Consistent implementation of the electronegativity equalization method in molecular mechanics and molecular dynamics

Konstantin S. Smirnovt and Bastiaan van de Graar" Laboratory for Organic Chemistry and Catalysis, Delft University of Technology, Julianalaan 136,2628 BL Delft, The Netherlands

An approach for the inclusion of geometry-dependent charges in molecular mechanics (MM) and molecular dynamics (MD) calculations on the basis of the electronegativity equalization method (EEM) is proposed and tested. It is shown that for a consistent implementation of EEM, the force field has to be extended with charge-dependent intraatomic terms. This extension simplifies M D and MM calculations on both molecular and supramolecular systems. Compared with calculations with fixed charges the present approach leads only to a modest increase in the computational overhead.

The accurate prediction, by computational methods, of the properties of very large molecules or complex many-particle systems is still beyond the capabilities of quantum mechanics and thus force fields are an indispensable tool. Although modern force fields can predict geometries, energies and vibra- tional frequencies with accuracies close to experimental error, there are limitations. One of these is that, generally, force fields use fixed charges, charges that depend only on atom type, or fixed dipole moments, dipole moments that depend only on bond type. Obviously such force fields do not produce charge distributions that depend on geometry or respond to an external potential. In reality, the charge distribution in a molecule is affected by conformational changes and the molec- ular vibrations and it can be strongly affected by a solvent or by being close to the surface of a solid. Inclusion of geometry- dependent charges in force fields would result in a more realis- tic description. Such charges would also be of use to understand and to predict physical and chemical properties.

A first attempt to include geometry-dependent charges in MM was made by DoSen-Mibovik et a!.' by modifying the Del Re method2 to take polarization into account. The method was not included in an energy-minimization scheme and it has the drawback that a large number of parameters, one for each atom type and six for each bond type, are required.

More recently, two schemes, both based on the electroneg- ativity equalization pr in~ip le ,~ were introduced4v5 to calculate geometry-dependent charges taking into account, if necessary, an external potential. The first scheme, called the electroneg- ativity equalization method (EEM), was proposed by Mortier et aL4 on the basis of density functional theory. The second scheme, called charge equilibration (QEq), was developed by Rappe and Goddard.' The main difference between EEM and QEq lies in the treatment of the interatomic interactions: EEM uses a coulomb potential between point charges while QEq takes shielding, important at small interatomic distances, into account explicitly. Fundamentally, QEq appears to be the correct approach but EEM has the advantage that it is com- putationally simpler. Recently, it has been shown6 that shield- ing can be incorporated into EEM whilst retaining its computational simplicitly. Applied with shielding, EEM demands only three parameters for each element in order to calculate the charges.

Applications of the EEM and QEq schemes have also been reported.'-' ' Rappe et a!.' have included QEq in their univer-

sal force field. Van Duin et al. have applied EEM, in its orig- inal version without shielding, in the Delft molecular mechanics (DMM) force field for saturated and non- conjugated unsaturated hydrocarbons' and tertiary carbo- cation^.^ Rick et a1." have used a QEq-like approach in an MD study of water. The authors treated the charges as dynamical variables using an extended Lagrangian method where charge degrees of freedom are propagated according to Newton mechanics together with atomic degrees of freedom. An application of EEM in Monte Carlo calculations was very recently presented by Toufar et ul." It is shown that these calculations give insight into the reactivity of supramolecular systems or one of more of its parts.

Two methods to derive geometry-dependent charges, methods in which electronegativity equalization is not used, have recently been p r o p o ~ e d . ' ~ , ' ~ In the model of Dinur and Hagler'' there is a charge transfer between all pairs of bonded atoms that depend on the length of the bond as well as on the magnitude of the valence coordinates that surround the bonds. The model needs a large number of parameters which were determined from quantum chemical calculations. The method of Sternberg et a l l 3 is based on the bond polarization theory.I4 Empirical parameters, calibrated also by ab initio calculations, are introduced for the polarity of an unpolarized bond and for the change of the atomic charge with n- and n-bond polarization and thus the method contains parameters for bonds rather than for atoms. The atomic charges are cal- culated by solving a set of linear equations which contain the bond polarization energies as additional parameters. These have to be computed for each configuration using Slater hybrid orbitals.

In this paper the implementation of EEM in MM and MD calculations is discussed. It is found that, for a consistent implementation, it is necessary to expand the force field with charge-dependent intraatomic terms. It is shown that these extra terms simplify both MM and M D calculations consider- ably.

EEM EEM has been developed4 in the framework of density func- tional theory with the assumption that, for a molecule in the ground state, the electronic density can be partitioned into spherical atomic contributions. The molecular energy at the Born-Oppenheimer surface is written as :

t Permanent address: Institute of Physics, St. Petersburg State Uni- versity, St. Petersburg 198904, Russia.

J . Chem. SOC., Faraday Trans. , 1996,92(13), 2469-2474 2469

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400 I I

300

200

-14100 k I Potential energy 1 I I 1 . I1

I -14200

-1 4300 11.1.I.III.I 0 2 4 6 8 10

ti melps

Time history of the potential, kinetic and total energy in MD Fig. 1 run (ii), see text for details

where qi is atomic charge of atom i, defined as qi = (Zi - Ni) where Zi and Ni are nuclear charge and number of electrons, respectively. The terms enclosed in the square brackets rep- resent the contributions to the intraatomic energy. E T , x r and qr are expansion coefficients of the intraatomic energy as a function of atomic charge. The last two coefficients are called intrinsic electronegativity and hardness of an atom in a mol- ecule and they can be related to the atomic electronegativity and hardness of the free a t ~ m . ~ , ' ~ , ' ~ The last term in eqn. (1) describes the interatomic interactions.

Partial differentiation of eqn. (1) with respect to qi defines the electronegativity xi of atom i in the m o l e c ~ l e : ~ , ~ ~

According to the electronegativity equalization principle all atomic electronegativities in the molecule must be equalized

20c

15C

- -13950 I - E" 3 -14000

F 5 -14050

\

Q,

-14100

-14150

-1 4200

, Kinetic energy I

Total energy

Potential energy, . . 1

I . I . 1 . I . I . I

0 2 4 6 8 10 time/ps

Fig. 2 run (iii), see text for details

Time history of the potential, kinetic and total energy in MD

to the global electronegativity of the molecule x . (2b)

1 qi = constant ( W

x1 = x 2 = . * . = x Eqn. (2a) together with the constraints (2b) and (2c)

1

can be used to calculate directly the atomic charges qi and the global molecular electronegativity x , if the atomic parameters

and q: are known. These are usually calibrated to mimic ab initio (STO-3G) atomic charges derived from Mulliken population analyses. Obviously, the molecular electronegat- ivity and the atomic charges depend on the molecular confor- mation because of the last term in eqn. ( 2 4 and thus, eqn. (1) and (2) provide a basis for an implementation of geometry- dependent charges in models used in classical computer- modelling techniques. A detailed description of EEM, the derivation of eqn. (1) from density functional theory as well as some applications can be found in the l i t e r a t ~ r e . ~ * ' ~ - ' ~

As eqn. (1) describes the molecular energy, the question arises whether this equation can be used directly as a model in MM and MD calculations. However, it is for a diatomic mol- ecule, straightforward to show that the energy is not a contin- uous function of the interatomic distance and no stable molecule is formed.

Simple implementation of EEM in a force field Generally, force fields used in MM and MD calculations describe the potential energy E , , in MM often called the steric energy, of the molecule as :

bonds angles torsions non-bonded

( 3 4

where &, , E , , E , , and EvdW stand for the individual contribu- tions to the bond, angle, torsion and non-bonded energy, respectively. They are calculated with potential functions for which the parameters are determined from experimental data or from quantum chemical calculations. Often cross-terms, like stretch-bend and torsion-stretch, are included in eqn. (3a). In most force fields the summation over the electrostatic interactions excludes atom pairs in bonds and valence angles (1,2- and 1,3-interactions). In MM, several force field^'*^.'^*^' allow the calculation of the heat of formation from the steric energy E , by :

A H , = E , + 4RT + 1 I i + POP + TOR (3b)

in which 4RT arises from pV-work and the translational and rotational degrees of freedom, Ii is an atomic increment depending on atom type, POP is a correction for the presence of higher energy conformations and TOR is a correction for low-frequency, large-amplitude vibrations. In the DMM force field8v9 the last three terms in eqn. (3b) depend only on the connectivities in the molecule.

The simplest implementation of EEM in a force field is the combination of eqn. (2) with eqn. (3). This has been done in the DMM force However, for a proper combination the summation of the electrostatic interactions takes into account all atom pairs.

In both MM and MD calculations, derivatives of the energy with respect to atomic coordinates are needed. Using geometry-dependent charges means that the first derivatives of the energy will contain the first derivatives of the charges with respect to the atomic coordinates while the second derivatives, which are used in Newton-Raphson minimization algorithms in MM, will require the second derivatives of the charges to be calculated. The necessary derivatives of the charges can be

atoms

2470 J . Chem. SOC., Faraday Trans., 1996, Vof . 92

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obtained analytically within the framework of EEM (see Appendix). However, the calculation of these derivatives, espe- cially the second derivatives, increases the computational costs considerably. In a first approach, therefore, these derivatives were neglected.8 The result is of course that energy 'minima' found are not consistent with the description of the force field [eqn. (3a)]. For neutral hydrocarbons the discrepancies are small and acceptable but for the carbocationsg the computa- tional costs, which result from the calculation of first and second derivatives of the charges with respect to geometry, had to be accepted.

A further drawback of the simple implementation of EEM in a force field was observed in M D calculations. When calcu- lations are carried out in the NVE ensemble, with calculation of first derivatives of the charges, an increase in the total energy and temperature of the system is observed along the trajectory (Fig. 1 ) . If EEM charges are used in the MD calcu- lations with neglect of the charge derivatives, the energy and temperature remain fluctuating near a mean value, but large fluctuations of the total energy indicate the lack of stability at integration (Fig. 2). The conclusion is that the simple imple- mentation of EEM results in non-conservative behaviour in MD calculations.

Consistent implementation of EEM in force fields (CIEEM) The non-conservative behaviour of the model obtained by a simple implementation of EEM in the force field during MD calculations prompted us to examine both the EEM [eqn. ( l ) ] and the force field expression [eqn. (3a)] for the molecular energy more closely, giving of course special attention to the charge-dependent terms. The last term in eqn. (1) describes interatomic coulombic interactions and it is completely equiv- alent to the last term in eqn. ( 3 4 . The two equations, however, differ in their treatment of the effect of the atomic charges on the intraatomic contributions to the molecular energy. Eqn. (1) includes charge dependent intraatomic contributions that are absent in eqn. (3a). Intraatomic contributions are obvi- ously included in eqn. (3b) by the atomic increments I , but these depend only on connectivity and are independent of the atomic charges. Quite probably, it is the lack of charge- dependent intraatomic contributions in the force field that causes the simple implementation of EEM to fail in MD cal- culations. To remedy this we propose to combine eqn. ( 1 ) and (3) to represent the molecular potential energy as:

N

1 bonds angles torsions

L 7 .- K.. non-bonded I ] # i 11

where the first sum is the charge dependent intraatomic con- tributions and Eb, E e , E , , EvdW have the same meaning as in eqn. (3a). For geometry-independent charges eqn. (4) is equiv- alent to eqn. (3) with the atomic increments I given by:

N

z i = 1: + c [xi* q i + r f q?] ( 5 ) 1

where I : is a charge-independent atomic increment for atom i .

Applying CIEEM in calculations on molecular systems Having obtained the representation for the molecular poten- tial energy in the form of eqn. (4), we can examine it for use in both MM and M D calculations. Again, we are interested in the derivatives of the potential energy with respect to the atomic coordinates (first derivatives in M D and both first and

second derivatives in MM). Differentiation of eqn. (4), with respect to a coordinate a (a = x, y , z ) of an atom k gives:

+- i f k R z i auk auk

q k q i a R k i I aENC

- z k xK a u k

where E N , includes all terms which do not depend on the atomic charges, i.e. Eb, E e , E, and EvdW. Now, according to EEM, the terms in square brackets in the first sum are equal to the electronegativity x i of the atom i in the molecule and each x i is equal to the global molecular electronegativity x [eqn. (2b)l. Thus, eqn. ( 6 4 can be rewritten as:

The first term in eqn. (6b) vanishes because a movement of an atom in the molecule does not influence the total molecular charge but only results in a change of the charge distribution, i.e.

(7)

Thus, when the potential energy of the molecule is represented by eqn. (4) the first derivatives of the potential energy and the corresponding forces are given by :

which is completely equivalent to the expression that can be derived from eqn. (3a) in the fixed-charge approximation. The geometry-dependent charges appearing in eqn. (8) can be directly computed with eqn. (2). The reader should be aware that eqn. (6) and (8) were obtained with implicit use of the Born-Oppenheimer approximation, i.e. it is assumed that the electronegativities of the atoms undergo fast equalization with respect to the movements of the nuclei.

To apply Newton-Raphson minimization in MM one needs the second derivatives of the energy with respect to atomic coordinates. These can be obtained by differentiation of eqn. (6a):

(9)

The first term in eqn. (9) vanishes because for all conforma- tional space of the molecule its total charge is constant and the sum of the first derivatives of the atomic charges is equal

J . Chem. SOC., Faraday Trans., 1996, VoZ. 92 2471

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to zero [eqn. (2b) and (7)]. Thus, the second derivatives of the molecular energy contain only the first derivatives of the charges with respect to atomic coordinates. In the Appendix we show how these first derivatives can be efficiently obtained within the EEM formalism.

Applying CIEEM in calculations on supramolecular systems The simplest way to include EEM into calculations of supra- molecular systems would be to treat the system as a united macromolecule. The results of the previous section would then be directly applicable for this case. However, such a treatment would allow charge transfer between molecules when they have different global electronegativities. In order to avoid pos- sible artefacts, it seems worthwhile to explore the situation in which charge transfer between the molecules does not occur.

Let us consider a system of M molecules each consisting of N atoms which interact with H atoms of a solid, e.g. adsorbed molecules. It is supposed that the charges of the atoms of the solid atoms are fixed. Extension, however, to variable charges is straightforward. Thus, in analogy to eqn. (4), the potential energy of the system can be written as:

4 i a 4 j f i l M x i * . q i a + V i * . q i 2 , + ? 1 1 -

a i fi j f i z i a R i a j f i

H

h Riah + 1 ".I + E N ,

where, as before, EN, includes all charge-independent terms. Taking the partial derivative with respect to coordinate a of atom k one obtains:

1 M N H

1 M N M N q i a q j f i a R i a j f i --ccc c --

a i fi j f i # i a R$f i auk

M N H

M N H

Eqn. (1 1) can be rewritten as : M N

a i f k Rk2ia

q i a q k aRkia a q i a - 1 1 - -

where we have made use of the constant charges of the solid. The parameter x-' is the global electronegativity of the mol- ecule o! in the presence of an extramolecular electrostatic potential due to the other molecules and the solid:

M N H

(13)

where the last two terms stand for the external potential. Without intermolecular charge transfer the electronegativities of the atoms are equalized within the molecules while the global molecular electronegativities can be different. The first term in eqn. (12) then vanishes and this equation again

becomes completely equivalent to the expression that can be obtained for the case of fixed charges. In order to compute the N x M atomic charges and M global molecular electronega- tivities, N x M , eqn. (13) together with the M constraints [eqn. (2b) and (2c)l are written in a form [N x (M + M)][N x M + M] matrix equation [see ref. 11, eqn. (16)] which is solved in the same way as for an isolated molecule. The charges now depend not only on the molecular conformations but also on the external potential, i.e. on the positions of the molecules with respect to each other and with respect to the solid.

Testing the CIEEM approach To see if the CIEEM approach performed as it should, two tests were carried out. The main test was an MD study of a system consisting of eight methane molecules adsorbed in zeolite silicalite (siliceous counterpart of ZSM-5 zeolite). Such MD calculations are sensitive numerical tests for theory because any mistake in theoretical derivations (and/or programming) results in the lack of energy conservation and/or stability of the integration. The other test was a partial reparametrization of the DMM hydrocarbon force field' to see if a high-quality force field can be constructed on the basis of eqn. (4).

In this paper we discuss the MD results only in their rela- tion to the computational approach. A detailed investigation of the system is the subject of the accompanying paper. The zeolite lattice was represented by two unit cells arranged along the c crystallographic axis (576 atoms). The framework was considered to be fixed with atoms at their crystallo- graphic positions.21 Methane molecules were treated as totally flexible with intramolecular potential taken from ref. 8. Interactions between the molecules, and between the mol- ecules and the zeolite atoms, were described with a com- bination of a Lennard-Jones (12-6) and a coulombic potential. Parameters of the (12-6) potential were equal to those used in ref. 22 for an MD study of methane and propane diffusion in silicalite. The non-bonded interactions were treated within a cut-off radius of 9.5 A. Atomic charges for zeolite atoms were calculated by EEM using the parameters and the approach proposed in ref. 6. Initially, the methane molecules were uni- formly distributed within the zeolite pore system and the velocities of the atoms were taken from the Maxwell- Boltzmann distribution at 300 K. The equations of motion were integrated with the velocity form of Verlet algorithm23 with a time step of 0.5 fs. The first 10000 time steps (5 ps) were used as an equilibration period. During that period the system was coupled to an external temperature bath24 with the relax- ation time of 0.2 ps. For the following 40480 time steps (20.24 ps) the calculations were performed in the NVE ensemble. The last 20480 time steps (10.24 ps) were used for collecting data. All calculations were carried out on a Silicon Graphics Indy workstation. Four different M D runs were carried out: (i) with fixed charges calculated by EEM for the free molecule and not changed during the calculation; (ii) with EEM charges calcu- lated at each time step with eqn. (13), (2b) and (2c) within the framework of the simple implementation of EEM in the force field; (iii) the same as (ii) but neglecting the derivatives of the charges with respect to geometry and (iv) following the CIEEM approach.

In order to compare the level of accuracy of the calcu- lations, two quantities were considered : an energy conserva- tion parameter AE

1 NRUN E(0) - E(i) =" ? 1 E(0)

where E(0) and E(i) are the initial total energy and the total energy at step i, respectively, and N,"N is the number of time steps; and the ratio ( R ) of the rms of the fluctuations of the

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total energy to the rms of the fluctuations of the kinetic energy is a measure for the stability of the integration:

rms(TE) rms(KE) R=- (15)

Values of AE < 0.001 and R 6 0.01 were shown to provide a sufficient criteria of the numerical accuracy of the integra- t i ~ n . ' ' . ~ ~ Table l lists the accuracy parameters obtained in the MD calculations. The results for runs (ii) and (iii) were presented in Fig. 1 and 2; those of runs (i) and (iv) in Fig. 3 and 4. One can see that only the values of the parameters computed in the MD runs with fixed charges [run (i)] and using the CIEEM approach [run (iv)] satisfy the above cri- teria. The CPU time for the CIEEM approach is only a factor 1.2-1.3 larger than that needed for the fixed-charge calcu- lations, making the CIEEM approach in MD computationally quite affordable.

The DMM force field for saturated and non-conjugated hydrocarbons' was reparametrized for a few cycles in a suc- cessive one-parameter optimization. Obviously, the atomic increments, in particular, changed considerably in the first

Table 1 M D runs

Comparison of the accuracy parameters obtained in the

- 4.7 1 - 2.25 - 3.34 -5.13

0.01 04

0.7245 0.0103

126.82

200 Kinetic enerw.

- -13850 Totalenergy I t - z L 3 -13900

F $ -13950

\ > aJ

-14000

-14050

0 2 4 6 8 10 time/ps

Fig. 4 Time history of the potential, kinetic and total energy in M D run (iv)

I Kinetic energy

"% , I

1 Total energy I

- -13850 - E 3 -13900

P \ > a

-13950

I Potential energy -1 4000

-1 4050 I

0 2 4 6 8 10 ti me/ps

Fig. 3 Time history of the potential, kinetic and total energy in MD run (i)

Table 2 framework with expeimental data

Comparison of the results of the DMM' force field and those of a partial reparametrization of the D M M force field within the CIEEM

average deviation

cycles (see eqn. 5). In Table 2 the results are compared with those obtained with the DMM force field. Although a final judgement has to await the results of a time-consuming total reparametrization, it is clear that the results obtained are satisfactory and indicate that it should be possible to con- struct a high-quality force field within the CIEEM framework.

Conclusions A consistent approach for the inclusion of geometry- dependent charges in MD and MM calculations on the basis of EEM is proposed. The approach implies an extension of a force field by charge-dependent intraatomic contributions. Owing to this extension, the first derivatives of the energy with respect to the atomic coordinates are shown to be com- pletely equivalent to those calculated with a fixed charge approximation. The second derivatives of the energy contain the first derivatives of the charges with respect to geometry but not the second derivatives. This simplifies MM calcu- lations with Newton-Raphson minimization algorithms. Testing the approach in MD calculations of a system in NVE ensemble has revealed a high level of accuracy of the calcu- lations at rather modest computational overhead. A partial reparametrization of DMM force field,' on the basis of eqn. (4), has shown that the approach can be used to construct a force field of high quality.

K.S.S. gratefully acknowledges a research fellowship from Delft University of Technology. This work was in part sup- ported by Russian Foundation for Basic Research (grant No. 95-03-0955 1).

bonds" valence anglesb torsion angles' IR frequencyd heat of formatione force field /A /degrees /degrees /cm - ' /kJ mol-'

DMM 0.0038 CIEEM 0.0053

0.65 0.63

1.36 1.6

19.7 20.6

1.29 1.31

~~~~~ ~

Number of data points: " 61, 100, 15, 484 and 89.

J . Chem. Soc., Faraday Trans., 1996, VoZ. 92 2473

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References Appendix Eqn. (2a) can be rewritten using matrix notation as:

Aq = x ( A 4 where q is the vector of the charges, and x has the elements xi = x - x:. Elements of the matrix A are:

a . . = 2 q ? * 1J 1 3 j = i

Differentiation of eqn. (A.l) with respect to atomic coordinates results in 3N matrix equations:

where the elements of the dA/acrk matrix are:

aaij 1 dRij. aa, R; auk ’ j # i -- _-- -

These, in fact, are not equal to zero except for kth row and kth column. The form of eqn. (A.2) is now completely equivalent to eqn. (A.1):

Aq’ = b (A.3) where the vector b denotes the right-hand part of eqn. (A.2). Eqn. (A.l) and (A.2), together with the constraints, eqn. (2b), (2c) and (7), can now be used to compute both the charges and their first derivatives with respect to atomic coordinates. Having obtained the solution of eqn. (A.l), the 3 N equations (A.2) can be efficiently solved by making use of the LU decom- position algorithm” because the most time-consuming part of the algorithm, decomposition of the A matrix into the product of lower and upper triangle matrices, has already been done in solving eqn. (A.1). Thus, one needs only to perform 3N back- substitutions with the corresponding right-hand part of eqn. (A.2). With this algorithm, for a molecule of 300 atoms, the C P U times needed for solving eqn. (A.l) and for solving 3N eqn. (A.2) have a ratio of ca. 1 : 6. Therefore, the algorithm is rather computationally efficient.

Note that the equation to compute the second derivatives of the charges is equivalent to eqn. (A.3) with of course another vector 6. This means that the second derivatives can only be calculated by making 3N(3N - 1)/2 back-substitutions.

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