The initial and final states of electron and energy transfer processes:Diabatization as motivated by system-solvent interactions

Joseph E. Subotnik,1,2,a! Robert J. Cave,3 Ryan P. Steele,4,b! and Neil Shenvi4,c!

1School of Chemistry, Tel-Aviv University, Tel-Aviv 69978, Israel2Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, USA3Department of Chemistry, Harvey Mudd College, Claremont, California 91711, USA4Department of Chemistry, Yale University, New Haven, Connecticut 06520, USA

!Received 26 February 2009; accepted 14 May 2009; published online 15 June 2009"

For a system which undergoes electron or energy transfer in a polar solvent, we define the diabaticstates to be the initial and final states of the system, before and after the nonequilibrium transferprocess. We consider two models for the system-solvent interactions: A solvent which is linearlypolarized in space and a solvent which responds linearly to the system. From these models, wederive two new schemes for obtaining diabatic states from ab initio calculations of the isolatedsystem in the absence of solvent. These algorithms resemble standard approaches for orbitallocalization, namely, the Boys and Edmiston–Ruedenberg !ER" formalisms. We show that Boyslocalization is appropriate for describing electron transfer #Subotnik et al., J. Chem. Phys. 129,244101 !2008"$ while ER describes both electron and energy transfer. Neither the Boys nor the ERmethods require definitions of donor or acceptor fragments and both are computationallyinexpensive. We investigate one chemical example, the case of oligomethylphenyl-3, and weprovide attachment/detachment plots whereby the ER diabatic states are seen to have localizedelectron-hole pairs. © 2009 American Institute of Physics. #DOI: 10.1063/1.3148777$

I. INTRODUCTION TO DIABATIC STATES

There are many different definitions of diabatic states inthe chemical literature.1–3 Within the context of nonadiabaticquantum dynamics, one historical definition is that diabaticstates are the many-electron states with zero !or minimal"nuclear derivative couplings.4–7 Within the context of a sys-tem that can undergo electron or energy transfer in a con-densed environment, another definition is that the diabaticstates are the initial and final states of the system before orafter the transfer process.8,9 Many other definitions exist.10–16

In this paper, we will adopt the second definition givenabove, appropriate for condensed environments, and we willpresent three new approaches for generating many-electronstates relevant to nonequilibrium processes.

A. The historical motivation for diabatic states

Before discussing diabatic states relevant to condensedenvironments, we review the original, historical motivationfor diabatic states. According to the standard dogma of quan-tum mechanics, the relevant stationary states of a physicalsystem are the eigenstates of the Hamiltonian H. The stan-dard route to these eigenstates is to invoke the Born–Oppenheimer approximation, whereby the full Hamiltonianis first partitioned into nuclear !R" and electronic !r" opera-tors,

H!r,R" = Hnuc!R" + Hel!r;R" . !1"

According to the Born–Oppenheimer approximation, oneshould freeze the nuclei, diagonalize the electronic Hamil-tonian Hel, and generate adiabatic states %&! j'(,

Hel!r;R"&! j!r;R"' = Ej!R"&! j!r;R"' . !2"

Next, one expands the full wave function in terms of theadiabatic electronic states and diagonalizes the full Schro-dinger equation,

&"i!r,R"' = )j

Cij&# j

i!R"' ! &! j!r;R"' , !3"

H&"i!r,R"' = Eitot&"i!r,R"' . !4"

Here %&"i!r ,R"'( are the eigenfunctions of the total Hamil-tonian H, and %&# j

i!R"'( are the nuclear wave functions thatdescribe the vibrations and rotations associated with a givenBorn–Oppenheimer electronic state.

According to Eq. !4", it can be shown that the differentadiabatic electronic states %&! j'( are coupled together bynuclear derivative couplings !*!i!r ;R"&"R&! j!r ;R"'r", wherethe subscript r indicates integration over electronic coordi-nates. The meaning of the derivative couplings is that be-cause the electronic adiabatic states vary when the nucleardegrees of freedom are changed, any representation ofnuclear motion in terms of adiabatic states must necessarilycouple nuclear motion with electronic transitions. For thisreason, diabatic states were historically defined4–7 asrotations of adiabatic states with zero nuclear derivativecouplings,

a"Electronic mail: subotnik@post.harvard.edu.b"Electronic mail: ryan.steele@yale.edu.c"Electronic mail: neil.shenvi@yale.edu.

THE JOURNAL OF CHEMICAL PHYSICS 130, 234102 !2009"

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&$i' = )j=1

Nstates

&! j'Uji, i = 1, . . . ,Nstates, !5"

0 = *$i!r;R"&"R&$ j!r;R"'r. !6"

Although diabatic states satisfying Eqs. !5" and !6" ex-actly do not usually exist,17 one can select diabatic stateswhere the derivative couplings above are minimized.6,7 Withthis definition, diabatic states are coupled together only veryweakly through the nuclear derivative, but they can now becoupled together moderately through the electronic Hamil-tonian *$i!r ;R"&Hel!r ;R"&$ j!r ;R"', resulting in the so-calleddiabatic couplings or HAB’s that appear in nonadiabatic elec-tron transfer theory.

B. The motivation for diabatic states in a condensedenvironment

When applying the Born–Oppenheimer treatment to asystem capable of electron or energy transfer in a condensedenvironment, there is a separate motivation for constructingdiabatic states. Namely, even if the system does not ex-change electrons with its surroundings, one can find that theimportant quantum mechanical states of the system dependas much on the environment as on the system, in particular,when the environment is polarizable.18 Thus, in many cases,the adiabatic states of the isolated system will not be therelevant stationary states of the solvated system, and they donot describe the system before or after an electron or energytransfer event.

For example, consider the standard example19 of twosolvated iron cations !Fe!H2O"6"2

+5. If we ignore solvent andconstruct the Hamiltonian for the 107 electrons belonging tothe two cation complexes in vacuum, we will find adiabaticstates with exactly 53.5 electrons attached to each of the twoiron centers !assuming the Fe–O distances are equal for bothcenters". In fact, because the electronic Hamiltonian is sym-metric between the left and right, all eigenfunctions of Hel!r"must also be eigenstates of the parity operator and thereforedistribute charge equally between the two iron centers. Theaforementioned electronic adiabatic states of !Fe!H2O"6"2

+5

do not represent the important stationary states of iron insolution because, in a strongly polarizable environment,equal sharing of charge between the two cations is unphysi-cal. In solution, the solvent will polarize around charge andthe system-solvent interactions will drive an asymmetry,forcing the extra charge to be localized on one of the twoiron centers. Indeed, if we relax the constraint of equal Fe–Odistances even in the gas phase, the odd electron may welllocalize.19

With this motivation, diabatic states for a system in acondensed environment can be defined as the projection ontothe system of the stationary states of the system plus environ-ment. With this definition, when appropriate, the diabaticstates of a system represent the initial and final states of thesystem before or after an electron or energy transfer event.There is a vast literature of research focused on computingthese diabatic states when the surrounding solvent is mod-eled as a polarizable continuous medium20–24 !PCM" de-

signed to account for nuclear polarizability of thesolvent.24,25 For a comprehensive review, see Ref. 18.

C. The connection between the two definitions

Although a priori there is no reason to expect that di-abatic states %&$i!r ;R"'( representing the initial and finalstates of an electron transfer process should obey the historicdefinition of diabatic states !i.e., *$i!r ;R"&"R&$ j!r ;R"'r=0",as a practical matter the two definitions do usually agree. Aspointed out by Atchity and Ruedenberg10,11 and extended byNakamura and Truhlar,12–14 diabatic states with minimal de-rivative couplings should have “configurational uniformity”whereby the dominant configurations are unchanged over theentire potential energy surface. Pacher et al.15,16 also inde-pendently published similar ideas of diabatization, proposinga “block diagonalization” algorithm that minimizes the dis-tance !in wave function space" between the target diabaticstates and a reference basis of states with fixed character!which were assumed always available". Because the initialand final states of an electron or energy transfer processshould have a fixed character !e.g., covalent, ionic, etc.", wemay expect that the historic definition of diabatic states willusually agree with the condensed environment definition.

D. Current algorithms for constructing diabatic statesfor electron and energy transfer

Unfortunately, including explicit solvent !beyond a con-tinuum model" is computationally intractable for most calcu-lations. There is an immense literature modeling the solventas a continuum that polarizes the system self-consistently foruse in electron transfer.18,24,26 Two challenges with con-tinuum models are sensitivity to cavity definition and thedifficulty calculating a complete set of diabatic states whenthe nuclear geometry of the system is not close to the tran-sition state and the effective solvent nuclear geometry needsto be very different for each of the diabatic states sought. Formodern electron or energy transfer calculations, the standardapproach to constructing diabatic states %&$i'( !and their di-abatic couplings" is to rotate the adiabatic states of the sys-tem in vacuum %&!i'( #as in Eq. !5"$ according to some physi-cal criterion that should mimic the effect of solvent.Examples of such methods in the context of electron transferinclude generalized Mulliken Hush !GMH",8,9 Boyslocalization,27 constrained density functional theory!CDFT",28–31 fragment charge difference !FCD",32 and in thecontext of energy transfer, fragment energy difference!FED".33–35

In some cases, the diabatic states are obvious. For in-stance, in the case of !Fe!H2O"6"2

+5, our intuition is to rotatetogether the adiabatic ground state and the adiabatic first ex-cited state so that the resulting two diabatic states wouldhave charge localized either on the left or on the right cation.We make this choice of diabatic states for !Fe!H2O"6"2

+5 be-cause, according to electrostatic theory, the solvent shouldchoose to localize the extra charge on one or the other ironcenter if the dielectric constant is sufficiently large and themetal-metal electronic coupling is not too strong. For a moregeneral system, however, there can be no unique definition

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for constructing such diabatic states because the need forsuch diabatic states is based on the initial state preparationand our lack of information about solvent position and ori-entation. For this reason, many electron-transfer theoristshave preferred to avoid calculating diabatic states altogether,choosing instead to extract the electronic couplings neces-sary for predicting electron transfer rates using indirect meth-ods that do not make explicit reference to diabatic states.36–43

Nevertheless, because the physical meaning of diabaticstates in condensed environments is paramount, it is worth-while and useful to construct approximate system diabaticstates even if these computed states are not unique. Ideally, adiabatization algorithm applicable to electron and energytransfer in condensed environments should

• treat the initial and final states of both electron andenergy transfer equivalently;

• apply to both inter- and intramolecular electron orenergy transfer;

• be computationally feasible for large molecules witharbitrarily many charge or energy excitation centers;

• apply to molecules with arbitrary amounts of electron-electron correlation;

• rely on as few parameters as possible !e.g., dielectricconstants, cavity sizes, charge fragments, etc.";

• allow for the calculation of diabatic states at all nucleargeometries of the system with or without solvent#so that the Condon approximation !e.g., see Ref. 44"can be tested$.

In Sec. II, we will present two new diabatization algo-rithms, namely, Boys and Edmiston–Ruedenberg !ER" diaba-tization. The Boys approach satisfies most of the above re-quirements !for electron transfer only" while the ERapproach satisfies them all !for electron and energy transfer".A third approach, von Niessen–Edmiston–Ruedenberg!VNER" diabatization, will be discussed in Appendix B. Be-fore introducing these new methods, we briefly review thefour modern algorithms commonly used to construct diabaticstates relevant to electron and energy transfer processes:GMH,8,9 FCD,32 FED,33–35 and CDFT.28–31

1. Generalized Mulliken Hush

The GMH algorithm8,9 is a simple approach that hasproved very popular for the specific case of describing theinitial and final states of an electron transfer process. Thegeneral idea behind GMH is to recognize that, when model-ing electron transfer, the diabatic states should correspond tocharge localized on different centers !donors and acceptors".The physical motivation underlying charge-localized diabaticstates is that solvent localizes charge on a system by reorga-nizing around it.

When seeking charge-localized diabatic states, accordingto GMH theory, one constructs diabatic states as follows !forthe two-state problem". First, one calculates all dipole matrixelements of the adiabatic states &!1' and &!2': %! 11, %! 22, and%! 12. Second, one recognizes that the important direction is

the direction of the dipole moment of the initial adiabaticstate minus the dipole moment of the final adiabatic state:v0!= !%! 11!%! 22" / &%! 11!%! 22&. Third, one projects all dipole ma-trix elements into the v0 direction and diagonalizes the dipolematrix. The motivation here originally was proposed by Mul-liken and Hush who reasoned that the transition dipole con-necting localized diabatic states should be zero for chargetransfer calculations due to the locality of the dipole momentoperator and the exponential decay of the localized donorand acceptor states. The rotation matrix that diagonalizes theprojected dipole matrix is taken as the GMH transformationmatrix from adiabatic to diabatic states.

GMH theory has been used to model several experimen-tal systems for two-state electron transfer. For systems withmore than two states but only two charge centers, GMHmakes the reasonable choice to diagonalize the Hamiltonianwithin the block of states for each charge center, thus gener-ating unique, locally adiabatic diabatic states. For systemswith more than two noncollinear charge systems, however,there is no unique charge transfer direction and GMH is un-satisfactory. GMH is also incapable of treating energytransfer.

2. Fragment charge difference

The FCD method32 is based on GMH, with the advan-tage that FCD can account for multiple charge centers. TheFCD approach works by associating each diabatic state &$i'with a given donor or acceptor fragment !indexed by i", andthen maximizing the sum over all diabatic states of thecharge density lying on the associated fragment. While thisapproach is general and can be applied to many charge cen-ters, the price for this generality is that a priori one mustdefine donor and acceptor fragments, rather than allowingthe diabatization routine to distribute charge naturally. Thus,it is difficult to rigorously justify the FCD algorithm onphysical grounds for intramolecular electron transfer. More-over, the FCD algorithm does not treat energy transfer.

3. Fragment energy difference

The FED algorithm by Hsu and co-workers33–35 extendsthe FCD approach to energy transfer by defining energy ex-citation density as the density of electron attachment plus thedensity of electron detachment, all relative to a molecularground state. Similar to the FCD approach for electron trans-fer, FED makes diabatic states by first associating each di-abatic state with one molecular fragment for energy excita-tion, and second maximizing the sum over all diabatic states&$i' of the energy excitation density associated with frag-ment i. Just like the FCD approach, the FED method cantreat multiple energy excitation centers. However, as forFCD, the price for this flexibility is that the algorithm de-pends on a priori definitions of molecular fragments, whichis difficult to justify on physical grounds for intramolecularenergy transfer. Moreover, it is unclear if the FED algorithmcan be directly applied to charge transfer.

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4. Constrained DFT

In the past decade, CDFT !Refs. 28–31" has evolved asan alternative approach to constructing diabatic states, en-tirely avoiding the adiabatic states normally produced bystandard electronic structure methods. The idea behindCDFT is to define donor and acceptor fragments, and to con-strain the Kohn–Sham wave function to have the correctcharge on each fragment. While CDFT has the strong advan-tage of being computationally inexpensive, because of thenecessary fragment definitions, the algorithm is more diffi-cult to justify on physical grounds for intramolecular !ratherthan intermolecular" electron transfer. To our knowledge,CDFT-like methods have not yet been applied to energytransfer. Formally, CDFT diabatic states are not rotations ofadiabatic states arising from standard ab initio quantumchemistry calculations.

E. A brief synopsis of orbital localization routines

In Sec. II, we will derive two new diabatization routinesbased on different models for the interaction of system withsolvent !and a third model is given in Appendix B". Thesenew techniques bear a striking resemblance to algorithmscommonly used for molecular orbital localization, includingthe Boys,45–47 ER,48 and VNER !Refs. 48–50" localizationroutines. For completeness, we remind the reader how thesedifferent localized orbitals are defined.

For a closed molecular system, we denote the canonicalmolecular orbitals &i , i=1, . . . ,Norbitals which are almost al-ways delocalized. One can construct localized orbitals 'i byapplying a rotation matrix U,

'i = )j=1

Norbitals

& jUji, i = 1, . . . ,Norbitals. !7"

Because the !Slater" determinant is multiplicative, rotatingoccupied molecular orbitals together does not change theoverall many-electron state of the system.

The rotation matrix U is usually defined by maximizinga localization function. For the three localization routinesmentioned above, these localization functions are

fBoys!U" = fBoys!%'i(" = )i,j=1

Nstates

&*'i&r!&'i' ! *' j&r!&' j'&2, !8"

fER!U" = fER!%'i("

= )i=1

Norbitals

!'i'i&'i'i" !9"

= )i=1

Norbitals+ dr1+ dr2'i!r1"'i!r1"'i!r2"'i!r2"

&r1 ! r2&,

!10"

fVNER!U" = fVNER!%'i("

= )i=1

Norbitals+ dr1'i!r1"'i!r1"'i!r1"'i!r1" . !11"

In Eq. !9", we use the chemists’ notation for the two-electronCoulomb integral !pq &rs".51 Equations !8"–!11" should becompared to Eqs. !22", !28", and !B4" below. There wouldappear to be a remarkable connection between orbital local-ization techniques and many-electron state diabatizationalgorithms.

II. CONSTRUCTING DIABATIC STATES BASEDON SYSTEM-SOLVENT INTERACTIONS

We now present a new approach for constructing diaba-tic states relevant to electron and energy transfer, beginningwith a standard model for a solvated system.

A. The difficulty modeling system-solvent interactionsexplicitly

As noted many times before !e.g., see Ref. 18", whentreating electron and energy transfer in condensed environ-ments, the need for diabatic states is motivated especially bythe interactions between the system and the polar solvent.The full Hamiltonian for a quantum mechanical system in-teracting with solvent can always be written as18

Hfull = Hsys!r,Rsys" + Hint!r,Rsys,rsolv,Rsolv"

+ Hsolvent!rsolv,Rsolv" . !12"

r ,R denote electronic and nuclear degrees of freedom, re-spectively. We restrict ourselves to cases where electrons arenot exchanged between system and solvent so that systemelectrons !denoted r" can be distinguished from solvent elec-trons !denoted rsolv".

We will also assume that for any nuclear configuration ofthe solvent !Rsolv", the solvent electrons are in their groundstate !independent of the system", which can be found by anelectronic structure calculation on the solvent alone. Thisallows us to collectively represent solvent electrons and nu-clei by the solvent nuclear coordinate !Rsolv" alone,

Hfull = Hsys!r,Rsys" + Hint!r,Rsys,Rsolv" + Hsolvent!Rsolv" .

!13"

For future calculations, we seek an algorithm to computemeaningful diabatic electronic states that are applicable atany fixed nuclear configuration of the system !Rsys".Such states can be found by minimizing the ground stateenergy of the system as a function of all solvent nuclearcoordinates,18

Hfullel !r;Rsys,Rsolv

!m" "&"0!m"!r;Rsys,Rsolv

!m" "'

= E0!Rsys,Rsolv!m" "&"0

!m"!r;Rsys,Rsolv!m" "' , !14"

$

$RsolvE0!Rsys,Rsolv"&Rsolv=Rsolv

!m" = 0. !15"

In Eqs. !14" and !15", we have labeled one optimal choice ofnuclear solvent coordinates by index m and index 0 denotes

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the fact that &"0!m"' is the electronic ground state. The semi-

colons in Eq. !14" emphasize that nuclear and electronic mo-tion are decoupled according to the Born–Oppenheimer ap-proximation and, as stated above, solvent electrons aretreated implicitly by assuming that they are in the solventground state.

Because there can be many optimal nuclear configura-tions satisfying Eqs. !14" and !15", in order for us to describecompletely the stable states of the system, we must find allsuch optimal nuclear configurations for the system and sol-vent. The resulting electronic states %&"0

!m"!r ;Rsys ,Rsolv!m" "'( are

the important set of wave functions for the system electrons,and they depend parametrically on all nuclear coordinates!system and solvent". These wave functions are neither theadiabatic states of the system plus solvent !because we sepa-rate solvent electrons from system electrons" nor the adia-batic states of the system !because we have included theeffect of solvent in our Hamiltonian". Instead, by our defini-tions above, the set %&"0

!m"!r ;Rsys ,Rsolv!m" "'( are exactly the di-

abatic states for the system. Our goal is to compute thesewave functions as easily as possible.

If one treats the solvent as a polarizable continuous me-dium !PCM", Eqs. !14" and !15" have been explored withgreat success over the past 30 years by Tomasi andco-workers.20–23,52–54 The caveats in using PCM-like modelsfor electron or energy transfer are !i" the dependence of theresulting diabatic states on cavity geometry and !ii" the dif-ficulty computing a complete set of diabatic states for thesystem when the system nuclear geometry !Rsys" is biasedtoward one state and the effective solvent nuclear geometryneeds to be very different for each of the diabatic statessought. In the future, when possible, it will be worthwhile tocompare the diabatic states and diabatic coupling matrix el-ements !HAB" produced with PCM with those produced bythe algorithms presented below. In Sec. II B, we will derivean approximate description of the diabatic states in a polarsolvent that requires neither modeling explicit solvent norassuming a fixed cavity geometry.

B. Necessary approximations for system-solventinteractions

We follow the formal procedure given above for con-structing diabatic states in a polar environment, attempting tosatisfy Eqs. !14" and !15" approximately. In the process, wewill eventually derive Boys and ER localizations, which re-quire only calculations of the system in vacuum. To makeany progress, we make several initial assumptions, which arediscussed in detail in Sec. IV.

1. Nearly degenerate adiabatic system states

Our preliminary assumption is that, in the absence ofsolvent, the adiabatic eigenstates of the isolated system invacuum will include some set of nearly degenerate levelswhich, in the presence of a solvent, are mixed together toform diabatic states. This assumption is clearly true in nu-merous physically relevant cases including, for instance,charge transfer systems between symmetric ions or energytransfer along a chain of chromophores.

Formally, we diagonalize the system Hamiltonian Hsysand generate a complete set of adiabatic states !denotedA and labeled by n" for the isolated system,A,%&!n!r ;Rsys"'(, with no reference to a solvent coordinate,

Hsys!r;Rsys"&!n!r;Rsys"' = En!Rsys"&!n!r;Rsys"' . !16"

Here, n=0 is the electronic ground state of the isolated sys-tem, n=1 is the first excited state, etc. We will assume thatthe important adiabatic states are nearly degenerate in en-ergy, with the energy scale to be determined below.

2. The system-solvent interaction is neither too strongnor too weak

For systems that can undergo electron transfer, movingthe solvent corresponds to an outer sphere reorganization,which should be small compared to the range of electronicenergies for the adiabatic states of the system. The magni-tude of a reorganization energy is usually between a fewtenths of an eV and a couple of eV,55 while quantum chem-istry calculations routinely calculate molecular excited stateswith energies 10 eV above the ground state. Thus, we expect

&(Hint& ) energy spread!A" . !17"

Given Eq. !17", when constructing diabatic states, it isreasonable to focus on a reduced subspace W!A whichincludes all system states that are spread out over an energyrange of no more than a 1–2 eV. W can include or excludethe ground state adiabatic wave function of the system, andwe define the dimension of W to be Nstates. We expect Wshould contain the nearly degenerate adiabatic system statespredicated above. The possibility that other, dynamically un-important system states are included in W is discussed inSec. IV A. Now, suppose we are given an optimal set ofsolvent nuclear coordinates Rsolv

!m" satisfying Eq. !14". By as-suming that (Hint should not be too large, the implicationis that in order to compute the electronic groundstate &"0

!m"!r ;Rsys ,Rsolv!m" "', we need only minimize

Hsys!r ;Rsys"+Hint!r ;Rsys ,Rsolv!m" " within the subspace W.

Next, after assuming an upper bound for &(Hint& in Eq.!17", we assume that in a polar solvent, &(Hint& is not toosmall and is larger than the spread of system eigenvalues inW,

energy spread!W" * &(Hint& ) energy spread!A" . !18"

Thus, the system-solvent interaction is assumed to be thedominant term in lifting the near degeneracy of the vacuumadiabatic states. The implication of this second and balancingassumption is that instead of diagonalizingHsys!r ;Rsys"+Hint!r ;Rsys ,Rsolv

!m" " directly in the W subspace, agood approximation is to diagonalize (Hint alone !in the Wsubspace" and look for the lowest eigenvalue. If the adiabaticstates in W were exactly degenerate, this would be equiva-lent to first-order degenerate perturbation theory.

3. The system-solvent interaction is electrostatic

Our last and biggest assumption is that the system-solvent interaction energy is based on electrostatics. With

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this in mind, we may write the electronic density at positionr!> as a dynamic variable and the system-solvent interactionHamiltonian as an integral over all space,

+!r!>" = )j

,!r!> ! r!!j"" , !19"

Hint!r;Rsys,Rsolv!m" " =+ dr!>&int!r!> ;Rsys,Rsolv

!m" "+!r!>" . !20"

Here, r!!j" represents the position of the jth electron and thisexpression is exact for arbitrary system-solvent interactionenergies that are based on electrostatics. The hat on + de-notes the fact that + is an operator, and &int is the electrostaticpotential caused by the solvent acting on the system. Werepeat that the index m stands for one configuration of sol-vent nuclei which is optimal for stabilizing the system.

C. Distinct models for the system-solvent interactionand the resulting diabatization algorithms

Thus far, we have outlined a general strategy for obtain-ing diabatic states based on assumptions about the strengthof the system-solvent interaction potential Hint relative toother energy ranges of the system. The derivation of differentdiabatization methods now hinges on our specific treatmentof the interaction potential Hint. As shown below, applyingdifferent models for Hint leads to distinct diabatization meth-odologies.

1. Multipole expansion: Boys diabatic statesand electron transfer

For a system located near the origin, one simple approxi-mation to the general expression in Eq. !20" is to representthe interaction energy as a multipole expansion of the elec-tronic coordinate r of the system around zero,

Hint!r;Rsys,Rsolv!m" " = Hint

!0,m" + Hint!1,m" + Hint

!2,m" + Hint!3,m" + ¯

= Q!0" + )i=x,y,z

Qi!1"!Rsys,Rsolv

!m" "ri

+ )i,j=x,y,z

Qij!2"!Rsys,Rsolv

!m" "rirj + ¯ .

!21"

Here, ri stands for a Cartesian component of the second-quantized electronic coordinate r! !i.e., ri=x ,y ,z". In theneighborhood of the system, we must assume that &Hint

!0,m"&- &Hint

!1,m"&- &Hint!2,m"&-¯. This multipole expansion works

best for small systems where the system-solvent interactionis dominated by the linear term.

Making all the assumptions in Sec. II B and approximat-ing Eq. !21" by the linear term !as would be expectedin the case of electron transfer", it follows that we can con-struct diabatic states by diagonalizing the operatorHint

!1,m"=)i=x,y,zQi!1"!Rsys ,Rsolv

!m" "ri in the subspace W. The prob-lem remains, however, that without actually constructingRsolv

!m" , which would be very costly, there is no straightforwardway to minimize Hint

!1,m" because the coefficientsQi

!1"!Rsys ,Rsolv!m" "ri depend on the solvent configuration. Thus,

any diabatization algorithm would appear to require model-ing the solvent, either explicitly as molecules or as a PCM.

Nevertheless, although computing Qx!1", Qy

!1", and Qz!1"

formally requires Rsolv!m" , note that if all three component op-

erators of r, i.e., x ,y ,z, were diagonalized simultaneously,then Hint

!1,m" would automatically be diagonal and minimiza-tion would be trivial. Now, even though x ,y ,z cannot besimultaneously diagonalized in the subspace W, an approxi-mate diagonalization of all three operators in the subspace Wcan be achieved by rotating the basis states %&!n'( into newstates %&$l'(, where the variances of the x ,y ,z operators areminimal. It is important to note that, for any operator A andstate ', if *'&A2&''! *'&A&''2=0, then ' is an eigenvectorof A.

Thus, we propose to rotate the adiabatic states into di-abatic states as in Eq. !5" where we fix the rotation matrix Uby minimizing the sum of the variances of each Cartesiancomponent in each of the different rotated states,

fBoys!U" = fBoys!%$i("

= )l=1

Nstates

)i=x,y,z

!*$l&ri2&$l' ! *$l&ri&$l'2" . !22"

Equation !22" is a reasonable minimization criterion becausethe linear expansion of a complex three-dimensional system-solvent interaction into three terms is a very compact repre-sentation, and minimizing the sum of the variances of allthree dipole operators !x ,y ,z" can be accomplished easilyand effectively. As a bonus, minimization of Eq. !22" is in-variant to translation of the origin or rotations of the coordi-nate axes #see Eq. !23"$.

Now, minimizing Eq. !22" generates approximate eigen-vectors of Hint

!1,m" without any detailed knowledge of the sol-vent coordinate Rsolv

!m" . Thus, for each solvent configurationRsolv

!m" , it would be pointless for us to find the exact &$m' thatminimizes *$m&Hint

!1,m"&$m'. Instead, we should assume thateach basis function in the set %&$l'( corresponds to the opti-mal electronic state for a different solvent configuration,Rsolv

!l" . This interpretation is particularly reasonable if each ofthe states in %&$l'( keeps excess charge localized on a differ-ent charge center. In that case, we will argue that the set%&$l!r ;Rsys"'(l=1

Nstates is a complete set of diabatic states for oursystem within the energy range defined by W. At the sametime, if two diabatic states have charge localized on the samecenter, the system Hamiltonian should be rediagonalizedwithin this two-dimensional subspace to generate unique, lo-cally adiabatic diabatic states. This completes our recipe forconstructing the diabatic states of the electron transfersystem.

Solving Eqs. !5" and !22" is exactly equivalent to theBoys algorithm, which was shown previously to give charge-localized diabatic states relevant to electron transfer.27 Thislocalization property of the Boys algorithm becomes mostobvious if, using the fact that the trace of an operator isinvariant to representation, we maximize Eq. !23" rather thanminimize Eq. !22",

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fBoys!U" = fBoys!%$i(" = )i,j=1

Nstates

&*$i&%! &$i' ! *$ j&%! &$ j'&2.

!23"

Moreover, if we were to pick a charge transfer direction v0and diagonalize the operator r ·v0 in the subspace W, wewould recover the GMH formalism. At this point, we haveapproximately justified the Boys and GMH algorithms onphysical grounds, and we have motivated why it is importantto rotate together only adiabatic states that are close inenergy.

One final point must be made about Boys localization.The form of Eq. !23" should remind the reader that Boyslocalized diabatic states are applicable only for electrontransfer and not for energy transfer.56 The optimization func-tion fBoys in Eq. !23" seeks to separate the centers of chargebetween different diabatic states and if we focus on a sub-space W of adiabatic states !within a certain energy range"that exhibits energy transfer and not electron transfer, we willfind that the sum fBoys is invariant to rotations in W andcannot be optimized. In such cases, the system-solvent inter-action is not linear in space and cannot be described by thelinear term in Eq. !21". Instead, we must work with a moregeneral expression for the system-solvent interactionpotential.

2. Linear response of the solvent: ER diabatic statesand electron/energy transfer

Beyond the case of solvent fields which are linear inspace, a more robust approximation to Eq. !20" is for us toassume that the response of the solvent is a linear function ofthe state of the system. With that in mind, we denote theelectrostatic potential at a point r!>1 in space caused only bythe system as

&sys!r!>1" =+ dr!>2*+!r!>2"'&r!>1 ! r!>2&

+ Vnuc!r!>1" . !24"

Here, the bracket * ' denotes that the density operator + mustbe evaluated in the state of the system.

We now assume that the solvent acts as a linear dielec-tric, with dielectric constant .. Thus, the electrostatic poten-tial from the solvent should be linearly proportional to theelectrostatic potential from the system, with a constant!1!." /.,

&int!r!>1" =1 ! .

.&sys!r!>1" . !25"

It follows that the total system-solvent interaction is nonlin-ear and can be written as

Hint =1 ! .

.+ dr!>1+ dr!>2

*+!r!>2"'+!r!>1"&r!>1 ! r!>2&

+1 ! .

.+ dr!>1Vnuc!r!>1"+!r!>1" . !26"

Equations !25" and !26" are commonly used in solid statephysics when computing solvation energies and screeningeffects for a system immersed in a linear dielectric.57 If weseek a diabatic state &$i' which minimizes the system-solvent interaction energy, we must minimize the energy,

Eint!i" =

1 ! .

.-+ dr!>1+ dr!>2

*$i&+!r!>2"&$i'*$i&+!r!>1"&$i'&r!>1 ! r!>2&

++ dr!>1*$i&Vnuc!r!>1"+!r!>1"&$i'. . !27"

Because we seek a complete set of orthogonal diabaticstates, it would not be fruitful to search for the global mini-mum of Eq. !27". Instead, we will search for the rotationmatrix U in Eq. !5" which minimizes the sum of the system-solvent interactions for each orthonormal diabatic state. Thesecond term in Eq. !27" !arising from the system nuclearpotential" is then a constant because the nuclei are fixed andthe resulting trace is invariant to representation. Hence, be-cause the dielectric constant of a condensed environment sat-isfies .-1, we must maximize

fER!U" = fER!%$i("

= )i=1

Nstates+ dr!>1+ dr!>2*$i&+!r!>2"&$i'*$i&+!r!>1"&$i'

&r!>1 ! r!>2&.

!28"

Because of its similarity to Eq. !9", Eq. !28" should be calledthe ER function for localizing diabatic states. In Appendix A,we show how to maximize the function fER in the context ofconfiguration interaction singles !CIS" excited states.

In Sec. III, we give chemical examples showing that ERlocalization successfully generates diabatic states applicableboth to electron transfer and energy transfer. In this sense,ER localization is much more powerful than Boys localiza-tion.

FIG. 1. !Color online" Detachment density plot for an adiabatic excited statein the trans conformer of OMP3. Note that the detached and attached den-sities are both delocalized over the molecule.

FIG. 2. !Color online" Attachment density plot for an adiabatic excited statein the trans conformer of OMP3.

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3. A delta function potential between freeand induced charges: VNER diabatic states

For completeness, in Appendix B, we present a thirdalgorithm for constructing diabatic states, the VNER formal-ism. The VNER approach can be derived by assuming thatthe solvent responds linearly to the system, but free chargesin the system interact with induced charges in the solvent viaa delta function in real space !rather than by Coulomb’s law".VNER diabatic states satisfy a maximally different densitycriterion and have elements in common with both Boys andER localized diabatic states.

III. A CHEMICAL EXAMPLE:OLIGOMETHYLPHENYL-3, ENERGYTRANSFER, AND THE CONDON APPROXIMATION

A previous article has already demonstrated that Boyslocalization leads to charge-localized diabatic states.27 In or-der to verify that ER localization does the same, we haveperformed straightforward CIS calculations on linear#Na–Be–Na$2+ in a 6-31G" basis, where the Na–Be distanceis 5 Å. All electronic structure calculations were performedusing the Q-CHEM package.58 In the ground state of the mol-ecule, after a HF calculation, the Be atom is neutral and bothNa atoms have charge +1. The first and second adiabaticexcited states of the system, &!1' and &!2', are nondegener-ate delocalized charge transfer states, wherein Be donates anelectron into orbitals on both Na atoms. Performing eitherER or Boys localization on the subspace W= &!1' # &!2'produces two energetically degenerate diabatic states, &$1'and &$2', each with excess charge localized on a separate Naatom. The diabatic coupling elements between the two statesare identical for the two algorithms. More details of thissimple case are given in the online EPAPS depository,59 in-cluding attachment/detachment plots. We conclude that,similar to Boys localization, ER localization can indeed de-scribe diabatic states appropriate for charge transfer. Themore challenging test, however, is whether ER can describediabatic states appropriate for energy transfer. We demon-strate this now for one chemical example.

Consider the molecule shown in Figs. 1 and 2, wherethree benzene rings are attached together by two CH2 singlebonds. For notational ease, we will call this moleculeoligomethylphenyl-3 !OMP3". CIS calculations for OMP3 ina 6-31G" basis in the HF optimized ground state geometryshow that the six lowest-lying excited states have energies6.137, 6.138, 6.267, 6.291, 6.427, and 6.553 eV relative to

the ground state. The next lowest excited states begin at7.652 eV and will be ignored. Figures 1 and 2 areattachment/detachment60 plots for one of these six adiabaticstates which shows that in the adiabatic picture, both thedetached and attached electron density are delocalized overthe whole molecule. According to Figs. 1 and 2, the effectivedetached electron comes from the / system and the attachedelectron goes into the /" system.

Given that the six lowest excited states of OMP3 are soclose in energy, we propose that these six states should bemixed together and diabatized according to the ER algorithmabove, which should approximately represent solvent !or vi-brational" effects. In order to check the Condon approxima-tion, we have performed this diabatization procedure both forthe gauche and trans conformers of OMP3, checking thesensitivity of the coupling element HAB to nuclear geometry.

In contrast to the delocalized picture of electronic exci-tation in the adiabatic basis, the attachment/detachment plotsin Figs. 3–6 show that the electronic excitation can beviewed as local in the basis of diabatic states. For each di-abatic state, the attached and detached electron appears to belocalized in the / and /" orbitals of one specific monomer.Note that after the initial ER localization, we find two diaba-tic states with electron-hole pairs on each benzene monomer.In order to generate the locally adiabatic diabatic ER statesshown in Figs. 3–6, we have rotated within each two-dimensional subspace so that the final Hamiltonian is diago-nal between states with excitation energy on the same center.As described above, our intuition is that states with excita-tions !or charge" on the same center should be uncoupled, asthe effect of solvent should be minimal between states soclose together. We admit, however, that this approach is notgeneral and cannot be rigorously justified. Nevertheless, it iscomforting that the attachment/detachment plots of the ERdiabatic states before and after this final diagonalization lookvery similar. Finally, we mention in passing that unlike ERdiabatization, Boys localization !with the dipole operator"produces erratic and unphysical diabatic states with delocal-ized electron-hole pairs for this molecule.56

In Table I, we give the diabatic coupling elements!HAB= *$A&Hel&$B'" between ER diabatic states. From thedata, the six excited states can be separated into two sets ofthree, those with even state numbers and those with odd statenumbers, and each set is nearly uncoupled from the other.Let us label the degenerate highest occupied molecular orbit-als of benzene /1 and /2, and the degenerate lowest unoc-cupied molecular orbitals /1

" and /2". Because the diabatic

states shown in Figs. 3–6 show no nodal plane perpendicularto the benzene plane, it is likely that the two diabatic states

FIG. 3. !Color online" Detachment density plot for an ER localized diabaticexcited state in the trans conformer of OMP3. Note that the detached den-sity is localized on one and the same benzene monomer as the attacheddensity. This state with a localized electron-hole pair could be called anexciton.

FIG. 4. !Color online" Attachment density plot for an ER localized diabaticexcited state in the trans conformer of OMP3.

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on each benzene monomer are approximately &$1,2'= !1 //2"!!/1

""†/10 !/2""†/2"&"HF'. In this second-quantized

expression, we imagine replacing /1 by /1" and /2 by /2

",and adding or subtracting the two singly substituted Slaterdeterminants. This rough picture of the diabatic states wouldfit the attachment/detachment density seen in Figs. 3–6, al-though this is not conclusive.

According to Table I for both groups of diabatic states,the nearest-neighbor diabatic couplings are very similar be-tween the trans and gauche conformers, agreeing with theCondon principle that HAB should be insensitive to nucleargeometry. Between diabatic states on opposite ends, how-ever, the diabatic coupling is dramatically changed, in onecase increasing by a factor of 3. This reflects the fact that theopposite benzene monomers become much closer when themolecule is rotated around the methylene group. Moreover,for the gauche conformation, the odd and the even groups ofstates are coupled more strongly than they are in the trans

conformation. Nevertheless, the nearest-neighbor diabaticcouplings within each group are still one to two orders ofmagnitude larger than all other couplings, and because thesecouplings vary only minimally from trans to gauche, theCondon approximation largely holds for the molecule.

Finally, note that we have calculated diabatic states andHAB elements at the ground state geometry of the molecule,rather than at the formal transition state geometry, where theedge and middle states would be degenerate. While the lattergeometry would be more common in typical Marcus theoryelectron or energy transfer applications, we point out thatwithin each group, the energies of our diabatic states arewithin 0.15 eV. By contrast, for the adiabatic states, thespread in energies is approximately 0.41 eV. We also notethat the on-diagonal energies of the diabatic states changeonly minimally !*0.01 eV" between the trans and gaucheconformations. In the end, these ER diabatic states conformto our expectations of the initial and final states of an energytransfer process.

TABLE I. The diabatic coupling elements !HAB" for the ER diabatic states of OMP3 in units of electron volts!eV". Here, we have included six adiabatic excited states in our subspace W and, because of symmetry, we findtwo groups of three states that are nearly uncoupled !i.e., the odd and even numbered states below". The sixadiabatic excited states were computed with CIS in a 6-31G" basis. Because the nearest-neighbor couplings arevery similar for both trans and gauche conformers, the ER diabatic states obey the Condon approximation.

State

Monomer

1 2 3 4 5 6

Center Center Left Left Right Right

!a" Trans conformation1 Center 6.1615 0 10.0394 1210!5 0.0396 10.00012 Center 6.2759 0.0002 0.1384 10.0003 0.13833 Left 6.2673 0 4210!5 0.00074 Left 6.4213 10.0007 10.00615 Right 6.2671 06 Right 6.4212

!b" Gauche conformation1 Center 6.1609 0 10.0395 0.0011 0.0391 10.00112 Center 6.2736 10.0010 10.1391 10.0011 10.13883 Left 6.2670 0 0.0001 10.00114 Left 6.4204 10.0009 10.00685 Right 6.2669 06 Right 6.4203

FIG. 5. !Color online" Detachment density plot for an ER localized diabaticexcited state in the gauche conformer of OMP3. Note that the detacheddensity is localized on one and the same benzene monomer as the attacheddensity. This state with a localized electron-hole pair could be called anexciton.

FIG. 6. !Color online" Attachment density plot for an ER localized diabaticexcited state in the gauche conformer of OMP3.

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IV. DISCUSSION AND SUMMARY

Rather than seeking diabatic states with zero derivativecouplings, in this paper, we have examined diabatic stateswhich describe a system interacting with a polar solvent be-fore or after an electron or energy transfer process. In prac-tical terms, we have proposed constructing such diabaticstates by !i" calculating the adiabatic states of the systemwithout solvent; !ii" grouping together those adiabatic statesthat fall within an energy window approximately equal to thereorganization energy of the solvated system; !iii" perform-ing either Boys or ER localization on this relevant subspaceof adiabatic states. Both the Boys and ER diabatizationschemes have very physical foundations, and we believeboth will be successful in describing nonequilibrium trans-port phenomena. Boys localization is based upon a system-solvent potential that is linear in space and is applicable toelectron transfer. ER localization is based upon linear re-sponse of the solvent to the system and is applicable to bothelectron and energy transfer. Overall, the ER method satisfiesall six of the desirable criteria for diabatization algorithmslisted in Sec. I D, and the Boys method satisfies five. Weemphasize that neither algorithm requires definitions ofcharge fragments !unless multiple diabatic states have chargeor excitation energy on the same atom center or monomerand we choose to rediagonalize the Hamiltonian within thissubblock". For the OMP3 problem, the compatibility of theER approach with the Condon approximation is encouragingand the attachment-detachment plots in Figs. 3–6 agree withour intuition for the shape of excitons in this system.

Although we have tried to be as rigorous as possible inderiving the Boys and ER diabatization algorithm abovefrom system-solvent interactions, we would now like to ana-lyze the approximations made in modeling the effect of sol-vent and make explicit the limitations of our approach.

A. Assumption of nearly degenerate, well-separatedsystem states in a polar solvent environment

Our methods of diabatization assume that the relevantadiabatic system states are nearly degenerate and can begrouped together in a subspace W. Furthermore, we assumethat changes in the system-solvent interaction (Hint are onthe order of the reorganization energy, which is small com-pared to the full energy spectrum of the system but largecompared to the energy width of the nearly degenerate states.It follows that diabatic states can be constructed by minimiz-ing Hint in the subspace W. These two assumptions haveclear limitations, however.

First, for a general molecule or system, there is no guar-antee that the adiabatic states of interest will be close inenergy but well separated from other states. Such a casewould appear to be the exception rather than the rule. More-over, in practice, when doing electronic structure calculationswith active space models !e.g., CASSCF/CASPT2", un-wanted “intruder” states can appear that are not physical ordynamically important.61 Thus, for real ab initio calculations,there is not always a black-box algorithm for picking adia-batic states to mix together in order to form meaningful di-abatic states, and some physical intuition may be necessary,

using properties other than just energy. For difficult cases,when choosing the appropriate adiabatic states to mix to-gether, investigating the sizes of derivative couplings may behelpful as a last resort.

Second, when diagonalizing Hint instead of Hint+Hsys inthis subspace, we assume that solvent effects are strong and,thus, the algorithm here can work only with a polar solventstrongly coupled to the system. For less extreme conditions,besides applying PCM for the solvent, one approach to bal-ance the quantum mechanical system energetics !Hsys" withthe electrostatic system-solvent interaction !Hint" would be tointroduce an empirical parameter 3, such that the diabaticstates are the minimal solution to either

fERcomp = 3 )

l=1

Nstates

!*$l&Hsys2 &$l' ! *$l&Hsys&$l'2"

+ !1 ! 3"+ dr!>1+ dr!>2*$l&+!r!>1"&$l'*$l&+!r!>2"&$l'

&r!>1 ! r!>2&,

!29"

fBoyscomp = 3 )

l=1

Nstates

!*$l&Hsys2 &$l' ! *$l&Hsys&$l'2"

+ !1 ! 3" )l=1

Nstates

)i=x,y,z

!*$l&ri2&$l' ! *$l&ri&$l'2" .

!30"

When 3=0, we recover either the Boys or the ER diaba-tic states. When 3=1, we recover the adiabatic states. Thecomputational cost of solving Eq. !29" is identical to solvingEq. !28", and the cost of solving Eq. !30" is identical to Eq.!22". For weak system-solvent interactions, it may be benefi-cial to see whether choosing 0*3*1 best matches experi-ment. Similar attempts to mix energetic effects with localityeffects within the context of molecular orbital theory havehad moderate success.62 A good choice for 3 would neces-sarily depend on the dielectric constant of the solvent anddiabatic states computed with Eqs. !29" and !30" should becompared to models using PCM for the solvent.

B. Electrostatic assumption

Both the Boys and ER diabatization schemes have as-sumed that Hint arises only from electrostatics. This allowedus to use one-electron operators #either r or +!r!>"$ to approxi-mate the effect of solvent on the system, which makes dia-batization practical and fast. It is worthwhile, however, toconsider under what circumstances the system-solvent inter-actions can be safely modeled by a one-electron electrostaticoperator.

Interactions with solvent may induce strong effects be-tween different system electrons, and these are not includedin our Hamiltonian #Eq. !21"$, especially given that we diag-onalize Hint instead of Hint+Hsys. We are also ignoring chargetransfer between system and solvent and induced dipole-dipole effects, which could both be important. In particular,for the problem of energy transfer, we have assumed thatlocal electronic excitations will lead to different charge dis-

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tributions, and that the solvent response to these electrostaticchanges is enough to stabilize excitons, which may not al-ways be true. Many different physical effects may force thesystem-solvent interaction to be neither electrostatic nor aone-electron operator, and for comparison future researchshould explore diabatization using alternative ansätze for theinteraction. In particular, for a higher computational cost, itmay be possible to construct better diabatic states using bothone-electron and two-electron operators to account more ac-curately for both external and internal interactions.

Beyond electrostatics, we remind the reader that Boyslocalization makes the additional assumption that the system-solvent interaction potential has a valid multipole expansionnear the system which is dominated by the linear term. In thefuture, it will be very interesting to compare ER results withBoys localization computations on charge transfer systems,and assess the relative accuracy of each method. If our ex-perience with orbitals is relevant, different localizationschemes usually agree qualitatively, except in cases of sym-metry !/ versus 4, etc.". For the trivial example of#Na–Be–Na$2+ in Sec. III, the ER and Boys localized diaba-tic states were identical.

C. Derivative couplings and potential energy surfaces

Beyond the limitations expressed above, there remainsthe question of how Boys or ER localized diabatic statescompare to more standard diabatization algorithms. To con-nect our diabatization scheme with other approaches, the sizeof the derivative couplings63 must be small. Second, near aconical intersection where analytical forms exist for exactdiabatic states, the behavior of diabatic couplings and deriva-tive couplings produced with Boys localized states mustmatch standard solutions. These computational benchmarkswill yield important information toward understanding andvalidating Boys localization and the ER approach. We arecurrently implementing the necessary code to perform suchcalculations, and we will report our results in a later commu-nication.

A separate concern is to whether the potential energysurfaces and off-diagonal coupling terms are smooth in thebasis of Boys or ER diabatic surfaces. In this article, we haveperformed diabatization at one geometry alone, which waschosen to resemble the nonadiabatic transition state. We havenot checked for smoothness of the global potential energysurface, which will likely be problematic because, as the mo-lecular geometry changes, different adiabatic states will nec-essarily enter and leave the energetic window defined by theW subspace. Moreover, we have not examined the shape ofthe potential energy curves in the vicinity of a crossing point.We presume that one can compute the analytical gradients!with respect to nuclear displacements" of the energies ofthese diabatic states, although we have not yet done so. Forthese reasons, use of the Boys or ER diabatic states as acomplete basis for chemical dynamics requires more bench-marking and may be challenging.

Ultimately, however, the usefulness of any diabatizationalgorithm for electron and energy transfer can be measured

only by how well it matches experiment. In the near future,we hope to use Boys localization and ER to generate diabaticstates for ab initio models of electron or energy transfer forcomparison with experimental data.

D. Computational cost

Before finishing this discussion, we want to emphasizethat if optimally implemented, both algorithms discussedhere are computationally inexpensive and will never be rate-limiting compared to the necessary electronic structure cal-culations. Like GMH, Boys localization requires as inputonly the dipole matrices in the basis of adiabatic states, andbecause these are one-electron operators, they can usually becomputed with effectively zero cost. Likewise, the ER ap-proach also requires matrix elements of one-electron opera-tors ar

†as in the basis of adiabatic states, which have effec-tively zero cost. See Appendix A for details on how the ERalgorithm is applied in the context of CIS excited states.

Although neither Boys localization nor ER have a trans-portable formula for predicting the HAB coupling elementssuch as GMH, the standard algorithm for optimizing quarticfunctions of rotation matrices converges rapidly using so-called Jacobi sweeps.47,48,64 Because Boys localization re-quires the storage of only a quadratic number of variables,the algorithm runs almost instantaneously compared to anyprerequisite electronic structure method no matter how bigthe system size. The ER algorithm scales worse than theBoys localization because one must work with the two-electron matrix elements of 1 /r12, i.e., !%5 &64" !see Appen-dix A". Nevertheless, if one uses the sparsity of the atomicorbital basis and one works in a subspace W with fewer thantwenty adiabatic states !as would be expected usually", anoptimal implementation of the ER algorithm should be fast,with only a marginal increase in computational cost after thenecessary electronic structure calculations. Recent advanceswith resolution of the identity !RI" methods have made ERlocalization for orbitals relatively inexpensive,64,65 and thesame should be true for ER localization of diabatic states.See Appendix A for more details.

Ultimately, both Boys localization and ER can be ap-plied to large chemical systems, where they can model eitherelectron transfer or energy transfer !in the case of ER". Thisapplicability to large systems is not true for other diabatiza-tion approaches involving analytical derivative couplings,which are very computationally expensive to calculate.

ACKNOWLEDGMENTS

We thank Alex Sodt for discussions regarding the con-nections, both theoretical and computational, between orbitallocalization and many-electron state diabatization. We thankProfessor Troy Van Voorhis for help in structuring this articleand useful suggestions for understanding this work in thecontext of a PCM model. J.E.S. was supported by a NSFInternational Research Fellowship !Grant No. OISE-0701345" and thanks Professor Abraham Nitzan, ProfessorMark Ratner, Professor Michael Baer, Sina Yeganeh, IgnacioFranco, and David Masiello for stimulating conversations.

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R.J.C. is pleased to acknowledge support from Harvey MuddCollege. N.S. and R.P.S. would like to thank Professor JohnTully for his support, the NSF !Grant No. CHE-0615882"and the Department of Energy Basic Energy Sciences !GrantNo. DE-FG02-05ER15677" for funding.

APPENDIX A: IMPLEMENTATION OF ERDIABATIZATION AS APPLIED TO CISEXCITED STATES

For concreteness, we derive the principal equations nec-essary for implementing ER localization of CIS excitedstates. Many of the equations below can be found also in thework of Hsu and co-workers.33–35 For the remainder of thissubsection, many-electron states are indexed with capital let-ters %I ,J ,K ,L ,M( and molecular spin orbitals are indexed bylower-case letters: %i , j( identify occupied orbitals, %a ,b(identify virtual orbitals, and %p ,q ,r ,s( identify either occu-pied or virtual orbitals. States are labeled by capital Greekletters !adiabatic states !, diabatic states $" and orbitals arelabeled by lower-case Greek letters !&".

ER localization for a group of adiabatic excited states%&!I'( is defined by the rotation U satisfying

&$I' = )J

&!J'UJI, !A1"

fER!U" = max%$I(

)I=1

Nstates+ dr!>1+ dr!>2*$I&+!r!>2"&$I'*$I&+!r!>1"&$I'

&r!>1 ! r!>2&.

!A2"

The one-electron density of diabatic state I at position r!>can be expressed using second-quantized operators in thebasis of molecular orbitals as

*$I&+!r!>"&$I' = )rs

*$I&cr†cs&$I'&r!r!>"&s!r!>" !A3"

=)JK

UJIUKI)rs

*!J&cr†cs&!K'&r!r!>"&s!r!>"

!A4"

For CIS excited states, the ansatz of the adiabatic wavefunction is

&!J' = )ia

tiJaca

†ci&!HF' . !A5"

According to Wick’s theorem, it follows that the density ma-trix D is

DrsJK = *!J&cr

†cs&!K' , !A6"

=0 )i

tiJrti

Ks, r,s = virtual

! )a

trJats

Ka + ,rs,JK, r,s = occupied.1 !A7"

Finally, using the chemists’ notation for the two-electronCoulomb integral !rs & pq",51 fER can be expressed as

fER!U" = )IJKLM

UJIUKIULIUMI )rspq

DrsJKDpq

LM+ dr!>1+ dr!>2

2&r!r!>1"&s!r!>1"&p!r!>2"&q!r!>2"

&r!>1 ! r!>2&, !A8"

= )IJKLM

UJIUKIULIUMIRJKLM , !A9"

RJKLM = )rspq

DrsJK!rs&pq"Dpq

LM !A10"

Equation !A9" shows that ER localization of diabaticstates is identical to ER localization of orbitals. All standardlocalization algorithms are quartic functions of the unitarymatrix U and can be solved with standard “Jacobi sweeps”48

or with other iterative approaches.64,65 Because we do notanticipate having too many states to localize !usually, under20" and computing an element of the tensor RJKLM will beexpensive, Jacobi sweeps should be the most attractive ap-proach for ER localization of states. In order to have a fastimplementation of ER diabatization, the key bottleneck toovercome will be computing the tensor RJKLM in Eq. !A10"quickly. Research is currently ongoing to develop an algo-rithm to make this contraction as fast as possible. In calcu-lations for this paper, we have computed RJKLM using theRI-approximation69 with an auxiliary basis !y , z",

RJKLM = )rspq

)yz

DrsJK!rs&y"!y&z"!1!z&pq"Dpq

LM . !A11"

In the future, we expect that this contraction can be mademuch faster.

Finally, we note that the form of Eqs. !A8"–!A10" iscompletely general and can be applied to arbitrary excitedstates, not just CIS states. The largest obstacle will be com-puting the tensor Drs

JK= *!J&cr†cs&!K' when &!J' and &!K' are

not CIS excited states. For CIS, Eq. !A7" is simple to deriveand the three terms are the density of attached electrons,detached electrons, and the ground state. The ground statedensity is just a constant and does not affect Boys or ERlocalization, suggesting a connection with the FEDapproach—FED only uses electron attachment/detachmentdensities.

APPENDIX B: THE MAXIMUM DENSITYDIFFERENCE CRITERION:VON NIESSEN–EDMISTON–RUEDENBERGDIABATIC STATES

In Secs. II C 1 and II C 2 above, we derived Boys andER diabatization on the basis of system-solvent interactions.Boys localization was based upon a linear expression for theinteraction potential in terms of the x ,y ,z operators, and ERwas based upon a linear response of the solvent to system.We now derive a third diabatization scheme, which sharessimilarities with both standard algorithms and may alsoprove useful. Because of the similarity to VNER localization#see Eq. !11"$, these diabatic states should be called VNERdiabatic states.

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Recall that in our derivation of Boys localization, al-though we wanted to minimize Hint in Eq. !21" directly, in-stead our approach was to minimize the sum of the variancesof the three dipole operators, x ,y ,z. This was a reasonableproposition because we did not know the coefficients in frontof the different dipole operators, but the linear expansion ofa complex three-dimensional system-solvent interaction intothree terms is a very compact representation, and minimizingthe sum of the variances of each operator was effective. As abonus, the resulting Boys localization algorithm was invari-ant to translation of the origin or rotations of the coordinateaxes.

Now, according to Eqs. !19" and !20", even when thesystem-solvent potential is not linear in space, the interactioncan always be expanded over an infinite number of operatorsindexed by r!> , namely, %+!r!>"(, where r!> can be any position inreal, three-dimensional space. Moreover, this expansion isactually a compact description of the system-solvent interac-tion because electrostatic potential energies are diagonal onlyin position space !and not in momentum space". Thus, justlike for Boys localization, we can attempt to diagonlize Hintin Eq. !19" by minimizing the integral of the variances of allof the %+!r!>"( operators,

fVNER!U" = fVNER!%$i("

= )l=1

Nstates+ dr!>!*$l&+2!r!>"&$l' ! *$l&+!r!>"&$l'2" .

!B1"

Because the trace of an operator is invariant to representa-tion, minimizing fVNER is equivalent to maximizing

fVNER!U" = fVNER!%$i("

= )k,l=1

Nstates+ dr!>!*$k&+!r!>"&$k' ! *$l&+!r!>"&$l'"2.

!B2"

Optimizing fVNER breaks the near energetic degeneracy in Wby rotating the adiabatic states so that their diabatic densitiesare maximally different. Thus, this approach has the flavor ofthe FCD approach32 and the FED approach,33–35 only withoutdefining any fragments. Moreover, from the form of thefunction fVNER in Eq. !B2", it is clear that similar to the Boysalgorithm, VNER diabatization is invariant to translationsand rotations of the origin.

Looking back at Eqs. !26"–!28", we see that the VNERapproach can also be derived by assuming that the responseof the solvent is linearly dependent on the system, but thatthe induced charges in the solvent interact via a delta func-tion ,!r!>1!r!>2" with free charges in the system !instead of theCoulomb operator 1 / &r!>1!r!>2&",

fVNER!U" = )l=1

Nstates+ dr!>1+ dr!>2*$l&+!r!>2"&$l'

2*$l&+!r!>1"&$l',!r!>1 ! r!>2" , !B3"

= )l=1

Nstates+ dr!>1*$l&+!r!>1"&$l'*$l&+!r!>1"&$l' . !B4"

Given the similarities between VNER and ER localiza-tion, VNER diabatic states should be applicable to both elec-tron transfer and energy transfer, but this must be checkedexplicitly. One potential advantage of the VNER approachwill be computational speed. Whereas ER localization re-quires the contraction of the two-electron Coulomb integral!pq &rs" in Eq. !A10", the VNER approach requires only theone-electron overlap of four orbitals,

Spqrs =+ dr!>&p!r!>"&q!r!>"&r!r!>"&s!r!>" . !B5"

In an atomic orbital basis, S%564 is much more sparse than!%5 &64", and the VNER approach will be faster than ERlocalization, although still slower than Boys localization. Ingeneral, we expect the VNER approach to be a compromisebetween Boys and ER localization and it may also find use infuture diabatization calculations.

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who proposed them in Ref. 48. However, ER orbitals currently refer to agroup of maximally self-interacting orbitals rather than a set of orbitalswith minimal density overlap. Moroever, von Niessen appears to be thefirst to implement the density overlap algorithm.

51 A. Szabo and N. Ostlund, Modern Quantum Chemistry: Introduction toAdvanced Electronic Structure Theory !Dover, New Jersey, 1996".

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the origin. For this reason, we have not pursued this approach here.57 N. W. Ashcroft and N. D. Mermin, Solid State Physics !Brooks Cole,

Orlando, 1976".58 Y. Shao, L. Fusti Molnar, Y. Jung, J. Kussmann, C. Ochsenfeld, S. T.

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59 See EPAPS Document No. E-JCPSA6-130-035923 for more details onthe BeNa2

+ calculation. For more information on EPAPS, see http://www.aip.org/pubserve/epaps.html.

60 Attachment/detachment plots are generated by subtracting the one-electron density matrix of the ground state from that of the excited state,and separating positive and negative eigenvectors. See Ref. 66 for moredetails.

61 B. O. Roos and K. Andersson, Chem. Phys. Lett. 245, 215 !1995".62 J. E. Subotnik and M. Head-Gordon, J. Chem. Phys. 122, 034109 !2005".63 The question has been posed before as to the magnitude of the derivative

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!1993".

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