6201
J. Phys. Chem. 1991, 95, 6201-621 1
model. This deficiency is sometimes balanced by the neglect of single, triple, and higher excitations, for example at the a-carbon. Proceeding to the next higher d e r of perturbation theory,UMP3, increases the doubles term as expected and spoils this fortuitous agreement, moving the calculated coupling constants away from the QCISD(T) values, except for the splitting due to the a-hydrogen. Table V collects the results of the QCISD(T) calculations for comparison with experiment. The DZ basis is of little use in forming quantitative prediction of unpaired spin density. The agreement observed for the averaged @-hydrogencoupling is fortuitous. The addition of polarization functions to give the DZP basis markedly improves this accord, and results within 10% of experiment can be expected except at &hydrogens. The much larger TZP basis set improves the level of agreement still further, especially when account is taken of vibrational corrections. These have been most thoroughly evaluated by Chipmanz' and are adopted here from that work. This implies that correlation effects are roughly constant across the regions of the energy hypersurface which are influential in developing vibrational correctionsdue to zero-point motion. Further exploration of basis set convergence with regard to the calculation of the isotropic coupling constants in the ethyl radical are precluded by current computational facilities; however, in the HzCN radical, extensions beyond the TZP level, including diffuse shells, additional polarization functions, and higher angular momentum functions on each atom produced little change in the computed a values." There is only one minimum of the UMP2/6-31G(d) surface for the ethyl radical (other than the dissociation asymptotes), and this is the structure employed above. However, Ruscic et al.29
located another stationary point of C, symmetry which has one imaginary frequency (and is thus a saddle point) at this level of theory. In this structure one methylenic C-H bond eclipses a methyl C-H bond, while the other bisects the remaining C-H pair. A very small energy difference, CO.1 kcal/mol, was found2% between these structures, and such a conclusion is also reached here. With the TZP basis set, the energy separation is computed to be just 0.08 kcal/mol. The frozenare QCISD(T)/TZP energy at the UMP2/6-31G(d) minimum is -78.9746 hartrees. We have also estimated the coupling constants in this low-lying transition structure, which is of 2A" symmetry. The QCISD(T)/DZP results for the isotropic coupling constants are a( 13Ca) = 85.2 MHz, a(I3Cg) = 4 5 . 1 MHz, a('H,) = -76.4 MHz, and a('Hg) = 66.8 MHz. Except at the a-carbon, these values are within a few percent of the corresponding splittings calculated at the global minimum. Even at the radical center the computed coupling constant is reduced by only 15 MHz. Changes, similar to those described above, occurring at the UMP2/6-31G(d) minimum upon improvement of the basis set, are also noted here. Correlation corrections are a few percent larger at this geometry, with the maximum computed decrement of 5 1% being in the UHF value of a("C,). Once again, the averaged isotropic coupling constant computed for the &hydrogens shows little sensitivity to the inclusion of electron correlation in the UHF wave function.
-
Acknowledgment. The research described herein has been supported by the Office of Basic Energy Sciences of the United States Department of Energy. D. M. Chipman is thanked for a preprint of ref 2 1. Registry NO. Eth~l-'~C,, 134629-43-9.
Charge Flux and Electrostatlc Forces In Planar Molecules Uri Dinur Department of Chemistry, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel (Received: February 7, 1991)
The electrostatic forces exerted by external point charges on model compounds are calculated ab initio and analyzed. It is shown that in general molecules cannot be represented as collections of isolated point charges interacting through vacuum. The distortion of the molecular geometry, caused by the external source, leads to an intramolecular charge redistribution ("charge flux*) that modifies the static pairwise force, in some cases drastically. In general, charge flux introduces a dynamical anisotropy into electrostatic forces that is usually much larger than the static anisotropy due to atomic dipoles and higher atomic multipoles. The effect of charge flux is also larger than that of polarizability. Because of charge flux, electrostatic forces on nuclei in molecules are, in general, nonlocal and nonpairwise. Rather, electrostatic forces on nuclei in molecules are determined by the response of all other nuclear sites to the external source, and particularly the sites that are closest to the perturbing source. Consequently, flux forces may oppose the forces from the static multipoles and even override them. The calculation of charge flux parameters is not straightforward with common methods of determining atomic charges. This problem is solved by using the recently introduced force related (FR) atomic multipoles which are well-defined derivatives of quantum mechanical expectation values. The FR atomic multipoles and their flux successfully reproduce the ab initio forces in the molecules discussed in this work.
1. Introduction The frozen point charges madel for inter- and intramolecular electrostatic interaction has proved to be very successful in simulations of rigid bodies, as practiced in calculations of crystal structures,I Monte Carlo calculations of condensed phase^,^^^ or ( I ) E.g.: Williams, D. E.;Houpt, D. J. Acra Crysrallogr. 1986, BIZ, 286. (2) E.g.: Brims. J. M.; Matsui, T.; Jorgensen, W. L. J . Compur. Chcm. 1990, 11, 958. (3) Jorgensen, W. L.; Chandrasekhar, J.; Madura, J. D.; Impey, R. W.; Klein, M. L. J . Chrm. Phys. 1983, 79, 926.
0022-3654/91/2095-6201$02.50/0
molecular dynamics of condensed phases with frozen b o n d ~ . ~ J Recently it was noticed, however, that when molecular flexibility is introduced into simulations, Le., the molecules are allowed to vibrate along with the motion of the center of mass, the conventional electrostatic model is no longer adequatea6In particular, the electrostatic forces on the nuclei are not properly described by frozen atomic point charges. The main reason is the fact that (4) Straatsma, T. P.; McCamon, J. A. J . Compur. Chcm. 1990, 1 I , 943. (5) Ryckaert, J. P.; Ciccotti, G.; Berendsen, H. J. C. J . Compur. Phys. 1977, 23, 327.
0 1991 American Chemical Society
6202 The Journal of Physical Chemistry, Vol. 95, No. 16, 1991 as the molecular internal coordinates vibrate the internal distribution of charge vibrates as well. Although the amplitudes of ground-state vibrations are small, so that the changes in intramolecular charge distribution are small, the rate of these changes, Le., the gradients of the molecular charge density with respect to the nuclear coordinates, may be large. Consequently, the forces that result from these gradients may be larger in magnitude than the forces that arise from the static charge distribution. Intramolecular charge redistribution that accompanies molecular flexibility is thus found to have a most significant effect on intermolecular forces. The effect is of first order and considerably larger than that of polarizability.6 The inadequacy of the static point charge model for representing intermolecular electrostatic forces means that current empirical force fields lack a whole new type of parameters that determine the change of atomic charges with the change in the molecular internal coordinates. On the other hand, the significance of such parameters is long recognized in the literature on IR intensities, where the internal redistribution of charge with the molecular vibrations is termed ”charge flux”. Recently, charge flux parameters were introduced by Miwa and Machida in an empirical force field for alkanes,’ although the focus in their work was on IR spectra and thermodynamic functions and not on explicit intermolecular forces. A proper description of intramolecular charge flux is in general a nontrivial task. In fact, one has to decide first on the technical definition of atomic charges before determining their gradients with respect to nuclear displacements. When this question is resolved, charge flux can be calculated. However, what is really needed is some general understanding of this phenomenon in order to implement it in force field calculations. Currently, the physics of charge flux is not well known and no model exists with respect to the way by which a displacement of a particular nucleus in the molecule induces a charge flow from one site to another. The purpose of the present work is to begin addressing this question, with the intention of incorporating charge fluxes in future empirical force fields. The paper is organized as follows. Section 2 briefly outlines the theoretical basis for the present work. The approach taken is based upon recent theoretical analysis of molecular response to an external electrostatic potential.&” For planar systems the analysis shows that the molecular electrostatic potential, as well as the forces on the nuclei due to an external electrostatic source (two different entities), are consistently characterized by “atomic” multipoles, (e.g., point charges, dipoles, quadrupoles, etc., located a t the nuclei) that are derived from the appropriate elements of the molecular multipolar tensors. The atomic multipoles thus defined8J0*I4are termed force related (FR) because they are directly related to theforces on the nuclei. The formalism has been developed for planar molecules and tested for perpendicular electrostatic forces which do not involve charge flux.” Section 3 implements this formalism with respect to six model compounds. Atomic charges, atomic dipoles, and their in-plane fluxes in the model compounds are obtained from ab initio calculations and used to analyze the electrostatic forces on the nuclei due to external point charges. 2. Theory
The interaction energy U of a neutral molecular system with an external electrostatic potential, under the conditions that the source of the potential is far enough and its field weak enough so that penetration and polarizability effects can be ignored, may be written in terms of the static molecular multipoles and the field derivatives: (6) Dinur, U. J . Phys. Chem. 1990, 94, 5669. (7) Miwa, Y.; Machida, K. J . Am. Chem. Soc. 1988,110.5183. For IR related charm and charge flux Darameters. in coniunction with ab initio calculations,scc for example: Rimos, M. N:;Castiglioni, C.; Gussoni, M.; Zerbi, G. Chem. Phys. Len. 1990, 170, 335. (8) Dinur, U.; Hagler, A. T. J . Chcm. Phys. 1989, 91, 2949. (9) Dinur, U.; Hagler, A. T. J . Chem. Phys. 1989, 91, 2959. (IO) Dinur, U. J . Compur. Chem. 1991, 12, 91. (11) Dinur, U. J . Compur. Chem. 1991, 12, 469.
Dinur
u = -PEA - I / J ~ ~ & / V &+E... A,J
(1)
k,l
In eq 1 p and 8 are respectively the permanent molecular dipole and quadrupole moments, and E A and VEA are the electric field and its gradient. Higher terms in the series involve higher order multipoles and field derivatives. The field and the multipoles are evaluated a t and relative to an origin that is chosen, as a matter of convenience, at the coordinates of a particular nuclear site A. The force on nucleus A in the molecule is obtained by differentiating eq 1 with respect to the Cartesian coordinates rA of the nucleus. For a neutral molecule one obtains FA,j = 4 U / d r A , i
= (VA,ip)*EA + p V A , i E A
+ hC(VA,iek/)VA,kEA,/+ O(VA2EA) k,l
(2) where i is x , y, or z and V Adenotes the derivatives with respect to the coordinates of nucleus A. As seen in eq 2, the local response of a molecule (i.e., the response of a particular nuclear site) to an external field depends on the derivatives of the molecular moments. The collections of these derivatives form what may be termed “atomic multipolar tensors” in generalization of the “atomic polar tensor” introduced by Morcillo and co-worker~.’~J~ Although eq 2 is a rigorous, well-defined, quantum mechanical statement, it is not so useful for empirical force fields and large-scale simulations, where one would like to write the interactions and the forces in terms of atomic multipoles. For the energy one writes
u = CqAVA - C I l l A ’ E A + A
A
(3)
where VAand EA are respectively the external electrostatic potential and its field at nucleus A. The force is given correspondingly by FA.i
= 4u/arA,i
= qAEA + C(VA,iqB)VB + mA’VA,IEA + x(VA,imB)*EB B
B
t4)
Since eqs 2 and 4 describe the same quantity, there must be a correspondence between the derivatives of the molecular moments in (2) and the atomic multipoles in (4), from which one can write the atomic moments in terms of the multipolar tensors. For planar molecules the analysis yields the following such definitions for the (force related) atomic charges and dipolessJO qA = (5a) mA.z
=
aaxz 6x2 %% =y 3G
where the molecule is in the xy plane, is the molecular second moment ( x z ) less the contrikution of the atomic charges as defined in (5a), and likewise for Myz. Higher order atomic multipoles can similarly be defined in terms of particular elements of the molecular octupolar and hexadecapolar tensors. A detailed discussion of the properties of these quantities is given in ref 10. Here we merely note that (a) the atomic moments thus defined are origin independent; (b) they exactly reproduce the molecular moments from which they are derived; (c) the expansion of the molecular electrostatic potential in terms of the FR atomic multipoles is usually a better convergent series than the expansion in terms of the molecular moment~;~Jl and (d) the FR atomic multipoles satisfy a “conservation” rule, e.g.: t 5c) ZmA, = T m A y = 0 A
(12) Morcillo, J.; Zamorano, L. J.; Heredia, J . M. V. Specrrochim. Acra 1966, 22, 1969. ( 1 3 ) Morcillo, J.; Biarge, J . F.;Hcredia, J. M. V.; Medina, A. J . Mol. Srrucf. 1969, 3, 77.
The Journal of Physical Chemistry, Vol. 95, No. 16, 1991 6203
Electrostatic Forces in Planar Molecules Equation 5c is a general propert that characterizes higher order FR atomic multipoles as well. I P An examination of eqs 3 and 4 reveals that the force on a particular nucleus is determined by elements, some of which are missing from standard empirical force fields. These are the derivatives of the atomic charges, dipoles, etc., with respect to the Cartesian coordinates, or, extending the nomenclatureused in IR spectroscopy, “multipolar fluxes”. These fluxes would be difficult to determine by means of common methods of obtaining atomic charges, such as fitting the molecular electrostatic potential, since the derivatives of the least-squares results may be ill-defined. In contrast, the multipolar fluxes are well-defined quantum mechanical quantities within the present formalism. Explicitly, they are the second derivatives of the molecular momentsI4
a2k
-aqA =axe
amA., -=y2-. axe
az,ax, amA, --
a2fixz aZAaXB’
axe
a2GYz
-y
2
a
(6)
aqA
23-I as,
axe
asi
axe
= CjIS/BiB
(7)
i
where /IS, denotes the charge flux aqA/as, and B,B denotes the transformation matrix between Cartesian and internal coordinates. The flux matrix j! has the dimension n X (3n 6) for a molecule with n atoms. It is obtained by inverting eq 7
-
j$ = (BtB)-’BtVq
(8)
where Vq is the matrix of Cartesian charge flux laqA/axB). A similar transformation can be applied to the magnitude of the atomic dipoles. Before concluding this section it is worth comparing charge flux with polarizability, since the latter seems to be a much more familiar phenomenon in the context of empirical force fields. Polarizability becomes important when the external field becomes significant and modifies the molecular moments. In such case eq 1 has to include also the interaction of the field with the induced moments. For the well-known leading term we have U s Uo- hE&E + .I. (9) were Uodenotes the interaction with the permanent moments (eq 1) and &E is the induced dipole moment. Higher order terms will include hyperpolarizabilityterms, proportional to higher powers of the field, as well as quadrupole polarizability, octupolar po(14) Dinur,
U. Chrm. Phys. h r r . 1990, 166, 211.
FA
= Fo,A
+ ‘/2VAaxxE2 +
(10)
where F0.A is the force due to the permanent multipoles as given by (2) and where x is the field direction. In simulations that treat molecules as rigid, the terms in eqs 2 and 4 containing the multipolar tensors vanish and the force depends only on the molecular moments and the derivatives of the fields. In such cases the next order of approximation after reproducing the molecular electrostatic potential involves the molecular polarizability. However, when flexibility is introduced into the molecular system charge flux appears, and whereas the polarizability force goes like lE21(eq 10) the flux force goes like IEI. This can be inferred from eq 4 and is easily seen in the case of a homogeneous field as follows. From eq 2 we have for the force F~~= (a~,/axA)Ex
(11)
Write pX in terms of atomic charges
with similar equations for dqA/dyB, etc. The second derivatives of the molecular moments can be obtained from ab initio calculations and may be regarded as “computational” observables. The physical origin of the charge flux is as follows. An external force tends not only to move the entire molecule but also to distort its geometry. This inevitably involves orbital rehybridization and reorganization of the intramolecular charge distribution. The “atomic charge” qA, which pertains to the electronic charge distribution surrounding a particular nucleus, changes with this distortion. Because charge is conserved, the charge that is being lost or gained must be exchanged with some other sites in the molecule. In general therefore the atomic charge of atom B changes as atom A is displaced. As a result, the electrostatic forces can be nonlocal-the force on A in eq 4 depends through the charge flux terms on the positions of all other atoms in the molecule. Because charge flux, as well as fluxes of higher order multipoles, are intrinsic molecular properties, it is useful to express them in terms of the internal coordinates, i.e., bonds and valence angles, rather than Cartesian coordinates. The charge flux with respect to an internal si is obtained from
-a q=A
larizability, etc. In the simple case of a homogeneous field the resulting force on nucleus A is
Mx
= CqAxA A
(12)
Substituting in (1 1) yields
Thus, both the charge and its flux contribute to the force to order E. (In general, the atomic multipole and its flux have the same order in the expansion, eq 4.) The charge flux is therefore a first-order effect while polarizability is a second-order effect. On a more intuitive ground we may consider any molecule and the change in its electronic structure due to a displacement of an adjacent molecule versus the change induced by elongating one of the molecule’s own bonds by 1 au. (The change in the atomic charges that is induced by such a bond elongation is a crude measure of charge flux.) In the latter case the molecule practically loses its identity and becomes a transition state. Clearly, the molecule’s own distortion changes its electronic charge distribution much more drastically than any neighbor. 3. Electrostatic Forces in Model Compounds In this section we implement the above-discussed formalism with respect to six model compounds. An external point charge qp is placed at a distance R from a given molecule and the forces on the nuclei as functions of R are calculated ab initio. To isolate the electrostatic interaction the molecules are taken at their ab initio equilibrium geometry, so that all forces on the nuclei result from the external point charge. The magnitude of the external charge was chosen to be 0.1 e. This reduces the effect of polarizability since forces due to static multipoles are proportional to q.p, whereas the polarizability is proportional to. :4 This in addition to the 1 /Pdependence of the polarizability vs the 1/R2 dependence of the atomic charges and l / R 3of the atomic dipoles makes the polarizability contribution at the distances discussed here negligible in comparison with the contribution of the permanent atomic charges and dipoles. The calculations reported below are of the SCF type at the 6-31G** level of theory, except for the case of CO where MP2 calculations were done due the large correlation effect in this case. For HF, H20, formaldehyde, and formamide the FR atomic moments have been reported and analyzed before. A partial list of flux parameters for the water molecule has also been published before. All other flux parameters are reported here for the first time. The calculations were performed with the program CADPAC 4 . 0 ’ ~on a Convex 220 at the Weizmann Institute. First derivatives of the molecular moments were obtained analytically while the second derivatives of these moments were calculated numerically (15) Amos, R. D.;Rice, J. E. CADPAC: The Cambridge Analytical Derivatives Package, issue 4.0, Cambridge, 1987.
6204 The Journal of Physical Chemistry, Vol. 95, No. 16, 1991 2.0
.l.S
,
I 2
. -
-E 0
1.0
between p and A. Figure 1 (top) shows the ab initio forces in the perpendicular approach, and the forces calculated with eq 14 and the FR atomic charges (Table I). Clearly, the ionic picture is fairly accurate for the perpendicular approach, and the external source “sees” two separate point charges. This picture changes somewhat when the external charge approaches the molecule along the bond. As seen from Figure 1 (bottom) the collinear ab initio forces on both nuclei are consistently smaller in magnitude than the perpendicular forces (top). This contradicts the simplistic ionic model, which predicts isotropic forces. An examination of eq 4 shows that the ionic model corresponds to the first term in the expression for the force, namely, FA = qAEA, but other terms exist that will in general lead to anisotropy in the force. Because some of these terms are novel in the context of current force fields we proceed to analyze eq 4 in some detail. Consider first the perpendicular interaction. Let the bond H-F lie along the x axis. The charge flux in the perpendicular direction z (or y) is necessarily zero
I
3
4
5
6
7
8
Dinur
9
1
-aqF = - = oaqH azF
8
*
-0.5
1
-1.0
I
m
2
3
” t ” F”.X
1
4
5
6
7
8
9
RFp ’ Rare 1. Forces on the nuclei of HF due to an external point charge of +0.1 e. The meaning of the symbols is the same for both panels.
from the analytical first derivatives. The numerical differentiation was done with increments of 0.005 A. The 6-31G** dipole moments are characteristically 10-1 5% too high as compared with experimental gas-phase values, and 6-3 lG** quadrupole moments may deviate from experimental data by 258.9 However, it was noted for w a t d that the 6-31G** FR electrostatic parameters agree fairly well with model potentials such as TIP4P that are parametrized for condensed phase.’ We find further that the FR atomic charges for formamide and formic acid (see below, and also refs 9 and 11) coincide surprisingly well with charges obtained from fitting experimental crystal structures of amides and carboxylic acids.’6J7 It appears therefore that, until a model for polarizability is developed, the 6-31G** FR charges and dipoles that are derived for isolated polar molecules may be used as effective electrostatic parameters in simulations of condensed phases of rigid molecules. This conclusion does not automatically apply to the flux parameters, of course. The accuracy of the latter is more difficult to assess because they do not correspond to a specific element of the polar tensor; rather, they contribute to these elements (see eqs 11-13). For the simplest cases, however, some comparison with experiment is possible and is given below. However, the focus of the present work is on qualitative features in electrostatic forces, and these are expected to be correctly reproduced by the ab initio calculations at the level used here. HF. This molecule is a simple prototype for polar molecules and will be discussed at some length. Its 6-31G** SCF electrostatic parameters are given in Table 1. The FR atomic charges in this case are sizable while the atomic dipoles are small. The flux parameters are also seen to be small. These results naturally lead to a simple “ionic” model which depicts HF as two independent point charges. According to such a model the interaction of each point charge with the external source is expected to be FA
qAEA
qAq$/R2
(14)
where qp is the external point charge and R is the distance vector ~
~
~~~~
(16) Hagler, A. T.; Huler, E.; Lifson, S. J. Am. Chcm.Soc. 1974.96.5319.
(17) Lifson, S.;Hagler, A. T.;Dauber, P.J . Am. Chcm. Soc. 1979, 101, 5111.
dZF
(15)
The reason for (15) is that, by symmetry, a perpendicular displacement of either H or F does not change the bond length to first order and is initially equivalent to a mere rotation of the H-F bond. A rotation does not involve work against the bond so that no rehybridization of orbitals occurs. Consequently, there is no charge flux force in the perpendicular interaction. The next term in eq 4 represents the interaction of the permanent atomic dipoles with the external field. This interaction is strictly zero for F, since the field is perpendicular to MF (Figure I , top), and is very small for H, since mHand aEx/dzHare small. Finally, there is a nonzero dipolar flux in the perpendicular direction, because when the fluorine nucleus is displaced vertically, the atomic dipoles-which, unlike the atomic charges, are vectors and have directions-rotate with the bond. The change in the direction of the atomic dipoles constitutes the dipolar flux, am& The exact expression for the perpendicular force due to this flux in planar and linear systems was derived in ref 10 and is given to order VE by FA,LU!L’)= ~ ~ A * V E A , ~
(16)
where A in the present case can be either H or F. Note that the flux force in (16) is determined by the static moment because the flux in this case is a mere geometrical effect that results from the rotation of the bond. For the fluorine nucleus the dipolar flux force is still zero since mF is perpendicular to VEF,z. For the hydrogen center eq 16 leads to only a small contribution to the force. It follows that the forces in the perpendicular direction are governed by the first term in eq 4; Le., they are predominantly ionic, as demonstrated in Figure 1 (top). Consider now the interaction of H F with an external charge that approaches the molecule along the bond. The static dipolar forces are no longer zero by symmetry. More than that, the effect of the force in this case is to distort the bond along with the displacement of the molecule as a whole, so that intramolecular redistribution of charge occurs and, consequently, the flux forces are no longer zero. The force on the fluorine nucleus is then given by FF.II = F F ~ aEF* qFEFs + j f , r ( vF - vH) + “ F , x X + j v ~ ( ~ F-, xEH.x) ( l 7,
--
where use has been made of eq 4 as well as the fact that qF = -qH and mF = -mH. From Table 1it is seen that the charge flux
is such that the fluorine site loses its electronic charge upon elongation of the bond. This affects the force in the following way. The external source attracts the negative fluorine center but as the attraction takes place negative charge is lost and the attraction is reduced. The charge flux generates therefore a repulsiue force that opposes the attraction to the static negative atomic charge. The interaction with the atomic dipole is also
Electrostatic Forces in Planar Molecules repulsive, with respect to both the static dipole and the dipolar flux. Similar considerations can be made with respect to the hydrogen. In particular, the charge flux force in this case is attractive. Displacing the hydrogen toward the external source contracts the H-F bond and this leads to a flow of negative charge to the fluorine center. This stabilizes the interaction with the external positive source since the fluorine is closer to the source than the carbon. This stabilization causes an attraction which is larger than the enhanced repulsion to the more positive, but more remote, hydrogen. Thus, for both nuclei the charge flux forces oppose the forces due to the static point charges. Figure 1, bottom, shows the various contributions to the total force. The static FR atomic charges (open circles) yield a force with magnitude larger than that of the ab initio one. When charge flux is included (open squares), the agreement with the ab initio forces increases, and further improvement is obtained by the addition of atomic dipoles and dipolar flux (open diamonds and filled circles, respectively). The dipolar contribution is felt only at short distances. Except at short distances the anisotropy in the electrostatic forces on HF, as shown by the difference between both panels of Figure 1. is mainly due to charge flux. This dynumic anisotropy will not be manifested in interactions between rigid bodies. The latter may still show static anisotropy due to static atomic dipoles and higher order multipole~.'~J~ Here we find, however, that the dynamic anisotropy is larger than the static anisotropy. This result is in fact general since, as seen in eq 4, the flux forces are proportional to the external field whereas the forces due to static dipoles are proportional to the gradient of the field. The actual ratio between these different components of the force, and the distance at which they may become comparable, depends of course also on the magnitude of the various atomic multipoles and the flux parameters. In the molecules discussed below the flux component is usually predominant. We now turn to discuss the pattern of charge and dipole fluxes in HF. It is useful in this respect to quote Herzberg's book on the spectra of diatomic molecules;" "For very large and very small internuclear distances the dipole moment (of a diatomic molecule) must clearly approach zero except when the molecule dissociates into ions". H F dissociates into neutral species and so its dipole moment as a function of the internuclear distance must show an extremum. However, the FR atomic charges that are derived from the molecular dipole moment do not have to show extrema along the internuclear coordinate. The atomic charges (in any heteronuclear diatomic molecule) are zero at infinite internuclear separation and increase in magnitude as the bond is formed. As the bond is shortened toward the "fused" united atom the individual atomic charges may decrease or increase. Similar considerations apply to the atomic dipoles. In the case of H F 141 increases monotonically from infinity to re so that at equilibrium dlql/dr < 0. On the other hand, dlmldr > 0 at equilibrium, indicating an extremum in Iml at a larger internuclear separation. (Le., the dipolar flux at equilibrium has the opposite trend to that of the charge flux in this case). The charge flux in H F may be explained by an interplay between three resonance structures2'
'HF' * H-F * H'F I I1 111
As the H F bond is extended the weight of the ionic structure (111) is reduced and the atomic charges diminish. The covalent structure (11), on the other hand, may be stabilized at equilibrium by a bond stretch, since, as the dipolar flux parameter in Table I indicates, a bond stretch increases the magnitude of the atomic dipoles. At any rate, there will be an internuclear distance where structure Price, S . L. Mol. Phys. 1987,62,45. (19) Stone, A. J.; Price, S. L.J. fhys. Chem. 1988, 92, 3325. (20) Herzberg, 0. Molecular Spectra and Molecular Srructure. I . Spectra of Dfaromic Molecules, 2nd ed.; D. Van Nostrand: Princeton, NJ, 1966; pp 96, 97. (21) Bruns, R.E . In Vibrational intensities in infrared and Ramon specfroscopy; Person, W.B.,Zerbi, G. Us.; Elsevier: Amsterdam, 1982. (18)
The Journal of Physical Chemistry, Vol. 95, No.16, 1991 6205
he
H,F
I
%
-0.4
2
3
4
5
6 RCP'
p-
-
1.5 1.0
-
5
0.s
-
0.0
-
2
.o.s
-5 L"
7
8
- - c=o
-1.0 -1.5 -2.0
ab initio
FO
! 2
9
A
3
4
5
6
1
8
9
Rep ' A
Figure 3. Forces on the two nuclei of CO exerted by an incoming point charge of +O. 1 e. The open diamonds in the upper panel denote forces that arise from the FR atomic charges and from perpendicular dipolar fluxes according to eq 16.
I will become more and more dominant until a complete dissociation to two neutral and spherical atoms has been reached. Figure 2 displays a schematic of these considerations with respect to the variation of the charge distribution in H F as a function of the internuclear distance. Finally, for HF we note that experimental FR atomic charges can be obtained in this case from the experimental bond length and dipole moment,22the charge flux parameters can be obtained from IR intensity of the single stretching ~ibration?~ and the FR atomic dipole can be obtained from the experimental quadrupole moment.22 For the experimental FR charges we find that 141 = 0.408, Le., -9% lower than the ab initio results. The charge flux parameter is -0,100 e/A compared with the calculated value of -0,098 e/A. The experimental FR atomic dipole of the fluorine atom as derived from the experimental molecular quadrupole adjusted to the equilibrium geometry2' is found to be 0.45 D compared with the value of 0.32 D calculated here. CO. This case is markedly different from the case of HF. First, as seen in Table I, the molecular dipole moment is small so that the atomic charges are small. Consequently, the interaction with (22) de Leeuw, F. H.;Dymanus, A. J. Mol. Spectrosc. 1973, 18, 427. (23) Sileo, R. N.;Cool,T.A. J . Chem. fhys. 1976, 65, 117. (24) Amos, R. D. Chcm. Phys. h i t . 1982,88, 89.
6206 The Journal of Physical Chemistry, Vol. 95, No. 16, 1991
Dinur
TABLE I: EkctmtaHc Parametersa$ of Model Systems H-F atom
X
9
H
0.0000 0.9005
0.4494 -0.4494
F
m A, -0.323 0.323
aqAtar -0.098 0.098
amA,lar
amA,lar -0.291 0.291
0.032 -0.032
C=O
atom C 0
X
4
mA.X
W a r
0.0000 1.1503
-0.0347 0.0347
0.892 -0.892
0.401 -0.401
atom
X
0
0.1022 -0.4655 -0.4655
H H
Y 0.7530 -0.7530
rOH
0
H
OHOH'
rOH'
a9Alas
0.260 -0.237 -0.023
H
9 -0.7875 0.3938 0.3938
O.oo00
0.260 -0.023 -0.237
mA.X
mA#
0.371 -0.186 -0.186
O.OO0 0.301 -0.301 rOH'
OHOH'
0.020 0.014 0.01 2
-0.342 -0.041 -0,041
rOH
-0.158 0.079 0.079
0
0.020 0.012 0.014
H H
almllas
Formaldehyde
atom C 0 H H
0.9257 -0.9257
4
mAJ
0.2679 -0.4022 0.0671 0.067 1
0.262 -0.304 0.021 0.021
mA#
r,
OH-C-0
rC-H
a4Alas -0.083 0.111 -0.158 0.130
-0.285 -0.289 0.287 0.287
H
O.oo00 O.oo00
1.1844 -0.5817 -0.5817 rC-0
C 0 H
Y
X
O.oo00
0.119 -0,013 -0.010 -0.097
-0.362 -0.135 -0,057 -0.057
C 0
H H
O.OO0 O.OO0 -0.150 0.150 @
rC-H
almllas
0.076 0.027 -0.094 0.027
H
a
0.078 -0.287 0.413 0.276
Q atom C 0 N H4 Hs
HC rC-0 C 0
rC-N
H4
-0.344 -0.393 0.444 -0.004
Hs
0.021
HC
0.276
0.204 0.299 -0.677 0.049 0.047 0.076
C 0 N H4 Hs Hc
-0.06 1 -0.294 -0.003 0.020 -0.033 0.018
-0.041 -0.066 -0.129 0.035 0.030 -0.01 5
N
4
Y 0.063 1 1.2553 -0.7480 -0.3448 -1.7349 -0.4872
X
0.0454 0.0888 1.1220 2.0305 1.0287 -0.8984 rC-HO
-0.129 0.125 0.098 0.026
0.141 0.039
-0.015
-0.104
0.018 -0.024 0.016 -0,029
r~-H, 0.017 -0.026 0.102 -0.119 0.002 0.025 -0.002 -0.007 -0,046 -0.029 0.008 -0.006
"A,
0.4935 -0.5062 -0.8108 0.3835 0.3882 0.0519
mA#
-0.OO0 -0.276 0.465 0.041 -0,264 0.036
0.317 -0.047 -0.540 0.279 -0.001 -0.008
@M-N
@%-c-N
@H,-N-c
@H~N-C
-0.021
-0.040 0.100 0.045 0.034 0.079 -0.21 7
-0.096 0.01 5 -0.030 0.017 0.042 0.05 1
0.161 0.017 -0.074 0.079 -0.210 0.023
0.208 O.OO0 -0,093 -0.210 0.096 -0.002
almlas -0.009 0.025 -0.048 0.001 -0.03I 0.010
0.829 0.091 -0.104 -0.107 0.102 0.265
0.400 0.306 -0,027 0.165 -0,037 0.441
0.342 -0.033 0.216 -0.142 0.100 0.029
0.236 -0.115 0.136 0.108 -0.189 -0.014
~N-H~
aqlas
0.01 3 0.015 0.121 0.009
-0.138
Electrostatic Forces in Planar Molecules
The Journal of Physical Chemistry, Vol. 95, No. 16, 1991 6201
TABLE I IContinuedl
8-
Formic Acid
atom C
o==
0HO HC
rC-0
C 00-
X
Y
-0.0220 1.1189 -1.0345 -0.6933 -0.3841
0.4184 0.1092 -0.4309 -1.3166 1.4413 k-0
-0.223 -0.381 0.328
0.101 0.336 -0.520
0.042
-0.036
0.233
0.119
C
-0.068
0
-0.295 O.OO0
-0.089 -0.049
HO HC
o= HO HC
0.016 0.028
-0.I26
0.044 0.01 6
4 0.5019 -0.4765 -0,4947 0.3970 0.0722 r*H
rC-H
adas
-0.169 0.118
0.074 0.023 -0,046 0.154
m u
-0.303 0.116 0.486 -0.334 0.035
004-H
etX-0
eC-0-H
0.055 -0,003 -0.023 -0,007 -0,021
-0.008 0.068 0.064 0.021 -0.146
0.037 O.OO0 -0.I57 0.112 0.007
O.OO0
-0.323 -0.282
-0.028 -0.005 -0.01 1
-0.I33 -0.455
0.502 -0.140 0.051 -0.220 -0.246
0.121 -0.004 0.107 -0.134 0.088
0.004 -0.039
0.070 -0,057 0.022 alml/as
0.045 0.01 1 -0,023 0.048
mA,
-0.072 -0.268 0.314 -0.003 0.029
-0.013
0.066
#Coordinatesin A, charges in e, dipoles in D, charge flux in e/A and e/rad, and dipole flux in e and D/rad. Values larger than 10.11 are italicized. b6-31G** SCF calculations for all cases except CO where it is 6-31G**/MP2. The static moments for HF, H20, H2C0, HCONH2,and HCOOH are taken from refs 9 and 1 1 . 'This column corrects an error in ref 6 with respect to the flux associated with the bending angle. The error occurred in the transformation of the results to internal coordinates. a perpendicular point charge is expected to be small. On the other hand, the charge flux, atomic dipoles, and dipolar fluxes are significant. Thus, large collinear forces are expected, and along with that a large anisotropy in the force. Figure 3 shows the ab initio forces on the two nuclei due to an incoming point charge of +0.1 e, in one case along the perpendicular to the bond above the carbon and in the other case along the bond. Whereas the perpendicular forces at 3 A are in the range 0.1-0.3 kcal/(mol A) the collinear forces at this distance are 1.0 and 1.5 kcal/(mol A), Le., up to 10 times larger. Figure 3 also shows the decomposition of the forces into the various contributions. As seen, the perpendicular forces are due mostly to the static charges. (The small deviations of -0.05 kcal/(mol A) at 3 A are due to the truncation of the dipolar interaction at order VE and large atomic quadrupoles. The latter were discussed in (1 1) and are not included here.) The much larger collinear forces, on the other hand, are almost entirely due to factors other than static charges. The following may be noted: (a) While the external positive source attracts the carbon from the perpendicular orientation, it repels the carbon in the collinear orientation! This is a result of the fact that the main drive for the collinear force is the charge flux and not the static charge. The stretching motion in this case pushes the negative charge from the carbon to the oxygen. The magnitude of this flow compared to the magnitude of the static charge is such that upon a short extension of the bond (-0.1 A) the carbon will become positive. The external positive source thus repels the carbon because in this way the carbon acquires extra negariue charge which stabilizes the interaction. Likewise, the oxygen is repelled from the external source in the perpendicular orientation and attracted to it in the collinear orientation. It is stressed that these are dynamic effects that will not be seen in interactions between a rigid CO and its surroundings. (However, to the extent that rigid molecules are assumed in calculations of crystal structures, charge fluxes may cause, in general, slight differences between bond lengths and valence angles in gas phase and in crystals). The role of dynamic charge flux force in the spectral shift of the stretching frequency of CO that is bound to a surface was recognized long ago by Efrima and Metiu.25 (25)
Efrima, S.; Metiu, H. Surf. Sct. 190, 92, 433.
Contributions to the Force$ Exerted on the Nuclei of CO by an External Point Charge of +0.1 e 3 A from the Carbon (Orientation as in Bottom Panel of Figure 3)b
TABLE II:
F(d C 0
-0.128 0.067
'In kcal/(mol A).
F(I') 1.229 -1.229 b&p
F(m)
F(P)
F,,
-0.456 0.172
0.513 -0.513
1.157 -1.503
= 3 A,
(b) The collinear force on the oxygen is larger than the force on the carbon even though the distance to the latter is shorter! This can be seen with the help of Table I1 which gives the breakdown of the forces in CO at 3 A into individual contributions. The static multipoles give rise to forces that are larger for the carbon, since it is closer to the source. The fluxes, however, operate in opposite directions to the static multipoles. Consequently, the total charge contribution, and the total dipolar contribution,each separately is larger for the oxygen than for the carbon. It is clearly the nonlocal aspect of the electrostatic interaction that brings this about. As for H F the above findings can be explained in terms of a resonance argument. As the carbon is attracted to the external source, the CO bond length increases. This destabilizes the triple bond (structure IV) and stabilizes structures V and VI. The latter, however, are repelled from the external positive source. The charge flux force on the carbon is therefore repulsive. The opposite holds for the oxygen. The atomic charge of the oxygen repels the external source. This will elongate the bond. However, the contraction of the bond is overall favorable because that stabilizes the interaction with the carbon, which is closer to the source. The positive oxygen is thus attracted to the positive source because of a nonlocal (and nonpairwise) flux effect. It may be seen also from Table I1 that the dipolar interactions-static and dynamics-behave in this case in parallel to the charges. The fluxes totally override the static multipoles in the collinear orientation, and the forces are flux controlled.
Comparing now the electrostatic parameters in CO with those in HF, it is seen that the atomic dipoles in CO are inversely oriented, which is the characteristic of a uporbital that is occupied
6208 The Journal of Physical Chemistry, Vol. 95, No. 16, 1991
c.0
I
6*, - R
co-+
Figure 4. A schematic of variation of the atomic charges in CO as a function of the internuclear distance.
in conjunction with a double bond.” The magnitude of the fluxes in CO is much larger than in H F and the directions are also different in the two cases. The FR atomic charges increase with bond stretch rather than decrease, and the atomic dipole decrease in magnitude upon bond elongation, unlike the case of HF. These results suggest that the dissociation of the bond in CO occurs in stages (Figure 4). The one manifested at equilibrium is the weakening of the triple bond. This is accompanied by an increase in the atomic charges and a decrease of the atomic dipoles (because the inverse polarization reduces to the magnitude that corresponds to C 4 ) . In this case the charge flux at equilibrium does not have the features of the limiting process of a dissociation into neutral atoms, unlike the situation in HF. Finally, as for HF, the FR atomic moments can be derived from the experimental dipole and quadrupole moments,%and the charge flux parameters can be derived from the experimental dipole moment derivative.*’a We find qc = 4 0 2 e, mc = 0.85 D, and aqclar = 0.56 e/A. H20. The anisotropy of the electrostatic interactions and the large charge flux along the OH bond in this case have been described before. As in HF, the O-H charge flux has the features of a dissociation process-the hydrogen regains its electron. Table I also indicates that the stretch of the OH bond hardly involves the other hydrogen; i.e., there is no exchange of charge between the two hydrogens. The two bonds act as isolated. Upon opening of the angle the hydrogens become more positive, in agreement with the common trend of increased acidity upon going from sp2 to sp hybridization of the heavy atom. The dipolar fluxes are relatively small except for the bending motion which affects the total quadrupole moment. (In the extreme case of a collinear molecule the atomic dipole on the oxygen is zero by symmetry, so that the dipolar flux Amo/A6 has to be nonzero. The opening of the HOH angle therefore reduces the atomic dipole of the oxygen.) Figure 5 shows the forces on the oxygen nucleus (top figure) and on the hydrogen (bottom) in the case of a negative external point charge that comes along the H-O bond. The atomic dipoles in H 2 0 are relatively small and the forces are mainly determined by the charges and charge fluxes. The latter contribute significantly, as already discussed elsewhere: although the qualitative picture appears to be local, in that the hydrogen is attracted to the external source more strongly than the oxygen is repelled. The novel feature here relative to the previous examples of H F and CO is the existence vertical components to both Fo and FH. This component results from the breaking of the pairwise symmetry by the second hydrogen. When the oxygen is repelled from the external source it exchanges charge with both of its neighbors, and the charge flux to the hydrogen below the source-oxygen line (denoted H’) generates a vertical force. Explicitly
where the water molecule is in the xy plane and y is perpendicular (26) Meerts, W. L.; de h w ,F. H.;Dymanus, A. Chem. Phys. 1977,22,
319.
( 2 7 ) Steele, D.Molecular Spcc?roscopy;Specialist Periodical Report; Chemical Society: London, 1978; Vol. 5, cited in ref 28. (28) Amos, R. D. Chem. Phys. Lrrf. 1919.68, 536.
-
0
.
3
2
5
5
5
4
6
’I
1 8
1
I
-2.5-
I
a3.0 1
I
I
2
3
4
1
I
5
6
7
8
Figure 5. Forces in H20exerted by an external point charge of -0.1 e. Top: the forces on the oxygen along the internuclear line (F,) and along the perpendicular to this line (FA).Bottom: the same for the hydrogen that is closer to the external source. O.W?
v‘-0
H 0.188
I
4.W
4.402
0.168
&04.36?
I
/
H
H
0.107
‘c L
0.182
0.172
H \C
H 4.213
/
H
l
H 4.34s
\ / H
0.007
(8)
0.201 0.527
H-
wo4.m
4.473
0 4.461
H
\&
o.O.251
0.050
F w 6. FR atomic charges in the equilibrium configuration, distorted structures,and fragments of formaldehyde. 6-3 IG**SCF calculations. to the horizontal H-O bond in the figure. The vertical force on the oxygen is in the direction of -j because a displacement of the oxygen in this direction increases BHOH. This makes the hydrogen closest to the source more positive (Table I) and stabilizes the entire system. Finally, the experimental gas-phase atomic charge of the oxygen is qo = 0.658, which is 20% lower than the calculated value. The experimental charge fluxes, based upon the experimental atomic polar tensor given by Zilles and PersonB are ago/& = 0.272 e/A and &/a9 = -0.171 e/rad, in close agreement with the SCF values reported in Table I. Formaldehyde. The charge flux along the carbonyl bond is similar in magnitude to the static charge on the carbon (Table I) so that the forces are expected to be highly anisotropic. The charge flux induced by the carbonyl stretch is such that both C and 0 gain electronic charge. This, again, suggests bond disso( 2 9 ) Zilles, B. A.; Person,
W.B. J . Chem. Phys. 1983. 79, 65.
Electrostatic Forces in Planar Molecules
The Journal of Physical Chemistry, Vol. 95, No. 16, 1991 6209 D
H. 4.03.0-
-
i
E
2.01.0-
0
2 -*
0.0-1.0-
IL
.1.0-
-3.0
I 2
I
3
4
5
6
1
8
9
*.O1
Figure 7. Collinear forces on the carbon and oxygen in formaldehyde.
ciation in stages, with the first stage being the weakening of the r bond. This yields a P-0-structure, with an increased negative charge on the oxygen. However, a complete dissociation yields a carbene fragment, with a negative carbon due to u charge transfer from the hydrogens to the carbon (Figure 6). The net result of these two opposing contributions is an increase in the electronic charge on the carbon. Weakening the r bond affects also the inverse polarization that is characteristic to double bonds, and, consequently, the atomic dipoles on C,O diminish as the C=O bond stretches (Table I). The large u-x interaction, which is seen in the strong participation of the hydrogens in the charge flux induced by the carbonyl stretch, also couples the C-H bond to the oxygen. The hydrogen that is being stretched withdraws electrons from the oxygen and other hydrogen rather than from the carbon. The C-H stretch is thus not isolated as the 0-H stretch is in the water molecule. Some further insight into the charge flux in formaldehydeis gained by considering the charges on the structures shown in Figure 6. As seen, a dissociation of a C-H bond yields HCO species with an acidic hydrogen, which corresponds to the finding that the CH stretch withdraws negative charge from the other hydrogen. Similarly, opening of the HCH angle (closing HCO) results in increased positive charge on the hydrogen, as observed for the HOH angle in H20. Figure 7 shows the collinear forces on the carbon and oxygen in formaldehyde. The large flux effect is seen in the large gap between the 4qJand flqj41. At 3 A from the oxygen the contribution of the atomic dipoles is already felt but the overall interaction is clearly controlled by the charge flux. The effect of the charge flux in this case is to enhance the forces on both nuclei. For example, the static charge on the carbon leads to a repulsion from the external source and therefore to a CO stretch. This repulsion is reduced by a flow of electronic charge from the hydrogens to the carbon that accompanies the CO stretch. However, the oxygen also gains electronic charge upon CO stretch. Since the oxygen is closer to the external point charge the overall effect is an increased repulsion with respect to the carbon and bond elongation. From this analysis, as well as from the previous examples, the following general pattern emerges: the force on a given molecular site due to an external source is largely determined by other sites that are closer to the source and can exchange charge with the site under discussion. Formaldehyde is one of a few polyatomic molecules for which attempts have been made to extract experimental polar tensors from IR intensities. Unfortunately, the results are not unambiguous because the sign of the dipole derivatives is underdetermined. Two possible sets of perpendicular elements of the polar tensor, which constitute the FR atomic charges, have been determined by Person and NewtonP qo = 4,327, qc = 0.162, and q H = 0.081, and, respectively, =0.396,0.385, and 0.004.The more recent intensity measurements of Reuter et ala3'for the dipole derivatives lead to similar values: -0.3164, 0.1408, and 0.0878 (30) Person, W. B.; Newton, J. H. J . Chem. Phys. 1974, 61, 1040. (31) Reuter, D. C.; Nadler, S.; Daunt, S. J.; Johns, J. W. C. J . Chem. Phys. 1989, 91, 646. (32) Cham, T. C.; Krimm, S. J . Chem. Phys. 1985,82, 1631.
-4.04 4
-
z
X
0.0
-1
J 5
6
8
7
9
1
.o
-1.5
10
ab initio I
~
5
6
7
8 RN$
9
10'
11
A
Figure 8. Forces on the three heavy nuclei in formamide exerted by an in-plane external point charge of +O. 1 e that approaches the oxygen along the CO bond. Shown are for each nucleus the components of the force along and perpendicular to the line connecting the external charge with the nucleus. A positive force points toward the external point charge. The orientation of the perpendicular axis can be deduced from the drawing at the top. The meaning of the symbols in the bottom applies to all panels.
and -0.4006, 0.3976, and 0.0016. Although the sign choice in ref 30 corresponds to the latter set, both sets are plausible and stand in qualitative agreement with the ab initio results in that the hydrogens carry only a small charge. Note, however, that the second set, in which the carbonyl group is virtually neutral, agrees with the empirical parameters of Lifson et al. derived from experimental crystal str~ctures.'~ F d k The HCO parametersstatic multiples and their fluxes-are qualitatively similar to the corresponding parameters in formaldehyde. The NH2 moiety is similar to H 2 0 in that the two N-H bonds are fairly isolated, so that an NH stretch leads to charge exchange mainly along the bond that is being stretched, and not significantly beyond it. Stretching of other bonds also does not affect the charges on the hydrogens linked to the nitrogen, However, the opening of &NH (via the two 6HNC) increases the positive charge on these two hydrogens, similar to the pattern observed for the opening HOH in water and HCH in formaldehyde, although the charge flows to the carbon rather than to the nitrogen. Finally, the CN stretch withdraws electrons from the carbonyl group to the nitrogen and reduces the atomic dipole on N, in agreement with the picture of r bond weakening and stabilization of VI1 at the expense of VIII.
VI1
VI11
6210 The Journal of Physical Chemistry, Vol. 95, No. 16, 1991 \
§A
'\, \
IFoli1.55 LcalimollA
Figwe 9. Forces (ab initio) in formamide due to an external point charge of 0.1 e, 5 A from the oxygen.
Figure 8 shows the forces on the three heavy atoms as an external charge of +0.1 e in the molecular plane approaches the oxygen along the CO bond. For each atom the figure shows the component of the force along the line connecting the external point charge with the particular nucleus, and the component that is perpendicular to that line. The overall flq3Qlmj")are shown along with the ab inito forces. As seen, the FR electrostatic parameters successfully reproduce the ab initio forces. (Although not explicitly shown in the figure, it is noted that the (mj"]contribution to the forces is significantly smaller than that of the atomic charges and their flux). Figure 9 illustrates the ab initio forces when the external force is 5 A away from the oxygen. Two points can be made with respect to these figures: (a) In all cases there is a vertical component to the force. This component is small for the oxygen, noticeable for the carbon, and
Dinur almost comparable to the parallel component for the nitrogen. It clearly demonstrates that the molecule is not a set of isolated points in vacuum. Rather, it is a complicated medium that relays the electrostatic interaction in a way that corresponds to the molecular structure. For both the carbon and the nitrogen the direction of the vertical component is determined by the CN shortening motion which makes the oxygen more negative and the interaction with the source more favorable. As in all previous examples the interaction is determined by the center closest to the external source, in this case the oxygen. (b) The force on the carbon is significantly lurger than the force on the oxygen, even though the latter center has a slightly larger FR charge (Table I) and is closer to the external charge! The reason is the fact that the repulsion of the carbon is enhanced by charge fluxes along two coordinates-the CO bond and the CN bond. A repulsion of the carbon stretches the carbonyl bond and shortens the CN bond; both stabilize the O--C-N+ structure (VIII). This is more consequential than just the CO stretch by the oxygen, or the CN stretch by the nitrogen. The carbon is thus different from its two heavy neighbors because of its location at the junction between two bonds. HCOOH. The last example is formic acid. The similarities to formamide and formaldehyde are now easily seen in Table I. With respect to the static moments we may note in particular that the two functional groups in formamide and formic acid are almost neutral: the carbonyl, which is common to both molecules, NH, in formamide and OH in formic acid. As already mentioned, similar patterns have been used in fitting experimental crystal structures of amides and carboxylic acids many years The flux parameters in formamide and in formic acid are also similar. Elongating the (2x0 bond makes the carbonyl group more ,p
z -.
i
2.0
-
1.0
-
0.5
-
1.5
m
*0
-.
2.0
3.0 2.5 3.5
0.0 2
3
s t
4
7
8
5
9
-. m
-
-1.0
-
-2.0
-
-3.0
-
4,
2
ILV - 4 . 0
I
I
I
9
10
0.0
- -0.1 --; ;- 0 . 2
2!
*0
--6 - 0 . 3
-0
Lx
-0.4 -0.5
-5.0
4
I
7
ROW)^'
ROPY
0.0
I
6
5
6
7 RCP'
8
9
I
*
10
A
Figure 10. Forces in formic acid exerted by an in-plane external point charge of +0.1 e that approaches the oxygen along the CO bond. Shown for each nucleus are the components of the force along and perpendicular to the line connecting the external charge with the nucleus. A positive collinear force points toward the external point charge. The orientation of the perpendicular axis can be deduced from the drawing at the top. The meaning of the symbols in the bottom right applies to all panels.
J. Phys. Chem. 1991,95,6211-6217
6211
4. Summary
Figure 11. Forces (ab initio) in formic acid due to an external point charge of 0.1 e, 5 A from the oxygen.
negative at the expense of the third heavy atom (through its ?r bond to the carbon), and at the expense of H, (through the u interaction). Likewise, elongating the single bond between the carbon atom and the heavy atom (N or 0)withdraws charge from the carbonyl oxygen through the r system (see VII, VIII, and IX, X). Another similarity is observed for the CH bond stretch which in both cases polarizes the carbonyl group in the same way, namely, negative charge flows from the oxygen to the carbon, which is also similar to the behavior in formaldehyde. The 0-H stretch in formic acid has a local effect and does not induce a significant flux even along the OH coordinate itself (unlike the HzO case), and, similarly to the N-H in formamide the hydroxylic hydrogen is not significantly affected by stretching other bonds. As observed for HOH in water, opening the C-O-H angle in formic acid increases the polarity of the 0-H bond.
o=c-0 IX
-
o~c=o+ X
Figure 10 shows the forces on the polar atoms of formic acid as an external point charge of +0.1 e in the molecular plane approaches the carbonyl group. Both the ab initio forces and the forces calculated with the (q#,mjm} set of electrostatic parameters are shown. The ab initio forces at 5 A are further illustrated in Figure 1 1. The general pattern is similar to that in formamide. The force on the carbonylic oxygen is practically central (pairwise point charge interaction). The carbon atom, being at the junction between two bonds, is subject to an enhanced force (larger than the force on the oxygen, which is closer to the source). The alcoholic oxygen, like the nitrogen in formamide, experiences a very noncentral force, because it is further removed from the source and separated from it by a “structured” medium that modifies the force significantly. The effect is smaller for the alcoholic hydrogen since the charge flux associated with this nucleus is relatively small.
We have shown in this work that electrostatic forces in molecules cannot be represented as arising from simple central Coulombic interactions, as commonly assumed in current force fields. Molecules are not collections of isolated atoms, interacting through empty space. Rather, molecules are structured media that relay the electrostatic interaction in a way that parallels the electronic structure and geometrical arrangement of the nuclei. Consequently, electrostatic forces in general are not isotropic. The anisotropy has a static part due to the nonspherical charge distribution around the nuclei, which is represented by atomic dipoles or higher order multipoles, and a dynamic part which is e x p r d during the vibrational motion of the molecules. Both on theoretical ground and on the basis of the cases discussed, the dynamic anisotropy is in general expected to be larger than the static anisotropy. It was found that charge flux is the main mechanism by which the central force is modified and anisotropy results. Because of this charge flux the forces are not local and depend on the response of other sites to the external perturbation. Finally, we have shown that the FR atomic multipoles reproduce the ab initio forces very well for the cases discussed. The results obtained in this work suggest that unsaturated compounds in general show large charge fluxes. The conclusion is less general for saturated systems. The two cases discussed here, HF and H20, behave differently in that respect. However, both are highly polar and one may plausibly assume that, for saturated nonpolar groups, such as methylene groups, charge flux is small.’ The systems dealt with in this work are all planar. Similar nonplanar molecules are not expected to behave qualitatively differently. For example, the C 4 stretch that induces a charge flux in the OCN moiety in formamide will do so in acetamide and N-methylacetamide as well. C h a m and K~-imm’~ have found that the polar tensor is fairly transferable among these molecules. Thus the qualitative behavior of electrostatic forces found in this work is likely to be general in nature. Further work is required, however, before a quantitative description of electrostatic forces in nonplanar molecules becomes possible. Acknowledgment. This research was supported by the Basic Research Foundation administered by the Israel Academy of Sciences and Humanities. The author is indebted to Dr. Marvin Waldman from Biosym Technologies for many useful discussions. Registry NO.CO,630-08-0;HF,7664-39-3; H20,7732-18-5; H2C0, 50-00-0; HCONHz, 75-12-7;HCOOH, 64-18-6.
A Flexlble/Polarlzable Slmple Polnt Charge Water Model S.-B. Zhu, S.Yao, J.-B. Zhu, Surjit Singh, and G. W. Robinson* SubPicosecond and Quantum Radiation Laboratory, Departments of Physics and Chemistry, P.O. Box 4260, Texas Tech University, Lubbock, Texas 79409 (Received: March 22, 1991)
The purpose of this paper is to report the properties of a flexible water model that is computationally simple enough SO that it can be applied to the molecular dynamics study of partial ensembles of liquid water in the presence of local perturbations, such as those arising from ions, surfaces, and electrical fields. This model is based on a 3-point version with both flexible geometry and polarization. The positions of the point charges are fixed, but their values are variable according to the local field. By use of this model, we have investigated a molecular dynamics simulation of the static and dynamic properties of liquid water at room temperature and normal density. Comparisons with experimental data and other model calculations are made.
1. Introduction ~~~i~~ the past
decades, a great number of computer simulation studies concerning the structure and properties of liquid Until recently, most of these have water have been 0022-3654/91/2095-6211$02.50/0
been carried out within the framework of pairwise additive intermolecular potentials. The influence of electrostatic induction ( I ) Barker, J. A.; Watts, R. 0. Chcm. Phys. Lett. 1969, 3, 144.
0 1991 American Chemical Society
