Qeq Method

7/28/2019 Qeq Method

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3358 J . Phys. Chem. 1991, 95, 3358-3363Charge Equilibration for Molecular Dynamics Simulations

Anthony K . Rap++ and William A. Goddard III*,lBioDesign, Inc., Pasadena, Cali fornia 91101,Department of Chemistry, Colorado State University,Fort Coll ins, Colorado 80523, and Materials Simulation Center, Beckman Institute (139-24),1 CaliforniaInstitute of Technology, Pasadena, California 91 I25 (Received: October 4, 1989)

We report here an approach for predicting charge distributions in molecules for use in molecular dynamics simulations. Theinput data are experimental atomic ionization potentials, electron affinities, and atomic radii. An atomic chemical potentialisconstructed by using these quantities plus shielded electrostatic interactions between all charges. Requiring equal chemicalpotentials leads to equilibrium charges that depend upon geometry. This charge equilibration (QEq)approach leads to chargesin excellent agreement with experimental dipole moments and with the atomic charges obtained from the electrostatic potentialsof accurate ab initio calculations. QEq can beused to predict charges for any polymer, ceramic, semiconductor,orbiologicalsystem, allowing extension of molecular dynamics studies to broad classes of new systems. The charges depend upon environmentand change during molecular dynamics calculations. We indicate how this approach can also be used to predict infraredintensities, dielectric constants, and other charge-related properties.

1. IntroductionKnowledge of the charge distribution within molecules ises-sential for determining the electrostatic energies (including hy-drogen bonding) in molecular mechanics and molecular dynamicscalculations.I4 Unfortunately, reliable charge distributions areknown only for a few organic molecule ^ ^ Thus, currently thereis no effective approach to estimate the charges for inorganicsystems (ceramics, zeolites, high-T, superconductors), and currentestimates of charges for polymers and large organic systems arequite uncertain. For biological molecules, the 20 standard aminoacids and four standard bases have been assigned charges2+ thatare expected to be reasonably accurate; however, charges are notavailable for nonstandard amino acids, unusual bases, and variouscofactors and substrates.An additional serious problem is that current approachesI4 tomolecular mechanics and molecular dynamics use fixed chargesthat cannot readjust to match the electrostatic environment. Sincethe charges are not allowed to respond to the environment, thetradition is to incorporate a dielectric constant in the interactionpotential, leading to additional uncertainties in the calculations.We propose here a general scheme for predicting charges oflarge mlecu es based only on geometry and experimental atomicproperties. The charge equilibration (QEq)approach allows thecharges to respond tochanges in the environment, including thosein applied fields, and can be applied to any material (polymer,ceramic, semiconductor, biological, metallic).In section 11, we derive the basic equations for the chargeequil ibration approach. The scaling parameter X relating atomsize to crystal atomic radii is determined in section 111 by com-paring theory and experiment for the alkali-metal halide diatomicmolecules. In section I V, we discuss hydrogen atoms, whichrequire an extension of the simple scheme of section 11. Finally,in section V we apply theQEq method to a number of moleculesand compare our results with experiment or ab initio theory.The concepts involved in the QEq approach rest upon earlierideas of Pauling, Mull iken, Margrave, Parr, Pearson, Mortier,and others. Section VI summarizes the relationship between QEqand some of these earlier ideas.In section VI I , we mention some possible extensions util izingthe ability of QEq to allow polarization of the charge distribution.11. Charge EquilibrationA. Charge Dependenceo Atomic Energy. In order to estimatethe equilibrium charges in a molecule, we first consider how theenergy of an isolated atom changes as a function of charge. Usinga neutral reference point, we can write the energy of atom A as7 ioDesign, Inc., and Colorado State University.*BioDcsign, Inc., and California Institute of Technology.(Contribution No. 8340.

Including only terms through second order in (1) leads to

so that ( )A o =y2(1P+EA ) =x i(3)

where IP and EA denote the ionization potential and electronaffinity and xA is referred to as the electronegatiuiry.To understand the physical significance of the second-derivativequantity a*E/dQZ, onsider the simple case of a neutral atom witha singly occupied orbital, 4A ,that is empty for the positive ionand doubly occupied for the negative ion. The difference betweenthe IP and EA for this system is(4 )

wheres), is the Coulomb repulsion between two electrons in the$A orbital (the self-Coulomb integral). We refer to this atomicrepulsion quantitypAAs the idempotential (self-Coulomb) forless awkward reference to it in later discussions. Of course, the

IP - EA =Jou

( I ) Williams, D. E.; Cox, S. R. Acta Crysrallogr.,Secr. B 1984, 40, 404.Williams, D. E. ;Houpt, D. J. Ibid. 1986, 42,286. Williams, D. E. ;Hsu, L.Y . Acra Crysral l ogr. ,Sect. A 1985. 41, 296. Cox,S.R.; Hsu, L. Y. ;Williams,D. E. Ibid. 1981, 37, 293.(2) Weiner, S. J .; Kollman, P. A.; Case, D. A.; Singh, U. C.; Ghio, C.;Alagona, G.; Profeta, S. ;Weiner, P. J .Am . Chem.SOC.1984, 106,765-784.Weiner, S. .; K ollman, P. A.; Nguyen, D. T.; Case, D. A. J . Compur.Chem.1986, 7, 230-252.(3) Brooks, R.; Bruccoleri, R. E.; Olafson, B. D.; States, D. J .; Swamin-athan, S. ; Karplus, M. J . Compur. Chem. 1983, 4, 187.( 4) orgensen, W. J .;TiradeRives, J . J .Am . Chem.Soc. 1988,110, 1657.( 5) Cox, S. T.; Williams, D. R. J . Compuf. Chem. 1981, 2, 304-323.(6) Chirlian, L. E.; Francl, M. M. J . Compuf.Chem. 1987,8,894-905.(7) Iczkowsky, R. P.; Margrave, J. L. J .Am . Chem.Soc. 1%1,83,3547.(8) Parr, R. G.; Pearson, R.G. .Am . Chem.Soc.1983, I 05, 1503-1509.0022- 3654/ 91/ 2095- 3358. $02. 50/ 00 1991 American Chemical Society


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Charge Equil ibration for M olecular Dynamics SimulationsTABLE I: Atomic Parameters"

element x . eV J . eV R. A L auLi 3.006 4.772 1.557 0.4174C 5.343 10.126 0.759 0.8563N 6.899 11.760 0.715 0.90890 8.741 13.364 0.669 0.9745F 10.874 14.948 0.706 0.9206Na 2.843 4.592 2.085 0.4364Si 4.168 6.974 1.176 0.7737P 5.463 8.000 1.102 0.8257S 6.928 8.972 1.047 0.8690C 8.564 9.892 0.994 0.9154K 2.421 3.84 2.586 0.452 4Br 7.790 8.850 1.141 1.0253Rb 2.331 3.692 2.770 0.516 2I 6.822 7.524 1.333 1.0726cs 2.183 3.422 2.984 0.5663H 4.5280b 13.8904b 0.371 1.0698"Reference9. bValues for QH=0; see eqs 20 and 21.

optimum shape of the orbital changes upon adding an additionalelectron, and an accurate description of the electron aff inityrequires configuration interaction so that the &A derived from(4) may differ somewhat from the & A calculated with a Har-tree-Fock wave function.Using (2) and (4) leads toEA(Q)=EAO X ~ Q A+' / ~ ~ A Q A ~ (1')

where the x i and &A can be derived directly from atomic data.However, the atomic I P and EA must be corrected for exchangeinteractions present in atoms but absent in molecule^^ (Theatomic states contain unpaired spins, whereas the molecules forwhich we will use xA and J Agenerally have all spins paired.) Thisleads9 to the generalized Mulliken-Pauling electronegativities andidempotentials in Table I .The idempotential is roughly proportional to the inverse sizeof the atom, and indeed, one can define a characteristic atomicsizePA y&A =14.4/RoA or RoA =14.4/&

where the conversion factor 14.4 allows R i to be in angstromsandPAo be in electronvolts. This equation leads toPH=0.84A , $ = 1 . 4 2 A , & = 1.22A,R $=1.08A, R ~i=2.06A ,@=1.60A, andPLi=3.01 A . Comparing with bond distances ofdiatomicsRL H=0.74 A , R& =1.23A , RONN=1.10A , =1.21 A , =2.20 A , R'& =1.63A , and P L i L i =3.08 A , wesee that this characteristic atomic distance corresponds roughlywith the homopolar bond distance.Use of a quadratic relation such as (1' ) is expected to be validonly in a restricted region. I n particular, thex and J are clearlyinvalid outside the range corresponding to emptying or filling thevalence shell of electrons. Thus we restrict the ranges to

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3360 The J ournal o Physical Chemistry, Vol. 95, No. 8, 1991 Rappi and GoddardTABLE 11: Charge EauilibrationResults

metal halide Q,Xp" Q Q M ~ QA-0.5NaCl 0.792 0. 766 0. 776NaBr 0.757 0.745NaI 0.708 0. 709KCI 0.800 0. 775KBr 0.783 0.768KI 0.740 0.754RbCl 0.784 0. 763RbBr 0.768 0. 757Rbl 0.753 0.747CSCl 0.743 0.769CsBr 0.134 0. 767CSI 0.735 0. 763LI F 0.837 0.791LiCl 0.731 0.939LiBr 0.694 0. 902L il 0.647 0.841Na F 0.879 0.665K F 0. 821 0. 662RbF 0.781 0.653CsF 0.697 0. 655

OReference 25. bFrom eq 17 with &,, =0. 4913.

0.7560.7200.7840. 7770.7640.7710.7660.7570. 7770.7760.7730.8030.9580.9210.8600.6710.6670.6570.660

9.0 ShieldedPotentials 1a* - - - - 1 4 . 4 R

3. 02 . 0 ~ ' " ' ' " ' 1

0. 0 1 .o 2.0 3. 0 4 . 0 5 . 0Distance (A )Figure 1. Shielded potentials for I s-7s Slater orbitals. Here wastaken from eq 17 with RA =0. 759~~carbon). Also included is theunshielded Coulomb potential, 14. 4/R.where N, is the normalization constant. From ( 15 ) , the averagesize of the atom is

RA (r) =(2n+1)/(2{A) (16)Consequently, we choose the valence orbital exponent {A for atomA by the relation

(17)where RA is the covalent radius in atomic units (ao=0.52917A ) for atomA, which we select from experimental crystal structuredata (seeTable I). An adjustable parameter X is included in (17)to account for the difference between an average atom size as givenby ( I 6) and the crystal covalent radius RA. We require that thesame X be used for all atom of the periodic table and in sectionI 11 determine I by comparing the predicted and experimentaldipole moments of the alkali-metal halide diatomics. The diatomicCoulomb integral JAB involving these Slater functions is evaluatedexactly for {A and tB t the various distances.111. Alkali-Metal Halides

In order to determine the scaling factor A that adjusts atomicradii to Coulomb shielding distance, we considered the 12alka-li-metal halide molecules MX, where M =Na, K , Rb, orCs andX =C, Br, or I . For these systems, (8) and (9) reduce toQx =-QM

{A =X(2n + I)/(%)

c = 14 = I c =2 =o0.420 2. 504 0.682 0.3950.391 2.561 0. 644 0.3680.350 2.663 0.585 0.3280.473 2.095 0.737 0. 4470.448 2. 165 0.708 0.4230.411 2.299 0.663 0. 3870. 484 1.918 0.741 0. 4590.460 1.977 0.713 0. 4350. 424 2.088 0.672 0. 4000. 505 1.874 0.763 0.4790.482 1.935 0.739 0. 4570.449 2.054 0.703 0. 4240.427 6. 033 0.748 0.3990.406 13. 513 0.737 0. 3790. 377 13. 563 0.685 0. 3510. 333 14. 936 0.608 0. 3100.434 1.751 0.666 0.4110. 473 1.530 0.695 0.4500.481 1.435 0.695 0.4580.496 1.427 0.711 0.473

We require that the calculated Q M lead to the experimental dipolemomentt0~ M X ( 1 /4.80324)QuR ~x (19)where RM x is the experimental bond distance (the constant4.803 24 allowsQ tobe in electron units, R in angstroms, and pin debyes). The only variable here is the scaling parameter A.The best value of X is 0.4913, which leads to an average error of0.0018 e (see Table 11). Rounding off to X =1/2 leads also toan average error of 0.0018 e, and hence (17) becomes

(17')(with RA in units of u,,).We did not use M =Li and X =F in the fits because the errorswere larger for these first-row elements. However, the resul ts forthese eight cases are also listed in Table 11. Including thesecases,the average error increases to 0.15 e.I V. HydrogenThe Mulliken-like definition' for electronegativity leads forhydrogen tox! ='/2(IP+EA ) =7.17 eV, which is not consistentwith the Paulint2or otherI3 empirical values for electronega-tivities. With xH,he hydrogen is more electronegative than C(x$J "P=5.34) or N (x&*P =6.90),whereas the Pauling scale(based on chemical experience) has h drogen much more el=-electronegative than boron ( x ' , =2.0). As discussed in ref 9, theproblem with x! is that the effective EA for H is much smallerthan the atomic value because the H orbital involved in a bondcannot expand to the value achieved in a free H- ion. Conse-quently, we redefine x and JOHH for hydrogen, allowing EAH tobe a variable.From an examination of the charges on H in the molecules L iH,CH4,NH3,H20, and HF, we find that an accurate descriptionof Q H isobtained if the effective charge parameter t H is allowedto be charge-dependent:

(20)

{A =(2n +1)/(4RA)

tropositive thanC (x; =2.1, whilexc2=2.5) and slightly more

{H(QH) = ZOH +QHHere s", =1.0698 is based on (17) whereRH=0.371A . Theidempotential JHH becomes charge-dependent:

J HH(QH) = ( 1 +Q H /& ) J & H (21)( I O) Huber, K.; Herzberg,G. K . Cohtranrsof Diatomic Molecules;Van(11) Mulliken, R. S. . Chem. Phys. 1935, 3, 573.(12) Pauling,L . Nature of rhe Chemical Bond,3rded.;Cornel University( I 3) Sanderson,S. T. Chemical Bondsand Bond Energy;AcademicPress:

Nostrand-Reinhold Co.: New York, 1979.Press: Ithaca, NY , 1960.New York, 1976.


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Charge Equil ibration for M olecular Dynamics Simulations The J ournal of Physical Chemistry, Vol. 95,No.8, 1991 3361

-0.050.04)024-

0.19 (0.16)

0.28 4.78 0.18029

Figure2. (a) Predicted charges for A la-His-A la. The N and0 termini are charged as appropriate for a peptide. Comparisons with charges fromAMBER^are given in parentheses. (b) Same as (a) except that His is protonated.TABLE 111: Char ges on Hydrogen'

compd exptl QEqb HFC QEqHFdH F 0.419 0.462 0.462 0.4570.3258 0.353 0.398 0.3460.2678 0.243 0.338 0.233HZO 0.150" 0.149 0.124 0.124NH,CHILiH -0.76v -0.767 -0.682c -0.679

Fitted to exper-imental charges; x: =4.5280,& =13.8904. CReference6. dFi tted toH F charges; XH =4.7174,&=13.4725. eCloslowski, J. Phys. Rev.L e f f . 1989, 62, 1469. /Reference 25. #Reference 26. "Reference 27.To determine the parameters x i and JOHH , we considered the fivecases in Table 111 and compared with experiment (where theexperimental charges are based on the lowest moments) or ac-curate theory. A least-squares fit leads to

' rom eq 23 for xH(Q)and eq 7 for other atoms.

x =4.5280 eV .&H(O) =13.8904 eV (22)and a good fit to the experimentally derived charges (see Table111, first two columns). Thus

EH+EHO+xOHQH +! @ ! I H Q H ' ( ~Q~/1.0698) (23)Instead of determining the parameters in (20) with experi-mentally derived charges, wecould use the charges calculated fromelectrostatic potentials of H F wave functions. This might bemoreappropriate for comparing with charges from HF calculations.This leads to

x i =4.7174 eV aH 13.4725 eV (24)and other results as in the last column of Table 111.Equations 20 and 21 are well-behaved in the range of QH

(5')corresponding to (5). (They would lead to unphysical results forsufficiently negative values of QH.)In solving (10) using (21), we use (20) with an estimated QHand iterate until all QH's are self-consistent. This converges rapidlyfor all cases tried (six to ten iterations for an initial guess of zerofor all cases discussed).V. ResultsTo test the utility of the charge equil ibration approach, cal-culations were carried out on a representative set of moleculesfor which ab initio partial charges or experimentally derivedcharges are available.s.6 The calculations were carried out at theexperimental using the electronegativities, idem-

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3362 The J ournal of Physical Chemistry, Vol. 95, No. 8, 1991 Rappi5 and GoddardI I ,0.143 ,

-0.284

I0.050 I

\ L?0 . 1 39 4 -0.5340.057Figure4. Predicted charges for (top to bottom) polyethylene, poly(vi-nylidene difluoride),poly(t etr af l uoroethyl ene), and poly(oxymethy1ene).

Figure5. Predicted charges for (a) Nylon 66and (b) PEEK. For eachcase, an additional unit was included on each end (to eliminate endeffects).puter program (in parentheses).2Figure 3 shows the QEq charges for a typical nucleotide(adenosine) from DNA. This is compared with the charges fromAMBER (in parentheses).2In Figure 4, we give the charges for the repeat units for (a)polyethylene, (b) poly(viny1idene difluoride), (c) poly(tetra-fluoroethylene), and (d) orthorhombic poly(oxymethylene), whileFigure 5 has those for (a) Nylon 66 and (b) PEEK (poly(eth-er-ether-ketone)) (poly(oxy- 1,4-phenyIeneoxy-1,4-phenylene-carbonyl - l , 4-phenyl ene)) . In all cases, we used the extendedconfiguration with three such units for which only the chargesof the middle fragment are shown. There is no standard ofcomparison for these systems; however, the charges are in rea-sonable agreement with analogous parts of Figures 2 and 3.In Figure 6, we show charges for NaCl , (NaC1)2,and (NaC1)4,where RNaCl=2.84A and the bond angles are 90 as in an NaClcrystal. The charges show a reasonable trend.VI . Comparison with Other M ethodsThe idea that the Mulliken electronegativity,x =1/2(IP+EA),is equal to the chemical potential p =+E /aQ was suggested byIczkowsky and M argra~e.~he relation between electronegativityand quantum-mechanical wave functions was established by Parr

0.826 -0.828 0.823 -0.823Figure6. Clusters of sodium chloride, all with R =2.84 A and bondangles of 90.et a1.,22who showed that p =dE(p) /Gp,whereE(p ) is the densityfunctional for the energy.The charge expansion of the total energy (1) was suggested byIczkowsky and M argra~e.~It was Parr and Pearson* who identified the physical significanceof the second-order coefficient I P - EA as an atomic hardness

T& =f/,(IP - EA ) =724~We agree with them that the quantity I P - EA is an importantchemical quantity. However, the identification with hardness inacids and bases is less obvious. Consequently, we refer9 to thisatomic Coulomb repulsion quantitypAAI P - EA =274 as theidempotential (self-Coulomb interaction) for less awkward ref-erence in discussions.Popular approaches for estimating charges in molecules havebeen the partial equalization of orbital electronegativities (PEOE)of Gasteiger17and the Del Re18 scheme. These methods dependupon the topology (bond connections) but not on geometry. FromTable IV , we see that these schemes generally lead to the propersign but underestimate the QH of alkanes by a factor of 3-6.A simplification of QEq theory would be to replace the Coulombinteraction J AB(R)by a shielded Coulombic term

J A B ( R) = 4*4/ (25)We show in Table I11 the effect of using this approximation (using(21) with the fit of xH and J H H in experiment). Using t =1.0often leads to net charges opposite those expected from electro-negativities and hence to very unreasonable charge distributions.Using dielectric constants around t =2.0 leads to the best self-consistent charges. For t =14, we obtain values that are generallythe right sign but a factor of 2-3 small. The results are verysensitive to geometry and to t, resulting often in singular pointsin the variation of Qwith R. Thus, we cannot recommend thesimple Coulomb potential with dielectric constant approximation( 25 ) .Mortier et aI.I9 derived equations analogous to (lo)-( 12) fromdensity functional theory and suggested that these equations leadto reasonable charges if the standard atomic electronegativitiesare modified for the molecular environment. The major differencefrom QEq is that Mortier used the unshielded Coulomb potential(25) with t = 14.4. Mortiers electronegativity equalizationmethod6 (EEM) starts with the Sanderson13values for electro-negativities (which arenot in electronvolts) and the Parr-Pearsonvalues* values of the Mull iken hardnesses (in electronvolts) andmodifies them so that the predicted charges best fit the Mullikencharges from STO-3G calculations on small molecules. Usingthe Mortier-modified Sanderson electronegativites(xH=3.832,xc =4.053, xN=5.002, xo =5.565) and Parr-Pearson hardnesses{qH =6.836, qc=5.617, qN =6.8158, qo=6.777, where q=/2J )with t =14.4 leads to the EEM results in Table 111. Thecorrect sign is generally obtained, but the magnitudes are low byfactors of 3-6. Mortier has applied this approach to the predictionof charges for a number of ceramic crystals.23All inger and co-workers21have developed the induced dipolemoment and energy (IDME) method for treating electrostaticeffects in molecules in terms of bond dipole moments and inducedbond dipole moments. This approach is geometry-dependent but

(22) Parr, R. G.; Donelly,R. A.; Levy,M;Palke, W. E J.Chem. Phys.(23) Uytterhoeven, L.; Mortier, W. J.;Geerling, P. J . Phys. Chem.Solids1978,68, 3801.1989, 50, 479.


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Charge Equilibration for Molecular Dynamics Simulations The Journal of Physical Chemistry, Vol. 95, No. 8, 1991 3363TABLE IV: Char ge EquilibrationResults

compd atom QEq QEqHF exptl HF MP2 PEOE Del Re I DM E EEM c =14b c = I b c =2bH F H 0.46 0.46 0.4Ic 0.46 ... 0.27 0.22 -1.00 0.36H,O H 0.35 0.35 0.33d 0.40 0.47 0.21 0.33 0.36 0.05 0.11 -1.00 0.23Ni-i3 H 0.24 0.23 0.27d 0.34 ... 0.11 0.24 0.02 0.05 -0.11CHI H 0.15 0.13 0.14e 0.12 ... 0.02 0.04 0.07 0.00 0.01 -0.02C2H6 H 0.16 0.13 0.171 ... 0.23 0.02 0.04 0.07 0.01 0.02 -0.05C2H4 H 0.15 0.13 0.17 0.22 0.05 0.03 0.01 0.02 -0.08C6H6 H 0.10 0.09 ... 0.06 0.03 0.01 0.03 0.040 -0.45 -0.45 -0.33 -0:45 ... -0.19 -0.14 -0.05 -0.1 1 - 0 . 1 50 -0.43 -0.44 -0.50 -0.45 -0.31 -0.14 -0.41 -0.10 -0.22 -0.34C 0.19 0.24 0.58 0.19 0.11 0.05 0.36 0.03 0.05 - 0 . 1 1H 0.12 0.10 -0.04 0.13 0.10 0.04 0.21 0.04 0.08 0.22H$OH H (0) 0.36 0.34 0.39 ... 0.21 0.30 0.36 0.03 0.07 - 0 . 5 10 -0.66 -0.66 -0.63 ... -0.40 -0.45 - 0 . 5 8 -0.11 -0.26 0.22C -0.15 -0.09 0.21 .., 0.63 -0.01 0.23 0.01 0.01 0.24H, 0.14 0.12 -0.01 ... 0.05 0.06 -0.03 0.02 0.06 0.06H, 0.18 0.16 0.04 ... 0.05 0.06 0.00 0.02 0.06 -0.06HzNC(0)H 0 -0.42 -0.43 -0.57 -0.28 -0.13 -0.09 -0.22 -0.24 -0.33C 0.39 0.42 0.66 0.48 0.20 0.17 0.03 0.06 -0.20N -0.63 -0.61 -0.92 -0.89 -0.37 -0.52 -0.05 -0.10 0.30Hc 0.29 0.28 0.42 0.42 0.16 0.22 0.04 0.09 0.04H 0.23 0.22 0.40 0.41 0.16 0.22 0.04 0.09 -0.21HOC(0)H 0 -0.44 -0.44 -0.60 ... -0.26 -0.12 -0.08 -0.18 ...C 0.56 0.58 0.78 ... 0.29 0.22 0.05 0.12 ...H(C ) 0.16 0.14 0.03 ... 0.15 0.05 0.05 0.12 ...0 -0.65 -0.65 -0.67 ... -0.48 -0.45 -0.08 -0.19 ...H(0) 0.38 0.37 0.46 ... 0.30 0.30 0.06 0.13 ...

C2H2 H 0.13 0.11 0.29 ... 0.12 0.03 0.01 0.03 ...co2H2CO

HlCCN N -0.24 -0.25 -0.43 ... -0.20 -0.07 -0.07 -0.14 0.15C 0.22 0.22 0.43 ... 0.06 0.08 0.01 0.00 -0.36C -0.37 -0.31 -0.39 ... 0.02 -0.12 0.00 -0.01 0.06H 0.13 0.11 0.13 ... 0.04 0.04 0.02 0.05 0.05HzC=C=O 0 -0.45 -0.45 -0.41 ... -0.23 -0.15 -0.10 -0.23 ...C 0.42 0.42 0.77 ... 0.12 0.14 0.02 0.05 ...C -0.23 -0.19 -1.08 ... -0.01 -0.06 0.02 0.03 ...SiH4 H 0.13 0.11 0.36 ... 0.07 0.03 0.03 0.07 ...H 0.08 0.07 0.06 ... 0.01 0.04 ... 0.02 -0.07H3 H 0.19 0.18 0.15 ... 0.10 0.04 ... 0.08 -1.00CI H H 0.32 0.31 0.25 ... 0.15 ... ... 0.16 -1.00'Carpenter, J . E. ;McGrath, M. P.; Hehre, W. H. J .Am. Chem.SOC. 989, I l l ,6154. *Fromeq 25. cReference25. dReference 26.27. Buckingham, A . D.; Disch, R. L .; Dunmur, D. A. J . Am . Chem.SOC.1968, 90, 3104.

assumes that electrostatic interactions can bebuilt upon a mo-lecular connectivity framework. Thus the extension to salts is notapparent. Partial charges reported by Allinger and co-workersare provided in Table 1V in the column labeled IDME. Thecharges calculated by this approach appear to underestimatecharge transfer from hydrogen by a factor of 2 (e.g., 0.07 on Hin CH, and CzH6compared with 015and0.17 for experimentallyderived charges).VI I . PropertiesIn addition to calculating electrostatic energies and multipolemoments, the self-consistent charges can be used to evaluate theother properties such as infrared or Raman intensities. For ex-ample, if we express the a component of the dipole moment as

~u =EQi& (26)ithen the dipole derivative can be written as

Using ( I O)ZAikQk =-Bkwe can write

where (for i # 1)(29)aAik aJ i k a J l k-.=---aRj, aRj6 aRj,

0.130.050.040.060.050.05-0.29-0.3 10.140.090.20-0.420.000.070.080.27-0.350.190.17-0.310.350.10-0.400.25-0.210.13-0.160.08-0.340.26-0.130.110.060.150.26

!ReferenceEquation 28 is solved to obtain the aQk/dRj5hat are substitutedinto (27),which is transformed to normal modes to yield the dipoleintensity of each mode. Similar formulas can be derived forRaman intensities and other charge-related quantities.For crystals, this approach could be used to predict dielectricconstants and changes of polarization with temperature (pyroe-lectricity) and stress (piezoelectricity).VI I I . SummaryThe charge distributions rom charge equil ibration (QEq) leadtogood agreement with experiment. TheQEq approach uses onlyreadily available experimental data (atomic IP and EA , atomicradius) and thus can be applied to any combinationof atoms. (Therelevant x and J values have been tabulated for all elementsthrough L W . ~~ ~~ )he results for simple examples of typicalorganic, inorganic, biological, and polymer systems seem rea-sonable, and we believe that this approachwill prove valuable insimulating biological, polymer, and inorganic materials.

Acknowledgment. These studies were initiated while A.K .R .was on sabbatical at BioDesign, and we thank the BioDesign stafffor useful discussions. We also thank BioDesign for the use ofBIOGRAF in carrying out these calculations. Partial funding forthis research (W.A .G.) was provided bya grant from the Air ForceOffice of Scientific Research (No. AFOSR-88-0051).

(24) Rap*, A . K.; Goddard,W. A., 111. Generalized Mulliken-PaulingElectronegativities. 11. J . fhys. Chem.,submitted for publication.(25) Lovas, F. J .; Tiemann, E. J . fhys. Chem. Ref. Duru 1974, 3, 609.(26) Hellwege, K.-H., Hellwege,A . M ., Eds. Moleculur Consrunrs;Lan-dolt-Bornstein, New Series, Group 11, Vol. 14; Springer-Vcrlag: New York,1982.(27) Amos, R. D. Mol. Phys. 1979,38,33. Based on CI calculations atRcH =1.102A and Qxvs =6.17 x IO cm2.

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Qeq Method