Basic time-independent density-functional theorems for ground states andexcited statesMel Levy Citation: AIP Conf. Proc. 577, 38 (2001); doi: 10.1063/1.1390177 View online: http://dx.doi.org/10.1063/1.1390177 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=577&Issue=1 Published by the AIP Publishing LLC. Additional information on AIP Conf. Proc.Journal Homepage: http://proceedings.aip.org/ Journal Information: http://proceedings.aip.org/about/about_the_proceedings Top downloads: http://proceedings.aip.org/dbt/most_downloaded.jsp?KEY=APCPCS Information for Authors: http://proceedings.aip.org/authors/information_for_authors

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Basic Time-IndependentDensity-Functional Theorems ForGround States and Excited States

Mel Levy

Department of Chemistry and Quantum Theory Group,Tulane University, New Orleans, LA 70118, U.S.A.

Abstract. Several basic time-independent density-functional theorems are reviewedfor ground states and excited states. In particular, the simple constrained-search for-mulation is utilized to prove the Hohenberg-Kohn theorem for degenerate as well as fornon-degenerate situations. Then, a time-independent Kohn-Sham theory is presentedfor an individual excited state, and first-order adiabatic connection perturbation theoryis compared with a common approximation within time-dependent theory for excitedstates.

INTRODUCTION

My main purpose here is to review a few of the proofs of fundamental time-independent theorems for ground states and individual excited states in density-functional theory (DFT). I have discussed time-independent ensemble theory [1-5]elsewhere [6].

Hohenberg and Kohn [7] proved their seminal theorems for non-degenerateground states. Subsequently these theorems were extended to include degen-eracies as well as non-degeneracies by utilizing the" constrained-search" approach[8-10]. This approach followed earlier work on non-interacting systems [11]. Thisconstrained-search formulation is so simple and transparent that I eagerly reviewit here. I hope this review appeals to non-specialists as well as to those special-ists who are not already familiar with the approach. I also list here general DFTreviews [12-20] that feature basic principles.

The time-dependent DFT formulation [21] has generated fruitful research onexcited-state properties within DFT [22-31]. What about the time-independentsituation for an individual excited state? Well, from the earliest days of DFT,excited-state energies have been approximated by simply utilizing the ground-stateexchange-correlation functional and by using holes below the highest-occupied en-ergy level, as needed, in the Kohn-Sham equations [32]. But, the proper time-independent exchange-correlation functional, for an individual excited state, is not

CP577, Density Functional Theory and Its Application to Materials, edited by V. Van Doren et al.© 2001 American Institute of Physics 0-7354-0016-4/01/$18.00

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the same as the ground-state exchange-correlation functional. With this in mind, Ihere review a recent exact time-independent Kohn-Sham theory for excited states[33], and properties of the effective potential are discussed. Then, a common ap-proximation within time-dependent excited-state DFT is compared with first-ordertime-independent adiabatic-connection perturbation theory [34-36].

Const rained-Search Derivation of First Hohenberg-KohnTheorem for Ground States

My purpose is to review the proof that a ground-state density, PGS, contains allthe information of the system. A constrained-search proof by construction shall beemployed,which encompasses both the non-degenerate and degenerate situations.The work in references [8-10] is presented.

Say that pcs is a ground-state density of

& = X>(ri)+f+ Vrec (1)

«=1

where T is the kinetic energy operator, Vee is the electron-electron repulsion oper-ator, and v is a local-multiplicative potential. The object, then, is to show thatH can be deduced from pGs5

and that H is unique within an additive constant.Accordingly, first note that PGS determines T and Vee because both operators areentirely determined by the number of electrons TV, which is obtained by simplyintegrating PGS- Next, whatever v is, observe that

N= f v(f)PGS(r)d3r (2)

J

with any antisymmetric wavefunction ^ that yields PGS- Consequently, since theright-hand-side of Eq. (2) depends upon only v and PGS, by the variational theorema ground-state wavefunction, ^GS, distinguishes itself, among those \£ which yieldPGS, as one that minimizes just

(3)

NSo, we have seen how PGS determines ^GS? which in turn determines ^ v(fi)uniquely, within an additive constant, though [10]

N

This completes the determination of H. (To arrive at expression (4), simply takeand divide by

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In summary, we have ascertained, as desired,

> H —* all the properties of H (5)

[Assume that more than one ground-state wavefunction yields our PGS- Then, eachof these wavefunctions has to be a ground-state of the in Eq. (1), because all thesewavefunctions clearly give the same value for expression (2) and the same valuefor expression (3)]. This concludes the proof [8-10] of the first Hohenberg-Kohntheorem. Observe that nowhere has it been assumed that DGS is a non-degenerateground-state density. Consequently, the above proof [8-10] has generalized the orig-inal Hohenberg-Kohn theorem [7] to include degeneracies.

Constrained-Search Derivation of VariationalHohenberg-Kohn Theorem for Ground States

Here I review the constrained-search proof [8-10] of the second, or variational,Hohenberg-Kohn theorem. Again, the constrained-search proof easily lifts the non-degeneracy requirement of the original Hohenberg-Kohn proof [7].

Start with the familiar variational theorem:

= min /* H* \ (6)

The above mimimization can be divided into an inner and outer one as follows:

or

EGS = minmin/* HP y-+p \

£Gs=min|min{* H

(7)'

(8)

Now assume that expression (1) is our Hamiltonian of interest. Then with recog-nition that

N

for all \£ —> p, it follows that

H

= / v(r)p(r)d3r

= jv(f)p(r)d3r + F(p]

with

(9)

(10)

(11)'

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The substitution of the right-hand-side of Eq. (10) into Eq. (8) gives

EGS = mm Uv(r)p(f)d3r + F[p}\ (12)

which completes the proof.For density p, F[p] is the universal Hohenberg-Kohn functional for the sum of

kinetic and electron-electron repulsion energies. The functional searches over all \I/that yields p and delivers the minimum in (T 4- Vee\. Hence, identification (11) isdesignated the "constrained-search" formulation of F. (The assertion of Eq. (11),taken from Refs. [8-10], was greatly influenced by a non-interacting counterpart ofEq. (11), involving just T, which was given on page 97 in Ref. [11]).

Eq. (11) assigns a number for each p. This number, in actual calculations,is of course meant to be obtained by approximating F without using many-bodywavefunctions. That is, accurate calculations currently approximate the exchange-correlation component of F by explicit functionals of D and employ non-interactingwavefunctions for the non-interacting Kohn-Sham kinetic energy component of F.(Ref. [37], in this volume, gives a valuable analysis of the present status of approx-imate exchange-correlation density functionals for the ground state.)

Universal Variational Bifunctional for Excited States

In this section we discuss a universal variational bifunctional [6] for excited states,FK[P,V], which is defined to be /\P T + Vee ^\ of that antisymmetric $ whichyields p, is orthogonal to the first K — 1 states of the Hamiltonian in Eq. (1), andsimultaneously minimizes (\? T + Vee ^V The functional Fx[p, v] always existsbecause there are an infinite number of different wavefunctions that yield an arbi-trary p and are simultaneously orthogonal to any finite number of excited states ofany Hamiltonian. Observe further that FK[P,V] is universal in K, p, and v. Withthe above definition of FK[P, v], it follows directly from the ground-state proof, inthe proceeding section, that

EK = mm || v(f)p(r)dsr + FK[p, v]} (13)

The bifunctional FK[P, v] generalizes earlier ones [38,39].In order to approximate Fx[p, v], without wavefunctions, as an explicit functional

of p, v, and Kj it is necessary to know its fundamental properties. The first propertycomes directly from the definitions of FK and F = FQ. We have

FK[p,v] > FK[p,v] > F[p] (14)

Notice that F2[p, v] is identical to the functional which gives the lowest energy of agiven symmetry [40], if the symmetry is different from the ground-state symmetry.

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A particular stringent requirement stems from [6,33]

EK > mmUv(r)p(r)d3r + FK[p,vf]\ (15)

where the strict inequality applies when vr differs from v by more than an additiveconstant. In other words, the idea is to create an approximate FK such that theright-hand-side of the Eq. (15) is maximized by v' = v. Related to requirement(15), it can be shown that

= 0 (16)6v'

where pvK is the K-ih eigenstate density associated with external potential v. Re-

quirement (16) should likely prove easier to invoke than requirement (15). I closethis section by observing that

= *W[p], (17)

so that

"^r"1d3r = 0 (lg)

where v is an arbitrary potential and c is an arbitrary constant. (In closing thissection note that the v in FK[P,V] may be replaced by the ground-state density ofv or a Kohn-Sham orbital that is obtained from a ground-state calculation).

Kohn-Sham Equations for Excited States

To facilitate the presentation, in this section we shall consider a non-degeneratefirst-excited state only, but the generalization to any state should be clear to thereader. To begin, we must define a proper non-interacting Kohn-Sham wavefunc-tion. With this in mind [33], and with consideration of the adiabatic connectiondiscussed later, define <£[/), v] as that wavefunction which is the first excited-stateof that non-interacting Kohn-Sham Hamiltonian whose ground-state density re-sembles PQ the closest in a least-squares sense (or in some other meaningful sense),where p$ is the ground-state density of our interacting H of interest [Eq. (1)]. Theni [p, v] is conveniently partitioned as

F^p, v} = Ts[p, v} + Vee[p, v} + Ec\p, v}, (19)

where

[M> (20)

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Ve (21)

The following corresponding Kohn-Sham equations follow:

1__9

where

v, =6Vee[p,v]

6pSEc\p,v]

P=PI 6p

(22)

(23)

Here pi is the first excited-state density of H. Simultaneously, pi is the first excited-state density of that non-interacting Hamiltonian with attractive potential va.

Requirements for Excited-State Kohn-Sham Potential

Consider the HamiltonianN

(24)

where va is constrained such that the first excited-state density of Ha is independentof a. This density is our p\. As a result, v\ is our v of interest in Eq. (1) and VQ isour excited-state Kohn-Sham vs in Eq. (23).

Since the excited-state density is fixed as a is varied, the long-range behaviorof the density is fixed. Consequently, the ionization energy from the first-excitedstate is fixed. From this fact, the formulation in Refs [34,35,15] lead to the factthat by taking the a —-> 0 limit [15], one obtains here [33]

(25)

where (pn is the highest-occupied Kohn-Sham orbital in system (22). For pointswith a > 0, one obtains relations comparable to Eq. (25), but instead involving theexcited-state Ec and its functional derivative, in addition to Vee and its functionalderivative.

Illustrative Calculations

In the exact case, it has been shown that [42-46]

/ = -6H (26)

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TABLE 1. lonization Energy (in Ry) from excited state, through high-est-occupied orbital energy (e#). (Ground-state energy of ion minus ex-cited-state energy of atom.)

AtomLi

Na

He

Ne

Configuration[He]2p[He]3s[He]3p[He]4s[He]4p[Ne]3p[Ne]4s[Ne]4p[Ne]5s[Ne]4dIs2p 3PIs2p IP[He]2p5 3s 3PfHeJ2p5 3s IP

-eH0.2620.1500.1160.0780.0650.2230.1440.1030.0760.0630.2700.2550.3570.346

Experimental0.2600.1480.1140.0770.0640.2080.1430.1020.0750.0630.2660.2480.3610.349

where CH is the highest-occupied orbital energy in equations (22). In other words,CH is the orbital energy of <pn- Further, I is the smallest ionization energy. Thatis, I is equal to the ground-state energy of the (N — l)-electron system minus theexcited-state energy of the ]V-electron system. Our calculations [33] in Table 1report e#.

In our calculations [33], we approximated vs by approximating the second termin Eq. (23) by zero and by approximating the first term by a modified KLI-type potential. That is ^f- was approximated by using holes, as necessary, belowthe highest-occupied orbital level, to give the correct excited-state configuration.Moreover, the potential was constructed to satisfy requirement (25), in the spiritof the KLI approach for ground states [47].

The results in Table 1 are encouraging. For He and Ne, a degenerate treatmentwas employed through a linear combination of determinants and the use of subspacetheory involving ensembles. Note that the exact highest-occupied orbital energiesgive the exact single-triplet splitting. That is,

E(triplet) — jE(singlet) = e(triplet) — e(singlet) (27)

where e(triplet) is e# in the triplet calculation and e(singlet) is CH in the singletcalculation. This splitting is approximated nicely in Table 1.

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TABLE 2. Root-Mean-Square Errors in Hartrees.

SystemHeLiBe

Ae0.00670.01670.0178

OPT0.01590.01480.0358

TD-ACPT0.00390.00560.0099

Connection Between Time-Dependent Approach andAdiabatic Connection Perturbation Theory for Excited

States

It has recently been shown [31] that excitation energies from the time-independent first-order adiabatic conneciton perturbation theory (ACPT), orGorling-Levy perturbation theory [34-36], are identical to those from time-dependent (TD) density functional theory when just the frequency-dependent ex-change kernel is employed in the Laurent expansion (the single pole aproximation)[22], thus generalizing the previously established two-electron equivalence [30] toany number of electrons. Table 2 illustrates the success of these two equivalent ap-proaches. The table, from Ref. [30], gives the root-mean-square errors (in Hartrees)from virtual and occupied orbital energy differences, Ae, and from ordinary per-turbation theory (OPT), where the perturbation is the H in Eq.(l) minus theground-state Kohn-Sham Hamiltonian. For ACPT in the table, the perturbation is

NH plus £} vc(fi) minus the ground-state Kohn-Sham Hamiltonian, where vc is the

1=1ground-state correlation potential. The presence of vc in the above sentence mayappear surprising. However, it is the very presence of this vc which is crucial for thesuccess of ACPT, which keeps the ground-state density fixed at each order in theperturbation expansion. First order results are shown for both OPT and ACPT.

The excitation energy, Ek — EQ, is usefully expressed as

Ek — EQ = IQ — Ik (28)

where Jj, equal to EQ~I — Ej with j = 0 or j = fc, is the ionization energy from thejth state of H. Also, EQ is the TV-electron ground-state energy (previously calledEQS) of our H in Eq. (1) while E$~lis the ground-state energy of H with oneelectron removed.

In terms of Eq. (28), the success of TD and ACPT stems, in part, from the factboth of these formalisms give IQ exactly, and IQ is larger in magnitude than /&.Further, in TD-ACPT the error in /& is essentially due to only correlation effects.In contrast, while Ae gives IQ exactly, the error in the Ik component with Ae arisesat the larger exchange level. With OPT, neither /& nor IQ is obtained exactly.

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