The ESFF Forcefield

Functional Form
ESFF Atom Types

Functional Form

While the CVFF and CFF91 forcefields derive the forcefield parameters by fitting ab initio and/or experimental data sets, ESFF relies on atomic parameters coupled with rules for generating explicit parameters. The rules embody physical reality and therefore tend to break the redundancies and guarantee transferability. As much as possible, the atomic parameters are directly determined from experiment or calculated rather than fit.

Valence Energy

The analytic energy expressions for the ESFF forcefield are provided in Eq. 3-3 along with schematic representations in Figure 3-5. As in AMBER, only diagonal terms are included.

Eq. 3-3:

The bond energy is represented by a Morse functional form, where the bond dissociation energy D, the equilibrium bond length r₀, and the anharmonicity parameters are needed. In constructing these parameters from atomic parameters, the forcefield utilizes not only the atom types and bond orders, but also considers whether the bond is endo or exo to 3-, 4-, or 5-membered rings. The rules themselves depend on the electronegativity, hardness, and ionization of the atoms as well as atomic anharmonicities and the covalent radii and well depths. The latter quantities are fit parameters, and the former three are calculated.

The ESFF angle types are classified according to ring, symmetry, and -bonding information into 5 groups:

The normal class includes unconstrained angles as well as those associated with 3-, 4-, and 5-membered rings. The ring angles are further classified based on whether one (exo) or both bonds (endo) are in the ring. Additionally, angles with only central atoms in a ring are also differentiated.
The linear class includes angles with central atoms having sp hybridization, as well as angles between two axial ligands in a metal complex.
The perpendicular class is restricted to metal centers and includes angles between axial and equatorial ligands around a metal center.
The equatorial class includes angles between equatorial ligands of square planar (sqp), trigonal bipyramidal (tbp), octahedral (oct), pentagonal bipyramidal (pbp), and hexagonal bipyramidal (hbp) systems.
The system class includes angle between pseudoatoms. This class is further differentiated in terms of normal, linear, perpendicular, and equatorial types.

The rules that determine the parameters in the functional forms depend on the ionization potential and, for equatorial angles, the periodicity. In addition to these calculated quantities, the parameters are functions of the atomic radii and well depths of the central and end atoms of the angle, and, for planar angles, two overlap quantities and the 1-3 equilibrium distances.

To avoid the discontinuities that occur in the commonly used cosine torsional potential when one of the valence angle approaches 180°, ESFF uses a functional form that includes the sine of the valence angles in the torsion. These terms ensure that the function goes smoothly to zero as either valence angle approaches 0° or 180°, as it should. The rules associated with this expression depend on the central bond order, ring size of the angles, hybridization of the atoms, and two atomic parameters for the central atom which is fit.

The functional form of the out-of-plane energy is the same as in CFF91, where the coordinate () is an average of the three possible angles associated with the out-of-plane center. The single parameter that is associated with the central atom is a fit quantity.

Nonbond Energy

Partial Charges

The charges are determined by minimizing the electrostatic energy with respect to the charges under the constraint that the sum of the charges is equal to the net charge on the molecule. This is equivalent to equalization of electronegativities. The derivation of the rule begins with the following equation for the electrostatic energy:

Eq. 3-4:

where

is the electronegativity and

the hardness. The first term is just a Taylor series expansion of the energy of each atom as a function of charge, and the second is the Coulomb interaction law between charges. The Coulomb law term introduces a geometry dependence that ESFF for the time being ignores, by considering only topological neighbors at effectively idealized geometries. Minimizing the energy with respect to the charges leads to the following expression for the charge on atom i:

Eq. 3-5:

where

is the Lagrange multiplier for the constraint on the total charge, which physically is the equalized electronegativity of all the atoms. The

term contains the geometry-independent remnant of the full Coulomb summation.

Eqns. 3-4 and 3-5 give a totally delocalized picture of the charges in a relatively severe approximation. To obtain reasonable charges as judged by, for example, crystal packing calculations, some modifications to the above picture have been made. Metals and their immediate ligands are treated with the above prescription, summing their formal charges to get a net fragment charge. Delocalized systems are treated in an analogous fashion. Finally, systems are treated using a localized approach in which the charges of an atom depend simply on its neighbors. Note that this approach, unlike the straightforward implementations based on the equalization of electronegativity, does include some resonance effects in the system.

The electronegativity and hardness in the above equations must be determined. Previous authors have often determined them from experimental ionization potentials and electron affinities; however, these spectroscopic states do not correspond to the valence states involved in molecules. For this reason, ESFF is based on electronegativities and harnesses, calculated using density functional theory as implemented in DMol. The orbitals are (fractionally) occupied in ratios appropriate for the desired hybridization state, and calculations are performed on the neutral atom as well as on positive and negative ions. From these curves the electronegativities and hardnesses can be obtained.

van der Waals Interactions

ESFF currently uses the 6-9 potential for the van der Waals interactions. Since the van der Waals parameters must be consistent with the charges, they are derived using rules that are consistent with the charges. Starting with the London formula:

Eq. 3-6:

where

is the polarizability and IP the ionization potential of the atoms. The polarizability, in a simple harmonic approximation, is proportional to n/IP where n is the number of electrons. Across any one row of the periodic table, the core electrons remain unchanged so that the following form is reasonable:

Eq. 3-7:

where a' and b' are adjustable parameters that should depend on just the period, and n_eff is the effective number of (valence) electrons. Further assuming that

is proportional to R³ and that another equivalent expression to that in Eq. 3-6 is:

Eq. 3-8:

where

is a well depth, the following forms are deduced for the rules for van der Waals parameters:

Eq. 3-9:

The van der Waals parameters are affected by the charge of the atom. In ESFF we have found it sufficient to modify the ionization potential (IP) of metal atoms according to their formal charge and hardness:

Eq. 3-10 :

and for nonmetals to account for the partial charges when calculating the effective number of electrons.

ESFF Atom Types

ESFF atom types (Table 3-3) are determined by hybridization, formal charge, and symmetry. In addition, the rules may involve bond order, ring size, and whether bonds are endo or exo to rings. For metal ligands the cis-trans and axial-equatorial positioning is also considered. The addition of these latter types affects only certain parameters (for example, bond order influences only bond parameters) and thus are not as powerful as complete atom types. In one sense they provide a further refinement of typing beyond atom types.

Table 3-3. Atom Types for the First Three Rows of the Periodic Table--ESFF

The ESFF forcefield has been parameterized to handle all elements up to radon in the periodic table. For the atom types that are not listed in this table, please refer to the $BIOSYM_LIBRARY/esff.frc file.

The format is:

atom type: description

and you may quickly jump to the classes of atom types by clicking:

hydrogen
carbon
nitrogen
oxygen
sulfur
phosphorus
other second-row elements
other third-row elements

hydrogen types

dw: deuterium in heavy water
h: generic hydrogen
hi: hydrogen in charged imidazole ring (equiv. to h*)
hw: hydrogen in water (equiv. to h*)
h*: hydrogen bonded to nitrogen, oxygen
h+: charged hydrogen in cations

carbon types

c: generic sp³ carbon
ca: general amino acid alpha carbon (sp³) (equiv. to c)
cg: sp³ alpha carbon in glycine (equiv. to c)
ci: carbon in charged imidazole ring (equiv. to cp)
co: sp³ carbon in acetals (equiv. to c)
coh: sp³ carbon in acetals with hydrogen (equiv. to c)
cp: sp² aromatic carbon with partial double bond
cr: c in neutral arginine (equiv. to c=)
cs: sp² aromatic carbon in 5-membered ring next to S (equiv. to cp)
ct: sp carbon involved in a triple bond
ct3: sp carbon involved in CO
c1: sp³ carbon with 1 H 3 heavies (equiv. to c)
c2: sp³ carbon with 2 H's, 2 heavies (equiv. to c)
c3: sp³ carbon with 3 H's, 1 heavy (equiv. to c)
c5: sp² aromatic carbon in 5-membered ring
c5p: sp² aromatic carbon in 5-membered big pi ring
c': carbon in carbonyl (C=O) group
c-: c in charged carboxylate
c+: c in guanidinium group (equiv. to c=)
c=: generic sp² carbon

nitrogen types

n: generic sp² nitrogen (in amides)
na: sp³ nitrogen in amines
nb: sp² nitrogen in aromatic amines
nh: sp² (3 [sp²] 2 [p]) nitrogen in 5-membered ring
nho: sp² nitrogen in 6-membered ring
ni: nitrogen in charged imidazole ring (equiv. to nh)
no: sp² nitrogen in oxides of nitrogen
np: sp² nitrogen in 5-membered ring
nt: sp nitrogen involved in a triple bond
nt2: central nitrogen involved in azide group
nz: sp nitrogen in N₂
n1: sp² nitrogen in charged arginine (equiv. to n=)
n2: sp² nitrogen (NH₂) in guanidinium group (HN=C(NH₂)₂) (equiv. to n=)
n4: sp³ nitrogen with 4 substituents (equiv. to n+)
n+: sp³ nitrogen in protonated amines
n=: sp² nitrogen in neutral arginine (double bond)

oxygen types

o: generic sp³ oxygen in alcohol, ether, or acid group
oa: sp³ oxygen in ester or acid
oc: sp³ oxygen in ether or acetals (equiv. to o)
oh: oxygen bonded to hydrogen (equiv. to o)
op: sp² aromatic in 5-membered ring
os: oxygen bonded to two silicons
ot: oxygen with hybridization sp
o1: oxygen bonded to oxygen
o': oxygen having a single double bond
o*: oxygen in water
o-: double bonded oxygen in charged carboxylate COO^- (equiv. to o')

sulfur types

s: sp³ sulfur
sp: sulfur in an aromatic ring (e.g., thiophene)
s1: sp³ sulfur involved in (S-S) group of disulfides
s2d: sulfur with oxidation number 4, two double sigma bond
s3d: sulfur with oxidation number 4, three sigma bond, (C_3v)
s4d: sulfur with oxidation number 6, four sigma bond, (T_d)
s4l: sulfur with coordination number 4 (C_2v)
s5l: sulfur with coordination number 5 (D_4h, C_2v)
s5t: sulfur with coordination number 5 (D_3h)
s6: sulfur with coordination number 6 (D_4h, D_2h)
s6o: sulfur with coordination number 6 (O_h)
s': S in thioketone group
s-: double bonded sulfur in charged phosphate PSS^- or PSO^- (equiv. to s')

phosphorus

p: general phosphorous atom
p4d: phosphorous atom with oxidation number 5 and 4 sigma bonds (C_Td)
p4l: phosphorous atom with oxidation number 5 and 3 sigma bonds (C_2v)
p5l: phosphorous atom with oxidation number 5 and 3 sigma bonds (D_4h, C_2v)
p5t: phosphorous atom with oxidation number 5 and 3 sigma bonds (D_3h)
p53: phosphorous atom with oxidation number 5 and 3 sigma bonds (planar)
p6: phosphorous atom with oxidation number 5 and 3 sigma bonds (D_4h, D_2h)
p6o: phosphorous atom with oxidation number 5 and 3 sigma bonds (O_h)
p': sp² phosphorous atom

other second-row elements

b: boron sp³ atom
bt: boron sp atom
b': boron sp² atom
Be: berillium atom
Be+: berillium cation
Be+2: berillium cation
f: fluorine atom
F: fluorine anion
Li: lithium atom with s orbitals involved in bonding
Li+: lithium ion
ne: neon atom

other third-row elements

ar: argon atom
Al: aluminum atom
Al033: aluminum atom with coordination number 3
Al034: aluminum atom with coordination number 4
Al035: aluminum atom with coordination number 5
Al035s: aluminum atom with coordination number 5 (D_4h)
Al035t: aluminum atom with coordination number 5 (D_3h)
Al036: aluminum atom with coordination number 6 (D_4h, D_2h)
Al036o: aluminum atom with coordination number 6 (O_h)
cl: chlorine atom
Cl: chlorine ion
cl': chlorine atom in oxo acid
he: helium atom
Mg: magnesium atom
Mg025: magnesium atom with 5 coordinations
Mg025s: magnesium atom with 5 coordinations (D_4h)
Mg025t: magnesium atom with 5 coordinations (D_3h)
Mg026: magnesium atom with 6 coordinations (D_4h, D_2h)
Mg026o: magnesium atom with 6 coordinations
Mg+: magnesium +1 cation (O_h)
Mg+2: magnesium +2 cation
Na: sodium atom
Na+: sodium ion
si: silicon atom
si4l: silicon atom (D_3h, C_2v)
si5l: silicon atom (D_4h)
si5t: silicon atom (D_3h)
si6: silicon atom (D_4h, D_2h)
si6o: silicon atom (O_h)
si': sp² silicon atom

Rule for Metal Atom Types

The names of the atom types for metals are based on the symmetry of the metal complex and on both the oxidation state and coordination number of the metal. For example, Ag024t indicates an Ag that has an oxidation level of 2+, is 4-coordinated, and has tetrahedral symmetry. The following table lists the abbreviations used for the symmetry types:

            symbol        symmetry
           ------------------------
              l            C_2v 
              s            D_4h  
              t            T_d 
              o            O_h 
              p            D_5h 
              h            D_2h, D_3h 
              d            D_oo

Some metals with differing oxidation numbers and symmetries may be handled with the same parameters. Here, a generic metal atom type is used.

The oxidation number of a metal is determined according to:

where Q_t is the total charge on the complex, and the sums over Fq_i and Fq_j are the sums of formal charges on atoms not bonded to metals and bonded to metals, respectively. N_m is the number of metal atoms in the complex, and N_b is the number of metal atoms bonded to the jth ligand atom.