Supplementary Material for: "Merck Molecular Force Field. III. Molecular Geometries and Vibrational Frequencies for MMFF94," T. A. Halgren, J. Comput. Chem., 17, 553-586 (1996).
This appendix discusses the format of the parameter tables (Tables III - VI) and describes in detail how requests for parameters are matched to entries in the tables.
A.1. Bond Stretching Parameters. The first column in Table III, labeled BTIJ, lists the bond type index discussed in the "QUARTIC BOND STRETCHING" subsection of the section on "Form of MMFF94 ...". The second and third columns list the atom types, and the fourth and fifth give the force constant in mdynes/Angs and the reference bond length in Angs. The last, labeled "Origin/Comment", distinguishes "C94" core MMFF94 parameters, derived from ab initio data as described in this paper, from parameters derived in the extended parameterization described in Paper V [1] (usually labeled "E94"). As can be seen, the bond-type index BTIJ normally takes the value "0" but occasionally takes the value "1". The latter value denotes a formally single bond between sp- or sp2-hybridized atoms of types that can participate in multiple (double or triple) bonding. Thus, the parameter whose three numerical indices are "0 2 2" describes a double bond between olefinic carbons (reference bond length, 1.333 Angs), whereas the next-listed parameter with numerical indices "1 2 2" describes the delocalized single bond between two such carbons in, for example, butadiene (reference bond length, 1.430 Angs). This last parameter, as it happens, is labeled "#C94", where the "#" sign indicates that a "compromise" value has been chosen for the reference bond length (cf. the discussion of 1,3-butadiene in the "EXPERIMENTAL MOLECULAR GEOMETRIES" subsection of the section on "Performance of MMFF94"). Other labels used in the full parameter file include "X94" for parameters whose reference bond lengths were derived from additional MP2/6-31G* data in MMFF's extended parameterization [1], "#X94" for parameters of similar computational origin but for which a compromise reference bond length was chosen, and "#E94" for bonds to hydrogen for which a poorly determined crystallographic bond length was replaced by an empirical rule value; see Paper V [1] for further details.
No wildcarded parameters are defined for bond stretching interactions. Rather, when a needed bond parameter is not found in the parameter file, MMFF94 employs an empirical rule [1] to assign a reference bond length using a modified form of the Schomaker-Stevenson equation [2], and assigns the force constant [1] from an inverse sixth-power relationship [3] . The bond parameters labeled "E94", for example, take their reference bond lengths from fits to cystallographically determined structures and use the empirical sixth-power rule to assign the force constant [1].
The canonical ordering reflected in the table is defined through an integer index, CXB, that is computed as
CXB = MC*(I*MA + J) + M,
where M denotes the bond-type index BT[IJ], MC is an integer equal to at least the maximum permitted bond-type index + 1, MA is an integer at least equal to the maximum numerical atom-type index + 1, and atom type I is less than or equal to J. Thus, I changes least rapidly and M changes most rapidly, with the result that parameters that reference the same atom types but different bond types appear consecutively in the listing. As for other terms in the MMFF94 energy expression, this ordering allows a rapid binary search algorithm to be used to locate the parameters but has no intrinsic effect on the matching of parameters to force-field interaction terms.
A.2. Angle Bending Parameters. The seven columns in Table IV define the angle-type index AT[IJK], the atom types I, J and K for the peripheral, central, and peripheral atoms in the angle, the force constant in md-Angs/rad**2, the reference bond angle in degrees, and the origin of the parameter, where "C94" and "E94" have meanings analogous to those discussed above. The full parameter file employs other labels as well [4]. The MMFF94 angle-bending parameters also include parameters that have zero values for atom types I and K and a zero value for the listed force constant; the first parameter listed in Table IV, for example, is of this type. These "default parameter" entries are used for interactions I-J-K of angle-type M = AT[IJK] when the parameter file contains neither the fully-qualified M:I-J-K parameter nor any related parameter obtained by successively equivalencing atom types I, J, and K to simpler atom types in the manner described in Paper I [5]. In such a case, the angle-type M and the central atom-type J are used to assign the reference angle, and an empirical rule based on previously published work is then employed to calculate the force constant; if no match on M and J is found, information relating to hydridization and ring size is used to assign the reference value [1]. The "E94" parameters combine reference angles obtained by fitting to experimental geometries with empirical-rule force constants. The default reference angles represent averages of explicitly parameterized reference angles for the same angle type M and central atom type J. See Paper V for further information [1].
The listed parameters employ angle-type indices ranging between 0 and 8. Their meanings are as defined below:
AT[IJK] Structural significance ----------------------------------------------------- 0 The angle i-j-k is a "normal" bond angle 1 Either bond i-j or bond j-k has a bond type of 1 2 Bonds i-j and j-k each have bond types of 1; the sum is 2. 3 The angle occurs in a three-membered ring 4 The angle occurs in a four-membered ring 5 Is in a three-membered ring and the sum of the bond types is 1 6 Is in a three-membered ring and the sum of the bond types is 2 7 Is in a four-membered ring and the sum of the bond types is 1 8 Is in a four-membered ring and the sum of the bond types is 2
The canonical ordering of the angle parameters is such that the atom-type index J changes least rapidly and the angle-type index M changes most rapidly; atom type I is always less than or equal to K and changes less rapidly. As a result, all angle interactions having a common central atom type appear consecutively in the listing. The canonical-order index that produces this ordering, CXA, is computed as:
CXA = MC*(J*MA**2 + I*MA + K) + M,
where MA is again at least the maximum atom type + 1, and MC is at least one greater than the maximum angle-type index.
A.3. Stretch-Bend Parameters. The format of Table V for stretch-bend parameters is similar to that employed for angle bending. For a given i-j-k interaction, two force constants, each given in units of md/rad, arise for each listing in the parameter file. The first, kba[IJK], couples i-j-k bending with the stretching of the i-j bond, whereas the second, kba[KJI], couples i-j-k bending to k-j stretching. The requisite reference bond lengths and reference bond angle (cf. eq 5) are taken from the respective bond and angle parameter sets, with the stretch-bend type index SBT[IJK] listed in the first column serving with the atom types to establish the proper connection. The stretch-bend types are defined as shown below:
SBT[IJK] AT[IJK] BT[IJ] BT[JK] ------------------------------------------------------------- 0 0 0 0 1 1 1 0 2 2 0 1 3 2 1 1 4 4 0 0 5 3 0 0 6 5 1 0 7 5 0 1 8 6 1 1 9 7 1 0 10 7 0 1 11 8 1 1Note that two stretch-bend types occur for each angle type for which the associated bond-type indices sum to 1.
Canonical order is established through an index, CXBA, that is computed from the stretch-bend type and the atom types in the same manner as was described above for the index CXA used for angle-bending parameters. The ordering of the tabulated parameters therefore follows the same prescription.
The matching of stretch-bend interactions to parameter file entries uses the stretch-bend type M = SBT[IJK] and the atom types I, J, and K. No equivalencies are employed, but default force constants are assigned whenever the fully qualified M:I-J-K parameter is not found (cf. Paper V [1]). All the parameters listed in Table V are labeled "C94", and thus represent core MMFF94 values derived by fitting to HF/6-31G* energy derivatives. The full parameter file includes some parameters labeled "X94"; these parameters were derived from HF/6-31G* data in a comparable manner to, but not as part of, the main "C94" parameterization described in this paper.
A.4. Out-of-Plane Bending Parameters. Of the four atom types that characterize the parameters listed in Table VI, the second, labeled J, corresponds to the central atom j to which the atoms i, k, and l of types I, K and L are attached. The force constant, koop[IJK;L], is given in units of md-A/rad**2. In the individual interactions that appear in eq 6, l represents the out-of-plane atom in the Wilson angle definition. This distinction does not affect the way in which the parameters are utilized, however, because MMFF94 employs the same force constant for all three out-of-plane angles. The canonical index, CXO, used to order the parameters is computed as:
CXO = J*MA**3 + I*MA**2 + K*MA + L,
where MA is as defined previously, L is greater than or equal to K, and K is greater than or equal to I. This ordering groups together all out-of-plane entries that share a common central atom type J.
Unlike the interactions discussed in the previous subsections, the four atom types suffice to characterize an out-of-plane parameter; no "bond type" or "angle type" information is utilized at present. The parameter search begins with an attempt to locate the fully qualified parameter. If this search is unsuccessful, a staged search using more generic parameter types is performed (cf. the "Implementation of MMFF ..." section in Paper I [5]). The entries for which atom types I, K and L are all zero serve as default values. The default force constants represent, roughly, the average of the force constants for the central atom type J derived in the fit to the HF/6-31G* data. All but two of nondefault entries in Table VI are labeled "C94" and were derived from the HF/6-31G* ab initio data as described in the section entitled "Derivation of Quadratic Force Constants." The two entries labeled "#C94" are modified values chosen to improve the representation of out-of-plane bending in amides (cf. the subsection on "GEOMETRIES FOR AMIDES AND UNSATURATED AMINES" in the "Performance of MMFF94" section). Some default force constants are labeled "E94" in the full parameter table; their values were assigned by analogy to computationally derived values found for similar atom types [1].
[2] R. Blom, A. Haarland, J. Molec. Struct., 128, 21-27 (1985).
[3] The relationship used is based on the sixth-power dependence employed by: R. S. Roy, J. Phys. B (Proc. Phys. Soc.), Ser 2, 445 (1968).
[4] See the "Extension of the Parameterization for Angle Bending" section of Paper V (ref. 1) for definitions for additional labels used for angle-bending parameters in the full parameter file.
[5] Paper I: T. A. Halgren, J. Comput. Chem., 17, 490-519 (1996).