# cff (c47b2)

Consistent Force Field (CFF)

* Usage | How to use CFF with CHARMM standalone

* Status | Current status of CFF implementation in CHARMM

* Theory | Basis for, parameterization and performance of CFF

* Funcform | Functional form of the CFF energy expression

* Refs | References to papers describing CFF

* Usage | How to use CFF with CHARMM standalone

* Status | Current status of CFF implementation in CHARMM

* Theory | Basis for, parameterization and performance of CFF

* Funcform | Functional form of the CFF energy expression

* Refs | References to papers describing CFF

Top

In order to use CFF in CHARMM, the user has to issue the following

commands:

1. use cff

2. read cff parameter file

3. (a) read rtf name <CFF-capable rtf file>, or

(b) read psf name <file_name>

4. read sequence ! if input is via the rtf route (step 3 (a))

5. generate

6. read coord, or ic build ! if input is via the read rtf/sequence route.

When using CFF95 or later Step 3a requires a CFF-capable rtf file. This means

a file in which BOND records have been replaced by analogous DOUBLE records for

cases in which the chemical structure has a double bond. Note that CFF-capable

rtf files are *back compatible*. That is, such rtf files can equally well be

used for calculations that utilize the CHARMM force field. Thus, it is *not*

necessary to maintain two versions of the rtf files.

NOTE: 1. no binary parameter files are supported for CFF.

2. CFF is an all hydrogen force field -- i.e., extended atoms

are not supported

Examples of CFF usage in CHARMM are given in the ccfftest directory.

In order to use CFF in CHARMM, the user has to issue the following

commands:

1. use cff

2. read cff parameter file

3. (a) read rtf name <CFF-capable rtf file>, or

(b) read psf name <file_name>

4. read sequence ! if input is via the rtf route (step 3 (a))

5. generate

6. read coord, or ic build ! if input is via the read rtf/sequence route.

When using CFF95 or later Step 3a requires a CFF-capable rtf file. This means

a file in which BOND records have been replaced by analogous DOUBLE records for

cases in which the chemical structure has a double bond. Note that CFF-capable

rtf files are *back compatible*. That is, such rtf files can equally well be

used for calculations that utilize the CHARMM force field. Thus, it is *not*

necessary to maintain two versions of the rtf files.

NOTE: 1. no binary parameter files are supported for CFF.

2. CFF is an all hydrogen force field -- i.e., extended atoms

are not supported

Examples of CFF usage in CHARMM are given in the ccfftest directory.

Top

Status of CFF implementation into CHARMM (May 2001)

=============================================================

This implementation of CFF in CHARMM is principally due to Rick

Lapp (MSI) and William Young (MSI).

Features currently supported in CHARMM/CFF

(1) energy and first derivatives

(2) minimization

(3) dynamics

(4) most ATOM based cutoff options

Major features NOT currently implemented in CHARMM/CFF:

(1) bonds between primary atoms and image atoms.

(2) Cutoff options currently not supported are group-based cutoffs,

distance shifting and force-based switching.

(3) Fast multipoles.

Other known limitations:

(1) correlation analysis tools have not been implemented for CFF specific

energy terms -- e.g. it is not possible to calculate the correlation

function for an out-of-plane bending angle, etc ...

(2) only all-atom models (no extended atoms)

There are probably other problems/limitations/bugs. Your comments about

limitations of the current CFF implementation in CHARMM (and bugs) will be

very valuable.

Please direct comments to:

William Young, MSI

e-mail: wyoung@msi.com

phone: (619)799-5348

KNOWN BUGS:

Status of CFF implementation into CHARMM (May 2001)

=============================================================

This implementation of CFF in CHARMM is principally due to Rick

Lapp (MSI) and William Young (MSI).

Features currently supported in CHARMM/CFF

(1) energy and first derivatives

(2) minimization

(3) dynamics

(4) most ATOM based cutoff options

Major features NOT currently implemented in CHARMM/CFF:

(1) bonds between primary atoms and image atoms.

(2) Cutoff options currently not supported are group-based cutoffs,

distance shifting and force-based switching.

(3) Fast multipoles.

Other known limitations:

(1) correlation analysis tools have not been implemented for CFF specific

energy terms -- e.g. it is not possible to calculate the correlation

function for an out-of-plane bending angle, etc ...

(2) only all-atom models (no extended atoms)

There are probably other problems/limitations/bugs. Your comments about

limitations of the current CFF implementation in CHARMM (and bugs) will be

very valuable.

Please direct comments to:

William Young, MSI

e-mail: wyoung@msi.com

phone: (619)799-5348

KNOWN BUGS:

Top

The aim of the CFF development is a force field that is:

* broad, covering a relatively large number of differing

functional groups,

* accurate, achieved via accurate reproduction of the

quantum mechanical energy surfaces,

* consistent between differing phases and molecular environments,

* applicable to a wide range of molecular properties,

* consistent between differing types of molecules,

such as interaction of protein active sites with ligands, or

assemblies of proteins with nucleic acids or with solvent.

Quantum mechanical forcefields

The intramolecular parameters constituting the current generation of

forcefields are based on the energies and energy derivatives computed by

ab initio quantum mechanical procedures for a series of model compounds.

CFF uses quantum computations in the Hartree-Fock approximation with the

6-31G* basis set to expand the wavefunctions [1][2]. The quantum

mechanical energies and the energy first derivatives (gradients) and

second derivatives (Hessians) were computed for the equilibrium molecular

structures, at conformational energy barriers, and for a set of distorted

structures. The distorted molecular structures were generated by randomly

deforming all the internal coordinates, as well as by systematically

rotating about individual bonds. These quantum observables were fit to

the energy expression to obtain the Class II parameters [3][4]. Many of

the atomic partial charges were also determined quantum mechanically.

The intermolecular parameters of the forcefield may also, in principle,

be computed quantum mechanically [5]. The remaining CFF forcefield

intermolecular or nonbond parameters were computed by fitting to

experimental crystal lattice constants and sublimation energies of

crystals [6][7][8].

Internal energy terms

The energy of the molecule or assembly is expressed in terms of internal

coordinates such as bond lengths, bond angles, and dihedral angles. For

Class II forcefields this set of descriptors is greatly expanded by

including cross terms, that is, the interactions between bond

lengths and angles, between pairs of angles, etc. CFF contains, in all,

twelve types of energy terms: bond stretching, valence angle bending,

valence dihedral angles, out-of-plane deformation, and eight cross terms.

The cross terms extend the accuracy and range of application of the

forcefield by including the effect of neighboring atomic positions on

each of the bond lengths, valence angles, and dihedral angles.

The aim of the CFF development is a force field that is:

* broad, covering a relatively large number of differing

functional groups,

* accurate, achieved via accurate reproduction of the

quantum mechanical energy surfaces,

* consistent between differing phases and molecular environments,

* applicable to a wide range of molecular properties,

* consistent between differing types of molecules,

such as interaction of protein active sites with ligands, or

assemblies of proteins with nucleic acids or with solvent.

Quantum mechanical forcefields

The intramolecular parameters constituting the current generation of

forcefields are based on the energies and energy derivatives computed by

ab initio quantum mechanical procedures for a series of model compounds.

CFF uses quantum computations in the Hartree-Fock approximation with the

6-31G* basis set to expand the wavefunctions [1][2]. The quantum

mechanical energies and the energy first derivatives (gradients) and

second derivatives (Hessians) were computed for the equilibrium molecular

structures, at conformational energy barriers, and for a set of distorted

structures. The distorted molecular structures were generated by randomly

deforming all the internal coordinates, as well as by systematically

rotating about individual bonds. These quantum observables were fit to

the energy expression to obtain the Class II parameters [3][4]. Many of

the atomic partial charges were also determined quantum mechanically.

The intermolecular parameters of the forcefield may also, in principle,

be computed quantum mechanically [5]. The remaining CFF forcefield

intermolecular or nonbond parameters were computed by fitting to

experimental crystal lattice constants and sublimation energies of

crystals [6][7][8].

Internal energy terms

The energy of the molecule or assembly is expressed in terms of internal

coordinates such as bond lengths, bond angles, and dihedral angles. For

Class II forcefields this set of descriptors is greatly expanded by

including cross terms, that is, the interactions between bond

lengths and angles, between pairs of angles, etc. CFF contains, in all,

twelve types of energy terms: bond stretching, valence angle bending,

valence dihedral angles, out-of-plane deformation, and eight cross terms.

The cross terms extend the accuracy and range of application of the

forcefield by including the effect of neighboring atomic positions on

each of the bond lengths, valence angles, and dihedral angles.

Top

Energy functional forms

The energy expression may be decomposed into diagonal terms that depend

on a single molecular internal coordinate such as a bond length, coupling

terms between internal coordinates, and nonbond internuclear distances.

This energy is fit to the quantum mechanical energy.

1. Bond stretching.

Ebond = K2 * (b - b0)^2 + K3 * (b - b0)^3 + K4 * (b - b0)^4 (1)

where K2, K3 and K4 are the quadratic, cubic and quartic forcefield

parameters or force constants, b is the bond length, and b0 is the

reference value of the bond length.

2. Angle bending.

Eangle = K2 * Delta^2 + K3 * Delta^3 + K4 * Delta^4 (2)

where Delta = Theta - Theta0 is the difference between the actual and

reference bond angles.

3. Out-of-plane bending.

Eoop = K * (Chi - Chi0)^2 (3)

where chi is an out-of-plane coordinate as defined by Wilson et al.[9]

4. Torsion energy, in order to reflect differing hybridizations about

the bonded atoms, must contain one-, two-, and threefold periodic terms:

Etorsion = SUM(n=1,3) { V(n) * [ 1 - cos(n*Phi - Phi0(n)) ] } (4)

where phi is a dihedral angle.

5. Stretch-Stretch interaction between two bonds in a valence angle.

Ebond-bond = K(b,b') * (b - b0) * (b' - b0') (5)

6. Stretch-Bend interaction between an angle and its bonds.

Ebond-angle = K * (b - b0) * (Theta - Theta0) (6)

7. Bend-Bend-Twist interaction between a dihedral angle and its

two valence angles.

Eangle-angle-torsion = K * (Theta - Theta0) * (Theta' - Theta0') *

(Phi - Phi1(0)) (7)

8. Stretch-Twist interaction between a dihedral angle and its end bonds.

Eend_bond-torsion = (b - b0) * SUM { V(n) * cos[n*phi] } (8)

9. Stretch-Twist interaction between a dihedral angle and its middle bond.

Emiddle_bond-torsion = (b - b0) *

{ F(1) * cos(phi) + F(2) * cos(2 * phi) + F(3) * cos(3 * phi) } (9)

10. Bend-Twist interaction between a dihedral angle and its valence angles.

Eangle-torsion = (Theta - Theta0) *

{ F(1) * cos(phi) + F(2) * cos(2 * phi) + F(3) * cos(3 * phi) } (10)

11. Bend-Bend interaction between two valence angles with a common

vertex atom.

Eangle-angle = K * (Theta - Theta0) * (Theta' - Theta0') (11)

12. Stretch-Stretch interaction between the two end bonds in a dihedral

angle.

Ebond-bond_1_3 = K(b,b') * (b - b0) * (b' - b0') (12)

Finally, the nonbond energy between atoms in different molecules or

between atoms separated by three or more bonded atoms is given by the

sum of the Coulombic electrostatic interaction and a van der Waals

energy of the 9-6 form:

13. Coulombic electrostatic interaction.

Ecoul = 332.0716*qi*qj/(D*Rij) (13)

where qi and qj are the atomic partial charges on atoms i and j,

Rij is the distance between them and D is the dielectric constant.

14. Van der Waals interaction.

Evdw = eps(ij) [2*r*(ij)/r(ij)**9 - 3*r*(ij)/r(ij)**6] (14)

where r*(ij) = [(r(i)**6 + r(j)**6))/2]**(1/6) (15)

eps(ij) = 2 sqrt(eps(i) * eps(j)) *

r(i)^3 * r(j)^3/[r(i)^6 + r(j)^6] (16)

where eps(ij) and r*(ij) are the negative of the minimum van der Waals

energy and that distance between atoms i and j where the minimum occurs,

respectively. Eps(ij) and r*ij are computed from the individual atomic

parameters eps(i), eps(j), r*i, and r*j by the Waldman-Hagler combination

rules [10].

The Hartree-Fock method, and to a lesser extent other quantum mechanical

methods, results in systematic deviations from experiment. For example,

bond lengths tend to be too short and bond-stretching vibrational

frequencies too high [11]. However, by comparison with experimental

gas-phase molecular structures and vibrational frequencies, these

deviations may be compensated for. In general, the energy expression

may be scaled using five constant factors, one for each of the classes

of energy terms: bonds, angles, torsion angles, out-of-planes and all

coupling terms [12]. The scaled energy is then:

Ediagonal = Sb * SUM{Ebond} + Stheta * SUM{Eangle} +

Sphi * SUM{Etorsion} + Schi * SUM{Eoop} (17)

Ecross = Sc * SUM{eight cross terms} (18)

The reference values b0 and q0 are also adjusted to fit experimental data.

All these values may differ among different types of bonds, bond angles,

and torsion angles. For the special case of hydrocarbons, the corrections

are especially well determined by gas-phase measurements. For hydrocarbons,

the best values of the scale factors are:

Sb(C-C) 0.88

Sb(C-H) 0.83

Stheta (all angles) 0.81

Sphi (all torsions) 0.84

Schi (all out-of-planes) 1.00

Sc (all cross terms) 0.87

The reference bond lengths for hydrocarbons were also adjusted.

Although the use of the quantum calculation greatly amplifies the

available data so that only a few such corrections are necessary for

the complete Class II forcefield, for the majority of functional groups

(molecular types) no accurate gas-phase data are available. However,

the Sb, Stheta, Sphi, and Sc constants are transferrable among different

types of bonds, bond angles, and torsion angles. Therefore, the same

scale factors are used in Eq. 17 and Eq. 18 in the final empirically

scaled forcefield. In general, the reference values b0 and theta0 are

determined from high-level quantum mechanical calculations on the model

compounds.

Validation of the CFF forcefield

Table 1 shows the accuracy of the CFF forcefield for several common

classes of molecules, compared with experimental gas-phase results.

Table 1. Summary of rms deviations between experimental and

CFF-calculated structural parameters, vibrational frequencies,

and energy differences.

bond valence torsion freq. energy

length angle angle diff.

(Ang) (deg) (deg) (cm-1) (kcal mol-1)

hydrocarbons 0.02 0.9 1.2 40 0.93

alcohols 0.02 1.7 1.7 37 0.71

aldehydes & ketones 0.01 1.1 2.3 32 0.62

amines 0.00 0.9 -- 18 0.62

carboxylic acids 0.02 1.6 1.0 34 0.78

esters 0.02 1.7 0.5 42 1.88

ethers 0.01 0.9 1.1 41 0.40

heterocycles 0.01 1.0 0.0 35 ---

sulfides 0.01 1.4 2.5 45 ---

disulfides 0.01 0.9 2.0 43 ---

thiols 0.01 1.6 1.0 -- 0.21

average 0.01 1.2 1.3 37 0.77

The frequencies are harmonic vibrational frequencies and the energy

differences include conformational energy differences and energy

barriers to internal rotation between stable conformers.

Energy functional forms

The energy expression may be decomposed into diagonal terms that depend

on a single molecular internal coordinate such as a bond length, coupling

terms between internal coordinates, and nonbond internuclear distances.

This energy is fit to the quantum mechanical energy.

1. Bond stretching.

Ebond = K2 * (b - b0)^2 + K3 * (b - b0)^3 + K4 * (b - b0)^4 (1)

where K2, K3 and K4 are the quadratic, cubic and quartic forcefield

parameters or force constants, b is the bond length, and b0 is the

reference value of the bond length.

2. Angle bending.

Eangle = K2 * Delta^2 + K3 * Delta^3 + K4 * Delta^4 (2)

where Delta = Theta - Theta0 is the difference between the actual and

reference bond angles.

3. Out-of-plane bending.

Eoop = K * (Chi - Chi0)^2 (3)

where chi is an out-of-plane coordinate as defined by Wilson et al.[9]

4. Torsion energy, in order to reflect differing hybridizations about

the bonded atoms, must contain one-, two-, and threefold periodic terms:

Etorsion = SUM(n=1,3) { V(n) * [ 1 - cos(n*Phi - Phi0(n)) ] } (4)

where phi is a dihedral angle.

5. Stretch-Stretch interaction between two bonds in a valence angle.

Ebond-bond = K(b,b') * (b - b0) * (b' - b0') (5)

6. Stretch-Bend interaction between an angle and its bonds.

Ebond-angle = K * (b - b0) * (Theta - Theta0) (6)

7. Bend-Bend-Twist interaction between a dihedral angle and its

two valence angles.

Eangle-angle-torsion = K * (Theta - Theta0) * (Theta' - Theta0') *

(Phi - Phi1(0)) (7)

8. Stretch-Twist interaction between a dihedral angle and its end bonds.

Eend_bond-torsion = (b - b0) * SUM { V(n) * cos[n*phi] } (8)

9. Stretch-Twist interaction between a dihedral angle and its middle bond.

Emiddle_bond-torsion = (b - b0) *

{ F(1) * cos(phi) + F(2) * cos(2 * phi) + F(3) * cos(3 * phi) } (9)

10. Bend-Twist interaction between a dihedral angle and its valence angles.

Eangle-torsion = (Theta - Theta0) *

{ F(1) * cos(phi) + F(2) * cos(2 * phi) + F(3) * cos(3 * phi) } (10)

11. Bend-Bend interaction between two valence angles with a common

vertex atom.

Eangle-angle = K * (Theta - Theta0) * (Theta' - Theta0') (11)

12. Stretch-Stretch interaction between the two end bonds in a dihedral

angle.

Ebond-bond_1_3 = K(b,b') * (b - b0) * (b' - b0') (12)

Finally, the nonbond energy between atoms in different molecules or

between atoms separated by three or more bonded atoms is given by the

sum of the Coulombic electrostatic interaction and a van der Waals

energy of the 9-6 form:

13. Coulombic electrostatic interaction.

Ecoul = 332.0716*qi*qj/(D*Rij) (13)

where qi and qj are the atomic partial charges on atoms i and j,

Rij is the distance between them and D is the dielectric constant.

14. Van der Waals interaction.

Evdw = eps(ij) [2*r*(ij)/r(ij)**9 - 3*r*(ij)/r(ij)**6] (14)

where r*(ij) = [(r(i)**6 + r(j)**6))/2]**(1/6) (15)

eps(ij) = 2 sqrt(eps(i) * eps(j)) *

r(i)^3 * r(j)^3/[r(i)^6 + r(j)^6] (16)

where eps(ij) and r*(ij) are the negative of the minimum van der Waals

energy and that distance between atoms i and j where the minimum occurs,

respectively. Eps(ij) and r*ij are computed from the individual atomic

parameters eps(i), eps(j), r*i, and r*j by the Waldman-Hagler combination

rules [10].

The Hartree-Fock method, and to a lesser extent other quantum mechanical

methods, results in systematic deviations from experiment. For example,

bond lengths tend to be too short and bond-stretching vibrational

frequencies too high [11]. However, by comparison with experimental

gas-phase molecular structures and vibrational frequencies, these

deviations may be compensated for. In general, the energy expression

may be scaled using five constant factors, one for each of the classes

of energy terms: bonds, angles, torsion angles, out-of-planes and all

coupling terms [12]. The scaled energy is then:

Ediagonal = Sb * SUM{Ebond} + Stheta * SUM{Eangle} +

Sphi * SUM{Etorsion} + Schi * SUM{Eoop} (17)

Ecross = Sc * SUM{eight cross terms} (18)

The reference values b0 and q0 are also adjusted to fit experimental data.

All these values may differ among different types of bonds, bond angles,

and torsion angles. For the special case of hydrocarbons, the corrections

are especially well determined by gas-phase measurements. For hydrocarbons,

the best values of the scale factors are:

Sb(C-C) 0.88

Sb(C-H) 0.83

Stheta (all angles) 0.81

Sphi (all torsions) 0.84

Schi (all out-of-planes) 1.00

Sc (all cross terms) 0.87

The reference bond lengths for hydrocarbons were also adjusted.

Although the use of the quantum calculation greatly amplifies the

available data so that only a few such corrections are necessary for

the complete Class II forcefield, for the majority of functional groups

(molecular types) no accurate gas-phase data are available. However,

the Sb, Stheta, Sphi, and Sc constants are transferrable among different

types of bonds, bond angles, and torsion angles. Therefore, the same

scale factors are used in Eq. 17 and Eq. 18 in the final empirically

scaled forcefield. In general, the reference values b0 and theta0 are

determined from high-level quantum mechanical calculations on the model

compounds.

Validation of the CFF forcefield

Table 1 shows the accuracy of the CFF forcefield for several common

classes of molecules, compared with experimental gas-phase results.

Table 1. Summary of rms deviations between experimental and

CFF-calculated structural parameters, vibrational frequencies,

and energy differences.

bond valence torsion freq. energy

length angle angle diff.

(Ang) (deg) (deg) (cm-1) (kcal mol-1)

hydrocarbons 0.02 0.9 1.2 40 0.93

alcohols 0.02 1.7 1.7 37 0.71

aldehydes & ketones 0.01 1.1 2.3 32 0.62

amines 0.00 0.9 -- 18 0.62

carboxylic acids 0.02 1.6 1.0 34 0.78

esters 0.02 1.7 0.5 42 1.88

ethers 0.01 0.9 1.1 41 0.40

heterocycles 0.01 1.0 0.0 35 ---

sulfides 0.01 1.4 2.5 45 ---

disulfides 0.01 0.9 2.0 43 ---

thiols 0.01 1.6 1.0 -- 0.21

average 0.01 1.2 1.3 37 0.77

The frequencies are harmonic vibrational frequencies and the energy

differences include conformational energy differences and energy

barriers to internal rotation between stable conformers.

Top

References

[1] Hariharan, P. C.; Pople, J. A. Theor. Chim. Acta 28, 213-222 (1973).

[2] Francl, M. M.; Pietro, W. J.; Hehre, W. J.; Binkley, J. S.; Gordon, M. S.;

DeFrees, D. J.; Pople, J. A. J. Chem. Phys. 77, 3654-3665 (1982).

[3] Dinur, U.; Hagler, A. T. In Reviews in Computational Chemistry, Vol. 2,

K. B. Lipkowitz; D. B. Boyd, Eds., VCH Publishers: New York, 99-164 (1991).

[4] Maple, J. R.; Hwang, M.-J.; Stockfisch, T. P.; Dinur, U.; Waldman, M.;

Ewig, C. S.; Hagler, A. T. J. Comp. Chem. 15, 162-182 (1994).

[5] Dinur, U.; Hagler, A. T. J. Amer. Chem. Soc. 111, 5149-5151 (1989).

[6] Hagler, A. T.; Huler, E.; Lifson, S. J. Amer. Chem. Soc. 96, 5319-5327

(1974).

[7] Hagler, A. T.; Lifson, S.; Dauber, P. J. Amer. Chem. Soc. 101, 5122-5130

(1979a).

[8] Hagler, A. T.; Dauber, P.; Lifson, S. J. Amer. Chem. Soc. 101, 5131-5141

(1979b).

[9] Wilson, E. B., Jr; Decius, J. C.; Cross, P. C., Molecular Vibrations;

Dover: New York, 1955, Chapter 4.

[10] Waldman, M.; Hagler, A. T. J. Comp. Chem. 14, 1077-1084 (1993).

[11] Michalska, D.; Schaad, L. J.; Carsky, P.; Hess, Jr., B. A.; Ewig, C. S.

J. Comp. Chem., 9, 495 (1988).

[12] Hwang, M.-J.; Stockfisch, T. P.; Hagler, A. T. J. Amer. Chem. Soc. 116,

2515-2525 (1994).

References

[1] Hariharan, P. C.; Pople, J. A. Theor. Chim. Acta 28, 213-222 (1973).

[2] Francl, M. M.; Pietro, W. J.; Hehre, W. J.; Binkley, J. S.; Gordon, M. S.;

DeFrees, D. J.; Pople, J. A. J. Chem. Phys. 77, 3654-3665 (1982).

[3] Dinur, U.; Hagler, A. T. In Reviews in Computational Chemistry, Vol. 2,

K. B. Lipkowitz; D. B. Boyd, Eds., VCH Publishers: New York, 99-164 (1991).

[4] Maple, J. R.; Hwang, M.-J.; Stockfisch, T. P.; Dinur, U.; Waldman, M.;

Ewig, C. S.; Hagler, A. T. J. Comp. Chem. 15, 162-182 (1994).

[5] Dinur, U.; Hagler, A. T. J. Amer. Chem. Soc. 111, 5149-5151 (1989).

[6] Hagler, A. T.; Huler, E.; Lifson, S. J. Amer. Chem. Soc. 96, 5319-5327

(1974).

[7] Hagler, A. T.; Lifson, S.; Dauber, P. J. Amer. Chem. Soc. 101, 5122-5130

(1979a).

[8] Hagler, A. T.; Dauber, P.; Lifson, S. J. Amer. Chem. Soc. 101, 5131-5141

(1979b).

[9] Wilson, E. B., Jr; Decius, J. C.; Cross, P. C., Molecular Vibrations;

Dover: New York, 1955, Chapter 4.

[10] Waldman, M.; Hagler, A. T. J. Comp. Chem. 14, 1077-1084 (1993).

[11] Michalska, D.; Schaad, L. J.; Carsky, P.; Hess, Jr., B. A.; Ewig, C. S.

J. Comp. Chem., 9, 495 (1988).

[12] Hwang, M.-J.; Stockfisch, T. P.; Hagler, A. T. J. Amer. Chem. Soc. 116,

2515-2525 (1994).