contour is a closed loop t in time domain,parametrized by 0 ,f, called pseudotime. A closedloop, means that t0 = tf. The use of Keldysh contours inthe context of memory functionals is explained, in detail, inRefs. 11 and 14. The action is formulated in terms ofpseudotime dependence . Any physical quantity is assumedto depend on . The potentials are then obtained as functionalderivatives at the physical time dependence. By that wemean that after the functional derivative is taken all quanti-ties are evaluated on the contour, replacing the variable byt.

GENERIC GALILEAN-INVARIANT EXCHANGE- PHYSICAL REVIEW B 72, 035106 2005

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a1mr, =1

2mul + lumiqli + qliilum

+ i mikl r,r,lukr,d3r+

1

lum + kukmliqli . 5.9Specializing to the homogeneous electron gas HEG, it ispossible to show that this form is compatible with the fol-lowing VK result for a HEG:10

ar, =1

2

fT u fL u , 5.10

where fL, n0 and fT, n0 are the HEG response kernels.This is achieved by taking

mikl r,r, =mi

kl r r 5.11

and having the kernels qli and

mikl obey

mikl + qlimk = mlik mkilfT fLimlk.

5.12

Note that for the HEG q is position independent since beinga zero order quantity, it depends only on n0 and n0 is positionindependent for the HEG.

Following the general ideas of Ref. 11 for constructing anaction beyond linear response, one can now choose a specificform for sn0 , g in Eq. 3.13 as

sg = gr,tNr,t,t tgr,td3rdtdt.5.13

By construction, this form is fully GI not just in the linearresponse limit. Furthermore, by choosing in conformity

with Eq. 5.12 we make this functional compatible with thelinear response properties of the HEG.

VI. SUMMARY AND DISCUSSION

Summarizing the present work, we have developed ge-neric forms for an action that yields potentials that are con-sistent with the zero force and zero torque condition. Theaction is based on the ELT metric tensor g. The metric tensoris also an important quantity when the Kohn-Sham equationsare presented in a Lagrangian system.16 The metric g is,therefore, emerging as an overall important quantity in anynonadiabatic TDCDFT scheme. The form of the action, Eq.3.13, may also include dependence on g, g, etc. In this casethe functional derivatives Q become differential operatorswith respect to time. More elaborate functionals can be builtfollowing similar ideas at the expense that additional gradi-ents are introduced the metric tensor already leads to poten-tials that use up to second gradients of the ELT Rr , t.

The present approach draws several ideas from the ap-proach of Dobson, Bunner, and Gross13 DBG, which alsoyields a GI method. However, our theoretical method and

results are more general. In particular, the DBG results areconsistent with only the longitudinal response of the HEG.Our method and results seem to differ significantly from an-other related work, namely, Tokatly and Pankratov17 TP.The TP approach constructs the GI XC potentials by usingthe concept of an XC Lorentz force that is the divergence ofa stress tensor. This latter property assures zero total forceand torque, thus constituting an elegant way of enforcingthese conditions. It is difficult to compare at this point theconnection between these theories. It seems that the func-tional derivative tensor defined in Eq. 4.1 is related to theTP XC stress tensor. Further work in assessing the two theo-ries needs to be done.

ACKNOWLEDGMENT

We gratefully acknowledge the support of the GermanIsrael Foundation.

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10 G. Vignale and W. Kohn, Phys. Rev. Lett. 77, 2037 1996.11Y. Kurzweil and R. Baer, J. Chem. Phys. 121, 8731 2004.12 G. Vignale, Phys. Lett. A 209, 206 1995.13 J. F. Dobson, M. J. Bunner, and E. K. U. Gross, Phys. Rev. Lett.

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