2 Force Field Methods
2.4 Differences in Force Fields
There are many different force fields in use. They differ in three main aspects:
(1) What is the functional form of each energy term?
(2) How many cross terms are included?
(3) What types of information are used for fitting the parameters?
There are two general trends. If the force field is designed primarily to treat large systems, such as proteins or DNA, the functional forms are kept as simple as possible.
This means that only harmonic functions are used for Estrand Ebend(or these term are omitted, forcing all bond lengths and angles to be constant), no cross terms are included, and the Lennard-Jones potential is used for Evdw. Such force fields are often called “harmonic”, “diagonal” or “Class I”. The other branch concentrates on repro- ducing small- to medium-size molecules to a high degree of accuracy. These force fields will include a number of cross terms, use at least cubic or quartic expansions of Estr and Ebend, and possibly an exponential-type potential for Evdw. The current efforts in developing small-molecule force fields go in the direction of not only striving to
Table 2.4 Comparison of functional forms used in common force fields;49the torsional energy, Etors, is in all cases given as a Fourier series in the torsional angle
Force field Types Estr Ebend Eoop Evdw Eel Ecross Molecules
AMBER 41 P2 P2 imp. 12–6 charge none proteins, nucleic
12–10 acids,
carbohydrates
CFF91/93/95 48 P4 P4 P2 9–6 charge ss,bb,st, general
sb,bt,btb
CHARMM 29 P2 P2 imp. 12–6 charge none proteins
COSMIC 25 P2 P2 Morse charge none general
CVFF 53 P2 or P2 P2 12–6 charge ss,bb,sb, general
Morse btb
DREIDING 37 P2 or P2(cos) P2(cos) 12–6 or charge none general
Morse Exp–6
EAS 2 P2 P3 none Exp–6 none none alkanes
ECEPP fixed fixed fixed 12–6 and charge none proteins
12–10
EFF 2 P4 P3 none Exp–6 none ss,bb,sb, alkanes
st,btb
ENCAD 35 P2 P2 imp. 12–6 charge none proteins, nucleic
acids
ESFF 97 Morse P2(cos) P2 9–6 charge none all elements
GROMOS P2 P2 P2(imp.) 12–6 charge none proteins, nucleic
acids, carbohydrates
MM2 71 P3 P2+P6 P2 Exp–6 dipole sb general
MM3 153 P4 P6 P2 Exp–6 dipole or sb,bb,st general (all
charge elements)
MM4 P6 P6 imp. Exp–6 charge ss,bb,sb, general
tt,st,tb, btb
MMFF 99 P4 P3 P2 14–7 charge sb general
MOMEC P2 P2 P2 Exp–6 none none metal coordination
NEMO fixed fixed none Exp–6 quad, polar none special
OPLS 41 P2 P2 imp. 12–6 charge none proteins, nucleic
acids, carbohydrates
PFF P2 P2 imp. 12–6 polar none proteins
PROSA 41 P2 P2 imp. 12–6 polar none proteins
QMFF 32 P4 P4 P2 9–6 charge ss,sb,st,bb, general
bt,btb
SDFF P4 P4 9–6 charge, ss,st,tt hydrocarbons
dipole, polar
TraPPE fixed P2 fixed 12–6 charge none C, N, O
compounds
TRIPOS 31 P2 P2 P2 12–6 charge none general
UFF 126 P2 or cos(nq) imp. 12–6 charge none all elements
Morse
YETI 17 P2 P2 imp. 12–6 and charge none proteins
12–10
Notation: Pn: Polynomial of order n; Pn(cos): polynomial of order nin cosine to the angle; cos(nq): Fourier term(s) in cosine to the angle; Exp–6: exponential +R−6;n−m:R−n+R−mLennard-Jones type potential; quad: electric moments up to quadrupoles;
polar: polarizable; fixed: not a variable; imp.: improper torsional angle; ss: stretch–stretch; bb: bend–bend: sb: stretch–bend;
st: stretch–torsional; bt: bend–torsional; tt: torsional–torsional; btb: bend–torisional–bend.
Table 2.5 Comparison of stretch energy parameters for different force fields
Force field R0(Å) k(mdyn/Å)
C—C C—O C—F C=O C—C C—O C—F C=O
MM2 1.523 1.402 1.392 1.208 4.40 5.36 5.10 10.80
MM3 1.525 1.413 1.380 1.208 4.49 5.70 5.10 10.10
MMFF 1.508 1.418 1.360 1.222 4.26 5.05 6.01 12.95
AMBER 1.526 1.410 1.380 1.220 4.31 4.45 3.48 8.76
OPLS 1.529 1.410 1.332 1.229 3.73 4.45 5.10 7.92
reproduce geometries and relative energies, but also vibrational frequencies, and these are often called “Class II” force fields. Further refinements by allowing parameters to depend on neighbouring atom types, e.g. for modelling hyperconjugation, and includ- ing electronic polarization effects have been denoted “Class III” force fields.
Force fields designed for treating macromolecules can be simplified by not consid- ering hydrogens explicitly – the so-called united atomapproach (an option present in for example the AMBER, CHARMM, GROMOS and DREIDING force fields).
Instead of modelling a CH2group as a carbon and two hydrogens, a single “CH2atom”
may be assigned, and such a united atom will have a larger van der Waals radius to account also for the hydrogens. The advantage of united atoms is that they effectively reduce the number of variables by a factor of ~2–3, thereby allowing correspondingly larger systems to be treated. Of course the coarser the atomic description is, the less detailed the final results will be. Which description, and thus which type of force field to use, depends on what type of information is sought. If the interest is in geometries and relative energies of different conformations of say hexose, then an elaborate force field is necessary. However, if the interest is in studying the dynamics of a protein con- sisting of hundreds of amino acids, a crude model where whole amino acids are used as the fundamental unit may be all that is possible, considering the sheer size of the problem.48
Table 2.4 gives a description of the functional forms used in some of the common force fields. The torsional energy is in all cases written as a Fourier series, typically of order 3. Many of the force fields undergo developments, and the number of atom types increases as more and more systems become parameterized, and Table 2.4 may thus be considered as a “snapshot” of the situation when the data were collected. The “uni- versal” type force fields, described in Section 2.3.3, are in principle capable of cover- ing molecules composed of elements from the whole periodic table, and these have been labelled as “all elements”.
Even for force fields employing the same mathematical form for an energy term there may be significant differences in the parameters Table 2.5 below shows the vari- ability of the parameters for the stretch energy between different force fields. It should be noted that the stretching parameters are among those that vary the leastbetween force fields.
It is perhaps surprising that force constants may differ by almost a factor of two, but this is of course related to the stiffness of the stretch and bending energies. Very few molecules have bond lengths deviating more than a few hundredths of an angstrom
from the reference value, and the associated energy contribution will be small regard- less of the force constant value. Stated another way, the minimum energy geometry is insensitive to the exact value of the force constant.