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Accurate Molecular ElectrostaticPotentials Based on Modified PRDDOrMWave Functions: III. Extension of thePESP Method for Calculation ofElectrostatic Potential-Derived AtomicCharges to Compounds Containing Li+,Na+, Mg2+, K +, Ca2+, Zn2+, and I

DENNIS S. MARYNICKValerian Software, 3058 Creekview Drive, Grapevine, Texas 76051

Received 16 December 1997; accepted 15 April 1998

Ž .ABSTRACT: The PESP Parameterized ElectroStatic Potential method forcalculating molecular electrostatic potentials, previously parameterized for H, C,N, O, F, P, S, Cl, and Br, is extended to molecules containing Liq, Naq, Mg2q,Kq, Ca2q, Zn2q, and I. For a collection of 166 molecules containing 1668 atomswith at least one metal or iodine atom, PESP achieves an average absolutedeviation in electrostatic potential-derived atomic charges of 0.042ey comparedwith ab initio MP2r6-31GUU calculations, with a correlation coefficient of 0.996.For a larger data set, consisting of 311 molecules encompassing all of the 16

Ž .elements just listed 2488 total atoms , PESP achieves an average absolutedeviation of 0.040ey and a correlation coefficient of 0.995. PESP calculations are

Ž .an order of magnitude faster than the simplest ab initio method STO-3G onlarge molecules, while achieving a level of accuracy that rivals much moreelaborate ab initio methods. Q 1998 John Wiley & Sons, Inc. J Comput Chem19: 1456]1469, 1998

Keywords: electrostatic; potential; PRDDO; PESP

Correspondence to: D. S. Marynick; e-mail: [email protected]

( )Journal of Computational Chemistry, Vol. 19, No. 13, 1456]1469 1998Q 1998 John Wiley & Sons, Inc. CCC 0192-8651 / 98 / 131456-14

ACCURATE ELECTROSTATIC FUNCTIONS. III

Introduction

Ž .he molecular electrostatic potential ESP is aT rigorously defined, real physical property thatcan be calculated directly from the charge distribu-tion or molecular wave function.1] 3 In terms of thewave function, the ESP at the point r is defined as:

U Ž X . Ž X . XZ c r c r drAŽ . Ž .V r s y 1Ý H X< < < <R y r r y rAA

It is now well-recognized4 ] 12 that electrostatic po-tentials play an important role in the analysis, andunderstanding of long range noncovalent interac-tions such as hydrogen bonding, solvation, crystalpacking, and protein]ligand interactions, and es-pecially in the definition of electrostatic potential-derived atomic charges. Because these charges maybe interpreted as the atom-centered monopole ap-proximation to the molecular charge density, theyrepresent one of the most unambiguous and rigor-ous definitions of atomic charge, and have gener-ally been considered to be highly suitable for de-scribing the long-range coulombic interactions inmolecular force fields.

Whereas the calculation of electrostatic-poten-tial-derived atomic charges for small molecules issimple, it is quite computationally intensive forlarge systems. A three-step process is required.First, the wave function must be computed. Sec-ond, the ESP is calculated from the wave functionfor a large number of points in the region outside

Žthe van der Waals envelope of the molecule wherethe monopole approximation to the charge density

.is valid . Third, a set of least squares equations issolved which defines the atomic charges that bestreproduce the quantum-mechanically derived ESPssubject to various constraints. Charges derivedfrom ab initio Hartree]Fock 6-31GU calculationsare generally accepted as being sufficiently accu-rate, even though it is clear that Hartree]Fockwave functions are in general too ionic. Even atthis level of theory, the calculation of wave func-tions for typical molecules of biological interest,with 50 or more atoms, is not a trivial task. For thisreason, a significant amount of work has appearedon the application of semiempirical methods tothis problem. A detailed discussion of the accuracyof various semiempirical methods for the calcula-tion of ESP-derived charges may be found in thefirst part of this series.13 However, it should be

noted that AM1, MNDO, and PM3 ESP-derivedatomic charges exhibit systematic errors whencompared with HFr6-31GU values, and require

Ž .large scaling factors 1.25]1.5 to bring the chargesinto reasonable agreement with the ab initio val-ues. Even with scaling factors, these methods typi-cally produce average absolute errors in the rangeof 0.07]0.10ey compared with HFr6-31GU calcula-tions.14, 15 Furthermore, the best scaling factor maydepend on the nature of the molecule. For in-stance, Merz15 examined a large data set of aminoacids and monosaccharides containing 1007 atoms.He compared charges derived from HFr6-31GU

calculations and various semiempirical ap-proaches. He found much better agreement with

Ž . ŽMNDO wave functions r s 0.96 than AM1 r s. Ž .0.81 or PM3 r s 0.70 , but even the MNDO

Ž .charges required a large scaling factor 1.29 toobtain optimum agreement with the ab initio val-ues. He also demonstrated that the best scalingfactor for MNDO charges is significantly different

Ž .for amino acids 1.26 than for monosaccharidesŽ .1.47 .

Most semiempirical approaches to the calcula-tion of ESP-derived atomic charges suffer fromanother problem: they require analytical evalua-tion of the electrostatic potential, a process thateasily dominates the cpu time requirements of theentire procedure. An encouraging approach to cir-cumventing this problem was recently proposed

Ž . 16, 17by Wang and Ford WF . In this approach, themolecular electrostatic potential is not calculatedanalytically, but is evaluated via new semiempiri-cal expressions that involve only two-center terms.This approach is approximately two orders ofmagnitude faster than methods that employ semi-empirical wave functions and analytic evaluationof the electrostatic potential, while achieving rea-sonable accuracy in both the high and low poten-tial regions surrounding the molecule. For a collec-tion of 21 molecules containing H, C, N, and O

Žatoms, WF found an average absolute error rela-U .tive to HF 6-31G of 0.097e and a correlation

coefficient of 0.963. When a uniform scaling factorof 1.208 was applied to the calculated charges, theerror was reduced to 0.073e. A second, very recentapproach to the rapid calculation of ESP-derivedatomic charges at the semiempirical level is theVESPA method of Beck, et al.18 In this approach,the ESP is evaluated directly from a semiempiricalwave function, but the charge distribution of eachheavy atom is represented by nine point chargesŽ .one of which is atom-centered and the chargedistribution of each hydrogen is represented by a

JOURNAL OF COMPUTATIONAL CHEMISTRY 1457

MARYNICK

single point charge. The ESP calculation is there-fore reduced to a trivial double sum that is ex-tremely fast. For 26 molecules containing H, C, O,

ŽN, S, and P, VESPA yielded the best results rela-U .tive to HFr6-31G calculations with AM1 wave

functions, where the overall correlation coefficientŽwas 0.934 and the average absolute deviation not

reported by them, but calculated here on the basis. yof their published data was 0.096e . Thus, both of

these semiempirical methods represent very effi-cient approaches to the calculation of ESP-derivedcharges, although they still suffer from significantinaccuracies likely to make them unacceptable forcalculation of high-quality, long-range electrostat-ics. In addition, neither of these methods havebeen evaluated for their ability to reproduce atomiccharges from correlated wave functions. The im-portance of electron correlation has been madeapparent in a recent study by Wampler,19 whoshowed that electron correlation accounts for abouta 10% change in the ESP-derived charges. This isnot a trivial effect, because ESP-derived chargesare often quite large.

In part I of this series,13 a new approach for thecalculation of ESP-derived charges was presented.

ŽDenoted PESP Parameterized ElectroStatic Po-.tential , this method is based on modified

PRDDOrM wave functions20 ] 25 and parameter-ized against correlated wave functions employingthe 6-31GUU basis set. Intermediate in computa-tional complexity between a fully ab initio and afully semiempirical approach, PESP is an order ofmagnitude faster than the simplest ab initio ap-

Ž .proach STO-3G , while achieving a degree of ac-curacy approaching that of the reference calcula-

Žtions which it is parameterized against ab initioUU .MP2r6-31G . Unlike previous approximate

methods, PESP achieves high accuracy through adirect parameterization of the wave function and

Ž .analytic but rapid calculation of the resultantelectrostatic potentials. Part I of this series de-scribed the PESP parameterization for organic andinorganic species containing H, C, N, O, F, P, S, Cl,and Br, including hypervalent species for P, S, andCl. In addition, the algorithm for the rapid near-analytic evaluation of the ESP over Slater orbitalswas described. An overall unscaled average abso-lute deviation relative to MP2r6-31GUU values of0.037ey for 820 symmetry unique atoms in 145molecules was achieved, with a correlation coeffi-cient of 0.990. In part II, this approach was ex-tended26 to the calculation of molecular electro-

Žstatic potentials in the high potential regions in-.side the molecular van der Waals envelope . It was

shown that simple modifications of the PESP pro-cedure result in highly accurate representations ofthe ESP inside the van der Waals envelope. Forinstance, the location and depth of lone-pair min-

Žima can be calculated with an average error rela-UU ˚.tive to MP2r6-31G of 0.03 A and 2.5 kcalrmol,

respectively. Furthermore, PESP maps of themolecular electrostatic potential are sometimesnearly indistinguishable from their MP2r6-31GUU

counterparts.Although the PESP method represents a signifi-

Žcant advancement in accuracy relative to semiem-.pirical approaches and in computational speed

Ž .relative to ab initio methods it still is not applica-ble to a broad enough range of elements to beuseful as a general tool for the exploration ofbiomolecular electrostatics. In particular, to be use-ful for protein electrostatics, PESP must be able tocalculate high-quality electrostatics for species con-taining a variety of metal ions. In this study, thePESP method is extended to molecules containingLiq, Naq, Mg2q, Kq, Ca2q, Zn2q, and I. For acollection of 166 molecules containing 1668 atomswith at least one metal or iodine atom, PESP

Ž .achieves without scaling an average absolute de-viation in ESP-derived charges of 0.042ey com-pared with ab initio MP2r6-31GUU values, with acorrelation coefficient of 0.996. For a larger data setconsisting of 311 molecules, taken from part I andthe current work, and encompassing all 16 ele-

Ž .ments just listed 2488 total atoms , PESP repro-duces the ab initio ESP-derived charges with anoverall average unscaled absolute deviation of0.040ey and a correlation coefficient of 0.995.

PRDDO/M and PESP Methods

A full description of the PRDDOrM methodmay be found in the original references.20 ] 25

PRDDOrM is an approximate molecular orbitalapproach designed to reproduce ab initio mini-mum basis set calculations with a Slater orbitalbasis set at a fraction of the computational cost.Because it is parameterized against ab initio calcu-lations and not experiment, PRDDOrM may beregarded as a nonempirical approach. The essenceof the PRDDOrM method may be summarized asfollows. First, almost all one-, two-, and three-center integrals are calculated explicitly to at leastthree-decimal-place accuracy. Second, the integralsare transformed to a Lowdin27 orthogonalized ba-¨sis. In this basis, four-function two-electron inte-

VOL. 19, NO. 131458


Ž < .grals of the form ij kl are small and may beneglected without significant loss of accuracy. Thisreduces the formal computational dependence ofthe method from n4 to n3. However, for computa-tional efficiency, the transformation from theatomic orbital basis to the Lowdin basis is not¨

Ž .done exactly: certain nonzero but small terms areneglected. This requires the introduction of a small

Ž .set four or five per atom of parameters, whichare derived by comparison to ab initio calculations,to correct the results to the full ab initio values.Unlike semiempirical methods, PRDDOrM is onlyweakly parameterized. Elimination of all parame-ters still results in reasonable molecular propertieswhen compared with the reference ab initio results.Indeed, PRDDOrM currently contains no parame-ters for the first transition metal series, but hasbeen used quite successfully to describe structureand bonding in a wide range of transition metalcomplexes.28

The essence of the PESP modifications to thePRDDOrM method is extremely simple. A PESP

Žcalculation is simply a PRDDOrMrFCP frozen-.core potential calculation with a new set of pa-

rameters chosen to minimize an error functionbased on ESP-derived atomic charges:

< ab i nit i o P ES P < Ž .« s q y q 2Ý i ii

For the elements in part I, the parameters con-sisted of the orbital exponents and two additional

Žvariables, which modified in a multiplicative fash-. Ž < 2 .ion two-center integrals of the form x x x1sH jA jA

Ž 2 < 2 .and one-center integrals of the form x x ,jA jAwhere j runs over a set of valence s and p orbitals.In this article, a similar treatment was used foriodine, where only the orbital exponents weretaken as free parameters. For the metals, however,it was found that varying only the orbital expo-nents did not yield satisfactory results, and thatmodifications of the above mentioned two-electronintegrals were ineffective in minimizing the errorfunction. Therefore, a new set of parameters had tobe developed.

Metal systems, even as simple as alkali metalcomplexes, proved to be a challenge for effectiveparameterization. The basic problem is easily un-derstood: PRDDOrM, like any ab initio or near abinitio minimum basis set method, overestimatesthe importance of the valence s and p orbitals onthe metal, and thus exaggerates metal]ligand co-valent bonding. Parameterizations that includeonly the orbital exponents compensate for this

effect by contracting the valence s]p shell. How-ever, this results in a metal ion with a far-too-highcharge:radius ratio, which in turn produces exag-gerated polarization effects on the ligands. Afterconsiderable experimentation, it was found thatthis problem could be dealt with by parameteriz-ing two center nuclear attraction integrals of the

Ž 2 2 < .form x x 1rR . Here B is a metal center,iA jA 1Band i and j run over the valence s and p orbitalsof atom A. Details of the computational proce-dures are discussed next.

Computational Details

For each atom, a reference parameterization setŽ . Ž .RPS of molecules was generated Tables I]VII .Geometries for these molecules were obtained fromHartree]Fock-level optimizations. Geometry opti-mizations were performed on an IBM RS-6000Model 250 computer using the program GAMESS.29

Most optimizations were performed at the 6-31GU

Ž . 30 31 Ž .6d level or SBK with NDFUNC s 1 as de-fined in GAMESS; however, some molecules wereoptimized at the 3-21G level32 in an effort to makethe parameterization largely independent of thetheoretical model used to obtain the geometry. Tomake the ab initio calculations tractable, simpleligands had to be employed in all of the metalcomplexes. This also had the advantage of keepingthe metal ion from being too buried in the molecu-lar interior to have well-defined atomic chargesŽ .see subsequent text . The ligands employed in-cluded H O, H N, H P, H S, CO, the formate2 3 3 2anion, formaldehyde, pyridine, OHy, HF, HCl,

Žand HBr the latter three as models of alkylhalide.systems . A range of coordination numbers, appro-

priate for each metal, was included.All ab initio ESP-derived charges were obtained

at the MP2r6-31GUU level using Gaussian-9433 onan NEC SX-4 computer. For all atoms except Zn,K, Ca, and I, the standard 6-31GUU basis set30 wasused. For K and Ca, the TZV basis set of Schafer,et al.34 was employed, supplemented with one

Ž .additional p and d function exponents s 0.10 .The same style basis set was used for Zn, exceptthat it was supplemented by two p functions, withexponents of 0.1430 and 0.0508. For iodine, theSBKJC35 basis set was used, with an additional dfunction having an exponent of 0.266. The as-sumed symmetries and the basis set employed forthe geometry optimizations of each molecule arelisted in Tables I]VII.


MARYNICK

TABLE I.Molecules in Lithium RPS.a,b

+ + c + +( ) ( ) ( ) ( ) ( ) ( ) ( )Li NH OCH Li NH Py Li NH Li H O NH3 3 2 3 3 3 4 2 2 3 2+ + + +( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )Li H O NH Li H O ClH Li H O PH Li H O SH2 2 3 4 2 2 2 2 2 3 2 2 2 2 2

+ + + +( )( ) ( ) ( ) ( ) ( ) ( ) ( )Li H O NH Li H O HCO Li H O ClH Li H O FH2 3 3 2 3 2 2 3 2 3+ + + d + c( ) ( ) ( ) ( ) ( ) ( ) ( )Li H O SH Li H O Li H O Py Li H O NH2 3 2 2 4 2 4 2 5 3

+ c + c + c( ) ( ) ( ) ( ) ( ) ( )Li H O ClH Li H O FH Li H O PH2 5 2 5 2 5 3

a Optimized at the HFr6-3 1GU level unless otherwise indicated.b All molecules are C symmetry unless otherwise indicated.sc Geometry calculated at the HFr3-21G level.d C symmetry.2v

TABLE II.Molecules in Sodium RPS.a,b

+ c + +( ) ( ) ( ) ( )( )) ( ) ( )Na NH Py Na NH Na NH H O Na H O HCO3 3 3 4 3 2 5 2 2 2+ c + + +( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )Na H O NH Na H O ClH Na H O FH Na H O PH2 3 3 2 3 2 3 2 3 3+ + + c + c( ) ( ) ( ) ( ) ( ) ( ) ( )Na H O SH Na H O Na H O PH Na H O SH2 3 2 2 4 2 4 3 2 2 4 2 2

+ c + c + c + d, e( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )Na H O ClH Na H O ClH Na H O FH Na H O FH2 5 2 5 2 5 2 5+ c + e + f( ) ( ) ( ) ( ) ( )Na H O PH Na H O Py Na H O2 5 3 2 5 2 6

a Optimized at the HFr6-31GU level unless otherwise indicated.b All molecules are C symmetry unless otherwise indicated.sc Geometry calculated at the HFr3-21G level.d ( )Geometry calculated at the HFrSBK + d level see text .e C symmetry.2vf D symmetry.2 h

TABLE III.Molecules in Magnesium RPS.a,b

2+ 2+ c 2+ 2+( ) ( ) ( ) ( ) ( ) ( )( )Mg NH OCH Mg NH Py Mg NH Mg NH H O3 3 2 3 3 3 4 3 2 52+ 2+ 2+ 2+( ) ( ) ( ) ( ) ( ) ( )Mg PH Mg SH ClH Mg SH Mg H O ClH3 4 2 2 2 2 4 2 2 2

2+ 2+ 2+ 2+( ) ( ) ( ) ( )( ) ( ) ( ) ( )Mg H O FH Mg H O FH ClH Mg H O Mg H O PH2 2 2 2 2 2 4 2 4 3 2+ 2+ 2+ +( ) ( ) ( ) ( ) ( ) ( ) ( )Mg H O Cl Mg H O ClH Mg H O FH Mg H O HCO2 3 2 3 2 3 2 3 2

2+ 2+ 2+ 2+( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )Mg H O PH Mg H O SH Mg H O NH Mg H O NH2 3 3 2 3 2 2 4 3 2 2 3 32+ e 2+ 2+ c 2+ c( ) ( ) ( )( ) ( ) ( ) ( ) ( )Mg H O Mg H O FH ClH Mg H O ClH Mg H O FH2 6 2 4 2 5 2 5

2+ 2+ c, d 2+ c, d 2+( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )Mg H O PH Mg H O PY Mg H O SH Mg H O SH PH2 5 3 2 5 2 5 2 2 2 2 3

a Optimized at the HFr6-31GU level unless otherwise indicated.b All molecules are C symmetry unless otherwise indicated.sc Geometry calculated at the HFr3-21G level.d C symmetry.2ve D symmetry.2 h

TABLE IV.Molecules in Potassium RPS.a,b

+ + +( ) ( ) ( ) ( ) ( ) ( ) ( )( )K NH OH K H O SH K H O ClH K H O PH3 5 2 2 2 2 2 3 2 3 3+ + + + c( ) ( ) ( ) ( ) ( )( ) ( ) ( )K H O PH K H O NH K H O PH K H O PH2 3 3 2 2 4 3 2 2 3 4 2 4 3 2

+ d + d + d + d( )( ) ( ) ( ) ( ) ( ) ( )K H O SH K H O K H O ClH K H O FH2 2 4 2 5 2 5 2 5d + d f f( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )K H O HCO K H O OCH K H O OH K H O OH2 5 2 2 5 2 2 5 2 5

+ c + d + c + g( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )K H O PH K H O Py K H O SH K H O NH2 5 3 2 5 2 5 2 2 6 3+ + g( ) ( )K H O K H O2 7 2 8

a ( )Optimized at the HFrSBK + d level unless otherwise indicated see text .b All molecules are C symmetry unless otherwise indicated.sc Geometry calculated at the HFr3-21G level.d C symmetry.2ve D symmetry.2 hf Two different structures with C symmetry were located for this species.sg C symmetry.1

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TABLE V.Molecules in Calcium RPS.a,b

2+ 2+ 2+ +( ) ( ) ( ) ( ) ( ) ( )Ca NH Ca NH Ca H O ClH Ca H O HCO3 4 3 6 2 3 2 2 3 22+ 2+ 2+ 2+( ) ( ) ( ) ( ) ( ) ( ) ( )Ca H O PH Ca H O Ca H O NH Ca H O ClH2 3 3 2 4 2 4 3 2 4 22+ 2+ 2+ 2+ c( ) ( ) ( ) ( ) ( ) ( ) ( )Ca H O SH Ca H O Ca H O NH Ca H O ClH2 4 2 2 2 5 2 5 3 2 5

2+ c + c 2q c +( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )Ca H O FH Ca H O HCO Ca H O OCH Ca H O OH2 5 2 5 2 2 5 2 2 52+ 2+ c 2+ c 2+ d( ) ( ) ( ) ( ) ( ) ( ) ( )Ca H O PH Ca H O Py Ca H O SH Ca H O2 5 3 2 5 2 5 2 2 6

a ( )Optimized at the HFrSBK + d level see text .b All molecules are C symmetry unless otherwise indicated.sc C symmetry.2vd D symmetry.2 h

TABLE VI.Molecules in Zinc RPS.a,b

2+ + 2+ d( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )Zn NH CH Zn NH H O Zn NH CH Zn NH H O SH3 2 3 2 3 2 2 2 3 3 3 3 2 2 2+ 2+ 2+ +( ) ( ) ( ) ( ) ( ) ( ) ( )Zn NH OH Zn NH H O Zn NH Zn NH OH3 3 3 3 2 3 4 3 42+ + 2+ c +( )( ) ( )( ) ( ) ( )Zn NH H O Zn CO H O Cl Zn CO Zn H O Cl3 2 3 2 2 4 2 32+ + 2+ c 2+( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )Zn H O FH Zn H O HCO Zn H O SH Zn H O BrH2 3 2 3 2 2 3 2 2 2 32+ 2+ 2+ 2+( ) ( ) ( ) ( ) ( ) ( ) ( )Zn H O ClH Zn H O PH Zn H O Zn H O NH2 3 2 3 3 2 4 2 4 32+ 2+ 2+ c 2+ c( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )Zn H O NH Zn H O PH Zn H O SH Zn H O SH2 4 3 2 2 4 3 2 2 4 2 2 4 22+ + c 2+ c 2+ c( ) ( ) ( ) ( ) ( ) ( ) ( )Zn H O NH Zn H O Cl Zn H O ClH Zn H O FH2 5 3 2 5 2 5 2 5

2+ c 2+ c 2+ c 2+( ) ( ) ( ) ( ) ( ) ( ) ( )Zn H O BrH Zn H O OCH Zn H O Py Zn H O2 5 2 5 2 2 5 2 6

a Optimized at the HFrSBK + d level.b All molecules are C symmetry unless otherwise indicated.sc C symmetry.2vd C symmetry.1e T symmetry.d

TABLE VII.Molecules in Iodine RPS.a

1,2-Bromoiodoethylene CH I CI COOH CI H3 3 3( ) ( )CI H CICl 1,1-Chloroiodoethylene C CH I OH4 2 3 2

( )1,1-Fluoroiodoethylene HC CH I HCOI HI3 21,1-Diiodoethylene HOI Iodobenzene CBrIH2Iodoacetylene ICN 2-Iodofuran p-IodopyridineNH I OI PH I ISH2 2 21,2-Bromoiodoethane Bromoiodoacetylene

a ( )Optimized at the HFrSBK + d level see text .

The ESP-derived charges at both the ab initioand the PESP levels were calculated using theMerz]Kollman algorithm.14, 36 ] 38 This method cal-culates the ESP at a set of points distributed onatom-centered spheres with radii that are some

Ž .multiples typically 1.4, 1.6, 1.8, and 2.0 of the vander Waals radius of the atom. Because recom-mended van der Waals radii for the Merz]Koll-man algorithm appear not to be available for K,Ca, and I, they were determined in this work.Although it is possible to estimate appropriateradii based on literature values or periodic trends,a more practical approach was taken here. For

Ž . Ž .q Ž .q Ž . Ž .2qK H O NH , K H O , Ca H O NH ,2 4 3 2 2 6 2 4 3 2Ž .2q Ž .Ca H O , CH I, H CO I, OI , and iodobenzene,2 6 3 2

the ab initio ESP-derived atomic charges were de-termined as a function of the assumed van derWaals radius. A radius was then selected for eachatom from plots of the calculated charge for theatom in question versus the assumed van derWaals radius. These plots are shown in Figure 1for potassium and calcium and in Figure 2 foriodine. The radius chosen, indicated by a verticalline, corresponds approximately to the value forwhich the calculated charge is least sensitive to theassumed radius. For potassium and calcium, any


MARYNICK

FIGURE 1. Plot of ESP-derived charges for( ) ( )+ ( ) ( )+ ( )K H O NH circles , K H O squares ,2 4 3 2 2 6

( ) ( )2+ ( ) ( )2+ (Ca H O NH triangles , and Ca H O inverted2 4 3 2 2 6)triangles as a function of assumed van der Waals radius.

( )The vertical line indicates the radius chosen see text .

˚value in the range of 1.0]1.4 A is acceptable, but,˚for simplicity, a common value of 1.24 A was

chosen. For iodine, the calculated charges are quiteinsensitive to the assumed radius over a wide

˚range, and a value of 2.5 A was chosen. All ofthese calculations were done at the Hartree]Focklevel, with the standard GAMESS TZV q d basisset39 for potassium- and calcium-containing mole-cules and the SBK q d basis set for the moleculescontaining iodine.40

Ž .PRDDOrM and, therefore, PESP is currentlyavailable only for the elements H]Ar and Sc]Kr.

FIGURE 2. Plot of ESP-derived charges for CH I3( ) ( ) ( )squares , iodobenzene inverted triangles , H CO I( ) ( )triangles , OI circles as a function of assumed van2der Waals radius. The vertical line indicates the radius

( )chosen see text .

The elements K and Ca are not implemented in themethod, because their unique orbital structure inthe m inim um basis set approxim ation,

Ž1s2 sp3sp4 sp as opposed to 1s2 sp3sp3d4 sp for.the remainder of the fourth row elements , has not

yet been programmed. Similarly, PRDDOrM cal-culations on iodine are not yet possible, becauseanalytic two-center coulomb integrals have beenprogrammed only through n s 4. However, it isstill possible to generate PESP parameters for K,Ca, and I. PESP is a valence-electron-only method,

Žand uses frozen-core potentials constructed from

TABLE VIIIA.Optimized Parameters.

Nuclear attraction parameters

N]F P]Cl

P C C C P C C Css sp pp ss sp pp

aLi 3.000 2.341 y8.758 0.824 } } } }Na 4.080 3.418 y1.418 2.010 } } } }

aMg 4.682 2.839 y9.281 0.753 3.000 } 10.905 }a bK 3.000 2.310 } = } } } }

Ca 2.992 1.949 y0.241 1.174 } } } }Zn 5.864 2.421 y3.531 0.993 } } } }

a ( )Constrained see text .b C = C .sp ss

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TABLE VIIIB.Optimized Parameters.

Exponents

1s 2sp 3sp 4sp 3d

Li 2.487 1.896 } } }aNa U 2.338 2.752 } }aMg U 2.894 2.832 } }

b aK U 3.717 1.757 } }c aCa U 5.668 1.788 } }

a aZn 17.076 U U 3.543 3.748d aI 5.726 10.220 8.093 2.692 U

a Unmodified.b ( )Treated as pseudo-Na atom see text .c ( )Treated as pseudo-Mg atom see text .d ( )Treated as pseudo-Br atom see text .

the core orbital electron repulsion integrals andcontaining the appropriate projection opera-

23 ] 24.tors . Thus, in the frozen-core approximation,potassium can be treated as sodium, calcium asmagnesium, and iodine as bromine, simply bysubstituting a different core potential and arescaled valence shell exponent. Because the corepotential is derived from the electron repulsionintegrals and the orbital exponents are taken asfree parameters in the electrostatic potential fits,the appropriate core potentials and valence shellexponent are found automatically during the mini-mization of the error function. This procedure

Ž .works extremely well see subsequent text .The parametric form for a s]s nuclear attraction

integral is:

1 1S S s S SA A A Až / ž /R R1 B 1 BP A R A E X ACT

Ž Ž rycr adA. . Ž .? 1 q C P 3ss

where crad is the covalent radius of atom A, r isAthe internuclear distance, and atom B is a metalcenter. This correction was applied to all nonzerocomponents of the integrals in a local diatomiccoordinate system, before rotation to the generalcoordinate system. For s]p and p]p components,a common value of P was used, and only one

Ž .additional parameter per integral C and Cs p p pwas introduced. In all elements, except magne-sium, parameterization was restricted to M]N,M]O, and M]F interactions and, in some cases,

Ž . Ž .the parameter C m, n s s or p in eq. 3 wasm nset to zero if the parameter did not strongly affectthe calculated error function. Parameter P wasnever allowed to take on a value of less than 3.0.

This insures the correct long-range behavior of theŽmodified nuclear attraction terms. An exception is

Ca, for which the optimized value for P of 2.992.was employed.

Only the charges of symmetry-unique atomswere considered, and the parameters previouslydetermined13 for H, C, N, O, F, P, S, Cl, and Brwere used without change. A simplex procedurewas used for optimization of the parameters.

Results and Discussion

The number of molecules in the ab initio RPSdepended in part on the number of independent

Ž .parameters optimized, and varied from 19 Li toŽ .32 Zn . The final parameters are reported in Table

VII, and the statistical results are summarized inTables IX]XV. For each element in each data set,the number of atoms, the average absolute chargeat the MP2r6-31GUU level, the average absolutedeviation and the average signed deviation aretabulated. In addition, the overall average devia-tion for each data set is reported, as is the correla-tion coefficient and the linear least-squares param-

Žeters m and b taking the ab initio values as the.dependent variable .

The results for the metal complexes may bediscussed as a group, because the general behaviorof the method is similar for all metal ions. Hydro-gen charges are very accurately determined, withthe largest average absolute deviation being only

y Ž . y Ž .0.026e Zn and the smallest being 0.016e Ca .This was also true for the molecules from part IŽ ywhere an average absolute deviation of 0.019e

.was found and is a reflection of the fact that


MARYNICK

TABLE IX.Statistical Results for Lithium RPS.

Avg. abs. Avg. abs. Avg. signedAtom Number charge deviation deviation

H 115 .376 .024 .000Li 19 .847 .090 y.003C 10 .185 .049 y.027N 16 .934 .079 y.008O 41 .913 .052 .022F 2 .514 .022 y.022P 3 .111 .179 y.179S 3 .402 .035 y.015Cl 3 .295 .019 .007

All 212 .043

No. of molecules = 19; r = 0.994; m = 1.008; b = 0.001.

TABLE X.Statistical Results for Sodium RPS.


H 119 .385 .023 .003C 9 .187 .041 .002N 9 .856 .052 y.033O 51 .923 .042 .003F 3 .532 .041 .011Na 19 .939 .061 y.001P 4 .229 .101 y.101S 3 .449 .013 .011Cl 3 .327 .056 .056

All 220 .035


TABLE XI.Statistical Results for Magnesium RPS.


H 155 .430 .023 y.008C 10 .150 .048 y.045N 13 .974 .063 y.023O 55 1.002 .042 .011F 6 .598 .041 y.041Mg 28 1.435 .133 .072P 8 .217 .084 y.084S 7 .313 .062 .031Cl 9 .221 .062 .040

All 291 .044


VOL. 19, NO. 131464


TABLE XII.Statistical Results for Potassium RPS.


H 158 .378 .022 .005C 5 .439 .142 .121N 8 1.007 .044 y.028O 78 .880 .048 y.002F 1 .570 .035 .035P 7 .265 .079 y.079S 3 .467 .034 .034Cl 2 .331 .050 .050K 22 .933 .036 .003

All 284 .035


hydrogen is the most common atom in both datasets, and is thus weighted more heavily in theerror function minimization. In addition, the metalcomplexes discussed here all have hydrogens onthe outside, whereas heavy atoms are buried in theinterior. This naturally leads to less well-de-termined charges for the interior atoms, becausethe ESP is sampled much more densely around theexterior atoms.41 Errors for nonhydrogen atomsare uniformly higher than in part I, but this is inpart due to the much higher absolute chargesfound in the metal complexes as opposed to thenonmetal systems dealt with earlier. If fact, if the

Ž . Žratio R s average absolute error r average abso-.lute charge is used as a measure of accuracy, the

charges for H, O, F, and S are significantly better

determined in the metal complexes than in themolecules in part I, whereas the errors are aboutthe same for N and Cl. Only C, P, and Br showsomewhat larger errors in the metal systems. It isgratifying that N and O exhibit reasonable behav-ior in these metal complexes, because they are byfar the most common atoms involved in bondingto metal ions in biological systems. For the metals

Ž .themselves, the ratio R defined earlier variesŽ . Ž .from 3.8% K to 10.6% Li and averages 7.4%.

Ž .The statistical results for iodine Table XV aresimilar in all respects to other halogens parameter-ized earlier.13 The average absolute error for io-dine in the iodine RPS is only 0.025ey, althoughthe R ratio is 20%, reflecting the fact that theaverage iodine charge is very small.

TABLE XIII.Statistical Results for Calcium RPS.


H 115 .457 .016 .001C 6 .366 .079 y.046N 9 1.051 .044 y.011O 56 .991 .062 y.016F 1 .681 .062 y.062P 2 .543 .029 .013S 3 .611 .065 .049Cl 4 .334 .034 .034Ca 20 1.744 .089 .028

All 216 .039



MARYNICK

TABLE XIV.Statistical Results for Zinc RPS.


H 169 .438 .026 y.002C 10 .426 .113 .041N 23 1.013 .089 y.077O 66 .923 .050 .005F 2 .609 .032 .032P 3 .283 .112 y.112S 4 .312 .118 y.065Cl 5 .386 .154 .141Zn 32 1.314 .119 .068Br 2 .273 .170 y.170

All 316 .053

No. of molecules = 32; r = 0.994; m = 0.981; b = y0.001.

The overall statistical results for all 166molecules in this article are reported in Table XVI.PESP achieves an overall absolute average devia-tion of 0.042ey, with a correlation coefficient of0.996. The linear least-squares parameters are m s0.991 and b s 0.003, demonstrating that scaling isnot required to bring PESP charges into optimalagreement with the underlying ab initio results.The use of a scaling factor is always a disadvan-tage, because one must assume a universal scalingfactor for all molecules, and there is good evidencethat such an assumption is unwarranted.15 TableXVII shows a similar statistical analysis, including

all of the molecules from part I and the currentwork. For 311 molecules containing 2488 atoms,PESP produces an average absolute error of0.040ey and a correlation coefficient of 0.995. Therange of ESP-derived charges for this data set isq2.04 to y1.53ey. A plot of this entire data set isshown in Figure 3.

Detailed timing statistics for several moleculeswere presented in part I of the series.13 Here, one

Ž . Žtypical example is presented. For Mg OH pyri-2 5.2qdine on an HP-735 workstation, PESP requires

19.4 seconds of cpu time for the calculation of thewave function and the ESP-derived atomic charges.

TABLE XV.Statistical Results for Iodine RPS.


H 42 .192 .024 .003C 39 .374 .054 .017N 3 .549 .059 y.021O 7 .378 .035 y.033F 1 .081 .001 .001P 1 .096 .076 y.076S 1 .169 .000 .000Cl 2 .010 .016 y.013Br 4 .072 .035 y.002I 29 .124 .022 y.009

All 117 .034

No. of molecules = 26; r = 0.990; m = 1.004; b = y0.009.

VOL. 19, NO. 131466


TABLE XVI.Statistical Results for All Molecules in Tables I–VII.


H 873 .401 .023 .000Li 19 .847 .090 y.003C 89 .318 .065 y.006N 81 .960 .068 y.035O 354 .925 .049 .002F 16 .548 .036 y.014Na 19 .939 .061 y.001Mg 28 1.435 .133 .072P 28 .245 .094 y.091S 24 .391 .056 .008Cl 28 .279 .065 .052K 22 .933 .036 .003Ca 20 1.744 .089 .028Zn 32 1.314 .119 .068Br 6 .139 .078 y.043I 29 .124 .025 y.009

All 1668 .042


TABLE XVII.Statistical Results for All Molecules.a


H 1172 .351 .022 .000Li 19 .847 .090 y.003C 307 .323 .059 y.005N 143 .822 .060 y.013O 463 .813 .045 .003F 51 .274 .030 y.014Na 19 .939 .061 y.001Mg 28 1.435 .133 .072P 49 .321 .083 y.056S 46 .386 .062 y.001Cl 61 .233 .052 .028K 22 .933 .036 .003Ca 32 1.314 .119 .068Br 27 .107 .040 y.015I 29 .124 .022 y.009

All 2488 .040

No. of molecules = 311; r = 0.995; m = 0.992; b = 0.001.a ( )Includes all molecules from part I see text .


MARYNICK

FIGURE 3. Plot of PESP atomic charges vs.ESP-derived atomic charges from ab initio MP2 / 6-31GUU

wave functions for all 2488 atoms in the data sets of thepresent work and part I.13

The corresponding times using Gaussian-94 areŽ .203 seconds HFrSTO-3G and 3924 seconds

Ž U . UUHFr6-31G . Calculations at the MP2r6-31Glevel would, of course, require much more cputime.

Conclusions

The PESP method has been extended to themetal ions Liq, Naq, Mg2q, Kq, Ca2q, and Zn2q. Itrepresents the first non-ab initio approach to beapplicable to a wide variety of elements and com-parable in accuracy to much more expensive abinitio methods.

Extensions of this method are now in progress.In particular, modifications are under develop-ment that should result in even more accuratecharges, especially for biological systems. This willbe accomplished by parameterizing against a new,larger data set with extensive additions of aminoacids, sugars, and biologically important anions. Inaddition, an approach for obtaining ESP-derivedcharges for whole proteins based on the PESPmethod has now been developed.42

Acknowledgments

The author thanks the Swiss Center for Scien-tific Computing and Dr. Djordje Maric for a gener-ous grant of cpu time required for the calculationof the ab initio ESP-derived charges at the MP2level.

References

1. E. Scrocco and J. Tomasi, Adv. Quantum Chem., 11, 115Ž .1978 .

2. P. Politzer and J. S. Murray, in Reviews of ComputationalChemistry, Vol. 2, K. B. Lipkowitz and D. B. Boyd, Eds.,VCH, New York, ch. 7.

3. P. Politzer and D. G. Truhlar, Eds., Chemical Applications ofAtomic and Molecular Electrostatic Potentials, Plenum, NewYork, 1981.

4. G. Naray-Szabo, Chemical Design Automation News, 8, 43´ ´Ž .1993 .

´5. J. G. Angyan and G. Naray-Szabo, in Theoretical Models of´ ´ ´Chemical Bonding, Part 4, Theoretical Treatment of LargeMolecules and Their Interactions, Z. B. Maksic, Ed., Springer,Berlin, 1991, p. 1.

6. J. E. Douglas and P. A. Kollman, J. Am. Chem. Soc., 102,Ž .4295 1980 .

7. G. Pepe, D. Siri, and J.-P. Reboul, J. Mol. Struct. Theochem,Ž .14, 289 1981 .

8. H. Weinstein, R. Osman, and J. P. Green, in ComputerŽ .Assisted Drug Design ACS Symposium Series 112 , E. C.

Olson and R. E. Christoffersen, Eds., American ChemicalSociety, Washington, DC, 1979.

Ž .9. F. A. Momany, J. Phys. Chem., 85, 592 1978 .10. S. R. Cox and D. E. Williams, J. Comput. Chem., 2, 304

Ž .1981 .11. C. M. Breneman and K. B. Wiberg, J. Comput. Chem., 11, 361

Ž .1990 .12. C. I. Bayly, P. Cieplak, W. D. Cornell, and P. A. Kollman, J.

Ž .Phys. Chem., 97, 10269 1993 .Ž .13. D. S. Marynick, J. Comput. Chem., 18, 955 1997 .

14. B. H. Besler, K. M. Merz Jr., and P. A. Kollman, J. Comput.Ž .Chem., 11, 431 1990 .

Ž .15. K. M. Merz Jr., J. Comput. Chem., 13, 749 1992 .Ž .16. G. P. Ford and B. Wang, J. Comput. Chem., 14, 1101 1993 .

Ž .17. B. Wang and G. P. Ford, J. Comput. Chem., 15, 200 1994 .18. B. Beck, T. Clark, and R. C. Glen, J. Comput. Chem., 18, 744

Ž .1999 .Ž .19. J. E. Wampler, J. Chem. Inf. Comput. Sci., 35, 617 1995 .

20. T. A. Halgren and W. N. Lipscomb, J. Chem. Phys., 58, 1569Ž .1973 .

21. T. A. Halgren and W. N. Lipscomb, Proc. Natl. Acad. Sci.Ž .USA, 69, 652 1972 .

22. D. S. Marynick and W. N. Lipscomb, Proc. Natl. Acad. Sci.Ž .USA, 79, 1341 1982 .

23. A. Derecskei-Kovacs and D. S. Marynick, Int. J. QuantumŽ .Chem., 58, 193 1996 .

VOL. 19, NO. 131468


24. A. Derecskei-Kovacs, D. E. Woon, and D. S. Marynick, Int.Ž .J. Quantum Chem., 61, 67 1997 .

25. A. Derecskei-Kovacs and D. S. Marynick, Int. J. QuantumŽ .Chem., 63, 1081 1997 .

Ž .26. D. S. Marynick, J. Comput. Chem., 18, 1682 1997 .Ž .27. P.-O. Lowdin, J. Chem. Phys., 18, 365 1950 .¨

28. See, for instance: F. U. Axe and D. S. Marynick,Ž .Organometallics, 6, 572 1987 ; F. U. Axe and D. S. Marynick,

Ž .J. Am. Chem. Soc., 110, 3728 1988 ; L. M. Hansen and D. S.Ž .Marynick, J. Am. Chem. Soc., 110, 2358 1988 ; C. A. Jolly

Ž .and D. S. Marynick, J. Am. Chem. Soc., 111, 7968 1989 ;L. M. Hansen and D. S. Marynick, Organometallics, 8, 2173Ž .1989 ; L. M. Hansen and D. S. Marynick, Inorg. Chem., 29,

Ž .2482 1990 ; M. Lawless and D. S. Marynick, Inorg. Chem.,Ž .30, 3547 1991 ; J. R. Rogers, O. Kwon, and D. S. Marynick,

Ž .Organometallics, 10, 2817 1991 ; M. Lawless and D. S.Ž .Marynick, J. Am. Chem. Soc., 113, 7513 1991 ; P. N. V.

Pavan Kumar and D. S. Marynick, Inorg. Chem., 32, 1857Ž .1993 ; J. R. Rogers, T. P. S. Wagner, and D. S. Marynick,

Ž .Inorg. Chem., 33, 3104 1994 ; J. R. Rogers, C. K. Johnson,Ž .and D. S. Marynick, Inorg. Chem., 33, 4566 1994 .

29. Original program: M. Dupuis, D. Spangler, and J. J. Wen-doloski, National Resource for Computations in Chemistry,Software Catalog, Program QG01, University of California,Berkeley, CA, 1980; This version: M. W. Schmidt, K. KBaldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H.Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su,T. L. Windus, M. Dupuis, and J. A. Montgomery, J. Comput.

Ž .Chem., 14, 1347 1993 .30. R. Ditchfield, W. J. Hehre, and J. A. Pople, J. Chem. Phys.,

Ž .54, 724 1971 ; W. J. Hehre, R. Ditchffeld, and J. A. Pople, J.Ž .Chem. Phys., 56, 2257 1972 ; J. D. Dill and J. A. Pople, J.

Ž .Chem. Phys., 62, 2921 1975 ; M. M. Francl, W. J. Pietro, W. J.Hehre, J. S. Binkley, M. S. Gordon, D. J. DeFrees, and J. A.

Ž .Pople, J. Chem. Phys., 77, 3654 1982 .31. W. J. Stevens, H. Basch, and M. Krauss, J. Chem. Phys., 81,

Ž .6026 1984 .32. J. S. Binkley, J. A. Pople and W. J. Hehre, J. Am. Chem. Soc.,

Ž .102, 939 1980 ; M. S. Gordon, J. S. Binkley, J. A. Pople, W. J.Ž .Pietro, and W. J. Hehre, J. Am. Chem. Soc., 104, 2797 1982 .

33. M. J. Frisch, G. W. Trucks, H. B. Schlegel, P. M. W. Gill,B. G. Johnson, M. A. Robb, J. R. Cheeseman, T. Keith, G. A.Petersson, J. A. Montgomery, K. Raghavachari, M. A. Al-Laham, V. G. Zakrzewski, J. V. Ortiz, J. B. Foresman, C. Y.Peng, P. Y. Ayala, W. Chen, M. W. Wong, J. L. Andres, E. S.Replogle, R. Gomperts, R. L. Martin, D. J. Fox, J. S. Binkley,D. J. Defrees, J. Baker, J. P. Stewart, M. Head-Gordon, C.Gonzalez, and J. A. Pople, Gaussian-94, Gaussian, Inc., Pitts-burgh, PA, 1995.

34. A. Schafer, H. Horm, and R. Alrich, J. Chem. Phys., 97, 2571Ž .1992 .

35. W. J. Stevens, H. Basch, M. Krauss, and P. Jasien, Can. J.Ž .Chem., 70, 612 1992 .

Ž .36. M. L. Connolly, J. Appl. Cryst., 16, 540 1983 .

37. U. C. Singh and P. A. Kollman, J. Comput. Chem., 5, 129Ž .1984 .

38. L. M. Chirlian and M. M. Francl, J. Comput. Chem., 8, 894Ž .1987 .

Ž .39. A. J. H. Wachters, J. Chem. Phys., 52, 1033 1970 .

40. Basis sets were obtained from the Extensible ComputationalChemistry Environment Basis Set Database, Version 1.0, asdeveloped and distributed by the Molecular Science Com-puting Facility, Environmental and Molecular Sciences Lab-oratory, which is part of the Pacific Northwest Laboratory,P.O. Box 999, Richland, WA 99352, and funded by the U.S.Department of Energy. The Paciffc Northwest Laboratory isa multiprogram laboratory operated by Battelle MemorialInstitute for the U.S. Department of Energy under contractDE-AC06-76RLO 1830. Contact David Feller, KarenSchuchardt, or Don Jones for further information.

41. M. M. Francl, C. Carey, L. E. Chrilian, and D. M. Gange, J.Ž .Comput. Chem., 17, 367 1996 .

42. J.-H. Wu and D. S. Marynick, manuscript in preparation;‘‘Quantum Mechanical Study of Polarization Effects onEnzyme-Substrate Interactions,’’ J.-H. Wu and D. S. Maryn-ick, abstracts of the 225th meeting of the American Chemi-

Ž .cal Society, Dallas TX, 1998 COMP Paper No. 125 .


accurate molecular electrostatic potentials based on modified prddo/m wave functions: iii. extension...

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