J. Phys. Chem. 1993,97, 66284636
6628
Transferable Net Atomic Charges from a Distributed Multipole Analysis €or the Description of Electrostatic Properties. A Case Study of Saturated Hydrocarbons Chistophe Chipot and J h o s G. hgyh Laboratoire de Chimie Theorique, Unit6 de Recherche Associb au CNRS nO 510, Universitb de Nancy I, BP 239, 54506 Vandoeuvre-12s- Nancy Cedex, France
Gytirgy G. Ferenczy Chemical Works of Gedeon Richter Ltd., H-1475 Budapest, P.O. Box 27, Hungary Harold A. Scheraga' Baker Laboratory of Chemistry, Cornell University, Ithaca, New York 14853-1301 Received: January 26, 1993
Point charges derived from electrostatic potentials and from distributed multipole expansions are compared critically throughout a full ab initio self-consistent-field study of several saturated hydrocarbons, using the split-valence type 6-31G, 6-31G*, and 6-31G** basis sets. Potential-derived net atomic charges failed to be transferable from one molecule to another structurally similar one, especially for aliphatic groups, hence constituting a major drawback for their use in force-field parametrization. Consideration of a limited number of extra-atomic charges is necessary to reach transferability and mimic the effects of higher order moments of the electron charge distribution. However, it has been shown that another method, recently proposed and based on the distributed multipole expansion of the molecular electrostatic potential, provides a reasonable compromise for both chemical transferability and accuracy to reproduce the electrostatic potential. Nevertheless, charges obtained from both methods have a tendency to reproduce the SCF potential with conspicuous errors, therefore indicating that the molecular charge distribution of nonpolar systems cannot be represented accurately by only atom-centered point charge models.
1. Introduction
Recent improvementsin computationalchemistry have enabled modern techniques to provide accurate wave functions, even for relatively large molecules. The complexity of the information hidden in the wave function makes it difficult to understand its physical and chemical significance (viz., molecular electrostatic properties). Simple chemical concepts, like atomic charges, are necessary to extract compact and practical representations such as the molecular charge density. Unfortunately, the notion of atomic charge is not a wellestablished one. Since it is not an observable, there is no rigorous way to define this quantity, and a wide variety of alternative definitions have proliferated in the literature. Perhaps the most well-known approach is the Mulliken population analysis,' which is based on a simple partitioningof the electrons among the atomic orbital basis functions. While it is a fairly reasonable approach, as far as the basis functions are really of atomic character (Le., minimal and small basis sets), extended basis functions can easily lead to charges that are contradictory to chemical intuition. In any case, it was recognized early that Mulliken charges reflect the polarity of atoms and functional groups reasonably (Le., correlation with electronegativity) but that they are basically inappropriateto describeelectrostaticproperties such as molecular multipole moments or electrostatic potentials. For these reasons, several methods have been proposedz-21for computing point charges, which reflect the electrostaticproperties correctly, but none of them turned out to be entirely satisfactory. In a recent paper," we calculated the net atomic chargesof various naturally occurring amino acid side chains from ab initio selfconsistent-field electrostatic potential and field. The lack of transferability insaturated hydrocarbons (viz.,valineor isoleucine side chains) suggested an obvious inadequacy of this technique to describe the molecular charge distribution of aliphatic compounds. Indeed, this method, which consists of fitting atom0022-3654/93/2097-6628S04.00/0
centered point charges to a quantum mechanically calculated potential or field on a grid of regularly spaced points surrounding the molecule, suffers from two major drawbacks. 1. The full potential, which is usually quite rich in details, cannot be fitted efficiently by a small number of point charges, unless the system is highly polar, and, as a result, low-order contributionsare much more important. Furthermore, tht shortrange effects of electron cloud penetration generally modulate the representation of the SCF electrostatic potential or field with respect to point multipole models. The region where penetration effects are nonnegligible overlaps with the range of action of high-order multipole moments (typically octopoles and higher moments). This "interference" makes it impossible to fit the quantum chemical potential properly by a simple atom-ccntered point charge (or point multipole) model. This problem is particularly criticalfor the description of saturated hydrocarbons, where higher moments are usually the first nonvanishing terms in the multipolar expansion. 2. The potential and/or the field has to be calculated from the wave function on a predefined regular or irregular grid. The judicious choice of the number and location of grid points may be crucial, although, as has been shown already,2°.22point charges derived from these electrostatic quantities are practically insensitive to the grid density, within reasonable limits. Nevertheless, since a fairly large number of grid points are needed to yield stable results, the computation of electrostatic potentials still requires a nonnegligible amount of CPU time (approximately the same amount as the direct-SCF procedure for molecules involving an average of 100 basis functions). These inconvenienceshave therefore led us to reinvestigatethe problem of atomic charge models from a different perspective. Since the manner of calculating the electrostatic potential, or the field, is the origin of the deficiencies of the method based on a fit to these quantities, it appeared necessary to have r w u r s e to Ca 1993 American Chemical Society
Distributed Multipole Analysis and Electrostatic Properties
The Journal of Physicaf Chemistry, Vof. 97, No.25, 2993 6629
another computationaltechnique better adapted to the description of nonpolar systems. The distributed multipole moment (DMM) a p p r o x i m a t i ~ n ~ ~ ~ ~ is known to give a good representation of the molecular electrostaticpotential. Since this procedure neglects penetration effects systematically (sothat the potential has a physical meaning outside the van der Waals volume only), the effect of higher multiples is not hidden any more by the tail of the charge distribution. Moreover, the calculation of the potential from DMMs is extremely rapid. It should be noted that the extension of Mulliken's partitioning' to multipoles of higher order than charges, as proposed by Sokalski et aLz6(Le.,cumulative atomic multipole moments (CAMM)), is a particular, nonoptimized case of the DMM scheme. Like ab initio molecular electrostatic potentials, DMM potentials and/or fieldsmay be derived to obtain point charge models 0 Center of multipole series (Q using a numerical least-squares-fit proced~re.~ This approach has been explored recentlyI8Bz0and led to promising results. 0 Pointcharge ( ) 9 In the present study, we have adopted another procedure, Figure 1 . (a) The distance between the multipole center and the point proposed recently by Ferenczy,19 avoiding any explicit use of a where the potential is to be calculated (r -io)is greater than the distance grid and therefore all the pitfalls that it represents. The theoretical between the multiples and thecharge (rto). The boundariesof integration background of this method, which replaces the grid of points by are pl and p2. (b) Definition of successivelayers of integration when (r an analytical integration over given regions of the space sur- r,,) < rl,,. The correspondingboundaries of integration are p p and p e l . rounding the molecule, is recapitulated in section 2. After summarizing the technical details of the computations C,,,,(8,4) denotes a renormalized spherical harmonic:33 in section 3, the results obtained with different basis sets are compared critically with those derived from the molecular (4) electrostatic potentia14*819J2-16*22 (section 4). The dependence of the charges on the basis set is analyzed by carrying out the where fi,,,(8,4)is a solid spherical h a r m ~ n i c .At ~ an ~ ~arbitrary ~~ computations at the extended split-valence type 6-3 1G, 6-3 1G*, point located at r, the difference between the potentials PMM(r), and 6-3 lG** basis setz7tz8 levels. The geometry of the molecules created by the DMMs (distributed multipole moments), and presented here has been fully optimized, using the same basis VQ(r), created by a set of point charges that is supposed to model sets. An attempt to improve the model, using a limited number the molecular charge distribution, may be written in terms of the of extra off-atom sites, is presented in section 5. The influence following function: of intramolecular electron correlation on the molecular charge distribution has also been evaluated by means of an MP2/63 1G**study on the same set of nonpolar molecules. These results, The electrostatic potential created by a set of point charges is together with a discussion, are reported in section 6.
,J
2. Derivation of Point Charges from Distributed Multipole series
This method, developed by Ferenczy,19consists of fitting point charges to the potentials created by a distributed multipoleseries. It has the major advantage of avoiding the explicit evaluation of the molecular electrostatic potential or field. The natural starting point of this study is a distributed multipole analysis (DMA),23J4-29*30 which provides a good and conciserepresentation of the molecular charge distribution. Unlike molecular electrostatic potential, or field, computations on a grid of points, this post-SCF calculation requires very little CPU time and is based on the useof a oneelectron density matrix over a basis of Gaussiantype orbitals (GTO)23*25 {x,). Such a matrix is readily available from any GAUSSIAN 9231checkpoint file after a simple singlepoint run. The electrostatic potentialcreated at an arbitrary point r around the molecule in the distributed multipole approximation is given by a
l,m
where Gm is the mth component of the rank f multipole moment centered at r, and obtained from a distributed multipole analysis and
is an irregular spherical harmonic32defined by
where qris a net point charge located at ri. Since the total potential is the sum of the individual potentials due to different multipole moment series centered at ra, the net point charge qi can be split into the corresponding contributions 4:: a
Provided that (r-rJ, the distance between the multipoles and the point where the potential is to be calculated, is greater than ria(see Figure la), the irregular spherical harmonic &(r) = l W ( r - rt) may be written as 1.m
1.m
where &(r) denotes a regular spherical h a r m o n i ~ ~defined ~v~~ bY
R,(r) = ?'+'IIm(r) In this case, eq 5 may be written as
The optimum set of net charges is obtained when the minimum of the integral of lf(r)I2 over an appropriately defined domain is reached. In other words, one must solve
-$
~ ~ ~ ~ ~ [ f (sin r 8) dr] de * d# ? =0
(3)
(9)
(1 1)
Chipot et al.
6630 The Journal of Physical Chemistry, Vol. 97, No. 25, 1993 which involves the quantity
and leads to the matrix equation
Aq=b
(13)
where the corresponding matrix elements are given by
cu BY1.W
The above elements involve the site4te couplings K&,,,,, defined by
This quantity can easily be evaluated analytically, using the orthogonality of the spherical harmonics, provided that a = b (the integral in eq 15refers to the same center) and the integration domain is a spherical layer (Le., 01 = 0, 02 = T , 41 = 0 and 92 = 2r). An analytical treatment of the nondiagonal coupling (a # b) is more cumbersome, but recent results have indicated that neglecting these couplings is a satisfactory appr~ximation.'~ In this case, the global fitting procedure is split into several independent fits to the site multipole series. For this purpose, one may define the following function: F(pl,p2)
= ~ PIp
2 ~ f ~ ( r ) ]sin 2 B3 dr dB d9
(16)
Performing the integration analytically over a spherical layer around point a, the grid construction involved in molecular electrostaticpotential-or field-derivedmethods is advantageously avoided:
where the Wp1,,,/ factors
weight the importance of the multipoles of rank 1. The least-squares-fit criterion is satisfied when the minimum of the quantity in eq 16 is reached. In other words, one must solve
yielding
m u Bmhne
Pl"l*W
Figure 2. Saturated hydrocarbons in the atom-centered approximation.
The set of partial point charges is given by qa = x ' z (23) The set of the net charges q1is obtained by merely summing up the contributions 4; (cf. eq 7 ) . As long as point charges are inside the spherical shell centered at the position of the multipole series, eq 16 is perfectly valid. However, when point charges are within the spherical layer (i..e., p1 5 ria 5 p2), it is necessary to define additional volumes of integration p (see Figure lb), in which eq 8, one of the basic assumptions of the model, is valid.
3. Computational deWLP Equation 17 indicates that the outcome of the least-squares fitting procedure depends on three parameters: the rank 1 of the distributed multipoleexpansion,together with the twointegration boundaries p1 and p2. Expansions carried out with 4 5 1 5 6 have been shown to yield very satisfactory results.19 In the present study, the expansion has been limited to the hexadecapole (1 = 4). The distributed multipole analysis was carried out with a program3&written in theOSIPE36benvironment,using theweight functions introduced by VignGMaeder and Cla~erie?~ However,determiningtheradiipland p2isa muchmorecritical task. Because the distributed multipole expansion converges formally to the correct potential only for points r located outside the charge distribution, the functional P(r)l2 must nacessarily be integrated beyond the van der Waals envelope. In order to ensure an acceptable fit of the charges to the multipoles, it is desirable that the lower boundary of integration p1 include a sufficient number of neighboring sited9 and that the upper boundary p2 cover the distance of typical molecular interactions. For these reasons, we have fixed the integration bounds at p1 = WdW + 2.0 A and p2 = rVdW + 5.0 A (where rVdW stands for the van der Waals radius of a given atom) so that molecules of significant size may be treated correctly. 4. ResolltsdDiscapsion
which may be written in terms of a matrix equation:
xqa= z where the corresponding matrix elements are
(21)
The classical method of electrostatic potential deri~ a t i o n , 4 . ~ . ~together J ~ - ~ ~with , ~ ~the distributed multipole series technique,19has been applied to reproduce the molecular charge distribution of several aliphatic compounds (see Figure 2), using
Distributed Multipole Analysis and Electrostatic Properties the extended split-valence type 6-31G, 6-31G*, and 6-31G** basis
The Journal of Physical Chemistry, Vol. 97, No. 25, 1993 6631 together with the corresponding mean absolute deviation
~ets.~'9~*
Net atomic charges derived from DMMs are gathered in Table I and compared with those obtained from the derivation of the SCF molecular electrostatic potential. In order to analyze quantitatively the discrepancies between these two techniques, we have computed the electrostatic potential created by both sets of charges. In each case, this quantity was evaluated on the grid of points generated for the SCF molecular electrostatic potential derivation procedureZZ(the exclusion radius and the grid step have been set at 4.00 and 0.5 A, respectively). A root mean square deviation (RMSD) between the ab initio quantum mechanicallycalculated and the point charge derivedelectrostatic potentials may thus be calculated
[1
RMSD =
NPl
112
(pcF(r,,)
- @(r,,))']
(24)
Npnt c-1
where Npnt denotes the number of points in the grid. VSCF(r,,)is the reference SCF molecular electrostatic potential evaluated at a point p, located at r,,, whereas W(r,,) is the potential created by net atomic charges. The values of the RMSD are given in atomicunits. Nevertheless,in order to compare the present results with those available in the literature, our values should be multiplied by a conversion factor of 627.5095 (hartree to kcal mol-'). The mean absolute deviation A is given in percent. This quantity is of great importance since low values of the RMSD resulting from small potentials (e.g.,for saturated hydrocarbons) may conceal nonnegligible errors arising from the fitting procedure. Since molecular electrostatic potential-derived charges are the most suitable to reproduce the SCF electrostatic potential, in a least-squares senseand for a particular grid of points, we naturally
TABLE I: Dependeace of tbe Net Atomic Chargesa of Several Hydrocarbonsb on the Basis Set and Comparison between Distributed Multipolec Potential-Derived (UDMM) and Molecular Electrostatic Potential-Derived (9,) Charges 6-31G atomd
90
*
6-31G* qDMM
90
6-31G** 4'DMM
9u
4'DMM
-0.533 0.133 0.038 33.65
0.037 -0.009 0.444 204.97
-0.558 0.139 0.044 38.80
0.012 -0.003 0.435 102.16
-0.561 0.140 0.049 37.34
0.172 -0.043 0.558 133.16
0.056 -0.019 0.268 167.93
0.122 -0.041 0.433 313.26
0.093 -0.031 0.309 217.51
0.120 -0,040 0.393 260.72
0.076 -0.025 0.387 305.11
0.158 -0.053 0.483 998.21
-0.231 0.339 0.040 0.045 -0.070 0.268 208.71
0.076 0.169 -0.029 -0.035 -0.061 0.384 283.71
-0.241 0.406 0.037 0.044 -0.088 0.309 199.98
0.071 0.170 -0.028 -0,033 -0.070 0.457 282.83
-0.249 0.385 0.041 0.048 -0.08 1 0.307 213.84
0.129 0.166 -0.044 -0.050 -0.067 0.479 307.65
-0.080 0.162 0.012 0.007 -0.054 0.375 334.96
0.108 0.087 -0.036 -0,039 -0.040 0.423 490.03
-0.065 0.200 0.005 -0.001 -0.069 0.432 192.40
0.098 0.094 -0,034 -0.037 -0.042 0.469 219.64
-0.079 0.189 0.010 0.005 -0.064 0.428 285.36
0.147 0.102 -0.048 -0.052 -0.049 0.507 691.90
-0.184 0.153 0.042 0.038 0.038 -0.047 -0.036 0.291 182.07
0.080 0.142 -0,029 -0.038 -0.036 -0.062 -0.057 0.376 371.10
-0.190 0.193 0.039 0.035 0.034 -0.062 -0.050 0.332 770.1 1
0.076 0.138 -0.028 -0.037 -0.034 -0.060 -0.055 0.411 678.80
-0.199 0.178 0.043 0.039 0.038 -0.056 -0.043 0.331 230.89
0.133 0.149 -0.045 -0.051 -0.054 4.069 -0.063 0.437 513.18
-0.250 0.224 0.126 0.059 0.046 -0.066 -0.055 0.320 11OOO.00
0.081 0.144 0.095 -0.029 -0.035 -0.059 -0.055 0.425 11OOO.00
-0.261 0.271 0.152 0.059 0.044 -0.083 -0.068 0.367 LlOOO.00
0.078 0.147 0.096 -0.029 -0.034 -0.060 -0.055 0.501 11OOO.00
-0.271 0.253 0.141 0.063 0.048 -0.075 -0.062 0.367 505.42
0.132 0.150 0.114 -0.045 -0.049 -0,066 -0.064 0.498 721.32
* In electronic charge units (em).Respectively, 6-31G,6-31G*,and 6-31G**full geometry optimization. Distributed multipole analysis camed out at atomic centera only. Indicts given in Figure 2. * Root m a n square deviation between the ab initio computed and the point charge derived electrostatic potentials. All values XlO-3 au. f Mcan absolute deviation expressed as a percent.
Chipot et al.
6632 The Journal of Physical Chemistry, Vol. 97, No. 25, 1993 TABLE E . Influence of the Inclusion of Additional Sites. on the Multipole Momentsb of Several Hydrocarbonsc Calculated from Molecular Electrostntic Potential-Derived Net Atomic Charges Using the Split Valence 6-316** Basis Set'
TABLE
II (Continued) atomic and atomic centers only bond V DMM V trans-Butane
atomic and
atomic centers only bond V DMM V Methane 1.228 0.389 1.655 -1.079 -1.048 0.690 0.573 -0.133 0.783 0.172 -0.745 -0.038
-0.376 -0.119 -0.507 0.331 0.321 -0,212 -0.176 0.041 -0,240 -0.053 0.228 0.012
0.089 0.089 -0.177 0.625 0.625 1.667 0.208 -0.834 -0.834
0.184 0.184 -0.368 1.300 1.300 3.466 0.433 -1.733 -1.733
0.055 -0.540 0.240 0.300 0.000 -0.010 0.004 0.577 1.420 -1.991 -3.046 -1.004 -0.140 1.955 1.09 1 -0.951
0.058 -0.289 -0.033 0.323 0.000 -0.886 -0.003 0.649 0.044 0.196 -2.722 1.774 1.745 1.346 1.376 -3.120
0.068 -0,352 -0.104 0.455 0.001 0.962 -0.003 -1.539 1.867 -1.288 7.103 7.124 3.734 -5.246 -1.856 -1.877
0.062 -0.253 -0.321 0.574 0.001 -0.527 -0.003 0.705 -0.150 -0.026 -3.693 4.777 1.677 0.296 3.397 -5.074
0.057 -0.362 -0,010 0.372 0.000 -0,454 0.000 -1.949 2.136 0.267 0.173 6.793 6.083
0.070 -0.194 -0.060 0.253 0.000 -0.677 0.000 0.961 0.156 -0.440 0.898 -2.218 -2.676
SCF
@zzzz
9,-
1.201 0.381 1.619 -0.834 -1.025 0.675 0.953 -0.222 1.302 0.285 -1.238 -0.064 0.291 0.291 -0.582 -2.228 -2.228 -5.940 -0.743 2.970 2.970
Propane 0.061 -0.557 -0.011 0.568 0.000 -1.342 0.010 3.407 -0.364 -1.707 3.087 -4.509 -4.432 -1.505 -1.582 6.014
0.059 -0.536 -0.018 0.554 0.000 -1.082 0.010 2.770 -0.004 -1.688 3.059 -7.847 -2.618 1.085 -4.144 6.762
4.316 0.289 1.929 2.389 0.649 -1.663
0.059 -0.929 0.201 0.728 0.003 1.215 0.000 -2.195 -1.435 2.415 0.349 3.161 2.245 -0.633 0.283 -2.528
0.052 -0.521 -0.124 0.645 0.001 1.595 0.000 -1.730 0.094 0.043 -10.404 2.943 -0.522 3.469 6.934 -6.412
6.682 -10.229 -2.631 -4.05 1 -10.716 10.637
7.980 -8.933 -2.693 -5.286 -13.248 10.665
0.062 -1.020 4.170 1.191 0.002 2.663 0.000 -5.526 0.673 2.190 -5.844 -8.788 -15.610 -0.488 6.333 9.277
0.060 -1.003 -0.139 1.142
Pentane
Ethane 0.305 0.305 -0.611 -3.207 -3.207 -8.552 -1.069 4.276 4.276
-2.848 -7.862 1.069 1.778 -5.641 6.313
SCF
0.004 2.487 0.000 -4.580 0.000 2.092 -5.265 -14.070 -11.130 4.103 1.163 9.968
C-C bond center only. b Traceless moments;'l p in D, 84 in D A, 6-31G** full geometry optimization. Comparison with distributed multipole potential-derived multipolar moments is also shown in this table. a
Qmin D A2,and @&sin D A'.
ELbaM
Propane
cis-Butane 0.079 -0.270 -0.595 0.866 -0,002 -1.035 0.006 1.907 -0.021 -0.853 18.581 -0.247 -7.217 -12.775 -5.806 13.023
0.076 -0,271 -0.559 0.830 0.003 -1.394 0.010 1.969 -0.006 -0.575 20.003 -0.632 -5.267 -12.319 -7.684 12.951
trans-Butane 0.072 -0.300 -0,126 0.426 0.000 -1.474 0.000 2.872 -0.493 -0.905 1.155 12.860 14.281
0.069 -0.303 -0.112 0.415 0.000 -1.554
O.Oo0 2.331 -0.002 -0.776 0.950 11.627 14.220
c u Bulanr
m,,s
BYI."(
Penlane
Figure 3. Inclusion of a limited number of off-atom sites X on saturated
hydrocarbons.
do not expect to obtain a better RMSD with the distributed multipole seriesderived charges. Nevertheless, a small difference in the values of the RMSD reflects the fact that distinct sets of charges are able to reproduce the electrostatic potential with comparable accuracy, as may be seen in Table I. Although the values of RMSD appear to be fairly small, the very significant mean absolute deviations reveal a clear inadequacy of both methods to describe the molecular charge distributionof saturated hydrocarbons satisfactorily. Table I shows that, for each basis set, the different sets of net atomic charges derived from the molecular electrostatic potential do not reflect any consistent transferability of the carbonhydrogen bond polarity; i.e., the magnitude and even the sign of the charges on C and H differ from molecule to molecule. In excellent quantitative accordance with the model of Rghini et aL3' the methane charges reveal a clear C6-Hd+ polarity, which, unexpectedly, turns out to be inverted in ethane. For the other aliphatic molecules presented here, only the terminal carbons
Distributed Multipole Analysis and Electrostatic Properties
TABLE IIk Influence of the Inclwion of A d d i t i d Sites. the Potential-Mved Net Point Charges' (q,) of Several HydrocarbonsCUs& Different Basis Setsc 011
~
40
atomd ethane c1, c2 HI, Hz, Ha, H4, H5, H6 X RMSIY
Af
6-31G -0.539 0.106 0.443 0.066 49.87
6-31G*
6-31G**
-0.601 0.114 0.517 0.070 62.55
-0.609 0.118 0.511 0.074 70.88
The Journal of Physical Chemistry, Vol. 97, No.25, 1993 6633
TABLEW. ~ c e o f ~ E & c t r o n Correlatiod on tbe Net Atomic Chargesbof scrcnl Hydrocubonscwith Comparison behreen Dlstrlbuted Moltipole Potentid-Derived (m) and Moleculu Electrodrtic Poteati.dDerived (9,) cllugea atomic and atomic centers only bond center# atomd qD WMM Qu methane -0.549 0.144 C 0.137 -0.036 HI, Hz, H3, H4 RMSIT
propane
A
cis-butane CI, c4 c2, c3 HI,HIO Hz, H3, Ha, H9 H4, H5, Hs, H7 XI, x3 XZ RMSD A
tram-butane CI, c4 c2, c3 HI, HIO Hz, Ha H3, H9 H4, H6 H5, H7 XI, x3 XZ RMSD A
pentane CI, c5 c2, c4 c3
HI, Hi2 Hz,H3, €310,HII H4, H5, Ha, H9 Hs, H7 XI, xc x2, x3 RMSD A
-0.551 -0.620 0.111 0.104 0.096 0.445 0.074 50.56
-0.612 -0.709 0.120 0.112 0.105 0.517 0.075 41.98
Af
-0.619 -0.725 0.123 0.116 0.112 0.514 0.078 44.77
-0.562 -0.617 0.111 0.118 0.090 0.440 0.463 0.070 7 1.82
-0.625 -0.703 0.120 0.116 0.097 0.5 13 0.535 0.072 50.63
-0.632 -0.713 0.123 0.120 0.101 0.508 0.541 0.075 109.08
-0.554 -0.636 0.111 0.107 0.103 0.099 0.093 0.448 0.456 0.072 102.02
-0.727 -0.615 0.120 0.115 0.1 11 0.108 0.102 0.521 0.531 0.073 483.09
-0.622 -0.744 0.123 0.119 0.115 0.114 0.108 0.519 0.534 0.076 100.08
-0.546 -0.652 -0.748 0.109 0.103 0.095 0.107 0.462 0.499 0.069 21OOO.00
-0.61 1 -0.743 -0.834 0.118 0.112 0.104 0.115 0.539 0.569 0.069 335.38
-0.617 -0.758 -0.849 0.121 0.115 0.111 0.122 0.535 0.570 0.072 219.76
ethane c1, cz HI, Hz, H3, H4, H5, H6 X RMSD A propane CI, c3
cz
Hi, Ha H2, H3, Ha,H7 H4, Hs XI, xz RMSD A cis-butane CI, c4 c2,
XZ RMSD A pentane CI, c5 cz, c4
0 Distributtdmultipoleanalysiscarried out at atomicand bondcentem. In electronic charge units (ecu). Respectively, 6-31G, 6-31G*, and 6-31G** full geometry optimization. Indices given in Figure 3. * Root mean square deviation between the ab initio computed and the point charge derived electrostatic potentials. All values XlW3 au.IMcan absolute deviation exprcased as a percent.
bear a negative charge. However, these compounds show an interesting common feature: their terminal methyl groups carry a net negative charge (uiz., qCH3 = -0.1 12 ecu (electronic charge unit) for propane,4.059 ecu for cis-butane,4.079 ecu for transbutane, and 4 . 1 12 ecu for pentane, using the 6-31G** basis set) which agrees with experimental data.38 This fact is obviously related to the inductive effects of the alkyl groups.39 For example, propane may actually be considered as an assembly of an acceptor methyl group and a donor ethyl group."o So far, we have seen that nonpolar systems are inadequately described by molecular electrostatic potential derived net atomic charges. This undoubtedly stems from the impossibility of such a method to take into account high-order multipole moments, uiz., hexadecapole (see Table 11), which are conspicuously predominant in these compounds.
c3
0.046 42.70
0.528 129.80
0.073 -0.024
0.134 -0.045
0.373 181.78
0.464 414.20
-0.238 0.365 0.039 0.046 -0.077
0.103 0.140 -0.036 -0.041 -0.056
0.291 465.86
0.492 709.75
-0.080 0.180 0.011 0.006 -0.061
0.117 0,090 -0.039 -0.042 -0.042
0.406 265.35
0.452 459.01
-0.192 0.168 0.042 0.038 0.037 -0.053 -0.041
0.108 0.127 -0.036 -0.043
0.309 401.49
0.475 21000.00
-0.257 0.245 0.127 0.059 0.046 -0.073 -0.058
0.101 0.123 0.092 -0.035 -0.038 -0.054 -0.052
0.347 270.28
0.533 617.42
-0.044
-0.059 -0.054
-0,590 0.1 14 0.493 0.066 64.53 -0.590 -0,689 0.118 0.1 11 0.106 0.488 0.069 134.69 -0.605 -0.680 0.119 0.115 0.097 0.483 0.519 0.064 64.55 -0,591 -0.690 0.118 0.113 0.110 0.101 0.106 0.493 0.487 0.066 150.36 -0.593 -0.713 -0.786 0.115 0.111 0.104 0.111 0.512 0.530 0.062 8 1.95
a MP2/6-31G** computation. In electronic charge unite (a) 6-. 31G** full geometry optimization. Indices given in Figures 2 and 3.
Root mean square deviation between the ab initio computed and the point chargederivtdelectrostaticpotentials.Allvaluee XlW3 au. /Mean absolute deviation expressed as a percent. C-C bond center only.
e
In contrast, the multipole series fitted charges are consistently transferable from one molecule to another structurally similar one, with no noticeablevariations in magnitude (uiz., qc = 0.130 ccu for the terminal carbons in propane, rrans-butane and pentane). But the most strikingresult is undoubtedly the invariant W - H b polarity, which, a priori, contradicts general chemical intuition, suggesting that hydrogen is more electronegative than carbon. Nevertheless, this important fact has already been mentioned in the l i t e r a t ~ r eand ~ . admittedly ~~~ corresponds to the most suitable picture for the classical interpretation of inductive effects in alkanes. The inuertedpolarity in these species
6634 The Journal of Physical Chemistry, Vol. 97, No. 25, 1993 does not affect, however, the net atomic charge of the terminal methyl groups, which consistently remains negative in accordance with the inductive effects observed experi~nentally.~~ Table I1 shows that, unlike molecular electrostatic potential-derived charges (qv),the present sets of qDMM charges are slightly more effective in reproducing the octopoles in both magnitude and direction (albeit still far from the reference SCF values). However, the description of the hexadecapoles is totally unacceptable and may be the origin of the large fitting errors encountered in this case study. Regarding the basis set dependence, Table I does not reveal any major changesin the potentialderived point chargescomputed with the three split-valence type 6-31G, 6-31G*, and 6-31G** basis sets. However, these charges generally have a greater magnitude with the 6-31G** basis set. The situation is more critical with thechargesderivedfromdistributed multipoleseries. Since individual multipole moments Q",, result from the contribution of the overlap densities x*,,xvand, hence, from the nature of the Gaussian primitive functions, the outcome of a distributed multipole analysis is likely to depend on the basis set.23,24DMMderived charges obtained with 6-31G and 6-31G* wave functions are similar, indicating that the inclusion of polarization functions on carbon atoms has a minor effect on the electron density distribution. On the contrary, the addition of pfunctions on hydrogen atoms modifies the multipole series derived charges markedly. It seemsworth noting that, for each species, the charge borne by the terminal carbon atoms is significantly greater with the 6-31G** basis set. However, central carbon atoms bear an equivalent net atomic charge, whatever the basis set. It may be observed that, although potential-derivedand DMMderived charges are substantiallydifferent,their respective RMSD values are somehow very similar (at least, more similar than with the charges obtained from a Mulliken population analysis; e.g., for trans-butane, using the 6-31G** basis set, RMSD = 0.779 X l e 3 for Mulliken charges). In addition, it is worth noting that, in most cases, unlike higher multipole moments (especially hexadecapoles),dipole and quadrupole moments computed from the DMM-derived charges are fairly close to their corresponding SCF values. This is mostly accounted for by the rather longrange effects of the dipole in the multipolar part of the potential. Nevertheless, the charges that we have computed from both methods are far from acceptable, principally because an atomcentered point charge model is not adapted to the description of the charge distribution of nonpolar molecules. Molecular electrostatic potential-derived charges are particularly critical since they do not exhibit any transferability and they lead to significant errors when trying to reproduce the ab initio SCF potential. Thus, it may seem that the only way to obtain reasonably low values of the RMSD, as well as mean absolute deviations, and reach transferability is to change the fitting procedure by adding a large number of off-atom point charges to yield a better reproduction of the higher multipole moments. That this is not completely true may be seen in detail in the next section. 5. Refmment of the Model
Up until now, we have seen that fixing point charges on the saturated hydrocarbon nuclei yields significant errors when attempting to reproduce the electrostatic potential of these nonpolar species. It is obvious that a thorough description of the molecular charge distribution requires the introduction of a substantial number of adequately located off-atom sites and therefore suffers from two nonnegligible drawbacks: the computational cost required by the fitting procedure and the fairly low tractability of the model for molecular dynamics (MD) or Monte Carlo (MC) simulation. Our intermediate approach consists of including one extra site per chemical bond involving two heavy atoms (Le., other than hydrogen). By doing so, the degrees of freedom of the fit are
Chipot et al. increased, so the procedure yields a more acceptable reproduction of the higher multipole moments. For practical reasons,dummy charges are located at the geometrical center of the bonds (see Figure 3). This arbitrary choice may lead to criticisms since the electronicdensity on a H 3 w H 2 bond and on a HzC-CH~bond are admittedly different. Nevertheless, as may be seen in Table 111, such an approximation appears to be justified and leads to very satisfactory point charge models. The first remarkable result is the transferability from one molecule to another structurallyrelated one. A consistent cb-H& polarity is obtained for each compound. It may also be observed that the charges of the terminal carbons, as well as the dummy charges, are almost constant from ethane to pentane (i.e., -0.632 I qc I 4 6 0 9 and 0.511 I qx I 0.570). Together with the charge transferability, a very significant decrease of the systematic errors in the fitting procedure is observed for each species. The simple inclusion of one extra site per carbon-rbon bond allowed both RMSD and mean absolute deviationsto be lowered by an averagefactor of 4. This interesting fact is undoubtedly related to the better description of the higher multipole moments, which now agree with their respective quantum mechanically calculated values, as shown in Table 11. So far, we have seen that the inclusion of a single additional site per C-C bond yields an acceptable charge transferability, as observed with simple DMM potential-derived atomic charge models. In addition, the errors in reproducing the electrostatic potential have decreased markedly. Nevertheless, these results may certainly be improved by means of supplementary off-atom point charges which will contribute to the enhancement of the description of the potential and, therefore, lower the value of the RMSD. Tests have been carried out with dummy charges at the center of C-H bonds as well as C-C bonds and showed indeed an additional decrease of the RMSD (e.g., for pentane, RMSD = 0.058 X lC3 au, A = 115.1096, and p = 0.061 D with a 17 atomic 16 off-atom point charge model, whereas a 17 atomic + 4 off-atom point charge model yields RMSD = 0.072 X l C 3 au, A = 219.76%,and p = 0.062 D). In view of these results, it may be concluded that it is still possible to improve the model by adding extra dummy charges (viz., two per chemical bond), but, by doing so, the system becomes less and less tractable. This problem is critical for the parametrization of force-fields, where it is desirable to avoid the use of off-atom point charges.
+
6. Influence of Intramokular Electron Correlation
The sensitivity of the electron density distribution to electron correlationhas been studied using second-order M~rller-Pleaset'~.~ (MP2) 6-31G** wave functions. As may be observed in Table IV, there is a slight and consistent decrease in the magnitude of both DMM- and electrostatic potential-derived net atomic charges. This interesting fact is related to the bonding-antibonding correlation, which causes the electron density to be drained out of the bonding region toward the n u ~ l e i . ~ Such ~ , ~ ' an effect may be illustrated by the following simpleexample: For each aliphatic species,the carbon-hydrogen chemical bonds are described in terms of the two approximate functions wonding and h n t i b ~ i n g rdefied by
'Pantibonding
= '2x1 - clxZ
(26)
At the SCF level, the Mulliken population is written as
When including intramolecular electron correlation, one should consider +%onding + Xpantibonding to evaluate the correlated charges
Distributed Multipole Analysis and Electrostatic Properties
TABLE V: Influence of Intramolecular Electron Correlation' on the Multipole Momentsbof Several Hydrocarbons' Calculated from Distributed Multipole Poteitthi-Derired and Molecular Electrwtntic Potential-Derived Net Atomic Charges atomic and atomic centers only bond V DMM V Methane 1.202 0.381 1.621 -1.057 -1.026 0.676 0.561 -0.131 0.767 0.168 -0.730 -0.037
-0.315 -0.100 -0.425 0.277 0.269 -0.177 -0.147 0.034 -0.201 -0.044 0.191 0.010
0.084 0.084 -0.168 0.594 0.594 1.584 0.198 -0.792 -0.792
0.159 0.159 -0,318 1.102 1.102 2.940 0.367 -1 -470 -1.470
0.047 -0.502 0.223 0.279 0.000 -0.001 0.004 0.542 1.359 -1.904 -2.868 -0.969 -0.139 1.849 1.019 -0.880 0.054 -0.305 -0.109 0.4 14 0.001 0.974 -0.002 -1.565 1.827 -1.234 7.123 6.832 3.650 -5.153 -1.971 -1.679
The Journal of Physical Chemistry, Vol. 97, No. 25, 1993 6635 TABLE V (Cmtioued) atomic and atomic centers only bond cente, V DMM V trow-Butane @m @rrrr
MP2 1.177 0.374 1.587 -0.8 17 -1.005 0.916 -0.213 1.252 0.274 -1.191 -0.061
Ethane 0.281 0.281 -0.562 -2.238 -2.238 -5.969 -0.746 2.984 2.984
0.053 -0.267 -0.014 0.282 0.000 -0.746 -0.002 0.563 0.042 0.143 -2.297 1.454 1.430 1.137 1.160 -2.590
0.053 -0.518 -0.016 0.534 0.000 -1.266 0.009 3.231 -0.336 -1.635 2.960 -4.300 -4.217 -1.439 -1.521 5.739
0.05 1 -0.501 -0.026 0.526 0.000 -1.019 0.009 2.645 -0.004 -1.626 2.706 -7.293 -2.732 0.927 -3.633 6.366
cis-Butane 0.052 -0.231 -0.255 0.486 0.001 -0.403 -0.003 0.532 -0.058 -0.070 -2.884 4.075 1.437 0.123 2.761 4.198
0.063 -0.229 -0.576 0.804 -0.002 -0.910 0.006 1.686 0.026 -0.805 18.073 -0.158 -6.749 -12.332 -5.741 12.490
0.061 -0.231 -0.545 0.776 0.002 -1.207 0.009 1.783 -0.005 -0.575 19.188 -0.437 -5.173 -1 1.962 -7.226 12.399
0.044 -0.287 -0.042 0.330 0.000 -0,371 0.000 -1.957 2.062 0.267 0.152 6.833
0.058 -0.229 -0.152 0.381
-0.234 -0.141 0.375
0.049 -0.816 0.155 0.661 0.003 1.034 O.OO0 -1.891 -1.442 2.298 0.114 2.952 2.101 -0.482 0.368 -2.470
0.03 1 -0.417 -0.105 0.522 0.001 1.208 O.OO0 -1.278 0.016 0.055 -8.338 2.377 -0.401 2.780 5.558 -5.157
13.681 6.277 -9.930 -2.526 -3.751 -10.263 10.234
13.689 7.222 -8.866 -2.399 -4.823 -12.368 10.180
0.053 -0.908 -0.192 1.100 0.002 2.473
0.050 4.893 -0.166 1.059
a
D
0.004 2.3 17
O.Oo0
O.OO0
-5.154 0.598 2.082 -6.148 -8.503 -14.970 -0.159 6.307 8.662
-4.313 O.OO0
1.996 -5.782 -13.049 -11.130 3.850 1.931 9.199
MP2/6-31GS* computation. Traceless momcntdl p in D. in in D A2, and a@, in D A). e6-31G** full geometry
A, , 0
optimization. d C-C bond center only.
Propane
rrans-Butane 0.054 -0.099 -0.105 0.204 0.000 -0.501
-1.955 -3.613 -0.156 1.503 2.1 11 0.360 -1.227
Pentane
0.662
0.294 0.294 -0.587 -3.106 -3.106 -8.284 -1.035 4.142 4.142
5.988 -2.576 -7.698 0.869 1.711 -5.485 6.129
MP2
0.055
0.OOO
O.OO0
-1.352
-1.423 0.000 2.150 0.000 -0.727 0.894 11.265
O.Oo0
O.Oo0
0.702 0.178 -0.319 0.748 -1.347
2.575 -0.385 -0.839 1.057 12.455
Qcpmlatd
and
Gmlated, so that
Thus, when the carbon net charge decreases, the charge borne by the hydrogen is expected to increase accordingly and vice versa (e.g., for methane, the potential-derived charges are #cF = -0.561 ecu, = 0.140 ecu, q y = -0.549 ecu, and = 0.137 ecu, whereas multipole series-derived charges are = 0.172 ecu, d","' = -0.043 ecu, qp = 0.144 ecu, and qH = -0.036 ecu). Table V reveals an average decrease of the multipole moments with respect to the corresponding SCF values. This phenomenon stems from the contraction of the orbitals allowed by electron correlati0n.2~ Regarding transferability from one compound to another structurally similar one, these charges show the same trend as the one computed at the SCF level. For this reason, we have applied the scheme described in Section 5 . The simple inclusion of one dummy charge per chemical bond involving only carbon atoms yields more satisfactoryresults than the conventionalelectrostatic potential-derived net atomic charge models.
&
dCF
gCF
"'
7. COnClUSiOM We have seen that distributed multipole expansions, just like molecular electrostatic potentials or fields, can also be applied to obtain point charge models. In the method described here, we evaluate the individual potentials created by multipole series centered at different locations of the molecule. Since the fitting procedure is carried out analytically, the use of a grid of regularly, or irregularly, spaced points is no longer necessary, and the computational time required for the fit is reduced substantially. The absence of charge transferability from one molecule to another structurally similar one is a major drawback of the potential-derived charges. On the contrary, DMM-derived
6636 The Journal of Physical Chemistry, Vol. 97, No. 25, 1993 charges are totally transferable and, thus, in better agreement with chemical intuition. Net atomic charges fitted to the molecular electrostatic potential are satisfactory, from a mathematical point of view. They are indeed the most suitable to reproduce this quantity, in a least-squares sense. Depending on the nature of the molecule and the adequacy of the point charge model, the accuracy of the fit is more or less reasonable. For very polar compounds, multipole series-derived and potential-derived charges are almost equivalent.19 In both cases, the fitting error is small because the first terms of the multipolar expansion are large enough (e.g., for acetate ion,22 potential-derived charges give RMSD = 0.325 X 10-3 au and A = 0.2296, whereas DMM-derived charges give RMSD = 0.681 X 10-3 au and A = 0.46%). Studies of such molecules are often reported in the literature, since they generally yield acceptable charge distributions. In the present study, we have considered a series of common saturated hydrocarbons. Because of their very nonpolarcharacter, these aliphatic species create problems. Indeed, in addition to their lack of transferability, potential-derived charges show no consistent polarity of the chemical bonds. Although values of the RMSD are reasonably low, mean absolute deviations are markedly large, indicating that the grid points are too far from the nuclei to take multipole moments of higher orders into account (avoidance of the effects of electron cloud penetration) and that a simple net atomic charge model is not adapted to the description of such a molecular charge distribution. Transferability of the potential-derived charges is reached when using off-atom charges located at the center of carbon-carbon bonds. A consistent Cb-Hd+ polarity is obtained for each compound, and the RMSD values, as well as the deviations, are lowered significantly. This, unfortunately, contributes to the increase of computational time and to the decrease of tractability. On the contrary, distributed multipole series-derived charges are perfectly transferable and exhibit a clear C*+-Hb polarity. This latter fact agrees with the literature," where it is stated that a generalized Mulliken scheme with an appropriate partitioning of the overlap populations should be applied for the computation of the net atomic charges of saturated hydrocarbons. Such a scheme is very similar to a distributed multipole analysis.23 The large errors in both RMSD and mean absolute deviations indicate again that a simple atomic charge model is incomplete for a correct reproduction of the molecular electrostatic potential. It should therefore be observed that a set of charges suitable for the description of such a quantity is rather unlikely to reproduce the polarity of the bonds and oice versa. The net atomic charges derived from the DMMs seem to correspond to the most acceptable compromise between polarity, transferability, and simplicity of the model.
Acbwledgment. We thank Dr. F. Colonna for many stimulating discussions and the continuous interest and encouragement of Profs. B. Maigret and J.-L. Rivail. C.C. is indebted to Roussel Uclaf Co. (France) for his doctoral fellowship. We thank the Groupement Scientifique "Modtlisation Molkulaire" IBM-CNRS for their generous support. The computations presented in this paper were carried out on the IBM-3090/600 and on the Serial IBM-RS/6000 cluster of the Cornel1 National Supercomputer, Cornel1 University, Ithaca, NY. This work was also supported by the U S . National Science Foundation (Grant numbers DMB-90-15815 and INT-91-15638) and the action incitative CNRS-NSF LET92 no 21 MDRI.
Chipot et al.
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-.-
