Computers Chem. Vol. 13, No. 2, pp. 123-128, 1989 Printed in Great Britain. All rights nwved

0097-8485/89 $3.00 + 0.00 copyright Q 1989 Pergamon Press plc

DETERMINATION OF APPROXIMATE FORCE FIELD PARAMETERS

MARKUS BRELJER and GERHARD H;~GELE*

Institut fiir Anorpnische Chemie und Strukturchemie I, Universitet Diisseldorf, Universitiitsstr. 1, 4000 Dtlsseldorf 1, F.R.G.

(Receiued 8 April 1988)

Abstrati-Methods are described which allow the addition of parameters to an existing force field in order to include new structural units. Data reouired for the oarameterization procedure are mainlv taken from quantum mechanical calculations. The methods have been applied ‘to add parametersV for amino- alkylnhosphonic acids to the GROMOS force field. A molecular dynamics (MD) simulation of 64 aminomeihylphosphonic acid molecules in eight unit cells has been carried .out.- The approximate parameters have also been used for a MD-study of aminomethylphosphonic acid in WC water at constant temperature and pressure

INTRODUCTION

Force field simulations including energy minimiz-

ation (EM) and molecular dynamics (MD) tech-

niques have been successfully employed with a variety of organic compounds. Nevertheless, the lack of force field parameters often prohibits the application of EM or MD to molecules containing less common functional groups. Since for such compounds often only few experimental data are available, a procedure to derive approximate force field parameters using the results of quantum mechanical calculations seems highly desirable.

PREVIOUSLY DESCRIBED PARAMETERIZATION PROCEDURES

Force field parameterization has been the subject of many papers but only a few can be mentioned here. A systematic approach to the problem was developed by Lifson and coworkers within their consistent force field (CFF) method (Lifson & Warshel, 1968). They used a linear least squares parameter optimization (Ermer, 1976; Niketic C Rasmussen, 1977). Some possible disadvantages have been discussed by Burkert & Allinger (1982). They suggest a combination of least squares methods and trial and error adjustments of parameters. The excel- lent results obtained with MM2 indicate how well this procedure works (Allinger, 1977).

Whereas the parameterization for standard organic molecules was based mainly upon experimental data, the inchrsion of unusual structural elements often requires quantum mechanical data. This is because there are not enough experimental data available.

*To whom all correspondence should be addressed.

Recently a method has been proposed to derive arbitrary force field parameters from ab initio calcu- lations. It has been realized in a program called “PROBE” (Hagler et al., 1987). Since reliable ab i&o calculations require much computational effort, PROBE might be quite demanding as far as com- puter power is concerned.

The most promising approach towards the devel- opment of exact potential energy functions and force field parameters seems to be the method outlined by Kollman et al. (1986) Using ab initio quantum mechanical calculations as an “essential guide” in the fitting of empirical data.

In order to allow an easy estimation of additional parameters for an existing force field Hopfinger & Pearlstein (1984) suggested use of semi-empirical quantum mechanical computations. They also pro- posed a correction of the results of such calculations when deriving force constants. However, with this method we were not able to reproduce perfectly the stretching force constant for a C-C single bond for the MM2 force field (Breuer et al., 1988).

Many other attempts have been made to find ad hoc parameters (Osawa, 1983, 1986).

NEW WAYS TO DETERMINE APPROXIMATE FORCE FIELD PARAMETERS

Generui considerations

The determination of approximate force field parameters has two purposes: first, these parameters may be used in simulations aiming at a qualitative ot semiquantitative discussion of molecular structures. Secondly, these data may be used aa a basis for a further optimization of the parameters. The overall procedure is shown in Fig. 1.

We consider here the determination of the five types of force field parameters which are used in most

123

124 MARKUS BREUER and GBRHARD H.&ELE

1 -me -r-r ---________,

QUANTUM MECHANICAL $XPERIMENNTAL DATA!

4

(APPROXIMATE) ~--__-----____,

FORCE FIELD t--:O~IMIZATION ;

PARAbtETERS I-___- -____)

7

I v 1

1 MM OR MD SI~lULATIONS I

I 6 1 1 1 I I I I L t

WITH EXPERIMENTAL ___Na_ ___:

.L

FURTHER SIMULATIONS

Fig. I. Approximate parameters and their refinement.

force fields. The parameters may be divided into two classes:

(i) Bonded interactions:

-harmonic bond stretching -harmonic angle bending

(ii) Nonbonded interactions:

-Coulomb interaction -van der Waals interaction -torsion potentials.

Bonded interactions

The procedure suggested in this paper for deter- mining approximate parameters for bonded inter- actions has two major advantages over other par- ameterization procedures: it can be carried out easily without great computational effort, even in cases where experimental data are lacking. The principles of the procedure are shown in Fig, 2.

Standard bond lengths and bond angles may easily be derived from quantum mechanical or experimental data. Since distances are not defined in the same way for the different methods available, it might be useful

to apply corrections. This ensures that certain system- atic errors which occur in quantum mechanical calcu- lations (Schifer, 1983; Van Hemelrijk et al., 1981) may be overcome.

If stretching and bending force constants are calcu- lated from quantum mechanical data a scaling factor can be introduced. This guarantees that the new force constants match those already available. The neces- sity of scaling quantum mechanically derived force constants in order to calculate reliable vibration frequencies has for example been discussed for CNDO/2 (Panchenko er al., 1976), for MINDO/3 (Van Hemelrijk, 1981) and for ab initio methods (Blom & Altoma, 1976; Blom et al., 1976).

Our parameterization procedure consists of two steps. First, a model is selected that is as closely related as possible to the compounds to be simulated with the force field method. Small model compounds reduce the computational effort needed for the quan- tum mechanical calculations. If for instance the P-O-stretching force constant for alkylphosphine oxides is to be determined, trimethylphosphine oxide may be an appropriate model compound.

In the second step, the geometry of the model compound is optimized using an ab initio or a

Approximate force field parameters 125

I S1RUCTURJ.L UNIT +

SUBSTITUENTS

SIMILARITY

QUANTUM MEXHANICS QUANTUM MECHANICS

L 1

WTENTIAL ENERGY POTENTIAL ENIZRGY

CURVE CURVE

LEAST SQUARES PIT L?ZAST SQUARES BIT

L- 1

CRUDE FORCE CONSTANT

K' KQ:q

I I

Fig, 2. Parameters for bonded interactions.

semi-empirical method. The energy is then calculated for fixed atom positions for varying bond lengths or angles so that a potential energy curve can be com- puted. The same procedure is also applied to a standard molecule. This standard should be struc- turally similar to the model compound. The standard force field parameter must be known for the force field being investigated. Corrections are derived from comparison of the standard force field parameter and the calculated potential energy curve. The application of these corrections to the calculated potential curve of the model compound allows the determination of reliable new force field parameters.

Force constants computed from ab initio (Hartree-Fock) or semi-empirical data should not be used as force field parameters without further cor- rections. One reason for this has been discussed by Hopfinger & Pearlstein (1984). The quantum me- chanical calculation of the potential energy curve includes all interactions, while in force field calcu- lations different kinds of interaction are treated indi- vidually. Thus additional contributions (i.e. non-

bonded interactions) to the steric energy arise which should not be incorporated into the force constant. However, these additional contributions are not significant for small model compounds. Deficiencies of the Hartree-Fock method are obviously of greater importance, Using a finite difference method to deter- mine the stretching force constants, one has to take into account that the Hat-tree-Fock potential energy curves are too steep for distances greater than the bond length (Pilar, 1968). Thus the scaling factor should be ~1, as we have found it to be. Never- theless, other influences on the force constant may also play a role.

Bending force constants must also be corrected. This is easily seen from Scheme 1 which shows an example of three ligand atoms attached to a central atom C. All atoms are in a plane.

2

B/ ‘B’

126 MARKUS BREUER and GFRHARD HAGELE

A variation of angle A-C-B also implies a vari- ation of angle A-C-B’. In this case the calculated bending force constant would be too large by a factor of 2. Similar considerations apply to other coordi- nation geometries.

To illustrate the procedure outlined above some results are presented concerning the force constant for angle N<-P in aminomethylphosphonic acid. In the solid state and in aqueous solution amino- methylphosphonic acid exists in a zwitterionic form (Wozniak et al., 1972).

H 0 1 I

H- N---Hz-- P--O--B

Ii I 0

We have simulated the structure of aminomethyl- phosphonic acid using the MNDO method (Dewar & Thiel, 1977; Dewar er al., 1978) available in the program AMPAC (Dewar, 1986). We have then calculated the potential energy curve for six values of the N-C-P angle keeping all other geometry para- meters fixed. The data chosen for the N-C-P angle were + 2, +4 and & 6” with respect to the optimized MNDO value. The harmonic force constant has then been determined by a least squares method.

By this procedure we obtained a force con- stant of 1.45 x 1O-‘8 J rad-’ for the betaine (NH$-CH,PO,H-). For the amino-acid form (NH&H,-POSH,) we obtained a force constant of 1.40 x 10-‘s J rad-‘. These values are substan- tially larger than the GROMOS value of 0.765 x 10-‘g J rade2 for all kinds of angles X-CHI-Y.

In order to correct the force constant for the angle N-P< we have applied the above procedure to two other molecules: to the zwitterionic form of glycine and to propane. These were used as standards. The angles N4-C and C-C-C have been investigated. The bending force constants found were 1.25 x lo-‘* and 1.38 x lo-l8 J rad-2. Applying a correction fac- tor according to Fig. 2 leads to factors Q of 0.61 in the case of glycine and of 0.55 in the case of propane. These results for the two quite different compounds indicate that a correction factor between 0.5 and 0.6 is appropriate. Thus a force constant of 0.765 x IO-” J Rad-* for X-CH,-Y seems to be reasonable also in the case that X = NH, and Y = PC$H.

In calculation of force field parameters for second row elements the AM1 method should be pre- ferred over the MNDO method. Geometry, heat of formation and vibrational frequencies can be calculated much more precisely for compounds con- taining silicon (Dewar & Jie, 1987) or phosphorus (Dewar, 1987).

Nonbonded interactions

van akr Wash parameters. van der Waals par- ameters can often be estimated easily from the parameters already available within the force field.

However, one should take care to note that the van der Waals radii used in force field calculations are often not identical with the van der Waals radii derived by X-ray crystallography (Burkert % Allinger, 1982; Kollman et al., 1986). Trial and error adjustments can be performed in order to improve the agreement between energy minimization results and experimental or quantum mechanical data (Breuer et al., 1988).

Torsions. Semi-empirical methods should not be used to calculate torsion potentials. Although the relative stability of possible rotamers is often pre- dicted correctly, considerable errors occur in the calculation of rotational barriers. Using these values to parameterize a force field as has been suggested by Hopfinger & Pearlstein (1984) would introduce arti- facts. Even ab initio methods cannot well reproduce rotational barriers unless very refined methods are used (Burkert & Allinger, 1982). Barriers calculated wiih the fixed rotator approximation should not bc used for parameterization (Stern et al., 1983).

Approximate dihedral angle potentials are calcu- lated from parameters already known by applying a correction factor if second row elements are involved. This is done in order to take into account the lower rotational barriers for bonds involving second row elements (Allinger, 1985). The use of a factor 2 does not mean that the rotational barrier is divided by 2. This indeed would not be justified. In fact, only a part of the rotational barrier stems from the torsional potential. An important contribution originates in the other nonbonded interactions that are calculated separately. However, the approximate parameter- ization procedure suggested here should not be used if the calculation of rotational barriers is desired.

Point charges. Several methods are available to determine atom point charges for calculating the Coulombic contributions to the steric energy:

-Parameters can be derived by fitting of observed quantities.

-Atomic populations can be determined experi- mentally by X-ray crystallography (Pearlman & Kim, 1985).

-A partial-charge model can be fitted to quantum mechanically calculated electrostatic potentials (Singh & Kollman, 1984).

-Atomic orbital populations can be calculated from molecular orbital wavefunctions. For this pur- pose several methods are available (Edgecomb & Boyd, 1987).

-Atomic charges can be calculated using elec- tronegativity concepts (Lindner, 1986; Gasteiger & Marsili, 1980; Shanker et al., 1986).

Only the last two methods are readily employable for an approximate parameterization. Some aspects of the determination of partial charges have recently been investigated by Damewood et al. (1988).

As a justification for the selection of the point charges for their SPC water Berendsen et al. (1981)

Approximate force field parameters 127

suggested that three body contributions should be tances are larger than the experimental values. The included in an effective pair potential. This indicates heavy atom skeleton bond angles are reproduced that even elaborate quantum mechanical or experi- within a mean absolute error of 2.9”. The dihedral mental methods may not lead to charges with good angle N-C-P-OH averaged over the last 5 ps of the accuracy. MD simulation is 61”, the experimental value is 59”.

Point charges added to an existing force field must match with other nonbonded parameters. Therefore we have used the existing GROMOS parameters (Herrnans et al., 1984) wherever possible. The charge on the phosphonate P atom has been adjusted to yield a net charge of zero for the total molecule.

Solution structure of aminomethylphosphonic acid

EFFICIENCY OF THE PARAMETERIZATION PROCEDURE

Some preliminary results are presented of simu- lations carried out using the GROMOS package (van Gunsteren & Berendsen, 1987). Our simulations for atninomethylphosphonic acid indicate that meaning- ful results may be obtained using our approximate force field parameters.

Aminomethylphosphonic acid was simulated to- gether with 194 SPC water molecules in a truncated octahedron. Periodic boundary conditions were ap- plied. The cut-off radius for aonbonded interactions was 0.9 mn. All bond lengths were fixed as in the case of the solid state simulation. The system was weakly coupled to a pressure bath (1 atm) and to a tem- perature bath (300 K). The integration time step was 0.002 ps. An equilibration period of 8 ps was applied. Then the system was followed for a further 8 ps.

Solid state structure of aminomethylphosphonic acid

In force fields which do not contain explicit hydro- gen bond potentials the correct representation of hydrogen bonds may be an appropriate way to test if nonbonded interactions are treated correctly. Moreover, MD simulations at constant pressure reveal whether intermolecular distances and experi- mental unit cell dimensions are reproduced (van Gunsteren & Berendsen, 1985).

The initial density was 1.014 g cms3. The mean density during the last 8 ps of the simulation is 0.98Og cme3. This is a decrease of 3.4%. Average bond angles and dihedrals have been calculated for the heavy atom skeleton. The mean absolute devi- ation from the X-ray values found for the bond angles is 3.6”. The conformation found in the solid state has been preserved. The dihedral angle N-C-P-OH is 63”, which is close to the X-ray value of 59”. In contrast to the situation in the crystal, a rotation of the NH, group occurred during the simulation of the solution.

Here we present some preliminary results from a MD simulation of the solid state structure of amino- methylphosphonic acid. The X-ray structure (Darriet et al., 1975) was used as input. After an energy minimization a MD simulation over 18 ps was carried out with a time step of 0.2 fs. Eight unit ceils contain- ing a total of 64 molecules were simulated applying periodic boundary conditions. A cut-off radius of 0.9nm for nonbonded interactions was used. Bond lengths were either fixed at their experimental values of to the GROMOS standard values in the case of X-H-bonds using the SHAKE-method (Van Gunsteren & Berendsen, 1977; Ryckaert et al., 1977). The CH,-moity was treated as a united atom. The system was coupled to a pressure bath and to a temperature bath. The reference temperature was 300 K, the reference pressure 1 atm. The results presented here were calculated from the trajectories of the last 5 ps.

Cyclic conformations for aminoalkylphosphonic acids in aqueous solution have been proposed on the basis of NMR data (Appleton et al., 1984). This idea is confirmed by the results of our simulation. How- ever, the structure we have found is quite different from the one suggested by Appleton et nl. According to our simulations a water molecule is involved in the cyclic structure. Hydrogen bonds between this water molecule and the aminomethylphosphonic acid betaine are predicted. As hydrogen bonding criteria the minimum donor-hydrogen-acceptor angle was set to 135”, the maximum hydrogen-acceptor distance was set to 0.25 nm. With these criteria a hydrogen bond P-O-H., OH, occurs in 98% of the configurations analysed. The average &H . . . 0 an- gle is 162’, the donor-acceptor distance is 0.269 run. Because of the rotation of the NH,-group N-H.. . OH, hydrogen bonds have only short life- times. A N-H. . . OHz hydrogen bond is observed in 43% of the configurations analysed. The mean angle is 155”, the mean donor-acceptor distance 0.305 nm.

The average volume of the eight unit cells in our simulation is 7.254 nm3. This means that the volume increased by 9.2% during the simulation. An increase of the intermolecular distances is also observed in the acceptordonor distances of hydrogen bonded sites. Every acidic hydrogen is involved in a hydrogen bond. This hydrogen bond network is well preserved during the simulation. The mean absolute error in the donor-hydrogen-acceptor angles is 5”, in the donor- acceptor distances 0.021 nm. All donor-acceptor dis-

CONCLUSIONS

It has been shown that approximate force field parameters for bonded interactions can easily be derived from quantum mechanical data. New par- ameters for nonbonded interactions can be. estimated from parameters already available in the force field under consideration. Preliminary results for amino- methylphosphonic acid in the solid state and in

128 MARKUS BREER and GERHAIUZI H&ELE

aqueous solution indicate that meaningful results can he obtained using approximate force field par- ameters, though a further optimization may improve the agreement between experimental and calculated data.

Acknowledgements-The Computer Centre of the Univer- sity of Diisseldorf has provided the means to carry out this research on the SIEMENS 7.580-S. M.B. wishes lo thank the Studienstiftung des deutschen Volkes for a personal grant. This study was supported by the Fonds der Chemischen Industrie.

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