A new hybrid approach for modeling reactions in molecular clusters:Application for the hydrogen bonded systemsAlexander V. Nemukhin, Bella L. Grigorenko, Ekaterina Ya. Skasyrskaya, Igor A. Topol, and Stanley K. Burt Citation: J. Chem. Phys. 112, 513 (2000); doi: 10.1063/1.480542 View online: http://dx.doi.org/10.1063/1.480542 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v112/i2 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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JOURNAL OF CHEMICAL PHYSICS VOLUME 112, NUMBER 2 8 JANUARY 2000

A new hybrid approach for modeling reactions in molecular clusters:Application for the hydrogen bonded systems

Alexander V. Nemukhin, Bella L. Grigorenko, and Ekaterina Ya. SkasyrskayaDepartment of Chemistry, Moscow State University, Moscow 119899, Russian Federation

Igor A. Topola) and Stanley K. BurtAdvanced Biomedical Computing Center, SAIC-Frederick, National Cancer Institute—Frederick CancerResearch and Development Center, Frederick, Maryland 21702-1201

~Received 29 July 1999; accepted 12 October 1999!

A hybrid quantum mechanical diatomics-in-molecules~QM/DIM ! method is formulated that aims todescribe chemical reactions in an environment within the framework of a discrete~or cluster!approach. Starting from the conventional DIM formalism, first-order intermolecular perturbationtheory is applied to calculate interactions between reacting particles and environmental molecules,and to introduceab initio QM energies for the central system. In this approach no boundaryproblems appear when combining two parts of the entire system. The pairwise contributions to theinteraction energy come from the true potential curves of the fragments. A proper treatment ofexcited electronic states is also provided in this approach. As a first application, we computepotential curves for the dissociation reaction of a single hydrogen fluoride molecule surrounded bya selected number of solvent HF molecules. ©2000 American Institute of Physics.@S0021-9606~00!30302-6#

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I. INTRODUCTION

Modeling chemical reactions in clusters is a new chlenging topic of modern researches in molecuphenomena.1 Chemical transformations in the gas phasevery important, but a vast majority of reactions occurscondense phases where, unlike for the gas phase, it is neasy to separate reacting particles and those of the envment. If, however, such a partitioning into the reaction cenand surrounding species is performed, it is assumed thainteraction between these two subsystems does not learedistributions of particles between them. Important eamples of such systems are solutions where a reaction ocamong solvent molecules. The simplest solvents are matrof rare gases, which are widely used for studies of reacspecies.2 At the other extreme are environments of biologicsignificance such as proteins and enzymes.3

Many aspects of solvation phenomena can be succfully treated within the frame of continuum models.4 Thediscrete representation in which environmental moleculesconsidered explicitly is a complementary part of the genestrategy to simulate reactions in the condense phase.treatment presented in this work is a step toward a devement of a discrete~or cluster! approach. In such modelsclusters of molecules play the role of fragments of an enronmental subsystem influencing the chemical reactionside the cluster. For brevity, we shall often use the followterminology: molecules in environmental clusters will bebeled as ‘‘solvent’’ molecules, and the particles of a reactsystem as ‘‘solute’’ ones, although the formalism is notstricted to solutions only. We are most interested in desc

a!Electronic mail: topol@ncifcrf.gov

5130021-9606/2000/112(2)/513/9/$17.00

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ing hydrogen bond networks in solvents and clusters becaof their importance in many biological processes.

There is no doubt that modeling properties of molecuclusters should be performed at the quantum level oftheory. However, at present, a completeab initio quantumtreatment of a molecular system experiencing chemtransformations inside the shell of a finite number of solvmolecules does not seem practical. One of the most diffiproblems, which preclude such an undertaking, is the nesity to find, for every value of the reaction coordinate, tcorresponding positions of solvent molecules. In turn, thadjusting solvation shells influence the energy profile ofreacting species. Since quantitative descriptions of intermlecular interactions such as solute–solvent and solvesolvent interactions generally require substantial computional efforts, it is clear that a straightforward applicationstandard quantum chemistry methods would lead to prohtively expensive calculations. The same problems arecountered in structural studies of molecular clusters, wheris well known that an appropriate treatment of electron crelation requires a very high level of theory and the useextended basis sets in order to predict positions of stationpoints on fairly shallow potential energy surfaces. Sometempts to rely entirely on quantum chemistry methodsstudies of reactions in clusters are described in the literatbut they usually consider very simplified models for the resystems.

Hybrid approaches, which use a quantum mechan~QM! description for a selected fragment of the entire systcombined with a lower theoretical level treatment of themaining part, is a more promising approach for practicalplications. The molecular mechanics~MM ! approach is oneof the obvious candidates for use in conjunction with qua

© 2000 American Institute of Physics

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514 J. Chem. Phys., Vol. 112, No. 2, 8 January 2000 Nemukhin et al.

tum calculations. Such hybrid QM/MM schemes5,6 are be-coming very popular for studies of extended systems aimto reproduce, for instance, structures and vibrational proties of the central subsystem.

Predominantly, the QM/MM approaches are basedthe use of empirical potential functions in the MM paSome attempts are known to apply the strategy of usingabinitio calculated points on the potential energy surfacecalibrate parameters of analytical functions describing inatomic interaction potentials; however, most applicatiorely on empirical MM potentials. There are many waysintroduce parameters and to select values of these paramof MM potentials, even if conventional programs likCHARMM7 or AMBER8 are employed, and additional corretions are required in applications to molecular aggregaThere is no guarantee that the MM potentials adjusted fodescription of the central system inside the cluster ofN mol-ecules will be valid for the next size (N11) cluster. It isdifficult to expect that the MM pair potentials for clustemay form a universal database. A general problem ofQM/MM approach is the linkage of the QM and MM subsystems, and treatment of these boundaries is essential,cially when molecular dynamics simulations are performon the QM/MM potential energy surfaces.6 Since the calibra-tion of empirical parameters is carried out for the groustate potentials, calculations for the excited electronic stare questionable within the conventional QM/MM schemFor this reason, a treatment of chemical reactions in thegions of transition states, which often correspond to thegions of avoided crossings of various potential surfacshould also present considerable difficulties for the QM/Mmodel.

A promising generalization of the MM scheme is an aproach that is supported by a quantum mechanical baground, namely, the diatomics-in-molecules~DIM ! theory.9

For every particular application the MM potential surfacan be obtained as a particular case of the DIM construafter a series of simplifications~e.g., Ref. 10!. The DIMmethod and its extensions allow one to generalize the pwise calculation schemes in such a way that many-body ctributions to interaction potentials are taken into accountshould be pointed out that the DIM approach is a semiemical valence-bond-type quantum chemistry method, asome adjustable parameters can be introduced into finapressions. Usually, the mixing coefficients of diatomic staof the same spin and spatial symmetry surve as paramespecific values of which are slightly varied until better agrement between DIM and reference potential energy surfacereached.10,11

In this work we suggest a new approach that canlabeled as QM/DIM. From this acronym it is clear that tidea is to combine the QM description of the central partthe entire system, which in our case includes the reacparticles, and the diatomics-in-molecules~DIM ! descriptionof the interactions of particles of the central part with pticles belonging to the surrounding molecular clusters. TDIM method, which originates from quantum chemistry vlence bond theory, allows one to establish a smoothbetween the QM part of the system and the environm

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Consideration of excited state potential energy surfacefeasible in this approach. Since the true diatomic potenenergy curves enter the calculation scheme, the creatiouniversal databases is possible. So we believe that theDIM scheme can overcome some of the shortcomings ofQM/MM approach.

More precisely, we use here the version of DIM, whiis known as the diatomics-in-ionic-systems~DIIS!approach.11 Retaining the essence of DIM as a scheme tcombines contributions from all possible diatomic fragmein the specific electronic states into total energy of the stem, the DIIS method accurately treats polarization energdue to partial charges on atoms. These energies are notwise additive and have, in essence, a many-body origin.historical reasons, we shall still use the acronym QM/DIkeeping in mind that actually the DIIS version of DIM iemployed.

Recently, the DIIS method has been successfully ufor simulations of structures and spectral properties of (HFN

and the reliability of its predictions has been proven.12 Here,we present a new application of this technique to a mchallenging problem of reactions in clusters. For the fiapplication we select the system of hydrogen bonded cters, and the (HF)N species present an important choice. Wshall describe the dissociation reaction HF→H1F of a singlehydrogen fluoride molecule surrounded by a selected numN of ‘‘solvent’’ molecules HF, in particular, forN54 andN58.

In Sec. II we present a general theory of the hybrid QDIIS approach. In Sec. III we give necessary details for hdrogen fluoride clusters. In Sec. IV we describe the mresults of simulations, and Sec. V is devoted to a discussand conclusions.

II THE QM/DIM FORMALISM

In this section we present the theoretical backgroundthe QM/DIM scheme for a reaction occurring in an enviroment of a selected numberN of similar molecules. Let usdenote the atom of typev of the i-th such molecule asWiv .Let symbolsAa distinguish atoms of the reacting center. Wshall use the symbol @ to separate two subsystems inding that the particles of the central zone are attached tomolecular cluster:

~A1••• Aa ••• !@~W1W2 ••• !1

3~W1W2 ••• !2•••~W1W2 ••• ! i••• . ~1!

Among the particlesAa , certain transformations may occubut no redistribution between subsystems is considered.address the following mutually related questions: how doenvironmental molecules modify the energy profile of treaction, and how does the arrangement of these molecadjust to the move of the particles of the reacting center.illustrative purposes an application of the formalism for tdissociation reaction of a single hydrogen fluoride molecinside the (HF)N cluster, namely (HF→H1F!@~HF)N , ispresented below.

We start by using the diatomics-in-molecules languaand then explore the advantages of this description, wh

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515J. Chem. Phys., Vol. 112, No. 2, 8 January 2000 QM/DIM for molecular clusters

allows us to raise the level of the treatment for the syst(A1A2¯) of a primary interest toab initio standards, and tolower it, if necessary, for other parts.

According to the DIM strategy9 the exact Hamiltonian ofthe system~1! containing a totalK atoms is partitioned to asum of diatomicHab and monoatomicHa contributions,

H5 (a,b

Hab2~K22!(a

Ha . ~2!

We can regroup the terms in Eq.~2! in order to distinguishthe subsystems:

H5H ~A1A2 ••• !1(i

H ~W1W2 ••• ! i

1(i , j

H ~W1W2 ••• ! i ~W1W2 ••• ! j

1(i

H ~A1A2 ••• !~W1W2 ••• ! i2Hat. ~3!

Treatment of the monoatomic HamiltonianHat ~and the cor-responding sum of atomic energiesEat is trivial, and onlydiatomic terms require attention.

If the ordinary DIM method of an approximate solutiofor the energy eigenvalues were used, this Hamiltonwould be converted into the matrix form with the chosenof polyatomic basis functions. A dramatic increase of tsize of matrices would prevent this approach from practapplications. Instead, we propose to use from the very insteps the intermolecular DIM perturbation theory.13 In thismethod the valence bond function is written in the formthe product

uO” &5uO~A1A2 ••• !&)i 51

N

uO~W1W2 ••• ! i&, ~4!

and within the first order of perturbation theory the energyestimated as follows:

E5E~A1A2 ••• !1(i

E~W1W2 ••• ! i1(

i , j^O~W1W2 ••• ! i

O~W1W2 ••• ! j

3uH ~W1W2 ••• ! i ~W1W2 ••• ! juO~W1W2 ••• ! i

O~W1W2 ••• ! j&

1(i

^O~A1A2 ••• !O~W1W2 ••• ! iuH ~A1A2 ••• !~W1W2 ••• ! i

3uO~A1A2 ••• !O~W1W2 ••• ! i&2Eat. ~5!

The energy of particles in the reaction centerE(A1A2 •••)

principally can be computed entirely within the DIM approach. However, a more reliable approximation arises wwe applyab initio methods of quantum chemistry~QM! forthis most important part of the system and then combinecomputed energy terms with the contributions from remaing subsystems. For the example presented in this panamely, the dissociation reaction of HF inside the molecucluster, this means that theab initio potential curve for thedissociating HF molecule is inserted into Eq.~5! as the en-ergy of a reacting subsystem. Generalization of the form

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ism for reactions with more than one degree of freedomstraightforward: the QM energiesE(A1A2 •••) for each geom-etry configuration along the chosen reaction coordinatecomputed and combined with the other contributions.

The interaction of environmental moleculebetween each other ( i , j^O(W1W2 •••) i

O(W1W2 •••) ju

3H (W1W2 •••) i (W1W2 •••) juO(W1W2 •••) i

O(W1W2 •••) j& can be

treated at various levels. In this work we apply the intermlecular DIM perturbation theory to estimate pair interactioof the HF molecules as in the previous works on the structof hydrogen fluoride clusters.12 For important solvent species, including water, reliable empirical potential functioare available, which can be used in this scheme. Semiemical quantum chemistry methods may be also helpful in soapplications.

The remaining problem is to estimate the contributiofrom the terms

(i

^O~A1A2 ••• !O~W1W2 ••• ! iuH ~A1A2 ••• !~W1W2 ••• ! i

3uO~A1A2 ••• !O~W1W2 ••• ! i&,

describing the cross interactions of atoms from the reaczone with the atoms from the environmental shells. The mconsistent way to estimate these important contributions iuse the DIIS approach directly. The Hamiltonian operatorsthese matrix elements are sums of the operators of diatospeciesAaWiv :

H ~A1A2 ••• !~W1W2 ••• ! i5(

a(v

HAaWiv. ~6!

By construction, the eigenstates in Eq.~4! are thevalence–bond wave functions and the componentsuOAaWiu

&referring to the pair of atomsAaWiv can be separated outthe proper transformations are performed. In the gencase, when we assume that neutral and ionic species occboth subsystems, the matrix elements needed to estimatteractions of the particles of the reaction center with thosethe environmental shells are expressed as follows:

^oAaWiuuHAaWiu

uoAaWiu&

5EAaWiuUnn1EA

a1/2Wiu

U in1EAaWiu1/2Uni

1EAa1/2W

iu2/1U ii . ~7!

Here we distinguish the energiesEAW of diatomic frag-ments depending only on theAa2Wiv internuclear distancesand the coefficientsU, which are responsible for proper mutual orientations of atomic functions on centersAa andWiv

~if they differ from theS-type functions!. These coefficientsalso can include the specific parameters of DIM theory tmix the electronic states ofAaWiv of the same spatial andspin symmetry. In other words, these are the coefficientsthe valence–bond expansions of wave functions,O. The en-ergiesEAW refer to the combinations of potential curvesspecific electronic states classified by the projection of anlar momentum and spin of a particular diatomic fragmen

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516 J. Chem. Phys., Vol. 112, No. 2, 8 January 2000 Nemukhin et al.

Allowance for the treatment of ionic particles~i!, alongwith the neutral~n! ones, is an important component of thapproach. If valence bond patterns introduce either neutraionic species some of the terms in Eq.~7! are zero, but, ingeneral, all contributions must be included in the scheBecause of the presence of ionic species, the DIM interacenergy should be augmented with the multicenter elecstatic contributions arising from interactions of a neutral scies with a polarizabilityan and point charges of ionic speciesqi ,

EES521

2 (n

anS (i

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r in3 D 2

. ~8!

Such an extension of the scheme leads to the diatomicionic-systems~DIIS! theory. A fraction of the electrostatienergy is taken into account by constructing the diatompotential curves, as seen from the following representati

EES521

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Long-range potentials of the type2aq/2r 4 are assigned tothe DIM part and the remaining many-body~MB! contribu-tions must be added to accomplish the construction.

The molecular mechanics approach can be simplyrived from this scheme if one ignores differences in potencurves of every diatomic fragment due to spin and angumomentum and, consequently, the couplings betweenstates of the same symmetry. Furthermore, one wouldhave to ignore approach anisotropy of pair interactionsmulticenter electrostatic contributions. As a result, only epirical pairwise potential functions can be assigned to ediatomic fragment, as in QM/MM schemes. On the contrain the QM/DIM scheme the well-defined potential curvesdiatomic molecules appear as building blocks. In macases, reliable potentials for these diatomic fragmentsknown from spectral or scattering experiments augmentedhigh level ab initio calculations and can form a universdatabase.

III. APPLICATION TO THE REACTION„HF˜H1F…@„HF…N

We shall follow the style of presentation of the DIImodel for the hydrogen fluoride systems as in our previpublications devoted to the modeling of structures ofhydrogen fluoride clusters.12 Every formula needed for calculations of energies of (HF)N is given explicitly in Ref.12~a!, and for brevity only essential details are reproduchere.

According to the general formalism described in the pceding section, we must specify for a particular systemvalence bond wave functions of Eq.~4!. In this applicationthe central subsystem consists of two atomsA15H, A2

5F, which at the beginning of the reaction are bound inmolecule, and at the end of the reaction are separated.

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valence bond functionO(A1A2)5OA in Eq. ~4! is written inthe basis set of atomic eigenstates2P(F), 2S(H), 1S(F),1S(H1):

uOA&5u2Pz~F!&u2S~H!&cos~bR!

1u1S~F2!&u1S~H1!&sin~bR!. ~10!

The mixing parameterbR , which depends on the internuclear distanceR ~i.e., the reaction coordinate!, governs thereaction channels. If sin(bR) goes to zero at largeR, themolecule dissociates to neutral atoms following the adiabground state potential curve. If sin(bR) goes to unity at largeR, then the molecule follows to the ion-pair limit along thexcited state potential. A mixture of the neutral and ionconfigurations at every value ofR allows one to describe thechanges in the electronic structure of HF upon the moalong the reaction coordinate. It is assumed that the loczaxis is the quantization axis of angular momentum. The fiterm on the right-hand side of Eq.~10! actually consists oftwo functions corresponding to different spin–couplinschemes.

The valence bond wave functions of the ‘‘solvent’’ Hmolecules O(W1W2) i

5Oi are formally constructed in thesame manner,

uOi&5u2Pz~Fi !&u2S~Hi !&cos~b i !

1u1S~Fi2!&u1S~Hi

1!&sin~b i !. ~11!

The mixing parametersb i depend on the Hi – Fi separations,but, unlike for the molecule in the reaction center, they oslightly vary around the values corresponding to the equirium geometries of the HF particles in the cluster.

The product of all theseN11 functions builds the wavefunction of the entire system within the first-order intermlecular DIM perturbation theory:

uO” &5uOA&)i 51

N

uOi&. ~12!

The corresponding expression for the energy@see Eq.~5!#includes the following contributions: energy of the centmolecule,EA , sum of the energies of solvent molecule( i 51

N EWi, interaction energies of solvent molecule

( i , j^OiOj uHi j uOiOj&, the solute–solvent interaction enegies,( i 51

N ^OAOi uHAiuOAOi&5( iVAi , and many-body elec-trostatic energy,EES

~MB! , of Eq. ~9!.For the termEA constituting the QM part of the project

we take the ground stateX 1S1 potential curve of the HFmolecule computed by us with the multiconfigurationquasidegenerate perturbation energy~MCQDPT! method14

with the aug-cc-pVTZ basis set for a series of pointsR be-tween 0.7 and 10 Å. The PCGAMESS version15 of theGAMESSpackage16 has been employed for these calculationA comparison of these data with the benchmark calculatresults of Feller and Peterson17 shows that this potentiacurve can be considered as very accurate.

The contributions from the solvent–solvent and solutsolvent interactions are computed with the DIIS approachthe same manner as in the previous studies of hydrogen

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517J. Chem. Phys., Vol. 112, No. 2, 8 January 2000 QM/DIM for molecular clusters

ride clusters.12 The summation in these expressions is pformed over the dimers of HF, and the final formulas of tDIM analysis are given in Eqs.~10!–~11! and in Appendix Bof Ref. 12~a!. For convenience, we reproduce here somethe expressions for the solute–solvent interaction showtheir relevance to a possible MM treatment.

First, we notice that each term in the sum over solvmolecules~j! consists of contributions of the types neutraneutral, neutral–ionic, ionic–neutral, and ionic–ionic, shoing which sort of the atomic species from theA and Wj

subsystems is considered in each case:

VA j5VA j~nn! cos2 bR cos2 b j1VA j

~ni ! cos2 bR sin2 b j

1VA j~ ln! sin2 bR cos2 b j1VA j

~ i i ! sin2 bR sin2 b j . ~13!

In turn, every termVA j(** ) includes the diatomic potentia

energy curves and the coefficients depending on the geetry of the dimer, for instance,

VFHj

~nn!5~ 34 V3 D—

(r FHj!1 1

4 V1 D—(r FHj

))U D—

1~ 34 V3O

~r FHj!1 1

4 V1O~r FHj

!!UO . ~14!

HereV1,3D—,O(r ) are the potential curves of the correspondi

symmetry types of HF, depending on the internuclear dtance between fluorine of the central solute moleculehydrogen of the solvent molecule,U D—,O are the combina-tions of angular parameters occurring in the HF..HjFj sub-system.

One can obtain from these expressions the formulathe MM approach if~i! some empirical functionsV(r ) areemployed instead of true diatomic potential curvesV1,3D,O

(r )

and ignoring differences between them;~ii ! empirical angu-lar functions are used~if used! instead of the coefficientsU D—,O dictated by rotational properties of angular momenteigenfunctions;~iii ! an empirical coefficient is introduced~ifintroduced! instead of mixing parameterbR responsible forreaction processing.

The above formulas are valid for the ground state pottial energy surface describing the reaction HF(X 1S1)→H1F in an environment. There is an easy way to switchthe excited state potential surface describing the reactwhich leads to creation of the ion pair: HF~2 1S1!→H11F2.In this case, we write the wave function of the central mecule as follows:

uOA* &5u2Pz~F!&u2S~H!&sin~bR!

1u1S~F2!&u1S~H1!&cos~bR!; ~15!

and, correspondingly, the wave function for the entire stems in the form

uO” * &5uOA* &)i 51

N

uOi&. ~16!

As a QM energy we use the potential curve of the secroot in the 21S1 symmetry block.Ab initio calculationshave been performed as in the case of the ground statusing the MCQDPT approach.

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IV. CALCULATION RESULTS

The strategy of the calculation of potential curves dscribing dissociation of HF in the (HF)n (N54 or N58)environment is as follows. We start from the arrangementthe clusters (HF)5 or (HF)9 corresponding to certain locaminima on the potential energy surfaces. In both casesselect those configurations among a variety of isomers whone monomeric species subjected to dissociation can besidered as being inside the cluster.

The planar configuration for (HF)5 shown in Fig. 1~a!obviously satisfies this criteria. This structure has beenscribed in previous publications as isomer 5ID of Ref. 12~a!or 3•3 of Ref. 12~b!. Its energy is about 0.45 eV above thenergy of cyclic (HF)5 corresponding to the global minimumon the ground state potential energy surface of the hydrofluoride pentamer.

FIG. 1. The arrangements of ‘‘solvent’’ molecules around the dissociaspecies for the reaction (HF→H1F!@~HF)4 . The internuclear distances argiven in angstroms.

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518 J. Chem. Phys., Vol. 112, No. 2, 8 January 2000 Nemukhin et al.

Figure 2~a! shows the configuration (HF)9 also referringto one of the points of local minima on the potential surfaof this oligomer. Among the structures (HF)N (n<10) pre-dicted by the DIIS method as the minimum energy pointspotential surfaces, this is one of the best candidates fodesired requirement for the HF molecule being surrounby the remaining species of the cluster. The central HF mecule resides at the axis that is almost orthogonal tohighly corrupted ring (HF)5 seen at the bottom of the pane~a!–~c! of Fig. 2, and three remaining monomers form a sof a chain around the central particle. The bonding in

FIG. 2. The arrangements of ‘‘solvent’’ molecules around the dissociaspecies for the reaction (HF→H1F!@~HF)8 . The internuclear distances argiven in angstroms.

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cluster is due to concerted contributions from hydrogbonded and van der Waals interactions.

We note that the H–F separations of the central HF mecule, which correspond to equilibrium configurations of tclusters, is 0.92 Å for (HF)5 @Fig. 1~a!# and 0.94 Å for (HF)9@Fig. 2~a!#. From these initial configurations, we create serof points along the reaction coordinateR by displacing theH–F distance of the central molecule from the equilibriuvalue, and adjusting positions of the surrounding molecufor every coordinateR. The contributions to the total energcome from theab initio potential energy curves assignedthe dissociating molecule~the QM part!, from the interac-tions between this molecule and surrounding species,solute–solvent interactions, as well as from the solvesolvent interactions, both treated by the DIIS method.

The same set of potential energy curves for H2 , F2 , HF,and for the corresponding ions, as in the paper18 describingthe hydrogen fluoride dimer, has been employed. Formixing coefficients,bR , b j ( j 51,N), depending on the in-ternuclear distancer in HF species, we use the universfunction b(r ). Parameters of the equation

sin2 b~r !5a0 exp@2a1~r 2r 0!2#, ~17!

namely, a050.41, a152.5 Å22, r 051.1 Å, have been selected in order to reproduce the known properties19 ~first ofall, the binding energies! of the main cyclic isomers of(HF)N (N53 – 6).

For every new value ofR (0.7,R,10 Å), the coordi-nates of the adjusting surrounding solvent molecules areculated by the steepest descent method starting from thefiguration of the preceding point. The energy derivativhave been estimated numerically by using the DIIS potensurfaces. Usually 20–30 cycles of geometry optimizationrequired to reach the minimum within the accuracy constent with the purposes of this modeling.

Figures 1 and 2 give an impression of how the arranments of solvent molecules are changing in the coursedissociation for several representative values ofR. The val-ues of reaction coordinateR are shown as underlined entriein Figs. 1 and 2. We clearly see that in both examplessolvation shells do not experience dramatic reconstructioalthough notable ‘‘breathing’’ of the cages takes place. Tcharacteristic interatomic distances shown near dotted lin the figures indicate that upon increasing the internucldistanceR of the dissociating species the cages expand@com-pare panels~a! and ~b! of Figs. 1 and 2#. However, afterescape of the hydrogen atom some contraction of the caalong two of three spatial directions is noticed@compare pan-els ~b! and~c! of Figs. 1 and 2#. The fluorine atom from thedissociating molecule is captured in the middle of almsymmetrically arranged base fragments of either two momers@Fig. 1~c!# or a five-member planar ring@Fig. 2~c!#.

The computed potential energy curves are shownFigs. 3 and 4. For convenience, in all cases the zero ofergy corresponds to the minimum of the ground state potials. The curves computed in the clusters (HF→H1F!@~HF)4 or (HF→H1F!@~HF)8 are compared to the freestate potential curves. First of all, we notice a remarkasimilarity of the pictures obtained for different environmen

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519J. Chem. Phys., Vol. 112, No. 2, 8 January 2000 QM/DIM for molecular clusters

namely, for four and eight surrounding molecules in the clter. Actually, it may occur that three upper~in Fig. 2! envi-ronmental molecules of (HF)8 perform almost the same action as four solvent molecules of (HF)4 , since theirarrangements look fairly similar. However, this may beindication that a small amount of solvent species bearsgreatest responsibility for major effects of environment.

The ground state potential curves in the solvent showfollowing distinctions from the gas-phase pattern. At tearly stage of dissociation (R,1.5 Å) the curve in the clus-ter goes lower that that in the free state, and thereforesolvent facilitates elongation of the H–F bond. Howevafter a critical point near 1.7 Å the curves reverse, and menergy is required to dissociate the molecule in the solvThe amount of extra energy is fairly substantial in this cathat is almost 1 eV~;20 kcal/mol!.

For the (HF→H1F!@~HF)4 reaction we have verifiedthese conclusions by repeating the calculations of the encurve by entirelyab initio tools. The coordinates of the tenatomic system for a series of points along the coordinatRhave been taken those as predicted by the QM/DIIS scheThe calculations for the system undergoing dissociatshould be carried out at the multiconfiguration self-consistfield ~MCSCF! level or higher in order to correctly describ

FIG. 3. Potential curves of the reaction (HF→H1F!@~HF)4 computed withthe QM/DIM approach.

FIG. 4. Ground state potential curves HF→H1F and (HF→H1F!@~HF)4

computed by theab initio tools.

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the changes in electronic configurations. We appliedcomplete active space SCF~CASSCF! approach for a smalactive space described a few lines below, and included ational correlation contributions into account by using tmulticonfigurational quasidegenerate perturbation the~MCQDPT! option. The basis sets of triple-zeta quality wutilized. The calculations have been performed with theGAMESS quantum chemistry package.15

The choice of the active orbitals for MCSCF requirspecial explanation. At this point we face a principal prolem of ab initio calculations of reaction energy profiles in aenvironment. If usual canonical molecular orbitals are eployed, it is extremely difficult to find out which of themshould be included into the active space of the MCSscheme. Enlarging the active space immediately leads tohibitively expensive calculations, but selection of truly actiorbitals for a concrete reaction among numerous delocalorbitals by a simple visual analysis is practically impossib

We have solved the problem by using the transformatto the natural bond orbitals~NBO!.20 In this case it is feasibleto recognize the pair of orbitals referring to thes~HF! ands* ~HF! patterns of the central HF molecule, which play tmain role in the dissociation process. In our calculatiothese two NBOs are populated by two electrons that contutes the essential description of the breaking chemical Hbond. As mentioned above, the MCQDPT scheme allowsto cover a substantial part of correlation energy beyondsmall MCSCF calculation.

The results of suchab initio calculations are shown inFig. 4, and a comparison of these data with those of Figindicates precisely the same trends in the interrelations of‘‘gas-phase’’ and ‘‘solution’’ potential curves as in the QMDIM model. Namely, forR(H–F!,1.7 Å, the dissociationcurve in the (HF)4 cluster goes lower than that for the puHF molecule, and forR(H–F!.1.7 Å the order of the curvesreverse. A direct comparison of the graphs in Figs. 3 andnot instructive, since in the QM/DIM treatment~Fig. 3! thequality of diatomic potential energy curves is considerabetter than in the case of MCSCF calculations shown in F4. In the latter case we ought to apply precisely the sascheme for both HF and (HF)5 systems, and the shape of thground state HF potential in the long-range limit is not pfect. However, the most important conclusion arising frothe comparison of Figs. 3 and 4 is that the interrelation ofdissociation curves of HF and HF@~HF)4 is correctly repro-duced by the QM/DIM method.

V. DISCUSSION AND CONCLUSIONS

In this paper we present the general description ofQM/DIM method for a calculation of energy profiles ochemical reactions in cluster environments as well as aticular illustration of a dissociating hydrogen fluoride moecule inside the (HF)N clusters. This method attacks thmost difficult problem of finding the arrangements of surounding species for every value of reaction coordinate.we pointed out before, there is no alternative from the sideab initio quantum chemistry since the required geometry

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520 J. Chem. Phys., Vol. 112, No. 2, 8 January 2000 Nemukhin et al.

timizations for molecular aggregates at a suitable accurlevel are prohibitively expensive with standard quantuchemistry tools.

The example considered in this work clearly demostrates the advantages of the QM/DIM approach, compato more conventional QM/MM treatments.

~i! No boundary problems arise in this method sincestart consideration entirely within the quantum mechandescription. We write the exact Hamiltonian for the tosystem, introduce the basis set corresponding to the valbond theory, i.e., use the steps of the conventional Danalysis. Then we perform a partitioning of the system itwo parts: reacting particles and particles of an environmand take into account interactions between them withfirst-order perturbation theory. At this stage we raiselevel of the treatment for the central part assuming thatabinitio energies instead of DIM energies contribute to the toenergy. The remaining contributions of the ‘‘solvensolvent’’ and ‘‘solute–solvent’’ types are computed herethe DIM ~more precisely, DIIS! level, but a wider class ohybrid approaches can be used in general. The reliabilitysuch an approach is verified here by a direct comparisoits results for (HF→H1F!@~HF)4 with direct ab initio cal-culations~cf. Figs. 3 and 4!.

~ii ! The pair potentials contributing to the total interation energies in this QM/DIM scheme are the true potencurves of diatomic molecules, here for H2 , F2 , HF, and cor-responding ions. These curves should be constructedonce and placed into the database. The coupling coefficiU in Eq. ~7! depending, in particular, on the geometry of tentire cluster, take into account the specificity of each pticular system composed of the elements H and F.

~iii ! We explicitly show how the excited state potentenergy surfaces can be computed with the QM/DIM meth

There are no experimental findings for a direct compason with these simulation results for the dissociation ofin the (HF)N clusters. From the theoretical side, the impotant results of relevance to the present simulations have bobtained by usingab initio molecular dynamics methods foliquid hydrogen fluoride.21 Reactions of a hydrogen fluoridmolecule inside water clusters have been modeled withhelp of an ab initio frozen density functionaapproximation.22 First-principles molecular dynamics methods have been applied to study the dissociation mechanof a single water molecule inside water clusters.23 The ap-proach described in this work provides a good alternativethe already known tools to model reactions in condenphases. As it is shown in Ref. 18, the DIM techniquecapable to describe correctly not only regions of potenwells, but also the repulsive parts of potential energy sfaces, and, therefore, large portions of configurational spcan be considered by the QM/DIM approach.

The data presented in this paper demonstrate a posscale of modifications of the reaction profile in real solvenWe can see in Figs. 3 and 4 that the changes due toenvironment for the ground state curves may account15%–20% of the total binding energy. It is also shown ththe effect of the solvent is not a uniform shift of the enercurves. The relative positions of the ‘‘gas-phase’’ and ‘‘clu

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ter’’ curves interchange along the reaction coordinate.More important modifications occur for the excited sta

ion-pair potential curves. The close approach of these potials to each other in the cluster environment is clear edence of the effect that has relevance to electrolytic dissotion phenomena of ionic substances in polar solvents.

Other applications and further developments of the QDIM approach are in progress, and we feel that this methwill be useful in various fields of molecular modeling.

ACKNOWLEDGMENTS

This research was supported in part by the Russian BResearch Foundation~Grant No. 99-03-33178!. We thankthe staff and administration of the Advanced BiomedicComputing Center where a part of the calculations has bperformed. The content of this publication does not necsarily reflect the views and policies of the DepartmentHealth and Human Services, nor does the mention of trnames, commercial products, or organization imply endoment by the U.S. Government.

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