qmse_35

JOURNAL OF CHEMICAL PHYSICS VOLUME 117, NUMBER 12 22 SEPTEMBER 2002

Combining ab initio and density functional theorieswith semiempirical methods

Qiang CuiDepartment of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138and Department of Chemistry and Theoretical Chemistry Institute, University of Wisconsin, Madison,Wisconsin 53706

Hong GuoDepartment of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138

Martin KarplusDepartment of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138and Laboratoire de Chimie Biophysique, ISIS, Universite` Louis Pasteur, 67000 Strasbourg, France

~Received 1 April 2002; accepted 25 June 2002!

For large reactive systems, the calculation of energies can be simplified by treating the active partwith a high-level quantum mechanical~QM! ~ab initio or density functional! approach and theenvironment with a less sophisticated semiempirical~SE! approach, as an improvement over thewidely used hybrid quantum mechanical/molecular mechanical~QM/MM ! methods. An example isthe interaction between an active region of an enzyme and its immediate environment. One suchmethod is the original ‘‘Our-own-N-layer Integrated molecular Orbital1Molecular Mechanics~ONIOM!’’ approach. In this paper, the interaction between the QM and SE region is describedexplicitly by two different schemes. In theiterative QM/SE schemes~QM/SE-I!, the electrostaticinteraction and polarization effects are introduced explicitly for both the QM and SE atoms by aself-consistent procedure based on either polarizable point charges or the electron density. In thenoniterative QM/SE scheme, based on the ONIOM model~QM/SE-O!, the exchange~Paulirepulsion! and charge transfer effects are taken into account at the SE level, in addition to theexplicit electrostatic interaction and polarization between the two regions. Test calculations aremade on a number of model systems~including small polar or charged molecules interacting withwater and proton transfer reactions in the presence of polar molecules or an extendedhydrogen-bond network!. The quantitative accuracy of the results depend on several parameters,such as the charge-scaling/normalization factors for the SE charge and the QM/SE van der Waalsparameters, which can be chosen to optimize the result. For the QM/SE-O approach, the resultsare more sensitive to the quality of the SE level~e.g., self-consistent-charge den-sity-functional-tight-binding vs AM1! than the explicit interaction between QM and SE atoms.© 2002 American Institute of Physics.@DOI: 10.1063/1.1501134#

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I. INTRODUCTION

Combined quantum mechanical and molecular mechcal ~QM/MM ! methods1,2 are powerful tools for the study oreactions in large systems. When carefully calibrated, thave been shown to provide useful insights into the mecnism of chemical reactions in solution, on surfaces andenzymes.1–5 Algorithms for computing molecular propertiesuch as infrared~IR! spectra6 and nuclear magnetic resonance~NMR! chemical shifts7 in the QM/MM frameworkhave recently extended the range of QM/MM application

Despite the overall success of the QM/MM approathere are several issues associated with the common immentations. One aspect concerns the appropriate treatmethe QM and MM interface, when it is inside a molecule.number of schemes have been proposed, which includestandard link-atom scheme, double link-atoms, frontier orals and effective boundary potentials.8 A comparison of sev-eral schemes showed that they gave satisfactory results wcompared to full QM calculations if ‘‘over polarization’’ o

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the QM boundary atoms is avoided.9 Another concern abouQM/MM methods is that the environment of the QM regiois usually treated with a non-polarizable force field~such asCHARMM!, in which atoms are represented byfixedpoint-charges. Although this appears to be qualitatively adequfor many properties, such as the determination of groustate structures, it is not quantitatively accurate, in geneespecially in cases where many-body effects are importAn example of the latter is the cooperativity inducedhydrogen-bond networks found in the active site of maenzymes.10 Although these nonadditive many-body effeccan be captured by extending the size of the QM region,cost of high-level QM methods often makes this choice ipractical. One possibility for improving the description of thinteraction between the QM region and its immediate enronment is to include polarization in the molecular mechaics method used to describe the latter. This can be done mstraightforwardly by the introduction of atomic polarizabities on the MM atoms.1~c!,11,12 Another possibility is to in-

7 © 2002 American Institute of Physics

IP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

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5618 J. Chem. Phys., Vol. 117, No. 12, 22 September 2002 Cui, Guo, and Karplus

clude polarization on the MM atoms by use of the principof chemical potential equalization.13 The concept has beeused in the development of polarizable force fields,14,15 andhas been introduced into the QM/MM framework.16 An al-ternative approach, termed the effective fragment poten~EFP! method, has been employed in the context of solutchemistry.17 There the solvent molecules are treated asrigidfragments, which are represented in terms of distributed mtipoles and polarizability tensors calculated withab initiotheory. An approximate formula for including the Paurepulsion18 and charge transfer19 between the solvent ansolute has also been developed. Applications to reactionsmall clusters20 and to enzymes~e.g., mechanism of chorismate mutase,21 residue pKa calculations22 and the absorptionspectrum of a vanadate complex at the RNase A active si23!where either the solvent molecules or the active site residare described by EFP have given promising results. A litation is that the geometry of the fragments has to be rigwhich makes it difficult to apply in molecular dynamics stuies of reactions in proteins and nucleic acids.

Another approach for improving the description of timmediate environment of the reactive region~e.g., the ac-tive site of an enzyme! is to employ multiple partitions, inwhich the core is described with a high-levelab initio ordensity functional theory~DFT! method, its immediate environment~first solvation layer! is treated with a lower-levequantum mechanical approach, such as a semiempirical~SE!method, and atoms further away are described at thelevel. This philosophy has been pursued by Morokumaco-workers,24 who pioneered a set of additive schemes, gerally termed ‘‘ONIOM’’ ~Our own N-layer Integrated molecular Orbital1molecular Mechanics!. In the originalONIOM implementation,24 the reactive region does not explicitly interact with the environment at the high QM leveand all the environmental effects such as polarization, Prepulsion, and charge transfer were described at a low qtum mechanical level~e.g., Hartree–Fock!. Therefore, theONIOM scheme tends to give reliable results only whenlow QM level captures the environmental effects in a safactory manner.25 In more recent developments, nonadditischemes were explored in which the wave function~or den-sity! of the core region at the high QM level is polarizeexplicitly by the environment~which is described at a lowquantum mechanical level!. In the method proposed bCarteret al.,26 the core region is described by a high-levelabinitio approach, such as configuration interaction~CI!, andthe environment is described with DFT; the two interact seconsistently through an embedding potential derived frthe charge densities of the two region. In the approachMerz and co-workers,27 the core region is described by DFand the environment is treated with a semiempiricalproach~AM1!; a consistent solution to the coupled electronstructure problem is found through a chemical potenscheme originally proposed in the context of a divide aconquer DFT treatment for large systems.28

Finally, we note that methods exist to treat complex stems in a modular fashion, similar in spirit to the multilayapproaches, but with the same level of QM method.example, Gao developed an approach~Ref. 42! for simulat-


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ing water, in which each water molecule is treated withsemiempirical method~AM1!. In the SCF cycle for eachwater molecule~‘‘solute’’ !, the other water molecules~‘‘sol-vent’’! are represented by partial charges derived from Acalculations; the SCF procedure is iterated until the ‘‘soluwater and the ‘‘solvent’’ water molecules are polarized cosistently. Another approach is based on the block-localiwave function method,29 which can treat the electrostaticPauli repulsion~exchange! and charge transfer between thmolecular fragments in a valence-bond framework.30 Thisapproach has been used for an energy decomposition otermolecular interactions31 and chemical reactions insolution.32

In the present work, we describe a number of multilaymethods that combineab initio or density functional theories~QM! with semiempirical~SE! methods; MM can be in-cluded as the outer layer. To simplify notations, we referab initio or DFT methods as the QM level and distinguithat from SE, although the latter is also quantum mechanin nature. The methods introduced here are different fromoriginal ONIOM24 approach in that interactions between dferent partitions are taken into account explicitly; the speccoupling schemes are also different from other methods cabove.26,27 In Sec. II, we describe the QM/SE theory annumerical algorithms. Section III presents several test cathat illustrate the features of the new combined method. Tinclude the interaction between several small polarcharged molecules and water, and the effect of polar mecules and hydrogen-bond networks on a proton transfeaction related to triosephosphate isomerase. The conclusare outlined in Sec. IV. More systematic explorations of tQM/SE methods and the necessary parameterization arefor a separate publication.

II. THEORY

The system is partitioned into three layers, as shoschematically in Fig. 1, though more general~multiple! par-titions are possible. The immediate neighborhood ofchemically important QM region is treated with SE, whi

FIG. 1. A schematic representation of a multilayer partition of a large stem in solution: the reactive region requires a high level quantum mechcal ~QM! description, its immediate environment can be described at asophisticated quantum mechanical level such as a semiempirical~SE! ap-proach, and molecular mechanics is sufficient for the rest systems. In scases, a continuum model can be used to describe effects due to thesolvent @characterized, for example, by the radial distribution functig(r )].


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5619J. Chem. Phys., Vol. 117, No. 12, 22 September 2002 Combining ab initio and semiempirical

the outer region is described with an MM force field. Ttotal Hamiltonian of the system can be written in a seevident notion as

H5HQM1HSE1HQM/SE1HQM/MM1HSE/MM1EMM.~1!

The expressions for the interactions between MM partthe QM ~SE! part have the usual form,1~b!

HQM(SE)/MM5 (JPQM(SE)

VJ,NucQM(SE)/MM

1 (i 51

NelQM(SE)

VelQM(SE)/MM1Evan

QM(SE)/MM

5 (I PMM

(JPQM(SE)

QIZJ

uRW I2 RW Ju

2 (I PMM

(i 51

NelQM(SE)

QI

uRW I2 rW i u1Evan

QM(SE)/MM, ~2!

where the first two terms on the rhs of Eq.~2! represent theelectrostatic interaction between the MM atoms and the Q~SE! nuclei and electrons, respectively; the last term isvan der Waals term between the two sets of atoms thatrespond to the Pauli repulsion at short range and disperat large distance. The interaction between the QM andregion is not as well defined and will be considered infollowing section. Two classes of methods will be describThe first uses iterative approaches that solve the coupled

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and SE problem, and includes only the electrostatic andlarization effects between the two regions. The second cfollows the spirit of the ONIOM method and is noniterativthey also include the charge transfer and exchange repubetween the QM and SE regions at the SE level.

A. The iterative QM ÕSE scheme „QMÕSE-I…:Electrostatic and polarization effects

In this section, we treat the QM/SE coupling problefollowing the framework that has been used in analyzweak intermolecular interactions. The methods obtainedquire iterative solutions and therefore are referred to as QSE-I ~I for ‘‘iterative’’ !. For simplicity, we assume that thQM and SE regions are not covalently bonded; more coplicated cases will be considered elsewhere. We start bymally considering both the QM and SE regions at tHartree–Fock~HF! level, and introduce necessary approxmations to the SE region later; it is straightforward to extethe formulation to cases where the QM is DFT~see below!.We label the QM part as regionA, and the SE part as regioB. Accordingly, the electronic structure problem of the coposite systemA1B is described by the standard generalizeigenvalue equations,

FA1BCmA1B5SA1BCm

A1B«mA1B ~m51, N!, ~3!

whereN is the number of atomic basis functions;F, C, andS are the Fock, eigenvector and overlap matrix, respectivand« is a diagonal matrix with eigenvalues on its diagon

The Fock matrix in~3! can be written explicitly in termsof intra-region andinter-region contributions

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2ZC

uRW C2 rWuUxnL 1K xmU (DPB

2ZD

uRW D2 rWuUxnL 1^xmuVelA,B/MM

uxn&1 (lsPA

Pls^mnuuls&

1 (l8s8PB

Pl8s8^mnuul8s8&1 (l9PA,s9PB

Pl9s9^mnuul9s9&1 (l-PB,s-PA

Pl-s-^mnuul-s-&, ~4!

wherexm(n) is the atomic basis function (m,nPA1B), and Pls is the one-particle density matrix element in the atomorbital ~AO! basis. Note that the electronic interaction with the MM atoms, if present, is included in Eq.~4!; in accord with Eq.~2!, it is directly separable and so is omitted for simplicity.

Now we formally introduce the NDDO~Neglect-Diatomic-Differential-Overlap! approximation33 used in AM1 and PM3for regionB, which is treated with the semiempirical approach. Furthermore, we modify the terms corresponding to thA–Binteraction. Specifically, we introduce the NDDO approximation not only for AO integrals involving only the SE regioalso for theinter-regionAO integrals,

^mKn KuulK1KsK1K&'0

K xmKU 2Z

uRW 2 rWu UxnKL , TmKnK'0$K,K%5$A,B%,KùK5B. ~5!

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edM

To further simplify the problem, we make the approximatithat the basis functions from the two regions do not mixinthe molecular orbitals, i.e., the off-diagonal block of the desity matrix in Eq.~4! was neglected,

Pl9s9[Pl-s-[PlKsK50. ~6!

-

This is a necessary simplification because usually quiteferent forms of basis functions are used in the QM~Gauss-ian! and SE~Slater! region. This would make the inter-regiotwo electron integrals, especially the exchange type integrrather difficult to evaluate and violates the simplicity offerby the SE approach. Moreover, explicit mixing of the Q


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and SE basis sets is likely to induce artifacts due to the ladifference in the two types of methods; e.g., artificial chatransfer between the two regions. The inter-region integapproximation@Eqs. ~5! and ~6!# is not expected to causlarge errors in describing the interactions between theregions when the separation is larger than a hydrogen-bdistance~i.e., longer than 2.5 Å!. This is based on the test osimilar approximations in the frozen density functionmethod of Wesolowskiet al.34 ~see below!. In Ref. 34, theyused the same basis sets for different regions and therecan actually check the validity of the approximation; in toriginal ONIOM,24 no explicit interaction between differenlayers is calculated.

With approximations Eqs.~5! and ~6!, the Fock matrixelement in Eq.~4! reduces to

FmnK,NDDO5FTmn1K xmU (

CPK

2ZC

uRW C2 rWuUxnL1 (

lsPKPls^mnuuls&1^xmuVE

K/MMuxn&G1F K xmU (

DPK

2ZD

uRW D2 rWuUxnL1 (

l8s8PK

Pl8s8^mnul8s8&G , m,nPK

5Fmn8 K1Fmn9 K/K. ~7!

Note that only the atomic basisfunctions centered on atoin region K (K5A,B) are present. The first three termsEq. ~7! represent the intraregion interaction, and the last tterms represent the inter-region electron–nuclei aelectron–electron Coulombic interactions, respectively.other words, only the Coulombic interaction betweenQM and SE region is taken into account explicitly. Thexchange-repulsion term and the two-center one-elecresonance terms that are present in the standard SE melements are not included. They are modeled by vanWaals terms between the QM and SE regions in analogwhat is done in the QM/MM interaction contributions@Eq.~2!, also see Eq.~9! below#. As a result, the eigenvalue problem Eq. ~3! becomes partially decoupled, in the sense tthe corresponding matrices are ‘‘region-diagonal,’’ althouthe matrix elements of each region depend on the solutiothe other through the last two terms in Eq.~7!. We have

S FA,NDDO 0

0 FB,NDDOD S CA 0

0 CBD5S SA 0

0 SB'I D S CA 0

0 CBD S «A 0

0 «BD . ~8!

The eigenvalue problem Eq.~8! has to be solved selfconsistently~Fig. 1!. One starts by solving the eigenvaluproblem for blockA ~or B!, which yields the~updated! den-sity matrix that enters the Fock matrix element for blockB~A! through the inter-region Coulombic terms in Eq.~7!. The


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eigenvalue problem for blockB ~A! is then solved which inturn updates the Fock matrix of blockA ~B!. The procedureis carried out until self-consistency is reached.

With the converged Fock matrix from Eq.~8!, the totalenergy of the system can be written

EQM/SE - ITot 5

1

2 (m,nPA

Pmn@hmnQM1hmn

QM/SE

1Vel,mnQM/MM1Fmn8QM1Fmn9QM/SR#

11

2 (lsPB

Pls@hlsSE1hls

SE/QM1Vel,lsSE/MM

1Fls8SE1Fls9SE/QM#1EvanQM(SE)/MM

1EvanQM/SE1ENuc

QM(SE)/MM1ENucQM/SE1EMM, ~9!

wherehmnQM(SE) are the one-electron integrals for the QM~SE!

region ~kinetic plus nuclear–electron terms! and hmnQM/SE

(hmnSE/QM) are the integrals for QM~SE! electron and SE

~QM! nuclei interactions@fifth term on the rhs of Eq.~7!#;the notation for the Fock matrix elements is indicated in E~7!. As mentioned above, the Pauli repulsion and chatransfer effects between QM and SE atoms are modeledthe effective van der Waals terms,Evan

QM/SE. In Eq. ~9!, theColoumbic interaction between the QM and SE electronsformally partitioned equally at the QM and SE levels, in tform of

1

2 (mnPA

PmnF (lsPB

Pls^mnuls&Gand

1

2 (lsPB

PlsF (mnPA

Pmn^lsumn&G ,respectively. In the practical implementation, however,polarization of the QM and SE wave functions was treatedan asymmetric manner because the QM region is moreportant~see Sec. II A 1!. An alternative expression was usein which the QM/SE electron–electron Columbic interactiwas evaluated at the QM level only. The required expresscan be derived from Eq.~9! through a straightforward rearrangement,

EQM/SE - ITot 5

1

2 (m,nPA

Pmn@hmnQM1Vel,mn

QM/MM1Fmn8QM#

1 (m,nPA

PmnFmn9QM/SE

11

2 (lsPB

Pls@hlsSE1Vel,ls

SE/MM1Fls8SE#

1 (lsPB

PlshlsSE/QM 1Evan

QM(SE)/MM

1EvanQM/SE1ENuc

QM(SE)/MM1ENucQM/SE1EMM.

~10!


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The derivations described above assumed a HartrFock QM level, but the formulation can be readily extendto perturbation theories such as MP2, which use the cverged MO coefficients from Eq.~9! or ~10!. An extension toother variational methods, such as CASSCF, can alsomade, but we do not describe it here. The correspondDFT treatment is straightforward, given the parallelism btween Hartree–Fock and the Kohn–Sham formalisms.35 Weconsider the Kohn–Sham matrix element for theA1B prob-lem at the DFT level,

Fmn5Tmn1K xmU (CPA

2ZC

uRW C2 rWu UxnL1K xmU (

DPB

2ZD

uRW D2rWu UxnL 1^xmuVelA,B/MM

uxn&

1 (lsPA

Pls^mnuls&1 (l8s8PB

Pl8s8^mnul8s8&

1 (l9PA,s9PB

Pl9s9^mnul9s9&

1 (l-PB,s-PA

Pl-s-^mnul-s-& 1Vmnxc ~r11r2!,

~11!

where the exchange integrals have been replaced byexchange–correlation potentialVxc(r). Invoking the NDDOapproximation for the two regions@Eqs. ~5! and ~6!#, theapproximate Kohn–Sham matrix element becomes

FmnK,NDDO5Tmn1K xmU (

CPK

2ZC

uRW C2 rWu UxnL 1^xmuVelK/MM

uxn&

1 (lsPK

Pls^mnuls&

1K xmU (DPK

2ZD

uRW D2 rWu UxnL1 (

l8s8PK

Pl8s8^mnul8s8&1Vmnxc ~r11r2!. ~12!

This expression is very similar to the one derived by Welowski et al.,34 except that the nonadditivity of the kinetifunctional is not considered here because one regiontreated with SE, and the associated effect~which is largely ashort-range repulsion! is modeled by a van der Waals terbetween the DFT and SE region. When the SE methoalso based on DFT~such as the SCC–DFTB approach36!, theexchange–correlation potential can, in principle, be evaated on a set of grids at both levels, given an approprrepresentation ofr1(r ) and r2(r ). When the SE level is aHartree–Fock based approach, such as AM1, althoughexchange–correlation potential can be evaluated at thelevel including a representation of the SE density~see Sec.II A 1 !, we choose to evaluate onlyVmn

xc (r1) in the currentimplementation, based on the consideration that the quaof electron density is different in DFT and SE approach


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This additional approximation is partially compensatedthe van der Waals interaction between QM and SE atomsthe DFT/AM1 combination, the AM1 Fock matrix takes thform of Eq. ~7!. With the expression of the Fock matrielements in Eq.~12! for DFT and Eq.~7! for SE, the totalenergy of DFT/SE/MM formulation is again given by Eq~10!.

The remaining ‘‘technical’’ problem is the numerical caculation of the inter-region electron–electron interactiterms in Eqs.~7! and~12!. This is not straightforward giventhat most QM methods use Gaussian-type basis functwhile SE uses Slater orbitals. In the following sections,describe two possible solutions.

1. The auxiliary basis method (QM ÕSE-I-A)—beyondthe point charge model

To compute the QM/SE Coulombic integrals in Eqs.~7!and ~12!, one can fit the Slater orbitals in the semiempiricregion with a STO–NG scheme, and then calculate theelectron integrals in the traditional manner.37 However, wepropose to use an alternative approach, which makes usthe fact that the required QM/SE Coulomb integrals canwritten in the form

I mn5E E xm~rW1!xn~rW1!1

r 12r K~rW2!d rW1 d rW2 ,

m,nPK, ~13!

wherer K(rW) represents the electron density for the atomsregion K. To computeI mn , one can use a set of auxiliarbasis functions of the same type as the atomic basis functin regionK to expandr K(rW) and then compute the resultinthree-center two-electron (3c,2e) integrals. This density-fitting scheme was chosen in the current work based oneral considerations. First, as shown in a number of previstudies,38 the density fitting technique is numerically verefficient and reduces the formal scaling of the Coulombtegral calculations~from N4 to N3). Second, as shown bSt-Amant et al.,39 linear scaling can be achieved with thdivide-and-conquer fitting scheme, which is an appealfeature for QM/SE calculations with a large SE region. Fnally, once an appropriate representation of the density~andits gradient, if required! is available, the exchangecorrelation potential can be evaluated as well.

As discussed by a number of authors,38 the density fit-ting procedure requires solving the following set of lineequations:

c5S21~ t1ln!, ~14!

where thec is the fitting coefficient vector,

r K~ rW!5(i

cix i K8 ~rW !. ~15a!

The other quantities are defined as the following:


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Si j 5E E xi K8 ~rW !x

j K8 ~rW8!

urW 2 rW8ud rW d rW8, ~15b!

t i5E E xi K8 ~rW !r K~rW8!

urW 2 rW8ud rW d rW8

5 (mnPK

PmnSTOE E x

i K8 ~rW !xm

STO~rW8!xnSTO~rW8!

urW 2 rW8ud rW d rW8,

~15c!

ni5E xi K8 ~rW !d rW, ~15d!

l5NK2nS21t

nS21n. ~15e!

The xi K8 (rW) in the above equations is the auxiliary bas

set centered on the atoms in regionK, which containsNK

electrons. Although one may follow the protocol to computhe inter-region Coulombic integrals in both the QM andFock matrices, we only applied it to treat the QM Fock mtrix in the current implementation and simply used partcharges on the QM atoms~see below! to polarize the SEwave function. This is based on the consideration thatpolarization effect of the SE region is less important than tfor the QM region, and point charge polarization is sufcient. The fitting basis was taken to be a set of uncontracGaussian orbitals developed by Salahubet al.;40 however,dfunctions are not included to avoid over fitting. The densmatrix for the semiempirical region is first transformed inthe STO-3G basis,Pmn

STO, such that the integrals in Eq.~15c!can be evaluated with the Gaussian orbitalxm

STO(rW8). Thefitted density@Eq. ~15a!# is then used to calculateI mn @Eq.~13!#.

2. The FLYQ method (QM ÕSE-I-F)—beyond the fixedpoint charge model

A simpler solution to avoid calculating the inter-regiotwo electron integrals is to consider the interaction withmultipole expansion of the electron density in regionK whenconsidering the Fock matrix of regionK; this was adopted inthe effective fragment potential approach.17 Only one-electron inter-region integrals are required, so thatmethod is faster than the auxiliary basis approach. Incurrent work, we truncate the multipole expansion at loworder, and uses partial charges to represent the electronsity distribution; Mulliken charges were chosen here for siplicity, although charges from other schemes, such as naorbital analysis41 can also be used. Since Mulliken chargfrom semiempirical methods are usually too small in magtude, we introduce a uniform scaling factor,f Q , which has tobe determined empirically. Although scaling can alsodone for the QM Mulliken charges, this was not done bcause their magnitude seems to be reasonable in the mtested here. Because the partial charges fluctuate as thometry of the system varies, the method can be considerean ab initio version of the FLUQ approach,14 with the fluc-


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tuating charges derived on the fly. We, therefore, termapproximation the ‘‘FLYQ’’ approach~on the ‘‘FLY’’ as de-rived charges!. The current approach does not require afitting parameters other thanf Q ~and the QM/SE van deWaals parameters!. A similar scheme at the semiempiricalevel has been reported recently by Gao,42 who proposed torepresent the electron density with either a set of parcharges or multipoles.

With the FLYQ approximation, the description of mutupolarization between the QM and the SE region is achieby iteratively polarizing the wave function of each part witthe other described byflexiblepoint charges. This is expecteto give rise to improvements over the standard QM/MM aproach in which the MM is described withfixedcharges. Theauxiliary basis approach~Sec. II A 1! introduces further im-provements by computing the QM/SE interaction beyondmonopole level. The quantitative difference between thmodels has to be established with test calculations, whichpresented in Sec. III.

B. ONIOM based QM ÕSE method „QMÕSE-O…: Chargetransfer and exchange effects

In the preceding sections, only the electrostatic andlarization effects between the QM and SE region are treaexplicitly. The exchange effect~Pauli repulsion! is replacedwith van der Waals terms, and charge transfer effect isincluded explicitly~although one may consider it as partthe effective van der Waals term!. Therefore, theiterativemethod~QM/SE-I! is applicable to systems with weak inteaction between the QM and SE regions. In cases where tare QM and SE atoms interact strongly~or even are co-valently bonded!, the charge transfer and exchange repulsbetween the two regions can become important and shbe accounted for. Even for relatively weak interacting stems, it is likely that a specific set of van der Waals paraeters has to be fitted for each QM and SE combinatiwhich is somewhat cumbersome for general applicatioGiven these considerations, we implemented the followapproach inspired by the ONIOM method of Morokumet al.,24 which will be referred to as QM/SE-O methods~Oas in ‘‘ONIOM’’ !.

We start by considering a two-layer system partitioninto A1B; including a third MM layer is straightforwardThe ONIOM energy for the system, for which regionsA andB are described with method I and II, respectively, is aproximated by24

EONIOMA1B 5EI

A2EIIA1EII

A1B

5^C IAuHAuC I

A&2^C IIAuHAuC II

A&

1^C IIA1BuHA1BuC II

A1B&, ~16!

where the superscript of the wave function indicates thegion, and the subscript denotes the level of calculation.noted by the original authors, the QM region atoms dointeract explicitly with the environment at the QM level, anthe environmental effect is fully described at the low~SE!level.24 This simplification has caused substantconcern,26,27and recent developments attempt to improve


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description by explicitly polarizing the QM wave functiovia an embedding potential26 or allowing electrons to flowbetween the QM and SE region according to a chempotential.27 Although the later scheme is of interest, the fathat high levelab initio or DFT are so different from SE interms of the orbital eigenvalues and qualities led us nopursue the idea. We continue to follow the philosophy thatexplicit basis mixture is allowed between the two regiobut allow the QM wave function to be explicitly polarized bthe SE atoms. This leads to the following modification of tONIOM approach. The total energy of the QM/SE systemfirst approximated by

EIA1B5^C I

A1BuHA1HA/B1HBuC IA1B&

;^C IAC I

BuHA1HA/B1HBuC IAC I

B&

5^C IAu@HA1^C I

BuHA/BuC IB&#uC I

A&

1^C IBuHBuC I

B&

5EIA1B , ~17!

where the formally ‘‘exact’’ wave function for the total system given in the first expression is assumed to be decouinto C I

A•C I

B in the second step. A corresponding express(EII

A1B) can be written for the energy ofA1B at the SElevel, EII

A1B . We formally addEIIA1B to and then subtrac

EIIA1B from Eq. ~17! to obtain

EIA1B5^C I

Au@HA1^C IBuHA/BuC I

B&#uC IA&

1^C IBuHBuC I

B&1EIIA1B

2^C IIAu@HA1^C II

BuHA/BuC IIB&#uC II

A&

2^C IIBuHBuC II

B&. ~18!

Assuming that regionB can be well described at the SE~II !level ~i.e., ^C I

BuHBuC IB&2^C II

BuHBuC IIB& is nearly constant,

we arrive at the expression for the modified ONIOM~QM/SE-O! energy,

EIA1B'^C I

Au@HA1^C IIBuHA/BuC II

B&#uC IA&

1EIIA1B2^C II

Au@HA1^C IIBuHA/BuC II

B&#uC IIA&

5EQM/SE - OA1B . ~19!

Comparing Eq.~19! with Eq. ~13!, it is clear that thedifference is that here the wave function of the QM regionpolarized also at the high level~I!. We note that to avoiddouble counting, the energy for the QM region at thelevel, also polarized by the environment, has to be stracted. Therefore, Eq.~19! includes the difference in theQM and SE calculations in terms of the response of the wfunction in the QM region to the environmental~SE! elec-trostatic polarization, while the original ONIOM representhe electrostatic and polarization effects only at the SE leThus, when the SE level is not satisfactory for describthese effects, the current QM/SE-O approach is expectegive better results than the original ONIOM formulation. Tcurrent scheme and the original ONIOM both describe


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change repulsion and charge transfer effects at the SE letherefore, no specific van der Waals parameters have tofitted for different QM and SE combinations.

The term that represents the electrostatic interactionpolarization between QM and SE atoms,Vpol

5^C IIBuHA/BuC II

B&, is similar in nature to the embedding potential in other developments.26 It can be evaluated at different levels, such as the schemed described for QM/SE-Sec. II A. Since it has to be evaluated at both the QM andlevels, we took the simplest choice for the current implemtation and used scaled Mulliken charge on the SE atoms fa calculation at the SE level for theA1B region. With sucha choice, the ‘‘embedding potential’’ is calculated only onand no iteration has to be performed. One short comingthat the Mulliken charges thus obtained do not sum exato the charge of regionB. This could cause problems whethere is substantial charge transfer between the QM andregions; no such cases were found in the current tests. Msophisticated schemes to construct the embedding termbe studied in the future.

Finally, it is straightforward to derive the expression fthe QM/SE-O total energy for a three layer system descriat the QM~I!, SE ~II !, and MM level, respectively. The resulting expression is

EQM/SE - OTot '^C I

QMu@HQM1^C IIBuHQM/SEuC II

B&

1HQM/MM#uC IQM& 2^C II

QMu@HQM

1^C IIBuHQM/SEuC II

B&1HQM/MM#uC IIQM&

1^C IIQM 1 SEuHQM 1 SE

1H (QM 1 SE)/MMuC IIQM 1 SE&1EMM. ~20!

C. Implementations

The algorithms described above were implementedthe program CHARMM, which is interfaced with GAMESand DeFT;43 the latter programs allow one to use theabinitio methods~Hartree–Fock, MP2, CASSCF, CI, etc.! anddensity functional theories, respectively. The DeFT code wchosen because it is highly paralleled, and employscharge-density fitting algorithm required for the auxiliary bsis scheme in QM/SE-I. The available SE methodsCHARMM include MNDO, AM1, and SCC–DFTB;36 thelatter method was recently introduced into CHARMM,44 andhas been shown to give superior results for proton tranreactions and metal ions than AM1 and PM3.36,44

There are a number of parameters that can be optimin the different QM/SE methods. In QM/SE-I, the Pauli rpulsion and charge transfer effects are not included expitly, and therefore requires a set of van der Waals parameit is likely that different sets would be optimal for differenQM/SE combinations as in different QM/MM calculations.45

Further, partial charges are used to polarize the SE atomQM/SE-I-A and to polarize both QM and SE atoms in QMSE-I-F. One has the freedom to choose the optimal separtial charges such as natural orbital charges, ESP chaor scaled Mulliken charges; the last were chosen heresimplicity. In QM/SE-O, no van der Waals parameters a


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needed because the Pauli repulsion and charge transfefects are taken into account at the SE level. In the evaluaof the embedding interaction terms, however, specific pareters such as a scaling factor or charge normalization fa~see Sec. II A! for the Mulliken charge would be needed.

When the QM and SE regions are covalently bondedMM regions, the interface can be treated with the standlink atom approach.1~b! If the QM and SE regions are bondetogether, a double link atom approach46 is the most straight-forward scheme to treat the boundary; i.e., link atomsadded to both QM and SE boundary atoms. Care is necesto avoid over polarization of the boundary QM or SE atomas in the QM/MM method analyzed previously.9 For ex-ample, the partial charge or the density fitting coefficientsthe boundary SE or QM atoms can be excluded inQM/SE interaction, similar to the exclusion convention usin CHARMM for QM/MM. 9 It is expected that the boundarissue is more delicate for QM/SE than QM/MM, and the bscheme has to be determined through test calculationsemphasize that the QM/SE-I approach is mostly suitedweak intermolecular interactions between QM andregions, and the QM/SE-O approach is more approprfor cases where the partition is done across a covabond. Consider the simple case of butanol, in which a pation is done between C2 and C3 atoms ~scheme I!. Note

Scheme I

that in the QM/MM calculations, the bonded terms betwethe two regions such as the C2–C3 stretch and C4–C3–C2

bending are included, which helps to maintain the ovestructural stability. Since no such terms are included in QSE-I, the butanol molecule would be described by separaethane and ethanol molecules, and thus would not bestrained to have the correct geometry. In the QM/SEscheme, however, there is a contribution correspondingthe entire butanol at the SE level@see Eq.~19!#, and thereforethe structure is expected to be qualitatively correct. Incurrent work, we restrict ourselves to systems where theand SE regions are not covalently bonded, and leavemore complicated cases to future studies.

III. TEST EXAMPLES

In this section, we illustrate the algorithms describedSec. II by a number of applications. We determine howdifferent QM/SE methods compare to full QM calculationand how sensitive the results are to the choice of thevolved parameters. Also we examine whether the QM/methods give significant improvements over QM/MM calclations. Finally, the QM/SE methods proposed here are cpared to determine which of them are most robust. Theset of models involves the interaction between small polacharged molecules with water. The second set deals witheffect of polar molecules on the proton abstraction fromCa position of formaldehyde by a base (HCOO2); this isinspired by the first reaction step catalyzed by trioseph


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phate isomerase,47 which has been studied extensively btheoretical methods.48 Finally, the effect of a hydrogen-bonnetwork on the same proton transfer reaction is studied. Tis based on the proposal of Guoet al.10 that cooperativityeffects due to extensive hydrogen-bond network may plarole in enzyme catalysis. The relatively small sizes of thomodels allow us to perform full QM calculations, to whicthe QM/MM and QM/SE results can be compared. A DFmethod with medium sized basis sets was used as QMcomparison, although more elaborate methods are likelybe explored in actual applications. The BP86 functiona49

was employed because it is the only nonlocal functioavailable in the DeFT program that we used.

A. Interaction between water and polar molecules

The first set of tests concerns the interaction betwwater and several polar or charge water molecules. Thsystems are analogous to studies in the validationQM/MM methods with SE1~b! and DFT50 QM methods. Themajor purpose is to test the robustness of the propoQM/SE methods, especially the schemes for calculatQM/SE interactions and the difference between the iteraand noniterative methods.

The first system is the complex between a Cl2 ion and awater molecule. This was chosen because it was noticeprevious studies50 that although full QM calculations gave aasymmetric optimal structure~i.e., the two Cl2¯H distancesare different, see Table I!, QM/MM calculations in which theCl2 was treated with QM and the water was treated wMM gave a symmetric structure~see Table I!. The result was

TABLE I. Water–Cl2 ion complexes calculated with different methods.a

Methodb

Cl2

DE RCl–H ~Å! RCl–H8 ~Å!

Full QM 215.3 2.128 3.283Full SE 213.8 1.987 3.022QM/MM 213.2 2.782 2.782SE/MM 213.6 2.765 2.765QM/SE-I-Fc 26.1 2.803 3.062

~27.9! ~2.688! [email protected]# @2.561# @3.239#

QM/SE-I-A 213.5 2.116 3.215QM/SE-O-0 213.8 1.987 3.022QM/SE-O-Mc 212.5 2.107 2.945

~212.4! ~2.132! [email protected]# @2.174# @2.874#

aThe QM is BPW86 with 6-311G** basis set, SE is AM1, MM is the TIP3Pmodel for water. In the QM/MM~SE! calculations, the Cl ion was treated aQM and the water was described with MM~SE!.

bThe QM/SE-I indicates the iterative QM/SE algorithms described in SII A; the QM/SE-O indicates the ONIOM based QM/SE methods describin Sec. II B. The suffix ‘‘-F’’ indicates that Mulliken charges from the Sregion is used to polarize the QM wave function, ‘‘-A’’ means that auxiliabasis functions were used to fit the SE electron density which interactsthe QM wave function~see text!. The QM/SE-O-0 method is the samONIOM approach originally proposed by Morokuma and co-workewhere no~thus ‘‘0’’ ! explicit interaction was evaluated between QM anSE atoms~Ref. 19!.

cThe numbers without and with parentheses were obtained with a scafactor ~see text! for the Mulliken charge of 1.0 and 1.2, respectively; thowith brackets were obtained with a scaling factor of 1.5.


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rationalized by the fact that the MM water~TIP3P! is notpolarizable, and the two hydrogen atoms have the sastrength of interaction which led to the symmetric structuWhen the water is polarized, the asymmetric structure isvored because the electrostatic interaction between Cl2 andone hydrogen atom can be maximized.

The binding energy and the optimal Cl2¯H distances at

different levels are shown in Table I; the Cl2 was treated byBP86/6-311G(d,p),51 and the water was treated by eithAM1 or TIP3P. Although the van der Waals parameters cbe optimized, no such attempt was made here for the purpof exploring the behavior of the methods. It is encouragthat the asymmetric structure was correctly reproduced athe QM/SE levels. When the polarization was modeledthe simple Mulliken charge in the iterative scheme~QM/SE-I-F!, the result depends sensitively on the scaling fac( f Q). Without scaling (f Q51.0), the Mulliken charges onthe water atoms are very small; i.e., the charge on the hygen is about 0.2, compared to the value of10.417 in theTIP3P model and the value of10.35 from a Mulliken analy-sis at the full QM level. With the small charge, the interation energy at the QM/SE-I-F level is substantially undertimated ~26.1 kcal/mol vs215.3 kcal/mol from full QMcalculations!, and the structure is only slightly different frombeing symmetric~the Cl H distances are 2.803 and 3.06Å, respectively, compared to the full QM values of 2.128 a3.283 Å, respectively!. As expected, the agreement with thfull QM results becomes much better when the AM1 Muliken charges are scaled. For example, withf Q51.5 ~e.g., avalue of 2.13 was found to be appropriate in Ref. 42!, theinteraction energy increases to210.9 kcal/mol, and the shorCl¯ H distance drops to 2.561 Å; better agreements canobtained if f Q is optimized~e.g., to obtain exact agreemewith the QM binding energy,f Q51.95 is required, whichgives a Cl H distance of 2.356 Å!. In contrast to the FLYQapproach, the auxiliary basis set approach~QM/SE-I-A!works rather well without the need of adjusting parameteThe interaction energy was calculated to be213.5 kcal/mol,and the Cl H distances are 2.116 and 3.215 Å, all in goagreement with the full QM results. This indicates that tdensity-fitting scheme is a robust approach to describeQM/SE interactions at hydrogen-bonding distances~but seebelow!.

With the original ONIOM model,24 the QM/SE-O-0~‘‘0’’ is used here to emphasize that no explicit interactibetween the QM and SE atoms was included at thelevel! gave exactly the same results as full SE~AM1! for theparticular case. This is true because both QM~BP86! and SE~AM1! gave zero gradient for a free Cl2 atom. With theembedding interaction included, the results become diffefrom full AM1 values. This is due to the fact that the currescheme includes a term reflecting the difference in thesponse of the QM~BP86! and SE~AM1! wave functions toexternal electrostatic interactions and polarization~see Sec.II B !. An interesting observation is that the QM/SE-O-F rsults are much less sensitive to the Mulliken scaling facthan QM/SE-I-F because both the QM~BP86! and SE~AM1! are polarized by the same~scaled! Mulliken charges.The results, however, are not significantly different from t


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original ONIOM values; in fact the binding energies asomewhat further away from the full QM values. The obsvation that the binding energy decreases and the shoCl¯H distance increases as the scaling factor increasesdicates that the BP86 wave function~density! interact lessstrongly than AM1 wave function with the external chargThis is consistent with the result that BP86/MM calculatipredicts a slightly weaker binding energy comparedAM1/MM ~Table I! with the same water model~Table I!.

To further test the robustness of the density-fittischeme, we studied a number of complexes between wand molecules representing the fragments of amino acThese include two neutral molecules, imidazole aN-methyl formamide~NMF!, which model the side chain ohistidine and the main-chain peptide unit, respectively, atwo charged species, NH4

1 and C~NH2)31, which model the

side-chain of lysine and arginine, respectively. The intertion energy between these models and a water moleculecalculated at the full BP86/6-31G(d,p), BP86/CHARMMand BP86/AM1 level with the auxiliary basis function aproach ~QM/SE-A!. Morokuma energy decomposition wacarried out with HF at the full QM level to reveal differencomponents; a full HF calculation was found to give vesimilar interaction energies as BP86 with the same basisIn the hybrid calculations, the water molecule was treawith BP86, and the fragment was treated with SE. In allmodels, the SE part is the hydrogen-bond donor and thepart is the acceptor; this is motivated by the fact in menzyme systems, protein residues serve as hydrogen-donors to polarize~or stabilize! charged substrates. In Fig. 2the electrostatic and polarization components of the intetion from the full HF calculations according to the Morokuma energy decomposition scheme are shown as a funcof the distance between the hydrogen-bond donor and actor. The corresponding components from BP86/MM aBP86/AM1 calculations, which are defined as the QM/M~AM1! interaction without the van der Waals contributioare also shown; the van der Waals term is not includedcause the dispersion energy is not described at the HartFock level. Clearly, there is striking agreement betweenfull QM and QM/SE calculations at all the distances consered here. The good agreement between QM/SE and fullcalculations is very encouraging, indicating that the SE dsity is reasonable for calculating hydrogen-bonding intertions. This is in accord with the previous observation thgood hydrogen-bonding interaction energies can be obtawhen the hydrogen-bond donor, such as water, is modelea charge density distribution fitted by a small set of bafunctions.52 On the other hand, we realize that the densdistribution in AM1 is far from optimal because the densitself was not directly involved in the fitting of AM1 parameters; in fact, many integrals were parametrized accordinexperimental values rather than calculated. Therefore, wepect that SE methods in which the density is well describsuch as the SCC–DFTB approach, will work even betthan AM1 in QM/SE-A calculations. Integration of DFT anSCC–DFTB in an iterative fashion with the auxiliary basapproach~Sec. II A 1! is currently underway. The QM/MMresults are very different at short distances. The observa


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as


FIG. 2. The electrostatic and polarization energy~inkcal/mol! between a water molecule and a hydrogebond donor that models a unit in an amino acid@imida-zole, N-methyl formamide, NH4

1 and C~NH2!31! as a

function of their separation. Such a quantity from a fuQM calculation was obtained from a Morokuma energdecomposition analysis at the HF/6-311G(d,p) level,which gives interaction energies close tBP86/6-311G(d,p) calculations. The latter was usein the QM/MM and QM/SE-I-A calculations, for whichthe electrostatic and polarization energy is definedthe interaction between QM and SE~MM ! atoms with-out the van der Waals contribution.

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does not imply directly that QM/MM results for the interation energy are poor, because the van der Waals contribuin the QM/MM calculations is empirical and does not neessarily reflect the full Pauli repulsion; in other words, ercancellation may lead to good interactions from QM/Mcalculations. Finally, we note that because the electrosand polarization contributions in QM/SE-A calculations avery different from the QM/MM values, different van deWaals parameters are required in the two sets of calculatiespecially at short distances.

B. The effect of polar molecules on a proton transferreaction

As the second set of tests, we study the effect of smpolar molecules on a proton transfer reaction inspiredtriosephosphate isomerase~TIM !.47,48As shown in Fig. 3, theproton transfer reaction takes place between formic acid

formaldehyde, which model the catalytic glutamic acid~Glu165! and substrate~dihydroxylacetone phosphate! in TIM,respectively. Several polar molecules hydrogen bonded tocarbonyl group in formaldehyde were studied. They areand two water molecules, one NMF, and one water molecplus one imidazole; the imidazole is chosen to mimic Hisin TIM, and the others are selected for an analysis of poization effects. For simplicity, the distance between the doand acceptor in the proton transfer is fixed to be 3.0 Å. Gometry optimization was carried out for the reactant, prodsystem in the presence of the constraint atHF/6-31G(d,p) level, and the transition state was detemined by an adiabatic mapping procedure at the same lecritical distances in the reactant, transition state and prod


on-r

tic

s,

lly

d

heele5r-r-t

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are shown in Fig. 3. Subsequently, single point energies wobtained by full QM (BP86/6-311G(d,p)), BP86/CHARMM, and BP86/AM1 calculations; in the hybrid caculations, the environment~the added polar molecules! wastreated with CHARMM or AM1 and the reacting groupwere described with BP86/6-311G(d,p). No specific vander Waals parameters were optimized for the MM or AMatoms, because our primary purpose is to examine the qtative behavior of different QM/SE methods relative to QMMM.

At the full QM level ~see Table II!, the effect of theenvironment increases in the order of one water, one NMtwo waters and one water plus one imidazole, and the pruct is stabilized more than the transition state. For examone water molecule stabilizes the transition state and proby 3.7 and 6.8 kcal/mol, respectively, and the valuesNMF are 5.0 and 9.2 kcal/mol, respectively. One imidazand one water has a larger stabilizing effect compared towater molecules, by;2 kcal/mol for the transition state an;4 kcal/mol for the product. These are consistent withfact that the carbonyl group in the product is more negativcharged than in the transition state; imidazole and NMF hlarger stabilizing effects than water because their dipole mments are better aligned with the lone pair of the carbooxygen.

At the QM/MM level, the same trend was observed,though the magnitude of the increase in the stabilizing effgoing from water to NMF and imidazole was reduced. Fexample, instead of a 2 kcal/mol increase in the stabilizeffect on the transition state for an imidazole comparedwater, a value of 0.6 kcal/mol was obtained from single poQM/MM calculations. Better agreements might be obtain


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FIG. 3. Model systems used to studthe effect of small polar molecules ona proton transfer reaction between fomic acid and formaldehyde. The distance between the proton donor anacceptor was fixed to be 3.0 Å durinthe geometry optimizations, whichwere carried out at the HF/6-311G(d,p) level. Critical distances inthe reactant, transition state~in paren-theses! and product~in brackets! areshown in Å. Selected Mullikencharges for the polar ligands are alsshown for full QM @BP86/6-311G(d,p)# and full SE single pointcalculation for the transition states tillustrate the difference between QMand SE Mulliken charges.

TABLE II. Comparison of full QM, QM/MM and different QM/SE algorithms for the effect of different molecules on the transition state and prenergetics of a model proton transfer reaction.a

Methodb

H2O NMF 23H2O Imd1H2O RMS error

TS Prod TS Prod TS Prod TS Prod TS Prod

Full QM 23.7 26.8 25.0 29.2 27.5 212.5 29.3 216.9QM/MM 22.8 24.8 23.3 25.6 26.1 210.3 26.7 211.0 1.8 3.8QM/SE-I-F 22.3 24.2 25.0 29.9 25.0 29.0 28.6 216.3 1.5 2.2QM/SE-I-Ab 24.5 29.0 26.7 214.2 29.1 217.7 212.4 225.0 2.0 5.5

~23.0! ~25.6! ~25.0! ~210.2! ~26.5! ~211.5! ~29.4! ~217.5! ~0.6! ~1.0!QM/SE-O-0c 21.6 23.3 23.1 26.0 24.6 28.1 26.7 211.9 2.4 4.1

~22.9! ~26.1! ~24.6! ~29.7! ~25.7! ~211.9! ~27.7! ~216.1! ~1.3! ~0.7!QM/SE-O-M 21.8 23.1 22.9 25.4 24.3 27.1 26.2 210.7 2.6 4.9

aThe QM is BPW86, SE is AM1, MM is CHARMM22 force field for proteins. The proton transfer fragments~formic acid and formaldehyde! were treatedwith QM, and the ligands were treated with either MM or SE in the QM/MM and QM/SE calculations, respectively. The values without ligands are 114.0 kcal/mol, for the TS and Prod, respectively. See footnote of Table I for notations for the different QM/SE methods, and Fig. 3 for the structurhichwere optimized with the HF/6-311G(d,p) method~see text!. In the QM/SE-I-F and QM/SE-O-M calculations, a scaling factor of 1.2 was used for theMulliken charges.

bThe values in parentheses were obtained by considering the fact that the QM/SE van der Waals contribution based on the standard CHARMM vaparameters is substantially underestimated, and an approximate correction was made based on the difference in the electrostatic and polarizationomponentsin QM/SE-I-A and QM/MM calculations for small molecules and water~Fig. 2!.

cThe values in parentheses were obtained with SCC-DFTB as the semiempirical level.

Downloaded 17 Sep 2002 to 128.104.71.49. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

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FIG. 4. Model system used to illustrate the performance of QM/SE methods in the presence of long-range noadditive polarization effects. See Fig3 for notations.

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thateringtuminE,

when the geometry is allowed to relax in the QM/MM caculations because it has been observed that shorter hydrbond distances~compared to full QM! are often required togive reliable QM/CHARMM interaction energies, due in pato the parametrization philosophy of the CHARMM forcfield to represent the situation in the condensed phase.53

At QM/SE levels, the same qualitative trend was otained although the quantitative results depend significaon how the QM and SE levels are combined. At the QM/SI-M level, a scaling factor of 1.2 was found to give gooresults for the effect of NMF and imidazole~see Table II!;the effect of water with the same scaling factor underemates the magnitude of interaction substantially becaAM1 gives a very small charge on the water molecules~seeFig. 3 for a comparison of QM and SE Mulliken chargesthe ligands from full QM and SE calculations!. At the QM/SE-I-A level, the effect of the environment was found tooverestimated, and the degree of overestimation is lawhen the interaction is stronger. For example, the stabilizeffect of one water on the product is 6.8 and 9.0 kcal/mothe full QM and QM/SE-I-A level, respectively, and the vaues for one water plus one imidazole are 16.9 and 25.0 kmol, respectively. These results can be understood whenrealizes that the van der Waals contribution in QM/SE-I-Aexpected to be very different from QM/MM or full MMcalculations, as discussed in the preceding section. Wheninteraction between the ligand and reacting group is strothe hydrogen-bond distance becomes very short; e.g., thedrogen bond between the imidazole and the carbonyl oxyin the product is only 1.75 Å. According to the calculatioin Sec. III A, the van der Waals contribution in QM/SE-I-Aexpected to be underestimated by as much as 7 kcal/msuch a distance~see Fig. 2!. Introducing this as an approximate correction to the van der Waals QM/SE-I-A contrib


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tions, the agreement with full QM becomes much better~seeTable II!.

Using the ONIOM approach, the results are not encoaging with both the original scheme of Morokumaet al.~RMS error is 2.4 and 4.1 kcal/mol for the TS and produrespectively! and the one proposed here which includesplicit polarization of the QM part~RMS error is 2.6 and 4.9kcal/mol for the TS and product, respectively!; the effects aretypically worse compared to QM/MM or QM/SE-I resultsThis observation reflects the fact that AM1 is rather poorrepresenting the interaction from the environment, and ater semiempirical approach has to be used. Indeed, very gresults~RMS error is 1.3 and 0.7 kcal/mol for the TS anproduct, respectively! were obtained when SCC–DFTB waused in the ONIOM calculations even when the QM regiis not explicitly polarized.

C. The effect of hydrogen-bond network on a protontransfer reaction

Although hydrogen-bonding interactions of mediustrength are dominated by electrostatic interactions, nonative effects have been shown to be involved in strohydrogen-bonding interactions.54 As discussed by Guo anSalahub,10 and by Luo et al.,55 the extensive hydrogenbonding networks found in the active site of many enzymcan induce such nonadditive effects. Since the effectssubstantially larger when the hydrogen-bond acceptorcharged compared to when it is neutral, it was proposeddifferential cooperative effects can play a role in enzymcatalysis where the charge of the substrate changes duthe reaction. Because the nonadditive effects are quanmechanical in nature, we expect that a hybrid calculationwhich the hydrogen-bond network is represented by S


n der Waalsc


TABLE III. Comparison of full QM, QM/MM and different QM/SE algorithms for the model proton transfer system plus a model di-peptide.a

Methodb

Model 1 His/H2O Di-peptide Effect of hydrogen bond network

TS Prod TS Prod TS Prod

Full QM 11.4 9.6 8.3 4.7 23.1 24.9QM/MM 13.3 11.7 11.9 10.6 21.4 21.2QM/SE-I-F 10.6 6.0 8.2 2.4 22.4 23.6QM/SE-I-Ab 11.3 5.4 8.3 1.6 23.0 23.8

~11.6! ~7.2! ~9.6! ~4.6! ~22.0! ~22.6!QM/SE-O-0c 12.5 9.7 10.9 6.9 21.6 22.7

~12.8! ~11.9! ~10.3! ~7.9! ~22.5! ~24.0!QM/SE-O-M 12.4 9.5 11.1 7.8 21.3 21.7

aSee footnote of Table I for the notations for the different QM/SE methods. See Fig. 4 for the geometries, which were optimized with HF/6-311G(d,p). Inthe QM/SE-I-F and QM/SE-O-M calculations, a scaling factor of 1.2 was used for the SE Mulliken charges.

bThe values in parentheses were obtained by considering the fact that the QM/SE van der Waals contribution based on the standard CHARMM vaparameters is substantially underestimated, and an approximate correction was made based on the difference in the electrostatic and polarizationomponentsin QM/SE-I-A and QM/MM calculations for small molecules and water~Fig. 2!.

cThe values in parentheses were obtained with SCC–DFTB as the semiempirical level.

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rather than by MM, will give a better description in this casThe proton transfer model shown in Fig. 4 was constructo examine this behavior. This model differs from those dcussed in Sec. III B in that the environment is replaced byextended hydrogen-bond network consisting of one imizole molecule and two diglycine peptides, which represthe backbone of a half-turn ‘‘helix;’’ a water molecule waadded on the proton acceptor end to stabilize the chareactant. Similar to Sec. III B, the geometries were optimizat the HF/6-311G(d,p) level with the distance between thproton donor and acceptor fixed at 3.0 Å. Single point engies were then calculated at the BP86/6-311G(d,p), BP86/CHARMM, and BP86/AM1 levels.

At the full QM level, the barrier for the proton transfersubstantially lower when the di-glycine is present to formextended hydrogen-bond network; the values are 8.311.4 kcal/mol with and without the di-glycine, respectiveThe distance between the imidazole and the carbonyl grin the reacting group is also much shorter in the presencthe di-glycine; in the transition state, the distance is 1.7and 1.842 Å with and without the di-glycine, respectiveThe product is also further stabilized by the di-glycine, dspite the fact that the latter is 6.7 Å~shortest distance! fromthe reacting center~the Cb in the formaldehyde!; the productis 4.7 and 9.6 kcal/mol above the reactant with and withthe di-glycine, respectively. In other words, the stabilizieffect of the di-glycine is 3.1 and 4.9 kcal/mol for the trasition state and product, respectively.

The QM/MM single point energetics are discouraginalthough a lower barrier~by 1.4 kcal/mol! was reproduced inthe presence of the di-glycine, the larger stabilizing effectthe product was not found~see Table III! and the error in theendothermicity of the proton transfer was nearly 6 kcal/mAs discussed in Sec. III B, the error might be reduced whgeometry optimization is allowed. The current results dodicate, however, that QM/MM calculations are not fully saisfactory for such nonadditive effects.

At the QM/SE levels, the qualitative trends in the stalizing effect are well reproduced. When the SE regionrepresented by the~scaled! Mulliken charges, as in QM/SEI-M, a scaling factor of 1.2 was found to give satisfacto


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results, as in the pair-wise type of models discussed in SIII B. When the SE density is represented by a set of auiary basis functions, as in QM/SE-I-A, the effects of the dglycine on the transition state and product are also descrwell. The absolute endothermicities of the reactions arederestimated. This is related to the fact that the van der Wcontribution has not been optimized for QM/SE and larerrors can arise at short hydrogen bond distances betwQM and SE atoms, as also discussed in the precedingtion. With the ONIOM approaches, the results are also safactory, especially when the environment was treated wSCC–DFTB, rather than AM1.

IV. CONCLUDING DISCUSSIONS

To improve the description for the interaction betwean active region in large systems, such as an enzyme, animmediate environment over that obtained with the standQM/MM methods,1 we have proposed several algorithms ftreating the active part with a high level QM~ab initio orDFT! approach and the environment with a less sophisticasemiempirical ~SE! approach. In the iterative QM/SEschemes~QM/SE-I!, the electrostatic interaction and polaization effects are introducedexplicitly for both the QM andSE atoms by a self-consistent procedure. In the noniteraQM/SE scheme based on the ONIOM model~QM/SE-O!,the exchange~Pauli repulsion!, and charge transfer effectare taken into account at the SE level, in addition toexplicit electrostatic interaction and polarization betweentwo regions.

From the test calculations, it is evident that the perfmance of QM/SE calculations are encouraging, and thatquantitative accuracy of the results depend on several paeters; they are the charge-scaling/normalization factors~seeSec. II A! and the QM/SE van der Waals parameters. Whthe SE region is represented by the Mulliken chargespolarize the QM atoms, the iterative QM/SE-I-M is sensitito the scaling factor because the SE Mulliken charges argeneral too small~at least for the cases tested here, see F3!. In the modified ONIOM approach, because the QMoms are polarized at both the QM and SE levels and on


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differential effect is considered, the results are less sensto the scaling factor. The auxiliary basis function approa~QM/SE-I- A! was found to be satisfactory for the descrition of hydrogen-bonding donor groups in all the test caexamined, including both neutral and charged species.is very encouraging considering the fact that density isoptimized in the standard SE approaches, such AM1,suggests that semiempirical approaches in which elecdensity is considered in the parametrization, such asSCC–DFTB method,36,44 are well-suited for QM/SE-I-Atype of calculations.

A practical issue related to the general applicationQM/SE-I-M~A! approaches is that the van der Waals paraeters have to be fitted to give reliable geometries and egetics for each level of implementation. This is more imptant for QM/SE-I-A, because the van der Waals contributis closer to that in the full QM calculation, which is verdifferent from the typical values in MM or QM/MM calculations. In other words, although using the standard MM vder Waals parameters in QM/MM calculations can gmeaningful results, very poor values result in QM/SE callations.

In the current studies, no dramatic improvements inONIOM results were found when the QM region is explicitpolarized. This is because the major part of the environmtal effects are described at the SE level, and explicit poization of the QM region only provides a correction to the Streatment of the QM atoms and their response to the elecstatic interaction and polarization. Therefore, the qualitythe SE description plays a more important role in tONIOM results, as illustrated by the observation that BPSCC–DFTB, in general, gives significantly better resuthan BP86/AM1. Finally, we note that although seconsistency in the iterative schemes is usually achieveless than six or seven cycles, the iterative procedure mathem more expensive~by a factor ofN, whereN is the num-ber of iterations! than the ONIOM based approach, whicdoes not require iteration in the current approximation forembedding polarization.

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