Shlomit Jacobi and Roi Baer- Variational grand-canonical electronic structure method for open...

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Variational grand-canonical electronic structure method for open systems

Shlomit Jacobi and Roi Baera

Department of Physical Chemistry and the Lise Meitner Center for Quantum Chemistry, the Hebrew

University of Jerusalem, Jerusalem 91904 Israel

Received 18 April 2005; accepted 17 May 2005; published online 3 August 2005

An ab initio method is developed for variational grand-canonical molecular electronic structure of

open systems based on the GibbsPeierlsBoguliobov inequality. We describe the theory and apractical method for performing the calculations within standard quantum chemistry codes usingGaussian basis sets. The computational effort scales similarly to the ground-state HartreeFockmethod. The quality of the approximation is studied on a hydrogen molecule by comparing to theexact Gibbs free energy, computed using full configuration-interaction calculations. We find theapproximation quite accurate, with errors similar to those of the HartreeFock method forground-state zero-temperature calculations. A further demonstration is given of the temperatureeffects on the bending potential curve for water. Some future directions and applications of themethod are discussed. Several appendices give the mathematical and algorithmic details of themethod. 2005 American Institute of Physics. DOI: 10.1063/1.1949202

I. INTRODUCTION

Electronic structure of open systems in contact with aheat and electron reservoirs is important in a variety of ap-plications in chemistry, biology, solid state, nuclear, and laserphysics as well as astronomy and materials science.121

Simulations, usually done using molecular dynamics MD,typically assume an integer average number of electrons con-fined to their ground state. At high densities and tempera-tures this is often unjustified. Another case of interest is inembedding theories; where assuming a closed system leadsto a slow convergence because of the need to considerablyincrease the size of the embedded cluster. The fact that veryfew studies allow for open system character is often due tolack of availability of reliable methods to compute electronicforces on nuclei in open system conditions.

This paper develops a theory analogous to the HartreeFock ground-state method,22,23 allowing a reliable variationalcalculation of the electronic structure for open systems. Weconsider a molecular system which is in thermodynamicequilibrium with a heat reservoir, characterized by a tem-perature T=kB

1, where kB is Boltzmanns constant, andan electron reservoir, characterized by a chemical potential. As discussed below, the average force on nuclei in suchsystems is conservative, as it is when electrons are confinedto their ground states within the BornOppenheimer ap-

proximation, but the potential function is not the internalenergy. Instead, it is the Gibbs free energy G, throughwhich all ensemble averages can be determined in Sec. III Bwe explain this.

The ab initio method presented here yields a variationalapproximation to the Gibbs free energy we call the optimizedeffective free energy OEFE. It can be straightforwardly ap-plied using standard ab initio quantum-chemical codes. The

OEFE is of comparable quality to the HartreeFock energy

for the electronic ground state. Both methods use an auxil-iary noninteracting system for computing observables andenergies. In that sense, we may say that the present methodneglects electron correlation. This effect can be introducedinto the framework we describe using correlation densityfunctionals, as in the KohnSham theory, which is valid foropen systems in thermodynamical equilibrium.24 Despite thislack of correlation, the attractive feature of the OEFEmethod is its variational weighting of all charged states andspin states. Thus, expensive excited-states methods are notneeded here.

We discuss the OEFE theory in some detail and deriveall results needed for an implementation using a standardquantum-chemical program. The formulation in terms of agrand-canonical ensemble greatly facilitates the derivations.It allows relatively simple and doable expressions for thevarious average quantities and gradients needed to minimizethe OEFE. Issues, such as calculation of gradients, ensembleaverages, and effective forces on nuclei, are all discussed atsome length. Several appendices, carrying some of the moretechnical derivations, are given. We also discuss an extensionof the method for computing an approximate Helmholtz freeenergy.

II. THE OPTIMIZED EFFECTIVE FIELD

The nonrelativistic Hamiltonian of electrons in a mol-ecule can be generally written in atomic units as

H= n

n22

+ extrn n,A

ZA

rnA+

1

2 nm1

rnm

T +1

2 nmrnm, 2.1

where lower and upper case indices are used to label coordi-

nates of electrons and nuclei, respectively. T is the one-bodyaAuthor to whom correspondence should be addressed. Fax: 972-2-

6513742. Electronic mail: [email protected]

THE JOURNAL OF CHEMICAL PHYSICS 123, 044112 2005

0021-9606/2005/1234 /044112/12/$22.50 2005 American Institute of Physics123, 044112-1

Downloaded 04 Aug 2005 to 169.232.76.24. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
http://dx.doi.org/10.1063/1.1949202


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part of the Hamiltonian, including kinetic energy and inter-action energy with external fields ext and the nuclei, whiler= r1 is the Coulomb repulsion potential between allpairs of electrons. We will use a second-quantized formula-tion in which all quantities are described with reference to anorthonormal basis of M real single-particle orbitals ir, i= 1 , 2 , . . . ,M. The basic operators are the electron creationand annihilation operators. The operator cjs

creates an elec-

tron of spin s or in orbital j, while cjs annihilates anelectron in this same state. Because electrons are fermions,these operators have the well-known anticommutationrelations25 cjs

, cjs+ =ssjj, cjs , cjs+ =cjs , c

sj

+=0.Using the creation and destruction operators, the Hamil-tonian can be written as

H = ij ,s

Tij cis

cjs +1

2 ijkl,s,s

Vijklcis

cks

clscjs . 2.2

The parameters of the second-quantized Hamiltonian arethen the one-electron integrals,

Tij = irTjrd3r, 2.3and the two-electron integrals,

Vijkl = irjrr rkrlrd3rd3r. 2.4Since the matrices T and V of Eqs. 2.3 and 2.4 defin-

ing the interaction do not reference spin, it is thus useful toeliminate explicit reference to spin from the formalism. Letus then introduce the density-matrix operator with elements

ij

=

s=c

is

c

js

, ij

= ji

. 2.5

The Hamiltonian of Eq. 2.2 can then be written directly interms of this density matrix as follows:

H = ij

tijij +1

2ijklij Vijklkl , 2.6

with

tij = Tij 1

2lVilkj , 2.7

Next, we introduce a shorthand notation defining the com-posite index I=i j sometimes we will also write K=kland write the Hamiltonian of Eq. 2.6 in the following way:

H = tT + 12TV. 2.8

Here, t and are considered column vectors of dimensionM2 in the index I, respectively, consisting of the elements,

tl = tij , I = ij . 2.9

The M2M2 matrix V elements are defined as

VIK = Vijkl , 2.10

and we use the mathematical notation,

ATB = I

AIBI. 2.11

A. The Gibbs free energy

The Gibbs free energy henceforth, simply free energyin the given basis26 is

G, = 1 ln Z,, 2.12

where is the inverse temperature, is the chemical poten-tial, and Z, is the partition function of the ensembledefined as

Z, = N=0

2M

eN trNeH = treH

N. 2.13

The trace trN is over all antisymmetric N-electron states. Thetrace with no subscript is over all 22M antisymmetric states

with all possible number of electrons the second equality isvalid for because H conserves the number of electrons. Be-cause of the Pauli principle, the maximal number of elec-trons is 2M, since at most two electrons can be inserted into

a single orbital. The ensemble average of an observable O

can be computed by

Oexact = Z1 treH

NO , 2.14

the symbol exact denotes an exact ensemble average in theinteracting system, as opposed to averages for a noninteract-ing system, denoted as discussed below. In particular, the

ensemble average number of particles N=i=1

Mii is given by

Nexact = G

, 2.15

and the average energy is

Hexact =G

+ Nexact. 2.16

The second derivative of the free energy is proportionalto the average fluctuation in the electron number,

2G

2= N Nexact

2exact 0. 2.17

Since the fluctuation is always positive, we find from 2.15and 2.17 that G as a function of is monotonic nonincreas-ing and convex upwards. Furthermore, Nexact is a mono-tonic nondecreasing function of . Similar considerationsapply to G as a function of.

The physical meaning of G is in the work done on theelectrons by external fields coupled through ext in Eq. 2.1.For example, suppose a nucleus is moved adiabatically veryslowly and at constants and in the molecule from aposition R to R+R. The work that needs to be invested forthis process is the scalar product of the average electronicforce and the displacement,

044112-2 S. Jacobi and R. Baer J. Chem. Phys. 123, 044112 2005



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W= HR

exact R . 2.18

Using Eqs. 2.122.14, it is straightforward to verify thefollowing relation:

HR

exact=

G

R. 2.19

Note, when taking the derivative, the fact that H/R and H

are noncommutative is not an issue because the trace of aproduct of operators is invariant to cyclic permutations. Wethus conclude that

W=G

R R , 2.20

showing that the average electronic force on the nuclei is aconservative field as it is the gradient of G. The differencein free energy of two completely different nuclear configu-rations G = GR2 GR1 is therefore equal to the workthat needs to be done on the electrons to adiabatically move

the nuclei from R1 to R2. This interpretation of the free en-ergy as the work function is standard in thermodynamics. Weuse it below to give a definition of the grand-canonical av-erages that can be generalized to cases where only the ap-proximate free energy is known.

B. The variational approximation

Because of the electron-electron interactions, calculatingthe grand-canonical partition function Z, is practicallyimpossible for any but the simplest model systems. However,if the electrons are noninteracting, there is a simple textbooksolution. Let us introduce a noninteracting electron system

with Hamiltonian,

H0 = hT tT + uT, 2.21

where H0 is composed of the same one-body term as H inEq. 2.8, but there is an additional one-body field U. If thematrix h = t+ u has eigenvalues 1 ,2 ,..., then the grand-

canonical partition function Eq. 2.13 with H replaced by

H0 is simply the FermiDirac product function,25

Z0, = l=1

M

1 + el2, 2.22

where the square of each term is a result of the maximaldouble occupancy of each state as dictated by the Pauli prin-ciple. This simple expression can be written in a compactform as17,27,28

Z0, = det1 + eh2. 2.23

Such a formula is extremely useful because computing thedeterminant and exponential of a matrix, as in Eq. 2.23, isconsiderably more economical than summing over 22M

terms, as suggested in Eq. 2.13. Using the determinantform, the ensemble averages of one-, two- and three-particleoperators are tractable, as shown below. It should be notedthat the grand-canonical trace automatically sums over all

spin states and charged states of the molecule here we usethe term molecule in a wider than usual sense; by it we meanall ionic electronic states for the same nuclear configuration.Note also that evaluating a canonical ensemble partitionfunction Q,N=trNe

H0 is actually more demanding thanevaluating the grand partition function. The latter step in-volves OM3 operations; the canonical trace sums over allN-electron states, which for a basis of 2M spin orbitals in-

volves 2MN operations. Nevertheless, many interesting prob-lems involve a system which has a well-defined electron

number, but is coupled to a heat reservoir. In this case, it mayadvantageous to obtain the canonical energy from the grand-canonical free energy, as discussed in Appendix A.

Since noninteracting fermions are easily handled, it isnatural to ask how to approximate the interacting system bya noninteracting one? One simple-minded approach wouldbe to have u relate to V through

H H0 = 012 VJKJK uJJ = 0. 2.24

Here and henceforth, the angular brackets are used to

denote average with respect to H0

in contrast withexact

in

Eq. 2.14 for H,

O = Z01 treH

0N

O, 2.25

where Z0 =treH0N

. Equation 2.24 is a problematicapproach. For one thing, the number of solutions is infinite;which one shall we use? Furthermore, it is desirable tosearch for some form of an optimizedapproximation.

Thus we go in a different route, invoking the GibbsPeierlsBoguliobov inequality2932 yielding an upper boundapproximation, denoted , for G,

G,, G0, + H H0, 2.26

where G0 =1 ln Z0 is the free energy of the noninteracting

electron system. The new quantity , is called hence-forth the effective free energy. By varying h or equivalently

u in H0, we can minimize , thus finding the optimized

effective Hamiltonian H0 =hT and the corresponding OEFE

the best variational approximation to G,. Once h isdetermined, u is defined and is called the optimized effective

field, allowing for the two-body interaction VIJ in H.

III. THE OPTIMIZED EFFECTIVE FREE ENERGY

A. Determination of the optimized effective field

We now turn to the practical issue of how to actuallycalculate the optimized effective Hamiltonian h. Since t isknown, it is more convenient to work in terms of the fieldu= h t. Thus we search for the field minimizing the effectivefree energy,

u = G0u + 12TV uT . 3.1

The first step is to compute the gradient of with re-spect to a field matrix element uI uii. It is shown in Ap-pendix B that such a derivative results in the following ex-pression:

044112-3 Grand-canonical electronic structure J. Chem. Phys. 123, 044112 2005



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uI=

1

2VJKIJK IJK uJIJ IJ .

3.2

Here and below we use the convention that repeated doubleindices are summed over. The operator I is a -averageddensity matrix given in Eq. B3. The actual evaluation of thecorrelated averages appearing in Eq. 3.2 is discussed in

Appendix C and the method of evaluating the various matri-ces is discussed in detail in Appendix D.

We write Eq. 3.2 in the equivalent, but more instructiveform as follows:

uI= IH H0 IH H0 . 3.3

At the minimum the following condition is met:

u= 0. 3.4

Thus, using Eq. 3.3, the optimized field is determined from

the following set of equations to be satisfied for each I:IH H0 = IH H0 . 3.5

Note that under the noninteracting ensemble, any one-body

operator OII which commutes with the optimized H0 is sta-tistically uncorrelated with the energy difference operator

H H0. This is an immediate consequence of the fact that in

this case O=O see Eq. B3. In particular, we have for the

number of electrons N and H0 itself,

NH H0 = NH H0, 3.6

H

0H

H

0 = H

0H

H

0 . 3.7In deriving the above results, we have here introduced u

as a convenience. We could have used h as the variationalfield. This would make no essential difference and, in fact,since h is directly determined by u, the following relationholds, in addition to Eq. 3.4:

h= 0. 3.8

At low temperatures and for a value of thatyields an integer average electron number N, h becomesequal up to a constant to the HartreeFock matrix the Fo-

ckian and u becomes equal to the matrix of the directHartree and exchange Fock operators. This is be-cause when the ground-state Slater determinants 0of H0 dominates the partition function trace. Thus the opti-mal h must yield the determinant that lowers the expectation

value of H to its minimum, i.e., the HartreeFock determi-nant. The relation between the optimized effective free en-ergy and the HartreeFock energy becomes in this limit,

u + N

0tT + 12

TV0. 3.9

Yet, at higher than zero temperatures, the method departsfrom the HartreeFock method.

The overall complexity of the method is OM4. In thispaper we developed the method in a orthonormal basis, sothere is an additional OM5 step to transform the two-electron integrals. However, a similar theory within theatomic basis can be developed which scales as OM4, i.e.,the same complexity of the HartreeFock method within asimilar basis set. This is attractive, given that the presentmethod treats all excited states and multiplies charged states

in a variational way.

B. Effective ensemble averages

What is the consistent way of evaluating expectation val-ues in the variational approximation? This is not a trivialissue. For example, which is a more consistent way of evalu-

ating the average number of electrons N ? Is it

1

Z0treH

0N

N or

? 3.10

The left option is the average value of the number operator inthe noninteracting system. The right option is the analog of

the exact relation given in Eq. 2.15.The way we approach this question is by making a con-nection between average values and the free energy. Oncesuch a connection is made, we can use the variational ap-proximation to compute averages. For this purpose, note that

the exact expectation value of an arbitrary operator O, givenin Eq. 2.14, can be interpreted in terms of work as follows.Let us introduce an external field s which couples to the

electronic system through the n-body operator O. Under agiven value of the field, the free energy is

G,,s = 1 ln treH+sON. 3.11

The work done on the electrons when the field ischanged adiabatically from value s1 to value s2 isG,= G, , s2 G, , s1. Therefore, from Eqs.2.14 and 3.11,

Oexact = Gs

s=0. 3.12

Thus emerges an interpretation of the ensemble average

value of O, it is the rate of work done on electrons whenturning on the field s. Once we have expressed any averagevalue in terms of work, we can use the optimized effectivefree energy, which we postulate to also have the meaning of

work, to produce an effective free-energy approximation foraverage values.We now show that it is consistent with this work inter-

pretation to simply take all averages in the noninteractingparticle system. When the energy due to the field s is intro-duced, the definition of the energy becomes

,,s = G0,,s + H + sO H0, 3.13

where H

0 = hT and h is the optimized Fockian matrix in the

presence of the perturbation. Of course, this optimized Fock-

ian now depends on s, however, because is insensitive to

changes of h at optimum see Eq. 3.8, we find




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s

s=0=

s

h

s=0

. 3.14

On the right, the derivative with respect to s is taken while h

is held constant at its s =0 value. Once h is held constant,only terms explicitly depending on s in 3.13 contribute to

the derivative. Since sO is the only one such term, we find

s s=0 = O, 3.15which is consistent with the exact expression in Eq. 3.12.

The conclusion from Eq. 3.15 is that all ensemble av-erage values in our approximation are to be taken in thenoninteracting system,

O =1

Z0treH

0N

O. 3.16

We now return to the question posed in Eq. 3.10;which of the two expressions are we going to take? Our

discussion above shows that it is the left one. What about theright one? Can we not use the analog of Eq. 2.15 in theoptimized effective free energy? Let us take the derivative of with respect to . We obtain from Eq. 2.26, after somemanipulations,

N +

= H H0N H H0N . 3.17

Using Eq. 3.6, we see that the left-hand side vanishes andindeed both expressions in Eq. 3.10 are equal and thuscorrect.

C. The effective force on nuclei

The fact that is a variational approximation to G al-lows us to use it as an approximate work function. When anucleus is pushed adiabatically from R to R+R the effec-tive average work done on the electrons is approximated by

W= HR R . 3.18

In the accurate calculation, the average force is conservative,i.e., the gradient of the exact free energy Eq. 2.20. Be-

cause is a variational approximation to G, the effectiveelectronic force on the nuclei H/R in Eq. 3.18 is con-servative as well. To see this, let us first note that the averageforce can be written as

HR = Z01 treH0H

R

=

RZ0

1 treH

0Hh, 3.19

where the last derivative is taken with h or equivalently H0held constant. We thus find

HR = E

R

h

. 3.20

Taking the derivative ofE= H at constant h is easily shownusing Eq. 3.8 to be equal to the derivative of the effectivefree energy, arriving at the result

H

R=

R

. 3.21

We thus find that the effective force is conservative and thework associated with it is the optimized effective free energy.

IV. EXAMPLES

After presenting the details of the theory and numericalalgorithms, we now turn to a demonstration of the formalismon actual molecular systems. We used standard Gaussian ba-sis sets and computed the T and V matrices Eqs. 2.4 and2.7 using the program GAMESS.33 Note that from these we

need to compute the matrix t using Eq. 2.7. The numericalsolution of Eq. 3.5 is obtained by invoking a minimizingalgorithm for , using the gradient of Eq. 3.2, based on theBroydenFletcherGoldfarbShanno algorithm BFGSmethod implemented in Ref. 34.

Since the scaling of the present approach is the same asthat of a HartreeFock calculation, the two methods havecomparable computation times. The HartreeFock Fockian isa good initial guess for the minimizing algorithm at high low temperature. Often it is necessary to perform severalminimizations, starting from high value of, gradually low-ering to the desired final value. At each minimization wetake the field obtained from the previous iteration as an ini-

tial guess. Thus, the total cost of an effective free-energycalculation is a few times that of a HartreeFock computa-tion.

We first treat in subsection IV A a system which can betreated exactly, i.e., H2 in a small double-zeta basis. Wethen describe in subsection IV B using the new method theeffect of temperature on the H2O bending motion.

A. The hydrogen system

In order to assess the quality of the approximation, weused it in a simple setting, where exact results are available.We examined the H2 molecule internuclear distance of

0.7 within a standard double-zeta DZV basis havingM=4 orbitals. The maximal number of electrons is thus2M=8 leading to a total of 22M=256 electronic states. Theexact within the basis set free energy G, is computedby full configuration interaction, i.e., calculations of all fullycorrelated 256 states using the GAMESS code.33 This free en-ergy is plotted in Fig. 1 as a function of the chemical poten-tial for several values of . The function is monotonicnonincreasing and convex upwards, since G/ is theelectron number, a non-negative quantity Eq. 2.15 andsince 2G/2 is negative Eq. 2.17.

The exact average electron number Nexact, obtainedfrom the full configuration-interaction FCI calculation is




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shown in Fig. 2. At low temperatures the staircase shape isevident where only quantized integer values of N are al-

lowed. The width of the stair multiplied by the corre-sponding quantized N is the free energy necessary to add anelectron going from N to N+1 electrons. At higher tem-peratures this interpretation is only approximate and gradu-ally the stair structure is destroyed.

The quality of the approximation can be assessed byconsidering the correlation free energy, G, =G,,. To facilitate the comparison, we plot G, asa function of N in Fig. 3. At the N0 and N2Mlimits, G0 and the result is exact, a general rule. As the N=1 and N=2M1 become exact as well, since fora single electron/hole there is no electron correlation in theground state. Furthermore, in the low-temperature limit , G approaches the ground-state correlation energy,i.e., the difference between the HartreeFock energy and theexact ground-state energy, which, for N=2 is around 0.03Ehfor this basis set. In general, when the temperature increasesthere is a larger difference even when the average electronnumber is N=1. We see in Fig. 3 that at not too high tem-

peratures, the maximal differences appear at odd integer val-ues of N and minimal differences at even values of N. Thisshows that structures with dominant open-shell character are

not described as well by the approximation. This is in spiteof our use of a spin-unrestricted formalism, that is, the and components of u are not constrained to be identical.

The difference between the exact average number ofelectrons and that derived from the approximation is shownFig. 4. At not too high temperatures the minimal differencesin electron count appear at integer electron count whilemaximal deviances appear at transition regions between thestaircase steps in Fig. 2, i.e., at half integer electron numbervalues. Here, the average charge of the system changesabruptly and thus there is high sensitivity to any approxima-tion. At high temperatures the errors in average charge aresmaller and smoother.

We now examine the potential-energy curve of H2. Wecompare several values of the temperature at a given chemi-cal potential in Fig. 5 top. At sufficiently high temperaturesthe bond length is affected as the temperature grows. At thehighest temperature shown =6Eh

1, T=52.6103 K thebond length has grown to 0.84 . As the temperature is

FIG. 1. The exact free energy for the H2 molecule in DZV basis as afunction of the chemical potential for several values of .

FIG. 2. The number of electrons as a function of chemical potential forseveral temperatures for the H2 molecule in a DZV basis.

FIG. 3. The difference between the exact and optimized effective free en-ergy for the H2 molecule in a DZV basis set.

FIG. 4. The difference between the exact and approximate number of elec-trons as a function of the number of electrons for several values of thetemperature same legend as Fig. 3.




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raised, the potential curve slightly flattens. This is due to thepopulation of the lower excited states which have antibond-ing or repulsive character. The electron count decreaseswith increasing bond length. Interestingly, for a given bondlength, the electron count as a function of temperature isnonmonotonic. At 32Eh

1 the electron count is practically2. When =16Eh

1, N is slightly less than 2 and continues todecrease, until 7.5Eh, where a further decrease in causes a fast growth in N. At =4Eh

1, N2.01 and at =2Eh

1, N 2.2. Initially, when the temperature grows, thehigher lying ionized H2

+ states become accessible and themolecule tends to loose electronic population. At still higher

temperatures a dense manifold of negatively charge states isbeing populated and thus the number of electrons grows.To summarize, we have shown that the OEFE is a rea-

sonable approximation to the exact free energy. Several ob-servables were studied and compared, showing a good agree-ment. In particular, we showed that high temperaturesstrongly affect the shape of the free-energy potential curve asa function of nuclear separation.

B. The water system

We now examine a slightly more complex system,namely, the water molecule. For this molecule a DVZ basis

was used, taking a fixed OH bond length of 1.8a0 and vary-ing the HOH angle . We chose a chemical potential of =0.1Eh, since it assures an average of ten electrons at lowtemperatures we discuss the electron number below. Wecomputed the effective optimized free energy as a function ofthe bonding angle and the temperature . The results are

shown in Fig. 6. The potential curve is essentially identicalwith the unrestricted HartreeFock ground-state potential at32Eh

1. As is reduced temperature grows the potentialchanges, at first flattening and then developing a new mini-mum at a HOH angle of 125. It is interesting to notethat the =4Eh

1 curve is more flat than the =2Eh1 curve,

although both show a minimum at the same place.The dependence of the number of electrons on the tem-

perature is examined in the bottom panel of Fig. 6. It is seenthat the average number of electrons grows with increasingtemperature. This behavior is different from the H2 resultwhere the number of electrons initially drops. The exact be-

havior ofN as a function of is sensitively dependent on thestructure of the neutral and ionic anionic and cationic en-ergy manifolds. Interpretation in simple terms of these re-sults is thus not an easy task.

The potential curves in Fig. 6 are made for constant . Itis interesting to compare with the potential for a processwhere N rather than is kept constant. The former is theapproximate Helmholtz energy ,N=+N, discussedin some detail in Appendix A. In Fig. 7 we compare the and potentials for =8Eh

1. The value of is computed bysearching for the value of for which N=10, using aninterpolation between three values of N, for =0, 0.1,and 0.2Eh. In general the and potentials can differ

FIG. 5. The effective free-energy potential curve top and average electronnumber bottom for H2 at =0.2Eh shown for various temperatureparameters.

FIG. 6. The bending potential curves top and average electron numberbottom for H2O for several values of. Chemical potential is =0.1Ehand OH distance is 1.8a0. Potential curves are shifted to a common minimalenergy to facilitate comparison.




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substantially, however, in this specific case it is seen thatboth are very similar, most likely because of the smallchange in number of electrons at =8Eh

1.

V. PROSPECTIVES

In this article we focused on a general account of theprinciples of the method. We have only given a small set ofdemonstrations. In future publications we intend to use thetheory for applications where the open system nature of themethod is important. Several such prospect cases are brieflydiscussed here.

A. Molecular dynamics at high temperatures

Molecular dynamics at high temperatures may be of in-terest in many applications, such as studies of shockwaveduring explosion, dynamics of hot plasmas and flames,1,2,6,7

and dynamics ensuing exposure to strong laser fields.35

Byhigh temperatures we mean those for which kBT is compa-rable to the energy differences between the ground and low-lying excited electronic states. If the electronic manifold issufficiently dense at these temperatures, the electronic relax-ation processes should be fast and it may be reasonable toassume that nuclei are acted upon by an average force of theelectrons. In such regimes the free energy GR1 ,R2 , . . . actsas an effective potential surface. We have seen that this po-tential surface is highly sensitive to the temperature.

B. Improving QM/MM and embedding in metals

The present method will be helpful for various embed-ding methods. For example, when a molecule is adsorbed ona metal surface, it is often modeled by a small cluster ofmetal atoms.10,36 By enlarging the cluster it is hoped that theadsorbed molecule properties will converge. However, usu-ally this is not the case and extremely large clusters need tobe used. Since the electronic screening is very effective inmetals, the convergence problems are not a result of long-range interactions. Instead, at least part of the problem hasmore to do with the nonlocal nature of electrons in the metal.The slow convergence of the adsorbate properties may bedue to the fact that any finite cluster, since it is part of alarger system, has a noninteger number of electrons on the

average. Most methods, however, enforce integer number ofelectrons for any finite cluster and therefore a huge cluster isrequired. One embedding method not suffering from this dis-advantage is the dipped adcluster model.12 The method pre-sented in this paper can be applied in a similar manner, andbeing a rigorous theory, it is likely to yield even better re-sults.

Quantum-mechanical/molecular mechanical QM/MM

methods3741

also use various types of embedding schemes.They have become a most popular approach for theoreticalstudies of biochemical systems. In the QM/MM approach, alarge biochemical molecule such as a protein is dividedinto two parts. A small active site which is of interest be-cause of its important reactive biological role and a muchlarger part which has non-negligible effects on the electronicstructure of the active site via direct structural interactionas well as long-range electrostatic interactions. In QM/MMthe bonds of the active site are disconnected from the rest ofthe molecule. The resulting dangling bonds are then satu-rated by, e.g., hydrogen atoms. The chemically isolated reac-tive site is still exposed to the existing external electrostatic

fields of the environment usually modeled as point charges.The active site electronic structure is then determined quan-tum mechanically by an ab initio or semiempirical electronicstructure method.

What present QM/MM approaches miss is once again,the fact that the isolated QM part should have the chemicalpotential of the environment. This means that it may be onthe average partially charged. Because the effect of chemi-cal potential is not taken into account in most QM/MM cal-culations researchers may have to increase the size of theQM part, making their calculation expensive.

C. Self-Interaction correction in density-functionaltheory

There are a few density-functional applications within anelectronic Helmholtz free-energy15,42,43 formalism. These arebased on the ensemble density-functional theory44 DFT andthe underlying exchange-correlation XC functional is localsuch as the local-density approximation45 or the generalizedgradient approximation46. The developments are consider-ably simpler than here due to the locality of the functional.The present method can be adapted quite straightforwardlyto deal with DFT. However, it is more powerful than that. Itis well known that excited states and anions cannot be rea-

sonably approximated within local DFT because of the self-interaction problem.4749 Correcting for this problem willtypically necessitate introduction of explicit so-called ex-act exchange into the XC functionals.49,50 The presentmethod has no problem in accommodating such a self inter-action correction. In DFT, such nonlocal functionals are of-ten treated within the optimized potential method of Talmanand Shadwick.51 The formalism in this paper can be changedto accommodate the search for an optimized effective poten-tial in these cases. This is done by constraining the search fora minimum ofu only to matrices u that represent poten-tials, i.e., commutative with the electron position operators x,y, and z. Such constraints can be enforced using, e.g., the

FIG. 7. Comparison of the potential curve approximations based on theGibbs free energy and the Helmholtz free energy . The curves areshifted to a common minimal energy.




9/12

augmented Lagrange multiplier technique.52 Thus, thepresent approach can be viewed as a precursor for a generalframework treating various functionals with local, nonlocal,and explicit exchange functionals at various temperatures.The ground-state results can be obtained by taking the zero-temperature limit.

D. Molecular electronics: Molecular bridges

under biasThe conductance of electricity through a single molecule

chemically connected to two metallic leads has been the sub-ject of intense research in recent years.13,14,19,5357 The com-mon theoretical description assumes that the macroscopicleads are each characterized by a chemical potential L forthe left lead and R for the right lead. The electricity flowsas a result of the difference in the chemical potentials. Thedifficult part of the calculation is to account for the chargedistributions on the bridge molecule under this chemical-potential bias. The present formalism makes a step towards aself-consistent approach in this problem. While here we aredescribing only a system with one chemical potential, theformalism has the capacity to be extended to describe thesystem with two chemical potentials. This will be done byselecting two sets of localized basis function for each sub-system.

E. Shift for the open system auxiliary fieldMonte Carlo

Since the present method already gives good free-energyestimates, the additional correlation-energy corrections canbe made using various many-body theories. One way is touse the shifted contour auxiliary field Monte CarloSCAFMC method.28,5860 In this method, a

HubbardStratonovich61,62 transformation is invoked, lead-ing to a multidimensional integral for the free energy, whichis then estimated using Monte Carlo methods. A critical ad-ditional component is the shifted contour,59 which reducesthe strong statistical fluctuations. In zero-temperature calcu-lations the shifting field is taken from the HartreeFock ap-proximation. Such an approach has lead to successful appli-cations of SCAFMC in a variety of cases.5860,6366 Themethod presented in this paper can give a good contour shiftfor application with SCAFMC in open system conditions.Application of this idea is currently being studied in ourgroup.

VI. SUMMARY

We have presented a variational method for treating theelectronic structure of molecules under conditions of an opensystem characterized by a chemical potential and a tempera-ture. The method builds an effective optimized free energyfor the electrons, which allows approximate determination ofthe average electronic properties and effective forces on thenuclei. We have demonstrated the theory on two small mo-lecular systems. It should be noted that both these systemshave rather large energy spacings excitation energies of theorder of 10 eV. Thus temperature effects are observedmainly in the T30 000-K regime 10Eh

1. Still, many

systems have a lower excitation energy manifold and thusstrong temperature effects can be observed even at muchlower temperatures. In metals, for example, temperature ef-fects will be strong even below room temperature.

The numerical effort in the method we presented scalesas OM4, similar to the complexity of the HartreeFockmethod within Gaussian basis-set methods. It is probable thatthe relatively high scaling can be reduced dramatically using

linear scaling techniques developed for HartreeFock anddensity-functional theory,52,6769 which are based on the spar-sity of the density matrix 1, a feature enhanced at hightemperatures.

ACKNOWLEDGMENTS

This work was supported by the Israel Science Founda-tion. We thank Professor Ronnie Kosloff for stimulating dis-cussions.

APPENDIX A: HELMHOLTZ FREE ENERGY

The grand-canonical ensemble only allows for a chemi-cal potential to be specified. The number of electrons is de-termined by the free energy. Such a situation is natural foropen systems, but there are physical problems where the sys-tem is closed and can only exchange heat but not particleswith an external reservoir. In such cases, the canonical en-semble is more useful and the appropriate free energy is theHelmholtz free energy defined by

A,N = ln Q,N, A1

with

Q,N = trNeH

. A2One can use the GibbsPeierlsBoguliobov inequality to de-rive a variational approximation for A, along with very simi-lar lines as before. However, this approach would not be easyto apply since the theory involves taking canonical tracesinvolving MN states for a system of N electrons. This highscaling is not present in the grand-canonical ensemble, asdiscussed in Sec. II B. As for the Gibbs energy, A has theproperty that it is the work potential function for processesdone at constants and N.

Since the canonical approach is complicated, an alterna-tive approach would be to obtain an energy function whichhas the property that it describes the work for processes un-der constant N and is compatible with the work derived fromthe OEFE. Consider the function,

,N = , + N. A3

For an infinite system this function is equal to the Helmholtzfree energy. But that is not so for finite systems. Neverthe-less, it is useful. In Eq. A3 is now a function ofand N,determined by the condition,

,,N

= N. A4

When R is changed by R under constant N, we have




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R=

R+ N

R. A5

Expanding the first term on the right, we find

R=

R+

R, A6

using A4, we find

R=

R. A7

Since /R is the effective force on the nuclei when thesystem has N electrons, we find that /R is also of thatsame force. We conclude that is a potential function for themotion of the nuclei at constant average number of electrons.This function as defined in A3 is such a potential which iscompatible with the OEFE . Computing is not as straight-forward as computing . It can be done by interpolationmethods to locate the that gives the required integer num-ber of particles.

APPENDIX B: GRADIENT OF THE EFFECTIVE FREEENERGY

In this appendix we discuss the formal development al-lowing the computation of the derivative of the effective freeenergy U, given in Eq. 3.1 with respect to UI where I isan index of an element ofU. The resulting quantities will beensemble averages, whose evaluation is described in Appen-dices C and D. In taking the gradient of Eq. 3.1, we find

uI =

G0

uI +

1

2V

JK

uI

J

K

I u

J

J

uI . B1

All these derivatives can be evaluated based on the for-

mal derivative for the exponential operator eH

0, where H0is the operator given in Eq. 2.21. Standard time-dependentperturbation theory shows that

uIeH

0 =

0

eH

0IeH0d eH

0I,

B2

where

I

= 10

eH0IeH0d. B3

From this, we find after a short calculation that

G0

UI= I = I. B4

Using Eq. B2, it is straightforward to prove also that

uIJ = IJ IJ , B5

as well as

uIJK = IJK IJK . B6

Plugging these identities into Eq. 3.1 yields Eq. 3.2.

APPENDIX C: ENSEMBLE AVERAGES

Here we speak of one-body Hamiltonians H0 =hII andshow how to compute grand-canonical averages of productsof operators with this Hamiltonian. We have

Z= N

eN trNeH0 = det1 + W , C1

where

W= eh. C2

Now, let A be another one-body operator. Its grand-canonical GC average

A = Z1 treH

0NA C3

will be calculated using the identity

A = deAtdt

t=0. C4

Thus

A = Z1d

dttreH

0N

eA

tt=0 . C5

Now, since eH

etA

is itself the exponent of a one-bodyoperator,

A = Z1d

dt

det1 + WetAt=0 . C6

Using the well-known relation,

det X= eTrln X, C7

we obtain after a short calculation,

A =d

dtTrln1 + WetAt=0 . C8

Note that we use the symbol tr to denote a trace over many-body states, while the symbol Tr is reserved to denote thematrix trace ofMM matrices. Expanding the exponent tolinear order in At, we have

eAt = 1 + At+ Ot2. C9

Using this, after a short manipulation, it is possible to show

A =d

dtTrln1 + W1 + DAt+ Ot2t=0 , C10

where

D = 1 + W1W. C11

Note also that

1 D = 1 + W1 . C12

To continue, we use the identity,




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TrlneXeY = TrX+ Y C13

this relies on the fact that deteXeY=deteXdeteY and

ln1 + Xt = Xt+ Ot2 . C14

in Eq. C10 giving

Trln1 + W1 + DAt+ Ot2

= Trln1 + W + DAt + Ot2 . C15

Finally, taking the derivative of C15, afterwards setting t=0 we find

A = TrDA . C16

Next we consider AB, where B is also a one-body operator.We will not present all stages of the calculation startingfrom,

AB = Z1d2

dtdsdet1 + WetAesBt,s=0 . C17

Working out the algebra, using the same ideas we presented

above, it is possible to arrive at

AB = AB + TrDA1 DB. C18

In the gradient expression 3.2 we need not the averageAB but the cumulate average,

AB AB = TrDA1 DB. C19

It is interesting to note that the matrix representing A appearshere in the following form:

A = DA1 D, C20

we term A as A smiley. This is an important observationwhich bares on a stable numerical evaluation of these expres-sions, as discussed below. Thus, we write

AB AB = TrAB. C21

The expectation value of a product of three operators can bedeveloped along similar lines to yield

ABC = TrDATrDBTrDC + TrDATrDB1 DC

+ TrDCTrDA1 DB

+ TrDBTrDA1 DC

+ TrA1 DB BDA1 DCD . C22In the gradient expression 3.2, we actually need only the

expression ABC ABC, finding a more compact expres-sion

ABC ABC = TrDCTrAB + TrDBTrAC

+ TrAB1 DC CDB . C23

Once again, the matrix representing A appears in this expres-

sion only in terms of the smiley-A matrix A again, thisis important for the stable numerical evaluation of theexpressions.

APPENDIX D: WORKING EXPRESSIONS

In this appendix we discuss the actual calculation of ex-pectation values of one-, two-, and three-body operators thatappear in the expressions derived in the previous appendix.We showed that in all expressions for expectation values ofsuch operators C8 and C22, we do not really need thematrix elements of the operator . Instead, we expressed therequired averages in terms of the smiley matrix elements of=D

1D, where D is the FermiDirac density matrix,

D = 1 + W1W=e

1 + e. D1

Since by itself might have huge and tiny matrix ele-ments causing numerical instabilities, we concentrate on thebetter behaved matrix , which is computed as follows. Wedefine K as the orthogonal matrix of eigenvectors of h and a diagonal matrix of its eigenvalues. Thus h =T+ U=KKT.Using these, we define

rI KTIK= K

TDKKTIKKT1 DK. D2

Plugging in D1, and using the definition of Eq. B2 wefind

rI =e

1 + e

1

0

eKTIKed

1

1 + e.

D3

Here I is the matrix elements of I. Because all matricesinvolving are diagonal, we can perform the integration ana-lytically, obtaining the relation,

rIlm = RlmKilKim , D4

where I= i , i and

Rlm =el

1 + elelm 1

1 + eml ml m

1

1 + ell = m .

D5

It is straightforward to verify that R is a symmetric matrix. Inactual calculations, to avoid numerical problems, we com-pute only Rlm when lm and then set RmlRlm. This way

of evaluating D5 leads to stable numerical results, becauseel/1+el1+ em is always a number be-tween 0 and 1, and so is elm.

In deriving Eq. D4 we also used the fact that

Ikk = ikik, D6

leading to

KTIKlm = KlkTIkkKkm = KilKim . D7

Building all matrices rI is a OM4 operation. Once eachmatrix rI is built, then it is possible to write




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I = KrIKT. D8

Since the overall computation of all I matrices scales asOM5, we do not actually build them. Instead, we use thefact that for any matrix X,

TrIX = TrrIKTXK , D9

and note that using the right-hand expression, the evaluationof each trace requires O

M2

operations, and since there are

M2 such traces, the overall effort is OM4. For example,consider the numerically expensive term in the gradient Eq.3.2, which using Eq. C23 can be written as

VJKIJK IJK

= VJK TrI2KDJ JK VJK2 TrIJTrDK .

D10

Computing, for example, VJK TrIKDJ scales as OM4,

as we show now. First, note that

VJK TrIKDJ = VJKInlKllDlijin . D11

Then, sinceJllVJKKii = Vllii, D12

it is possible to write

VJKIJK IJK = TrrIW D13

with W= KTWK and

Wil = Vijkl2Djk jk 2ViljkDjk. D14

Working along similar lines, the second term in the gradientof Eq. 3.2 is simply evaluated to

TrrIKTuK. D15

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