Energy dependence on fractional charge for strongly interacting subsystems

0.28 MB·English

by  Steven M. Valone

Energy dependence on fractional charge for
strongly interacting subsystems
Steven M. Valone
Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 and
Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131
7 Susan R. Atlas
0 Center for Advanced Studies and Department of Physics and Astronomy,
0 University of New Mexico, Albuquerque, New Mexico 87131
2
(Dated: February 6, 2008)

n
a The energies of a pair of strongly-interacting subsystems with arbitrary noninteger charges are
J examined from closed and open system perspectives. An ensemble representation of the charge
dependenceisderived,validatallinteractionstrengths. Transformingfromresonance-stateionicity
1
to ensemble charge dependence imposes physical constraints on the occupation numbers in the

] strong-interaction limit. For open systems, the chemical potential is evaluated using microscopic
r and thermodynamic models, leading to a novel correlation between ground-state charge and an
e
electronic temperature.
h
t
o PACSnumbers: 71.15.-m,31.10.+z,34.20.-b,34.70.+e
.
t
a
m Associating fractional charges with individual atoms assumed quadratic dependence. A direct consequence of
- is a habitual part of our every-day thinking about con- Eq. (1) is that the associated microscopic chemical po-
d
densed and molecular matter. Indeed, characterizing tential µ = −dE (q)/dq exhibits a discontinuity with
A
n
the energetics of such systems in terms of dynamically- respect to charge q [3]. The veracity of both limits
o
c evolvingchargesisnowrecognizedaskeytounderstand- has been confirmed numerically by Cios lowski and Ste-
[ ing the atomic-scalebehaviorofcomplexprocessesrang- fanov [6]. In this Letter, we construct a novel, ana-

ing from alloyingto motor protein function to molecular lyticmodelexplicitlylinkingtheweak(linear)andstrong
3
v logic gate operation [1]. Compounding the difficulty of (quadratic) interaction limits, and characterize the cor-
9 the problem is that the instantaneous redistribution of respondingbehaviorofµwithrespectto q. As discussed
0 charge occurs under the influence of strong interactions below, this requires the definition of interacting “atom-
0
among subsystems of atoms and molecules. in-molecule”(AIM)subsystemsandassociatedfractional
1
Historically, it was assumed that quadratic electro- charges. While these definitions are not unique [7],
0
6 static interactionsconstitute areasonablerepresentation and many physically-reasonablealternatives are possible
0 of the energy dependence on fractional charge [2], in- [8, 9, 10, 11], our results are independent of the details
/ dependent of the strength of interaction. However, the of a specific AIM approach.
t
a inadequacy of the historical assumption had been high- Previousattempts to extend PPLB’s formal results to
m
lightedbytheworkofPerdewet al.(hereafterreferredto moderate and strong interactions have recast the prob-
d- as PPLB) [3], where a linear dependence was found for lem in terms of charge resonances [12, 13, 14, 15] of a
n weakly interacting subsystems. This was found by con- closed system composed of A+R, where R represents
o sidering an open subsystem that was allowed to weakly the electron reservoir. In the charge resonanceview, one
c interact and exchange electrons with a reservoir of elec- supposes that special wavefunctions can be constructed
:
v trons [4]. For a diatomic molecule, the weak interaction such that the charges on the subsystems are integers
Xi restriction implies that the theory is valid only at large [16, 17, 18, 19, 20]. Interpreting PPLB in the language
internuclearseparationsR. PPLBconsideredanatomA of resonances, the state designated as M for A corre-
r
a as a subsystem in its neutralstate with M electrons and sponds to both A and R being neutral (covalent reso-
energy EM that becomes anionic by fractional charge q, nance), while M+1 corresponds to A being anionic and
where the anionic state with M +1 electrons has energy R being cationic (ionic resonance),leaving the total sys-
EM+1. The energy EA(q) was shownto be the ensemble tem neutral.
average
Extending this picture to the intermediate interaction
regime [12, 21], R ceases to be a structureless reservoir
E (q)=E +ω(E −E ), (1)
A M M+1 M
of electrons, but must instead be viewed as a structured
with ω ∼ ω = −q ≥ 0. This result established a subsystem B at a finite separation R from A. This sub-
PPLB
seminal extension of density functional theory (DFT) to system may be an atom, molecule, or bulk material. Al-
fractional numbers of electrons [5]. The energy E (q) though severaltwo-state valence bond models have been
A
is manifestly linear in q, in contrast to the historically- developedinthislimit[12,13,14,15],previousattempts
2
to extendthe valencebond approachto the stronginter-
action limit have proven unsuccessful.
A complete generalization of PPLB to the strong in-
teractionlimit canbe attainedby transformingfromthe
resonance-state basis to an ensemble or spectral repre-
sentation [4]. Let Hˆ denote the hamiltonian for the
closed system AB with eigenstates {Ψ } and eigenen-
k
ergies {E }. Let Ψ be any arbitrary trial wavefunction
k
with hΨ|Ψi = 1. The variational energy can then be
expressed in a spectral representation,
hΨ|Hˆ|Ψi= ω E , (2)
k k
k FIG.1: (Color) Energy of a heteronuclear diatomic molecule
X
asafunctionofbondlengthandchargeasmodeledbyEq.(4).
where the ωk =|hΨ|Ψki|2 are occupation numbers. Now The charge dependence is linear as the molecule dissociates
we express Ψ as a linear combinationof resonancestates [3],while being quadraticneartheground-statechargewhen
{ψ } with coefficients {c }, such that Ψ = c ψ . The R is near equilibrium values [2, 6]. The constituent atoms
i i i i i
eigenvectorsC arethe specificvaluesofcoefficientscor- interact strongly near equilibrium.
k
P
responding to the eigenstates. The basis rotation from
the {ψ } to the {Ψ } in terms of the C lead to occupa-
i k k 1.0
tion numbers 1.5 S dN
TinhtehseeoecnceurpgaytiwohnennωukmA=Bbe|rdsXisigsiovcce∗iiCartiisekes|2t[o.3,p2ie2c]e.wSispeelciinaelaizri(in3tyg) (,) (arb. units)wgggs 0000....8642 ggs 0124 q (arb. units) 100...050 In000te...c447rim---000e...d233cii005,aAt*e
0.0
toatwo-statemodel,wedefinetheionicityγ astheratio -10 -5 0 5 10 -2 -1 0 1 2
c /c , where c and c are the coefficients for the cova- g (arb. units) g (arb. units)
1 0 0 1 (a) (b)
lent and ionic resonances, respectively. Let γ and γ
gs xs
denote the eigenionicities corresponding to the ground- FIG.2: (a) (Color online) Intermediatelimit of Eq.(5). The
andexcited-statevalues ofc1/c0,atsomechosensepara- extremacorrespondtoγ =γgsandγ =γxs=−1/γgs. (b)Re-
tion R [12]. The dependence of the energy E(γ) on an lationship between charge and ionicity in the strong interac-
tionregimeforthetwo-statemodelandtheatom-in-molecule
externally-imposed ionicity γ is then given by
modelofRef.[12]. Sci andδNc∗i areAIMparametersentering
theγ−q relation in the strong interaction regime. The gray
E(γ;R) = E (R)
gs zonecorresponds to thephysicalrange of q between 0and 1.
+ω(γ;γ (R),γ (R))(E (R)−E (R)) , (4)
gs xs xs gs
with
Itreducestoγ =± q/(1−q)intheintermediateinter-
(γ−γ )2
gs actionregime[12,18,19],yieldingω(q) ∼ ω (q,q (R))
ω(γ; γ ,γ )= . (5) int gs
gs xs (γ−γ )2+ 1−γg2s (γ−γ )2 = ( (1−q (R))qp± q (R)(1−q))2 [12]. Although
gs γ2−1 xs gs gs
xs details of the γ−q relation will be affected by the spe-
(cid:0) (cid:1) p p
This occupation number governs how the energy of AB cificchoiceofAIMchargedefinition,wecanimmediately
changes when its ionicity is forced to deviate from its identify two quite general consequences of the transfor-
ground-state value (Fig. 1). Importantly, this result is mationtoq. First,inspecifyingtheγ−qrelation,choos-
independentofthestrengthoftheinteraction. Asisnec- ing a root, and restricting the charge to the physical
essary physically, ω = 0 when γ = γ , and ω = 1 when rangebetween0and1,somevaluesoftheionicityareex-
gs
γ =γ . Moreover,both extrema are represented, as the cluded. This outcome wasanticipated by Panet al. [24].
xs
derivative of ω with respect to γ is zero when evaluated Asaresult,ω(q)—incontrasttoω(γ)—maynolongerbe
at either γ or γ (Fig. 2a). a proper occupation number, since it does not span the
gs xs
Equation(5)isexpressedintermsoftheionicityγ. In full range from 0 to 1. Second, the presence of the two
order to complete the model it is necessary to eliminate rootsresultsintwobranchesintheE(q)surface(Fig.3a;
thisquantummechanicalparameterinfavorofthephys- only one was shown in Fig. 1), provided that the gap in
ical charge q [23]. This can be accomplished through an Eq.(4) is nonzero. The branchesaredegenerate for allq
AIM decomposition [6, 12]. The relation between γ and in the weak limit, degenerate only at the integer charges
qinthestronginteractionregimeisillustratedinFig.2b. in the intermediate limit, and completely nondegenerate
3
in the strong limit. and thermodynamic definitions [14, 26, 27]. The ther-
As the drivingforceforsuchchargetransfers,the con- modynamic definition is relevant because the number of
ceptofchemicalpotentialpresupposesanabilitytospec- statesinthespectralrepresentation,Eq.(2),becomesex-
ify subsystems. However,this concept requires the iden- ponentially large with system size. The grand canonical
tification of the subsystem energies, as well as charges. ensemble is most appropriate here [3, 28]. For an en-
As with the AIM charge, the subsystem energy is not semblecharacterizedbychemicalpotentialµandinverse
uniquely defined. Regardless, certain general features temperature β = 1/k T, the grand partition function
B
are valid for any definition of subsystem energy, pro- is exp(βµN∗−βE∗), where N∗ denotes the number
k k k k
∗
vided that the total energy of the parent closed system of electrons in state k with energy E [29]. Considering
P k
is preserved. That is, for closed system energy E , the again a two-state model, the average electron number is
AB
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
open subsystem energies E and E must sum to E : hN i=N +(N −N )exp(βµ(N −N )−β(E −
A B AB gs xs gs xs gs xs
∗ ∗ ∗ ∗ ∗ ∗ ∗
E +E = E . (Asterisks indicate open-system sta- E ))/(1+exp(βµ(N −N )−β(E −E ))), whose
A B AB gs xs gs xs gs
∗ ∗
tus with respect to energy and electron transfer.) This solution for µ=µ(q) with q =N −hN i [3, 4, 28] is
gs
proviso is necessary because we require the stationary
E∗ −E∗ +k T ln( q )
properties of the eigenstates of the closed system. xs gs B N∗−N∗−q
µ(q)= gs xs . (8)
Now, if EAB is either the ground or an excited state Nx∗s−Ng∗s
energy, it must follow that the subsystem energy defini-
tion applies to these as well, thereby defining E∗ and At T = 0 K, µ = (Ex∗s−Eg∗s)/(Nx∗s−Ng∗s) for all q. By
A,gs
E∗ . Thus, for subsystem A∗, its energy E∗ with an comparing Eqs. (7) and (8), we observe that the T = 0
A,xs A
arbitrary chargeq becomes K result corresponds to qgs = 0 (Fig. 3b). On the other
hand,atemperatureof34800Kcorrespondstoq =1/2,
gs
E∗(q) = E∗ +ω(q)(E∗ −E∗ ), (6) ifenergyE∗ −E∗ =3eVandN∗−N∗ =−1arechosen.
A A,gs A,xs A,gs xs gs xs gs
Because there are no other degrees of freedom with
where ω(q) comes from Eq. (5) and the exact γ−q rela- which to equilibrate, it is clear that T corresponds to an
∗
tion. An analogous expression holds for B . It is trans- electronic temperature. In fact, this is the grand canon-
parentthatthesumofthetwoAIMenergiesrecoversthe ical (open system) analog of the electronic temperature
totalenergyofthe closedsystem, independent ofthe de- introduced by Kohn [30] in the context of calculating
tailsofthechosensubsystemenergydefinition. Likewise, excitation energies using closed system ensemble DFT
ifEAB representsasmallchangefromaground-stateen- [31, 32]. The corresponding canonical ensemble is de-
∗ ∗ ∗
ergy,then oneconcludes that −dE /dq =dE /dq ≡µ , fined in terms of a spectral representation, as here, and
A B
∗
whereµ isthemicroscopicchemicalpotential. Differen- is characterized by a temperature θ. θ is defined implic-
tiatingEq.(6),andusingthesimplerintermediateregime itly through a self-consistent relation equating the total
with the negative root of the γ−q relation, yields entropy of the system to the integrated local entropy of
thesystem’selectrondensitydistribution,inturndefined
∗ ∗ ∗
µ (q) = −(E −E )
xs gs throughasetofstate-andtemperature-dependentKohn-
Sham equations [30, 31]. There is thus a direct analogy
(1−q )q
gs gs
× (1−2qgs±(1−2q) ), (7) between our electronic temperature for an open system
s (1−q)q
1.0
where the R dependence and atom subscript have been 20
∗
sitdnrueatpgenepgsnrefeeersrrsacreihcndyagrifgnaoerstfuh,cletllahbcrerhirtaaeynbrcgy(heF.episrgTeo.hvf3ieEsbnt)b(i.qen)hgaaµvtthiioneirstsecungoboemrstyeqdsstaefenfirmodnmeifdsrtoihnamet- nergy (arb. units) 000...864 gxxgssss ---- isisnntttrtreeoorrnnmmggee llddiimmiiaaiittttee (eV)m 100 ttmmhhiieeccrrrrmmoossooccddooyyppnniiaaccmm:: qqiiccgg::ss ==TT10 ==/2 03 4K800 K
E
timately related to the derivative discontinuity found by ed 0.2 -10
PPLB. At q =qgs with the negative root, µ∗ =0, in ac- Scal
0.0 -20
cordance with our zero of energy. Nonetheless, chemical 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
potential equalization [25] does hold. q (arb. units) (a) q (arb. units) (b)
Bycontrast,forstronginteractionswiththeexactγ−q
relationship, µ∗ is continuous at 0 and 1. Smoothing of FIG. 3: (a) ω(q), corresponding to branches of the energy
surfaces, Eq. (4), in the strong and intermediate limits for
tohfethcheedmeirciavaltpiovteendtisiaclonatriinseusitdieirsectotlynofrnoimntetgheermqovoeumtseidnet e−a0c.h20ro(osttroonfgt)h;eSγci−=q0raelnadtioδnNsch∗iip=: 0Sc(iin=ter0m.2e5diaantde).δNqgc∗si ==
of the physical range of [0, 1] (Fig. 3a). The subsystem 0.25 in both cases. Gaps form at integer charges. (b) Two-
cannow transfera full charge,but the drivingforce may statemicroscopicandthermodynamicmodelsofthechemical
be very large.” potential. The microscopic model is plotted for qgs = 0 and
To complete the discussion of chemical potential and 1/2, and the thermodynamicmodel for T =0 and 34800 K.
in analogy to PPLB, we compare microscopic (Eq. (7))
4
withchargetransfer,andthatofKohnetal.,eventhough phys.stat. sol. (b) 217, 41 (2000).
theyrestrictedtheirattentiontopolarizationexcitations [2] A. K. Rapp´e and W. A. Goddard, III, J. Phys. Chem.
ofaclosedsystemwithfixedtotalcharge. Inbothtreat- 95, 3358 (1991).
[3] J.P.Perdew,R.G.Parr,M.Levy,andJ.L.Balduz,Jr.,
ments, the temperature corresponds to deviation of the
Phys. Rev.Lett. 49, 1691 (1982).
chargedistribution awayfromthe groundstate distribu-
[4] J. P. Perdew, in Density Functional Methods in Physics,
tion. From a thermodynamic point of view, the excita-
NATO Advanced Science Institute Series, Vol. 123,
tions lead to anincrease in entropy. The correspondence edited by R.M.Dreizler and J. daProvidˆencia (Plenum
observed between T and qgs stems from the fact that Press, New York,1984).
both quantities encode information about the energetics [5] Y. Zhang and W. Yang, Theor. Chem. Acc. 103, 346
of the microscopic states of the interacting subsystems. (2000).
[6] J. Ciosl owski and B. B. Stefanov, J. Chem. Phys. 99,
Equivalently, both T and q reflect the strength of the
gs
5151 (1993).
coupling between a subsystem and a reservoir.
[7] R. G. Parr, P. W. Ayers, and R. F. Nalewajski,
In conclusion, we have shown how charge dependen- J. Phys.Chem. A 109, 3957 (2005).
cies in a pair of subsystems can be described by appeal- [8] Ph. Ghosez, J.-P.Michenaud,and X.Gonze, Phys.Rev.
ing to an ensemble representation of the closed system B 58, 6224 (1998).
[9] W.A.Harrison, Electronic Structure and the Properties
energy, even in the strong interaction limit. The an-
of Solids (Freeman, San Fransisco, 1980).
alytical form of the charge-dependent system energy is
[10] M. H. Cohen and A. Wasserman, Isr. J. Chem. 43 219
determined through a basis rotation involving the reso-
(2003).
nance states that embody charge transfer between sub- [11] Q. Wu and T. Van Voorhis, Phys. Rev. A 72, 024502
systems. When a decomposition method is applied to (2005).
closedsystemeigenenergies,charge-dependent,opensys- [12] S.M. Valoneand S.R.Atlas,J. Chem. Phys.120, 7262
tem energies can be defined, leading to a microscopic (2004).
[13] R. F. Nalewajski, Int.J. Quant. Chem., 69, 591 (1998).
model of the chemical potential. When the subsystems
[14] J. Morales and T. J. Martinez, J. Phys. Chem. A 105,
interactstrongly,thederivativediscontinuitiesinthe en-
2842 (2001).
ergyfoundbyPPLBmovetononintegerchargesthatare [15] J. Morales and T. J. Martinez, J. Phys. Chem. A 108,
outside of our physically allowed range, leading to con- 3076 (2004).
tinuouschemicalpotentialsattheintegercharges. When [16] A. Warshel and A. Bromberg, J. Chem. Phys. 52, 1262
compared with a thermodynamic model of the chemical (1970); A.WarshelandR.M.Wiess,J.Am.Chem.Soc.
potential, a correlation is observed between the ground- 102,6218(1980);J.˚AqvistandA.Warshel,Chem.Rev.
93, 2523 (1993).
statechargeandanelectronictemperature,thusdefining
[17] R. S. Mulliken, Phys. Rev. 50, 1017 (1936); ibid., 50,
complementary measures of the interaction strength.
1028 (1936).
The ensemble variational energy defined here may be [18] J. C. Phillips, Rev. Mod. Phys. 42, 317 (1970).
regardedas the wavefunction predecessor of the excited- [19] C. A. Coulson, L. B. R`edei, and D. Stocker, Proc. Roy.
state density functionals of Gross et al. [32]. However, Soc. (London) 270, 357 (1962).
[20] S. A. Adelman and D. R. Herschbach, Molec. Phys. 33,
the formal connection of our strongly-interacting, open
793 (1977).
system results to density functional theory remains an
[21] “Strength”isdefinedthroughtheenergiesofandoverlaps
openquestion,requiringtheconstructionofavariational
betweenthechargeresonancesasdefinedinRef.12: “In-
principle for open system excited states. To the best of termediate” means zero overlap but nonzero resonance
our knowledge such a variational principle has not yet energy. “Weak” means that both quantitiesare zero.
been established. [22] W. Yang, Y. Zhang, and P. W. Ayers, Phys. Rev. Lett.
84, 5172 (2000).
TheworkofSMVwasperformedinpartatLosAlamos
[23] U. Fano, Rev.Mod. Phys. 29, 74 (1957).
National Laboratory under the auspices of the US De- [24] X.-Y.Pan,V.Sahni,andL.Massa, Phys.Rev.Lett.93,
partment of Energy, under contract No. W-7405-ENG-
130401 (2004).
36, and funded through the Advanced Fuel Cycle Initia- [25] R. T. Sanderson, Science 114, 670 (1951).
tive. SMVthankstheUniversityofNewMexico,Depart- [26] P.K.Chattaraj,A.Cedillo,andR.G.Parr,Chem.Phys.
mentofPhysicsandAstronomyforitshospitalityduring 204, 429 (1996).
the2003-2004academicyear. TheauthorsthanktheNa- [27] P. W. Ayers, R. G. Parr, and A. Nagy, Int. J. Quant.
Chem. 90, 309 (2002).
tionalScience Foundationforsupportofthisworkunder
[28] E. P. Gyftopoulos and G. N. Hastopoulos, Proc. Nat.
grants DMR-9520371 (SRA) and CHE-0304710. SMV Acad. Sci. USA 60, 786 (1968).
dedicates this work to his sister Susan. [29] R. C. Tolman, The Principles of Statistical Mechanics
(Dover, New York,1979).
[30] W. Kohn, Phys.Rev.A 34, 737 (1986).
[31] L.N.Oliveira,E.K.U.Gross,andW.Kohn,Phys.Rev.
A 37, 2821 (1988).
[1] See,e.g.: Th.Frauenheim,G.Seifert,M.Elstner,Z.Ha- [32] E.K.U.Gross,L.N.Oliveira,andW.Kohn,Phys.Rev.
jnal,G.Jungnickel,D.Porezag, S.Suhai,andR.Scholz, A 37, 2809 (1988).