a charge analysis derived from an atomic multipole expansion

A Charge Analysis Derived from anAtomic Multipole Expansion

MARCEL SWART, PIET TH. VAN DUIJNEN, JAAP G. SNIJDERSTheoretische Chemie, Materials Science Centre, Rijksuniversiteit Groningen, Nijenborgh 4,9747 AG Groningen, The Netherlands

Received 12 February 2000; accepted 28 June 2000

ABSTRACT: A new charge analysis is presented that gives an accuratedescription of the electrostatic potential from the charge distribution inmolecules. This is achieved in three steps: first, the total density is written as asum of atomic densities; next, from these atomic densities a set of atomicmultipoles is defined; finally, these atomic multipoles are reconstructed exactly bydistributing charges over all atoms. The method is generally applicable to anymethod able to provide atomic multipole moments, but in this article we takeadvantage of the way the electrostatic potential is calculated within the DensityFunctional Theory framework. We investigated a set of 31 molecules as well as allamino acid residues to test the quality of the method, and found accurate resultsfor the molecular multipole moments directly from the DFT calculations. Thedeviations from experimental values for the dipole/quadrupole moments arealso small. Finally, our Multipole Derived Charges reproduce both the atomic andmolecular multipole moments exactly. c© 2000 John Wiley & Sons, Inc.J Comput Chem 22: 79–88, 2001

Keywords: density functional theory; charge analysis; multipole expansion;force field

Correspondence to: M. Swart; e-mail: [email protected]/grant sponsors: The Netherlands Foundation for

Chemical Research, The Netherlands Organization for ScientificResearch, and Unilever Research Vlaardingen

This article includes Supplementary Material available fromthe author upon request or via the Internet at ftp.wiley.com/public/journals/jcc/suppmat/22/79 or http://journals.wiley.com/jcc/

Journal of Computational Chemistry, Vol. 22, No. 1, 79–88 (2001)c© 2000 John Wiley & Sons, Inc.

SWART, VAN DUIJNEN, AND SNIJDERS

Introduction

T he electrostatic potential in a point rs from thecharge density ρ(ri) of a molecule is obtained

as:

VC(rs) =∫

ρ(ri)|rs − ri| dri (1)

however, this is not well suited for standard usein molecular simulations. One could simplify thisby using a one-center multipole expansion, but thiscannot be applied for larger molecules, because oneneeds to go to high order multipoles. Associatedwith this is a large convergence radius, outsidewhich the expansion is valid. A better method (forinstance, the Distributed Multipole Analysis1, 2) isto use multipoles located at several centers (usu-ally the atoms) with two advantages: the order ofmultipoles needed is smaller, and the convergenceradius is smaller (and per center). However, eventhese methods are cumbersome to use in simula-tions: all multipole moments should be transformedfrom the local (i.e., the one in which they were ob-tained) to the global coordinate system (i.e., the onein the simulation). This involves a number of sub-sequent [3 × 3]-transformations for each moment(one for dipoles, two for quadrupoles, etc.). A fur-ther simplification is therefore worthwhile: takingonly (atomic) charges into consideration. This re-duces not only the number of interactions per pair ofcenters (from 10–35 to 1, depending on the order ofmultipoles located at the centers), but also removesthe need for transforming the moments. The poten-tial is then obtained from the charges qi at positionsri as:

VC(rs) =∑

i

qi

|rs − ri| (2)

Therefore, assigning charges to atoms is an im-portant problem in computational chemistry,3 espe-cially in view of constructing accurate force fieldsfor simulations using Molecular Dynamics or MonteCarlo techniques.4, 5 Our interest is in the field ofbiochemistry, in particular copper proteins,6 wherethe standard force-field parameters, if present,are not accurate enough to model the chemicalprocesses going on in the active site of the proteins.This is especially true while studying properties likethe redox potential, where one wants to see the be-havior of the protein when an electron is removedfrom the system. The first objective of this study isto obtain atomic charges that give an accurate rep-resentation of the electrostatic potential outside the

quantumchemical system under study, in our case,the active site of copper proteins.

Many methods to obtain atomic charges from aquantumchemical calculation are available (detailedsurveys can be found elsewhere3, 7, 8), where thereis a general preference in the molecular modelingworld for Potential-Derived (PD) charges.3, 9 How-ever, associated with this class of charge analysesare several major drawbacks all of which are relatedmore or less to its need of a grid of points wherethe quantumchemical potential is fitted by the elec-trostatic potential from the charges. This results in:(a) a strong dependence on the choices made fordetermining the grid; (b) an arbitrary change in pa-rameters result in different atomic charges; (c) somemethods are even orientation-dependent;10, 11 (d) astrong dependence on the method how to fit thecharges; (e) underdetermined set of equations, mul-tiple “solutions” possible; (f) numerical (in)stability;(g) great uncertainties in assigning charges to atomsburied within the molecule, because the potentialoutside the molecule is mainly determined by theatoms near the surface;3 and (h) an enormous in-crease in CPU-time, because lots of 1/r terms re-sulting from charge-gridpoint distances should beevaluated many times.

Several authors have extended these PD methodsby introducing constraints on the atomic charges;usually the molecular multipole moments up to someorder should be conserved. This seems to make themethods less grid dependent, but they are still lim-ited to a relatively small number of atoms (approx.20, depending on the order of the multipoles9) andstill suffer from the other drawbacks.

We present here a new charge analysis based onideas used for the Dipole Preserving Charge analysis,12

but formulated in another way and using moreaccurate atomic multipoles. There are three stagesinvolved in this method: first, we write the mole-cular charge density as a sum of atomic densities.Second, from these atomic densities a set of atomicmultipoles can be defined, which can be used toget the electrostatic potential outside the charge dis-tribution. Third, these atomic multipoles are recon-structed by using a scheme that distributes chargesover all atoms to reproduce these multipoles ex-actly. Therefore, this method does not suffer fromthe drawbacks of the PD methods.

An important thing to notice here further, is theadvantage of using an atomic over molecular multi-pole expansion. Close to any atom the electrostaticpotential is mainly determined by the charge dis-tribution around that atom; or within the atomicmultipole expansion, by the atomic multipoles near

80 VOL. 22, NO. 1

A NEW CHARGE ANALYSIS

to that point. This is one of the main advantagesover using molecular multipoles only, when oneneeds to go to high orders to get a good repre-sentation of the potential in that point (if at all).Moreover, because the molecular multipole moments(up to order X) are reproduced necessarily by theatomic multipoles (up to order X), our charges havethe nice feature that they do not only represent theatomic but also the molecular multipoles.

This article consists of three parts: the first dealswith how to get from a molecular density to a set ofatomic multipoles; in the second, we explain the dis-tribution scheme; while in the third, we give someresults of the method.

Atomic Multipoles fromMultipole Expansion

A molecular charge density ρ is usually obtainedin a basis set expansion (with atom indices A, B, ba-sis functions χj, basis function indices i, j and PAB

ijelement ij of the density matrix):

ρ =∑AB

ρAB =∑ABij

PABij χ

Ai χ

Bj (3)

which defines the density as a sum of atom pairdensities ρAB. Next, we fit these atom pair densitiesby using atomic functions fi (without specific detailsabout the functions to keep the discussion general;our choice and motivation will be specified in theResults section):

ρ̃AB =∑

i

dAi f A

i +∑

j

dBj f B

j (4)

where the coefficients can be obtained from mini-mization of the density differences:∫

|ρAB − ρ̃AB| dτ (5)

The total density can now be written as a sum ofatomic densities:

ρ̃ =∑AB

ρ̃AB =∑AB

(∑i

dAi f A

i +∑

j

dBj f B

j

)=∑

Ai

cAi f A

i =∑

A

ρ̃A (6)

Furthermore, using these functions fi, the electro-static potential in a point s is given by

VC(rs) =∑

A

VAC (rs) =

∑A

∑i∈A

cAi

∫f Ai (r)|rs − r| dr (7)

Next, an atomic multipole expansion of the r−1

term can be used:

VAC (rs) =

∑l

m= l∑m =−l

4π2l+ 1

MlmZlm(R̂sA)

Rl+1sA

(8)

with the real spherical harmonics Zlm13 and the mul-

tipole “moments” Mlm where RsA is the distancevector from nucleus A to a point rs. These “mo-ments” can be obtained from the coefficients as:

MAlm =

∑i∈A

cAi

∫f Ai (r2)rl

2Zlm(r2) dτ2 (9)

These multipole “moments” are not equal tothe multipoles from the Buckingham convention(see Computational Details), but they can be trans-formed into them easily by inserting the Cartesianexpressions for the Zlms.

Atomic Multipole-DerivedCharge Analysis

We start from the atomic multipoles as they areobtained from the multipole expansion. Then, foreach atom, we reconstruct the set of multipoles (upto some order X) located on that atom by addingcharges qs,A to all atoms that preserve (up to orderX) that particular set of multipoles. The representedmultipole moments of the charges qs,A are obtainedwith the position vectors relative to atom A, ris,A, as:

QreprA =

∑s

qs,A

µrepri,A =

∑s

qs,Aris,A

2reprjk,A =

∑s

qs,A( 3

2 rjs,Arks,A − 12δjkr2

s,A

)�

reprlmn,A =

∑s

qs,A( 5

2 rls,Arms,Arns,A − 12δlmrns,Ar2

s,A

− 12δnlrms,Ar2

s,A − 12δmnrls,Ar2

s,A

)(10)

When the number of atoms is larger than the totalnumber of multipole moments (per atom) to be re-constructed, there is, of course, more than one wayto distribute the charges. We, therefore, use a weightfunction that falls off rapidly to keep the atomic mul-tipoles as local as possible, i.e., as close as possibleto the atom where the multipoles are located:

ws = exp(−ζ |rs − rA|

dA

)(11)

where dA is the distance from atom A to its near-est neighbor, ζ an exponential prefactor, and ws the

JOURNAL OF COMPUTATIONAL CHEMISTRY 81


weight for atom s, when distributing the multipolemoments of atom A.

Now, we want the redistributed charges to be assmall as possible, and at the same time constrain therepresented multipoles (Qrepr

A ,µrepri,A ,2repr

jk,A ,�reprlnm,A) to

be equal to the atomic multipoles from the multipoleexpansion (QMPE

A ,µMPEi,A ,2MPE

jk,A ,�MPElnm,A). We achieve

this by minimizing the following function where theconstrains are met by using Lagrangian multipliersαA, βi,A, γjk,A,1lmn,A:

gA =∑

s

q2s,A

2ws,A+ αA

(QMPE

A −QreprA

)+∑

i

βi,A(µMPE

i,A − µrepri,A

)+∑

jk

γjk,A(2MPE

jk,A −2reprjk,A

)+∑lmn

1lmn,A(�MPE

lmn,A −�reprlmn,A

)(12)

With this choice of function, we ensure that thedistribution mainly takes place close to the atom Awhere the multipoles are located. For the redistrib-uted charges qs,A we obtain the following equation:

qs,A = ws,A

(αA +

∑i

βi,Aris,A +∑

jk

γjk,A( 3

2 rjs,Arks,A

− 12δjkr2

s,A

)+∑lmn

1lmn(. . .))

(13)

that clearly shows that points far away from atom A(and thus a small weight ws) get a small redistrib-uted charge.

Using the constraints, we obtain for the La-grangian multipliers:

QA −∑

s

qs,A = 0 ⇔

QA = αA

∑s

ws,A +∑

i

βi,A

∑s

ws,Aris,A

+∑

jk

γjk,A

∑s

ws,A( 3

2 rjs,Arks,A

− 12δjkr2

s,A

)+ · · · (14)

µt,A −∑

s

qs,Arts,A = 0 ⇔

µt,A = αA

∑s

ws,Arts,A +∑

i

βi,A

∑s

ws,Arts,Aris,A

+∑

jk

γjk,A

∑s

ws,Arts,A( 3

2 rjs,Arks,A

− 12δjkr2

s,A

)+ · · · (15)

2tu,A −∑

s

qs,A( 3

2 rts,Arus,A − 12δtur2

s,A

) = 0 ⇔

2tu,A = αA

∑s

ws,A( 3


s,A

)+∑

i

βi,A

∑s

ws,A( 3


s,A

)ris,A

+∑

jk

γjk,A

∑s

ws,A( 3


s,A

)× ( 3

2 rjs,Arks,A − 12δjkr2

s,A

)+ · · · (16)

and a similar equation for the octupole moments.This results for each atom in a set of linear equations(of size 4 when distributing up to dipole moment[Q,µi], 10 up to quadrupoles [Q,µi,2jk] and 20 upto octupoles ([Q,µi,2jk,�lmn]) for the Lagrangianmultipliers, which are solved by a standard Ax = broutine.

Finally, the values obtained for the Lagrangemultipliers are used to get the redistributed atomiccharges, which when summed, result in the Multi-pole Derived Charges (up to some order X):

qMDCs =

∑A

qs,A (17)

In the following, we shall refer to the charges asMDC-D charges if the multipoles are reconstructedup to the dipole moment, MDC-Q if up to quadru-pole (recommended to be used), and MDC-O if upto the octupole moment.

Computational Details

MOLECULE SET

We investigated a set of 31 molecules14 to ob-tain a good test of the quality of the method, thatwere taken predominantly from ref. 9, and extendedwith some that were of interest to us. Further-more, as a second test set, we used all amino acidresidues. Because we are, in this article, not inter-ested in creating a generally applicable force fieldfor amino acid residues for use in MD simulationsof proteins, we took as a model for the aminoacid residues a reduced conformation as they ap-pear in proteins, i.e., replacing the NH+3 and COO−groups by NH2 and CHO. In fact, this means wecut off the backbone and replaced it by hydro-gens. We are aware of the fact that this is not thestandard model being used for getting amino acidresidue charges, but we like to keep the modelas simple as possible. Moreover, for a few aminoacid residues, we also tested the so-called dipeptide

82 VOL. 22, NO. 1


model CH3CONHCHRCONHCH3, which resultedin virtually the same charges in the side chain.

For all molecules, we first optimized the geome-tries, then we performed single-point energy cal-culations to get the molecular properties (energies,multipoles). All calculations were carried out withinthe Density Functional Theory framework3 withthe ADF program15 – 18 using the Becke–Perdewexchange-correlation19, 20 potential in the TZ2P (Vin ADF terms) basis set. In the following, we shallmake a distinction between the fitted and exact mole-cular multipole moments: the former refers to thevalues obtained from the fitted density (or from theMDC charges, because they represent them exactly),while the latter refers to the values from the exactdensity.

EXPONENTIAL PREFACTOR

This left us the task to find a value for the ex-ponential prefactor ζ , which we want to make aslarge as possible to keep the multipoles local. Onthe other hand, if we make it too big, the weightfunction will approach a delta function, leading tohardly any freedom to distribute the charges overthe other atoms and numerical instabilities. We tookas “optimal” value the largest value where the er-rors in the represented multipoles (due to machineprecision and numerical accuracy) of the amino acidresidues set were smaller than the required accuracyof the numerical integration in the calculations (10−6

in these cases), which turned out to be 3.0.

BUCKINGHAM CONVENTION

In the literature, several conventions are be-ing used for the multipole moments. In this arti-cle, we use the most commonly used Buckinghamconvention.21 – 23 This convention has the followingexpressions for the multipole moments:

µi =∫ρri dτ

2jk =∫ρ( 3

2 rjrk − 12δjkr2) dτ

�lmn =∫ρ( 5

2 rlrmrn − 12δlmrnr2 − 1

2δmnrlr2 (18)

− 12δnlrmr2) dτ

...

In this convention, the electrostatic potential isobtained as follows:

VAC (rs) = QA

rsA+∑

i

µirsA,i

r3sA

+ 12

∑jk

2jkrsA,jrsA,k

r5sA

+ 16

∑lmn

�lmnrsA,lrsA,mrsA,n

r7sA

+ · · · (19)

Inserting the expressions for the Zlms in the mul-tipole expansion [eq. (8)], and rewriting them intoCartesian components, the relation to the Bucking-ham multipoles can easily be derived.

All molecular multipole moments reported herehave been obtained relative to the center of mass ofthe molecule.

POINT CHARGES IN NONATOMIC(DUMMY) POSITIONS

For small systems, a proper charge distributioncannot be represented by assigning charges onlyto atoms.3 This is most easily seen for a homonu-clear diatomic molecule like hydrogen. When usingcharges on the two atoms only, all methods shouldresult in charges of exactly zero, because of sym-metry. Adding a third (dummy) point, for instancein the center, should then result in a much bet-ter description of the charge distribution within themolecule (see also Allen/Tildesley,3 who use nitro-gen as an example and use five points with ratherlarge charges). In practice, adding dummy points isonly necessary for small molecules, and in all casesadding one point already suffices to reproduce themultipole moments up to the quadrupole moment.

Results

DENSITY FUNCTIONAL THEORY INADF-PROGRAM

The method described in this article is generallyapplicable; however, we use the Density FunctionalTheory3 as incorporated in the ADF program15 – 18

and take advantage of the way the electrostatic po-tential is being calculated there. The program uses anumerical integration scheme, by employing a gridaround the atoms to do the integration.15 It uses abasis set of Slater type orbitals:

χµ(r,ϑ , ϕ) = rnµ−1e−αµrZlm(ϑ , ϕ) (20)

centered on each atom, with an auxiliary set of(Slater-type) fitfunctions (called fit set) to approxi-mate the density ρ by expansion in these one-center



functions fi with coefficients ais:

ρ(r) =∑µν

Pµνχµχν ≈∑

i

ai fi(r) (21)

The atomic multipoles are obtained from the co-efficients:

MAlm =

∑i∈A

ai

∫fi(r2)rl

2Zlm(r2) dτ2 (22)

where the a-coefficients are obtained from a leastsquares minimization of the difference1AB betweenthe exact and fitted density:

1AB =∫|ρAB − ρ̃AB|2 dτ (23)

The electrostatic potential is now obtained as:18

VC(rs) =∑

A

VAC (rs) =

∑i∈A

ai

∫fi(r′)|rs − r′| dr′ (24)

or, by using the expansion of |rs−r′| in spherical har-monics and using the exponential form of the STOfit functions (with the exponent αi and the principalquantum number ni):

VAC (rs) =

∑l

m= l∑m =−l

4π2l+ 1

Zlm(rsA)IAlm(rsA)

(25)IAlm(rsA) =

∑i∈A

δ(l, li)δ(m, mi)aiI(ni, li, αi, rsA)

The function I is obtained from incompleteGamma functions, and can be written as the sum ofa multipolar and exponentially decaying part:18

I(ni, li, αi, rsA) = 1(rsA)li+1

(ni + li + 1)(αi)ni+li+2

+ e−αirsA J(ni, li, αi, rsA) (26)

The function J consists of a power series in rsA, withni as highest power, and serves as a screening for theshort-range behavior of the multipole expansion.Because expansion with the screening is correct in-side the molecule, outside the molecule, where thescreening is absent (due to the short range of action),the expansion is also valid, and gives thus the cor-rect potential.

A lot of effort has been put in constructing ap-propriate fit sets, and the currently standard setsare very well qualified to reproduce electrostaticpotentials inside the molecule accurately. Usually,the set of fit functions is larger than the set of ba-sis functions, and sufficiently large to give a gooddescription of the electrostatic potential within themolecule. However, comparing the fitted molecularmoments with the exact values from the basis func-tions, we found some differences, which were dueto fit set incompleteness. To remedy this effect, weconstructed a new set of fit functions by adding p-and d-functions for the atoms involved in our testset of molecules, except for hydrogen, which wasnot altered at all. With this new set, the differences

TABLE I.Comparison of Standard and New Fit Set.

Standart New

Average Maximum Average Maximum

Qtot 0.000 0.000 0.000 0.000µx 0.004 0.039 0.004 0.060µy 0.002 0.023 0.001 0.014µz 0.038 0.303 0.026 0.2582xx 0.490 3.640 0.105 0.2672xy 0.047 0.807 0.035 0.4132xz 0.000 0.000 0.000 0.0002yy 0.373 1.788 0.102 0.2512yz 0.000 0.000 0.000 0.0002zz 0.685 3.577 0.139 0.502

Fit test 5.05 · 10−5 3.95 · 10−5

Energy difference 0.0289 kcal/mol

Absolute differences between exact and fitted molecular multipole moments, fit test difference, and absolute difference in total energy,all averaged over the set of 31 molecules14(a.u.).

84 VOL. 22, NO. 1


between the exact and fitted molecular multipolemoments were much smaller (see Table I for valuesaveraged over the set of 31 molecules14). From thistable, it can be seen that while the new fit set givesbetter molecular multipole moments, there is hardlyany influence on the total energy.

EXTENSIONS OF THE IMPLEMENTATION

The standard fit sets in the ADF program havebeen constructed in order to minimize the differ-ence between the exact and fitted density, therebyobtaining from both accurate electrostatic potentials

TABLE II.Atomic Charges and Molecular Multipole Moments (a.u.).

Name Charges Multipole Moment DFT Exact DFT Fitted Experimental

Benzene C6H6qC −0.123 2xx = 2yy 2.78 2.79 3.23qH 0.123 2zz −5.55 −5.59 −6.47 ± 0.3725

Carbonmonoxide COqC −0.537 µz −0.07 −0.09 −0.0426

qO −0.852 2xx = 2yy 0.76 0.74qXX 1.389 2zz −1.52 −1.48 −1.86 ± 0.2226

Carbondisulfide CS2qC −0.358 2xx = 2yy −1.35 −1.58qS 0.179 2zz 2.71 3.15 3.17± 0.2226

Fluorine F2qF 0.186 2xx = 2yy −0.38 −0.33qXX −0.371 2zz 0.76 0.66 0.5626

Hydrogen H2qH 0.610 2xx = 2yy −0.25 −0.31 −0.24qXX −1.220 2zz 0.51 0.61 0.47± 0.0326

Hexafluorobenzene C6F6qC 0.094 2xx = 2yy −2.99 −2.78 −3.53qF −0.094 2zz 5.99 5.56 7.07± 0.3725

Hydrogen fluoride HFqH 0.915 µz −0.69 −0.69 −0.7226

qF 0.128 2xx = 2yy −0.87 −0.95qXX −1.043 2zz 1.74 1.91 −1.75 ± 0.0226

Water H2OqO 0.458 µz −0.71 −0.73 −0.7326

qH 0.892 2xx −1.82 −1.93 −1.86 ± 0.0126

qXX −1.121 2yy 1.91 2.01 1.96± 0.0126

2zz −0.09 −0.08 −0.10 ± 0.0226

Water dimer (H2O)2qH 0.902 µx −1.38 −1.41qXX −1.161 µy −0.58 −0.59qO 0.264 2xx 2.68 2.84qXX −0.748 2xy −3.24 −3.35qH 0.699 2yy −2.08 −2.19qO′ −0.261 2zz −0.60 −0.65qH′ 0.930qXX′ −1.188

MDC-Q charges for some molecules out of the set of 31.14, 24

Molecular multipole moments from exact and fitted density from DFT calculations.



in the system under study. However, no specificconstraints were put on the functions to conservethe molecular multipole moments. Moreover, in thestandard versions of ADF, the expectation value ofonly the dipole moment (from the exact density) isbeing calculated. We implemented the calculation

of the expectation value of the quadrupole momenttensor, to check the represented quadrupole mo-ments from both the exact and fitted densities.

In Table I, we give the absolute differences be-tween the molecular multipole moments from theexact density on one hand, the fitted density on the

TABLE III.MDC-Q Atomic Charges and Molecular Multipole Moments (a.u.).

Name Charges Multipole Moment DFT Exact DFT Fitted

Cysteine (Cys, no H)N −1.028 Q −1.00 −1.00H 0.299 µx −1.85 −1.93H 0.350 µy 2.23 2.17C 0.461 µz 2.73 2.76H −0.139 2xx −17.78 −18.86C 0.319 2xy 5.23 5.34O −0.536 2xz 5.09 4.64H −0.014 2yy 14.92 15.60C 0.398 2yz −8.94 −9.05H −0.106 2zz 2.86 3.26H −0.117S −0.886

Glycine (Gly)N −0.798 µx −0.37 −0.39H 0.237 µy −0.56 −0.58H 0.232 µz −0.05 −0.04C 1.076 2xx 2.46 2.50H −0.249 2xy −1.94 −2.39C 0.316 Hxz 4.91 4.96O −0.517 Hyy 0.80 0.70H −0.051 Hyz 7.07 7.27H −0.247 Hzz −3.26 −3.23

Histidine (His-A)N −0.882 µx 2.20 2.15H 0.287 µy −0.15 −0.16H 0.291 µz 0.63 0.63C 0.505 Hxx 15.07 14.86H −0.087 Hxy −1.79 −1.41C 0.390 Hxz 6.49 6.36O −0.471 Hyy −8.51 −8.45H −0.039 Hyz 0.47 0.91C 0.440 Hzz −6.56 −6.41H −0.109H −0.178C 0.062N −0.613C 0.382H 0.081N −0.537H 0.381C −0.035H 0.132

86 VOL. 22, NO. 1


TABLE III.(Continued)

Name Charges Multipole Moment DFT Exact DFT Fitted

Methionine (Met)N −0.882 µx 0.58 0.57H 0.332 µy −0.99 −1.10H 0.280 µz 0.10 0.18C 0.361 2xx 0.86 1.01H −0.103 2xy 2.26 1.97C 0.367 2xz 1.27 1.62O −0.434 2yy 2.65 3.16H −0.021 2yz −0.50 −1.24C 0.496 2zz −3.52 −4.17H −0.113H −0.203C 0.176H −0.033H −0.057S −0.298C −0.142H 0.079H 0.104H 0.089

MDC-Q charges for some amino acid residues.24

Molecular multipole moments from exact and fitted density from DFT calculations.

other, averaged over the set of 31 molecules (seeComputational Details). We report the values forboth the standard fit set (TZ2P basis set, V in ADFterminology) as well as a new one, where we addedp- and d-functions to the fit set for the first-rowatoms (available as supplementary data). We alsogive the values of the fit test for both fit sets, aswell as the averaged absolute energy differences be-tween the two fit sets. These numbers give a clearindication that there is hardly any influence on thetotal energy and exact/fitted density difference. Thereis, however, a clear improvement of the representedmultipole moments, with an average absolute devi-ation of 0.10 a.u. for the quadrupole moments.

MDC CHARGES

The MDC-Q charges as well as the calculated ex-act multipole moments, fitted multipole moments,and experimental values of some of the 31 smallmolecules14, 24 and the amino acid residues are col-lected in, respectively, Tables II and III.

The other data are available as supplementarydata.24

In these tables, we give the charges on all atoms,and added points (indicated as dummy atoms XX),

as needed to give an accurate description of thecharge distribution. These dummy points were cen-tered on the bond midway point between the atomswhere necessary. It proved necessary not only forthe diatomic molecules, but also for some othersmall molecules like, for instance, water. There weput two equivalent dummy atoms in the middlebetween the oxygen and each hydrogen atom. Weincluded them at the same positions in the waterdimer. Comparing the atomic charges, it is interest-ing to see the charge shift away from the hydrogeninvolved in the hydrogen bond (H3), as was tobe expected. It can be seen immediately from thedummy atom next to it, which shifts from −1.12 to−0.75.

A classical case is the carbonmonoxide molecule,with an experimental dipole moment of −0.04 a.u.leading to a small negative charge on the carbon andan equally small but positive charge on the oxygenatom if only reconstructing up to dipole; MDC-Dcharges −0.0402 (C), +0.0402 (O). However, theydo not give a good representation for the molecularquadrupole moment 〈2zz〉: −1.522, MDC-D −0.026.This is corrected by adding a third point in the



bond middle point, which again reproduces the fit-ted value: MDC-Q −1.475.

Conclusions

The charge analysis presented here gives an ac-curate description of the charge distribution whenobtained with an appropriate set of fit functionsas we have introduced in this article. There are afew advantages of the method: the atomic multi-poles that are reconstructed exactly by our MultipoleDerived Charges, give by construction the best fitto the electrostatic potential around the molecule.Furthermore, with the appropriate set of fit func-tions the molecular multipole moments from thefitted density can be made arbitrarily close to theones from the exact density. Our Multipole DerivedCharges reproduce the fitted multipoles exactly, ex-cept in few cases (like diatomics) when there are notenough degrees of freedom to reproduce all multi-poles. Usually it is sufficient to add one extra point,for example, in the bond middle point of a diatomicmolecule.

The molecular multipole moments, both fromthe fitted and the exact density, have a small devi-ation from experimental values. Because the MDCcharges reproduce the fitted values exactly, theMDC charges give an accurate representation of thecharge distribution in the molecule and result in agood electrostatic potential.

References

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14. The set of 31 molecules consisted of: benzene, ethyl-ene, methylcyanide, methyllithium, methanol, methanol an-ion, methanethiol, methanethiol anion, methane, chlorine,carbonmonoxide, carbondioxide, carbondisulfide, fluorine,hydrogen, formaldehyde, hydrogenchloride, formamide,formic acid anion, hydrogenfluoride, hexafluorobenzene,hydrogen fluoride, lithiumhydride, nitrogen, ammonia,oxygen, dimethylether, hydroxyl anion, phosphine, thio-phene, water, water dimer.

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88 VOL. 22, NO. 1

a charge analysis derived from an atomic multipole expansion

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