Selected New Developmentsin Computational ChemistryThomas A. Darden,' L. Bartolotti,2 andLee G. Pedersenl'31National Institute of Environmental Health Sciences, Research TrianglePark, North Carolina; 2North Carolina Supercomputing Center, ResearchTriangle Park, North Carolina; 3Department of Chemistry, University ofNorth Carolina-Chapel Hill, Chapel Hill, North Carolina

Molecular dynamics is a general technique for simulating the time-dependent properties ofmolecules and their environments. Quantum mechanics, as applied to molecules or clusters ofmolecules, provides a prescription for predicting properties exactly (in principle). It is reasonable toexpect that both will have a profound effect on our understanding of environmental chemistry inthe future. In this review, we consider several recent advances and applications in computationalchemistry. Environ Health Perspect 104(Suppl 1):69-74 (1996)

Key words: molecular dynamics, particle mesh Ewald, quantum mechanics, transition states,rate constants, potential energy surfaces, fast multipole expansion

Introduction

Due largely to concurrent advances intheory, computer technology, numericalalgorithm development, and the rapidinflux of experimental information aboutthe structure of molecules, computationalchemistry stands poised to make significantcontributions to our understanding of bio-logical, and ultimately environmentallysignificant, processes. In this review we willconsider some of the recent advances thatseem most promising for the near future.We initially outline the recently developedparticle mesh Ewald (PME) method. PMEprovides a mathematical approach for accu-rate evaluation of the Coulomb interactionsin macromolecules. The development nowmakes possible the dynamical study of largeenvironmentally important molecules bythe well-established rules of classical mole-cules. We then discuss density functionaltheory (DFT). DFT is a relatively newmethodology for investigating the next levelof complexity-the electronic structure of

Manuscript received 21 August 1995; manuscriptaccepted 17 October 1995.

Address correspondence to Dr. Lee G. Pedersen,MD A3-06, NIEHS, P.O. Box 12233, ResearchTriangle Park, NC 27709. Telephone: (919) 5414630.Fax: (919) 541-7887. E-mail:pedersen@niehs.nih.gov

Abbreviations used: PME, particle mesh Ewald;DFT, density functional theory; PDB, protein data-bank; HOMO, highest occupied molecular orbital.

molecules. As with the PME method,significantly larger molecules of environ-mental interest can now be studied at theelectronic level. Although both methods areglobal with respect to the expected impacton physical chemistry, it is in the area ofenvironmental chemistry that profoundapplications may ultimately be found.Accurate Representation ofLong-range Coulomb ForcesThe major source of information forthe three-dimensional structures of bio-molecules is that derived from X-ray crys-tallographic or from nuclear magneticresonance (NMR) studies. The resultinginformation is normally tabulated asatomic Cartesian coordinates and isotropicthermal B factors. The latter are measuresof the degree of thermal motion and aredirectly proportional to the mean squaredeviation from the average structure. Forinstance, a B factor of 5.0 for an atom indi-cates little motion, whereas a value of 75.0indicates such large motion that the valueof the coordinates can be taken to be quiteuncertain. The structure files are ultimately(in most cases) deposited in databases suchas the Brookhaven Protein Databank(PDB). These structures, universally dis-played on two-dimensional graphics moni-tors, are static, unmoving representationsof the molecules that do little to reveal the

underlying motion of the atoms. Theindividual atomic motions can, however,be simulated by the technique of mole-cular dynamics (1). The underlying ideais that one assumes a potential energyfunction, or force field, and then solvesNewton's equations of motion for the sys-tem. If there are N atoms in the system,then there are 3N equations, one for eachCartesian coordinate:

Fx, i Mi dv,,i dt--E(xj,..., ZN)I61,i i = 1 to 3N.

Here FXi is the x force of the ith particle,vx,i is the corresponding velocity (vx,i =dxildt), and E(xi,..., ZN) is the potentialenergy function from which the forcederives. The potential energy function istaken to approximate the more rigorousquantum mechanical solution. The func-tion in its simplest form is a sum of qua-dratic terms in the displacement fromclassical equilibrium of bond stretches(A-B), angle bends (A-B-C), impropertorsion angle contraints, a truncatedFourier expansion for the proper (bonded)torsion angles (A-B-C-D), Lennard-Jonesterms for the nonbonded pairwise interac-tions (Alr'2-Blr6), which accounts forshort-range repulsion of the outer electronsin a pair of interacting nonbonded atomsand for the long-range attractive dispersioninteraction, and finally, a Coulomb's lawterm (qAqBlr) for the interactions of chargesqA and qB. Its most important terms arethose that affect space exclusion (atomscannot be in the same place at the sametime) and the Coulomb interactions thatare long range (- l/r). The potential energyfunction is parameterized by establishingvalues for the constants in the function forthe defined atom types. Parameterizationderives from spectroscopic and thermody-namic measurements or from accuratequantum mechanical calculations, usuallyon small model compounds. In mostcurrent molecular dynamics programs,for instance, atoms are assigned staticcharges determined from accurate quantummechanical calculations (2). The interac-tion between these charges is normallyhandled by Coulomb's Law, with interac-tions larger than a particular distance (8 Ais popular) being truncated to make thecalculations tractable. The truncation ofthe charged interactions, however, hasbeen shown to lead to artifactual (3-11)

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behavior for simulations of proteins andnucleic acids. In this section we describerecent progress in one approach for accu-rately accommodating long-range interac-tions between charges. Consider a crystalthat is made of identical unit cells. Eachunit cell holds a finite number of sym-metry-related molecules. If we assign eachatom in the molecule a charge, then we candetermine a given atom's electrostaticinteraction energy with all other chargedatoms in the crystal by summing overCoulomb interactions in the unit cell andsubsequently summing over all unit cells inthe crystal. These summations are veryslowly converging. In 1921, the Prussianphysicist Paul Ewa!d (12) showed that thesums could be rewritten as a set of alternatesummations, a direct space sum andan indirect space sum, that had muchimproved convergence properties (13).The indirect space sum, which has the nat-ural co rdinates of the theory of X-raycrystallo raphy, was recently effectivelysolved by an implementation of fastFourier transforms (5,7,14). The key ideawas to approximate the charges as if placedon a regular grid using interpolation orspline formulae. The procedure, the PME,leads to a numerical algorithm with a timerequirement of Nelog(N) rather than N2for the formal Ewald procedure (the proce-dure can actually be optimized to behave asorder N312) (Nis the number of atoms inthe unit cell). For large macromolecules,the time speedup with PME is impressive,and thus we can now perform accurate cal-culations on systems that were previouslyunreachable. Although numerical, the errorin the procedure is well defined in terms ofthe parameters of the method (14). Themethod also recently has been tested forperiodic box images of single moleculessurrounded by solvent and counterions (9)(Equation 1).

Applications of the Ewald sum tosimulation problems in other laboratoriesinclude studies of salt solutions (15), pla-nar interfaces (16), mobility of ions insolution (17), and DNA triple helices (18).Theoretical details of the finite size effectthat occur in Ewald-based simulations havebeen considered by Figueirido et al. (19).

An alternate approach for performingthe long-range Coulomb summation, thefast multipole method (20-22), proceedsby first dividing a volume into approximateM subvolumes. An external potential foreach subvolume is determined that theninteracts directly with particles in nearbyboxes but with a Taylor series expansion

for particles in distant boxes. The overalltime requirement can be shown to go as N,the number of charges, with the economybased on representing the interactionsbetween volumes distantly spaced by trun-cated multipole expansions. A version ofthe fast multipole method has recentlybeen applied to large macromolecularsystems (23).

It is clear that our understanding ofhow to accurately simulate the motions ofmacromolecules, or clusters thereof, isincreasing at a rapid pace. The applicationof these new techniques to problems inenvironmental chemistry should soonfollow. It is now known (24), for instance,that exposure of animal cells to low levelsof toxic compounds such as arsenicals,mercury derivatives, dimercaptans, perox-ides, quinones, or diphenols leads to eleva-tion of glutathione levels and induction ofdetoxification enzymes, e.g., glutathioneS-transferases, quinone reductase, epoxidehydroxylase, and UDP-glucuronosyl-transferase. The application of the newlyemerging techniques should have asignificant effect on our abilities to provideaccurate representations of the time-dependent interactions of drugs or toxicmolecules with proteins such as those justdescribed (once their three-dimensionalstructures are known), nucleic acids, ormembranes.

Applications ofQuantum Mechanics toEnvironmental ChemistrySince its inception in 1926, quantummechanics has held great promise forall areas of chemistry. Again, it has beenthe evolution of high-speed computingmachines with massive storage devices andcreative algorithm development that hasled to the belief that this promise will bekept (25). The last few years have seen theimplementation of the alternate densityfunctional approach (DFT) (26) to theSchrodinger wave function method. In thelatter, one (in principle) writes down a wavefunction for a chemical system that is com-posed of atomic orbitals of the individualatoms and then finds by application of thevariation theorem (27) the lowest energy(best) wave function for computation of theproperties (energy, geometry, vibrationalfrequencies) of the system. In the former(26), the energy is determined by the elec-tron density (not a wave function); i.e., ifone knows the electron density, the energy(and other measurables) can be calculated.

The DFT approach appears to besuited for large systems, since its computertime requirements scale with the numberof particles N as N3, while the wave func-tion approach scales as N4 or higher(Equation 2).

Equation 1.

If a system of N-charged particles is repeated on a regular cubic lattice 1, the electrostaticenergy is

E =1-'XIX.es 2E1 j=j=1|I r + Iwhere the charges are (qj, qj), r,j is the distance between charges, ctnd the sum over thelattice is defined by the integers I,, IV, I, where I= [/IL, I/L, /ZLI (1=0 is omitted for i=j).Ewald (13) showed that this term could be written as

N N ( oo, erfc(, r,i + 1l2 i=l, j=l Ill1=0 rii +

~q 47r2 [ ..2o(k,.))irL3k.o* kk2 exp 4p2 1J)

3L Niri3L3 F

erfc(z) = complementary error function

2icr

Lk2self-energy term

unit cell dipole term

The second term in this equation is amenable to a fast Fourier transform solution oncethe appropriate grid for evaluation is established. The parameter ,B is chosen to be largeenough so that the minimum image convention can be adopted.

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Equation 2.

In conventional Schrodinger quantum mechanics, the wavefunction can be found from the variational theorem by minimiz-ing (with respect to the parameters) the function (25)

E- fJTt HTtJtdrf ` t tdr

where xVt is the trial wave function and H is the electrostaticHamiltonian. From this function, estimates can be computed forany measurable property of the system. Most modern methodol-ogy involves choosing the wave function as an antisymmetrizeddeterminant of molecular orbitals, which can hold a maximum oftwo electrons and which themselves are linear expansions ofatomic orbitals centered on the atoms of the system. The solu-tion of the Hartree-Fock equations gives the expansioncoefficients. The wave function can then be improved to accountfor the dispersion energy (25) by configuration interaction orperturbation techniques.

In density functions theory (26), the orbital-density descrip-tion of the energy is given by

E --12 IJ(i()l I Oj(1)dT - | rZaP(1) dv2 a Jn21 ,(,)P(2) dv1dv2 + E Xj[P]2LJ rl2

and the equation that is solved is2 Z2-,ra +f|P( v2+Vr1 - Ei ) Oi(l) = 0

where= 3 E.,,[p]/Sp = exchange - correlation potential,

p = i2 electron density, andE-1

E Jc[p] = exchange - correlation energ.

radical (homolytic) attack. Plots of the Fukui function about amolecule can show positions susceptible to electrophilic ornucleophilic attack. The importance and usefulness of the reac-tivity index is demonstrated in Figure 1D. Here we display f(r),which identifies sites in a molecule that are susceptible to attackby soft electrophiles. This is in contrast to hard electrophiles inwhich the principal interaction proceeds under charge control(electrostatic interaction). The density functional computer pro-gram DMOL (Biosym Technologies, Inc., San Diego, CA) wasused to determine the HOMO (highest occupied molecularorbital) and the electron density. This computer package usesnumerical techniques to solve the Kohn-Sham orbital-densityequation (28). The local exchange-correlation functional devel-oped by von Barth and Hedin (29) is used in conjunction with atriple numerical plus double polarization basis set. The self-con-sistent field solutions are required to converge to 108 hartreeand the geometries are optimized until the gradient is near 0.001hartree/bohr. The topographical format for visualizing the possi-ble reactivity sites is generated by displaying an isosurface ofthe electron density (an isodensity value of 0.002 in atomic units)that encloses the van der Waals volumes of the individualatoms. On this surface, we have mapped the positive contribu-tion of the reactivity index onto the isodensity surface and havecolor coded the surface from red (most positive) to blue (zero).All pictures were generated with AVS (Advanced Visual Systems,Inc., Waltham, MD). Figure 1D shows that for azulene, the mostlikely position of attack by an electrophile is the carbon atom inpositions 3 and 5 of the 5-membered ring. This is apparent fromthe topographical display of f(r), and, in this case, the display ofthe HOMO2, in agreement with the experiment.

There is no wave function in DFT; the electron density is every-thing. The orbitals ¢j that result from the solution are used todefine the electron density and do not have the same meaningas in Hartree-Fock theory. Similarly, the orbital energy is not anapproximation for the ionization energy. Powerful interpretiveconcepts have arisen from the development of DFT (26):

X_ E ] = I +A v=IZaL3N 2 ar-ia A

the absolute softness S = 1/ r

and the Fukui function f(r) =[ ]

which is a chemical reativity index. Since p is not a smooth func-tion of N (when N is an integer), the derivative 8pI8N is discontin-uous. For integer N there exist three possible reactivity indices.The left-hand derivative f(r) provides information about sites ina molecule susceptible to attack by electrophiles. The right-handderivative f+(r) is a reactivity index for attack by a nucleophile,and the average of f+(r) and f(r) gives information about free

Figure 1. Reactivity index L(r) for azulene. A, a ball-and-tube model of azulene.Carbon atoms are colored green and hydrogen atoms are white; B, the electrostaticpotential of azulene mapped onto the isosurface of the electron density that justencloses the van der Waals volumes of the atoms, an isodensity value of 0.002. Theresulting surface is colored from most negative (blue) to most positive (red). C, theHOM02 mapped onto the isosurface. The resulting surface is colored from zero (blue)to most positive (red). D, the Fukui function L(r) is mapped onto the isosurface. Theresulting surface is colored from zero (blue) to most positive (red).

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the electronegativity:

the absolute hardness

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DARDEN ET AL.

Table 1. Recent quantum mechanical studies on environmentally pertinent molecules and reactions.

Molecule Reference Methoda Comments

C102 (32)* Extended Cl Quantum mechanical calculations form a basis with which to understandthe photochemistry of chlorine dioxide.

S-trans-l ,3-butadieneHCI02 isomers (5 total)

FOOCI and possibledissociation fragmentsN + 02-> NO +O

CF3+OH-CF3+H20

Formamidine +1-, 2-,3-water molecules

Vinyl bromide decomposition,many channelsCl + HCI

N2+ 02, H2+12Norbornadiene-> quadricyclane

FO + HCI-* HOF+ ClHO + HCI-* H20 + ClRH + Cl-> R + HCI

CH4 + Cl- CH3 + HCI

OH-+ CO2 -* HCO3-

Abstraction reactions of CF30

S03 + H20, or + 2H20

OH + HFCs, OH + HFEs

Electrophilic attack onmonosubstituted benzenes

Chemical reactivity of allorganic compounds

Chemical transition statesof reactions or equilibriumproperties of molecules

(33)* (6-311 G(2d,p)] and [6-31 G*I/MP4(34)* Several high-level HF/MP2,

highest level = CCSD(T)/AN04

(35)* Several high-level HF/MP2,highest level = 6-311 G(2df,2p)

(36)* Basis =five different contractionsof the ANP 14s9p4d3f primitivesCASSCF and MR-SDCI

(38)* MP2/6-31 1 G**QCISD

(39)* MP2/6-31 G(d,p),MP4 single point,CCSD(T) single point; many differentDFT forms

(40)* MP4 single points on 6-31 G(d,p) basis

(41)* Quantum reaction scattering

(42)* Classical trajectory calculation

(43)* QCISD(T)/6-31 1+G(3df,3pd)

(44)* 6-31 G** corrected via Melius (45)and new method

(46)* TZ+2P (very extensive), QCISD(T) andCCSD(T); some DFT also

(47) 6-31 1+G**/ Mp2 and DFT B-LYP

(48)* 6-31 g*/MP4 (FC)SDTQ + higher basis

(49)* 6-311 + G(d,p) with Mp2

(50)* KS DFT

(51)* 3-21G and Pearson parameters

(52)* DFT

(53)* QM/MM

The structure, vibrational spectra, and force constants.The HOOCI isomer is found to be the lowest HF/MP2 energy form on thegeneral potential energy surface using very high-level QM calculations; heatsof formation, vibrational frequencies and intensities are also computed.The stability of FOOCI with respect to the FO + CIO, FOO + Cl, and CIOO + Fdissociation limits is examined by very high-level computations on FOOCI.A novel gridding technique is defined for computing the potential energysurface. The Polyrate code (37) is then used to compute the thermaldependence of the rate constants with good comparison to experiment.Classical transition state barriers were computed for the reaction and then thereaction rate estimated with tunneling corrections using an Eckart barrier;excellent agreement with experiment is observed.The transition state for tautomerization of formamidine is studied as a functionof hydration, conventional basis/method and methods within DFT; the BH&H-LYP method is found to be roughly equivalent to the Mp2 calculations.Classical trajectory calculations on a potential energy surface parameterizedto fit the quantum mechanical starting, transition state, and end orientations.The reaction is studied on two PESs reactive (one excited) to determine thefine structure effect; the overall rate constants are found to be determinedby the ground-state path.Parameterized London-Eyring-Polanyi surfaces are used to describe thereactive surfaces; the reactants are embedded in cold inert gas clusters.The cluster plays a significant role in activation of the reactants on impact.Thermodynamic and activation barriers are computed for these potentiallyimportant atmospheric reactions.Activation barriers are computed for a number of hydrogen abstractionreactions. A simple method is proposed for correcting the energies andfrequencies; transition state calculations give reasonable agreement withtheory once tunneling corrections are made.Similar to previous work, except more extensive basis used; reasonableagreement found at T> 300 K for the calculated rates (transition state theory)and experimental rates.The reaction is studied with several different methods and with solvationcorrections made with two different continuum methods.Hydrogen abstraction by CF30 from CH4, C2H6, and H20 is studied; thetransition state barriers and thermodynamic properties for the gas phasereactions are found.The activation barrier for the formation of sulfuric acid is much lower with twcwater molecules in the transition state complex than with one; in agreementwith experiment.The experimental rate constants compared well with a correlation modelbased on the DFT highest occupied MO energy.A new global reactivity index is proposed based on the Pearson hard andsoft acid theory; the intrinsic parameters are computed from standardwave functions.Absolute hardness, softness, and electronegativity as derived from DFT areargued to be the integrative parameters for describing the chemical behaviorof organic compounds.The newly developed integrated molecular orbital molecular mechanics(IMOMM) method is tested on the reaction of H2 oxidative addition toPt(PR3)2 R=(H,Me,t-Bu, and Ph).

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Abbreviations: GM, quantum mechanical; MM, molecular mechanical; PES, potential energy surface; HF, Hartree-Fock; Cl, configuration-interaction; DFT, density functionaltheory; MPX, Moller-Plesset Theory (x= order of perturbation); CCSD(T), coupled cluster singles, doubles with estimate for triple excitation; ANO, atomic natural orbital basisset; CASSCF, complete active space self-consistent field; MR-SDCI, multireference single and double configuration interaction; QCISD, quadratic configuration interactionsingle and double exitation; BH&H-LYP, Becke half-and-half Lee-Yang-Parr exchange correlation functional; B-LYP, Becke-Lee-Yang-Parr; KS, Kohn-Sham; HFCs, hydrofluoro-carbons; HFEs, hydrofluoroethers. 'Basis of method from Hehre et al. (30) and Frisch et al. (31). *Indicates direct environmental application.

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In this section we will consider newdevelopments in the application of quan-tum mechanics to problems of immediateor potential environmental interest in theareas of single molecule properties and/orreactivity. The goal of Table 1 is to givethe flavor of modern theoretical workwhich has as its lofty aim the computationof accurate reaction rates from knowledgeof only the system molecules. The litera-ture cited is not exhaustive but illustrative.

ConclusionThe future appears bright for the applica-tion of sophisticated computational chem-istry techniques such as those consideredin this review to enviromentally relatedquestions that have a chemical basis. Theunion of molecular mechanics/dynamicsand quantum mechanics is being intenselystudied in several laboratories (53-56).These techniques make possible the intro-duction of time-dependent phenomena

such as polarization and charge exchangeand are thus more realistic in accommo-dating actual electronic behavior; newgeneral tools that make available quantummechanical dynamics should appear soon.The direct application to molecular prob-lems of environmental interest will followin short order.

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74 Environmental Health Perspectives * Vol 104, Supplement * March 1996