MOLECULAR MECHANICS
Empirical Energy Functions
Typical Empirical Energy Functions (potential energy functions,
empirical potentials, force fields, molecular mechanics functions) contain
two categories of terms to treat the
(a) bonding interactions (covalent); short range
(b) non-bonding interactions (non-covalent); long range
A little more detail:
(c) Ucov = Ubond + Uangle + Udihedral + + +
(d) Unb = UVdW + Uelec + + +
Bonding Interactions: Covalent structure; Local shape
Required Terms:
Ucov = Ubond + Uangle + Udihedral
Ubond: 1-2 Interactions (A,B; atoms A-B directly bonded), "Hooke's Law",
harmonic approximation
Ubond = b 0.5 kAB(RAB - RAB,eq)2
or, including anharmonicity,
(3)
Ubond = bonds 0.5 kAB(RAB - RAB,eq)2 + bonds 0.5 kAB (RAB - RAB,eq)3
(4)
[ + bonds 0.5 kAB (RAB - RAB,eq)4 ]
(3)
(Note: kAB is positive, but kAB
is negative)
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Uangle: 1-3 Interactions (A,B,C; atoms A-B-C directly bonded); "Hooke's
Law", harmonic approximation
Uangle = angles 0.5 kABC (ABC - ABC,eq)2
or
Uangle =
angles 0.5 kABC (ABC - ABC,eq)2
+
(3)
angles 0.5 kABC
(ABC - ABC,eq)3
(3)
(kABC is positive, kAB
is negative)
Udihedral: 1-4 Interactions (A,B,C,D; A-B-C-D directly bonded); functional
form based on ethane rotational barrier
Udihedral = dihedrals 0.5 V3[1 + cos(3)]
 (labeled  below) = angle required to rotate plane defined by atoms A(I),
B(J), and C(K) onto the plane defined by atoms B, C, and D(L).
I 

I
K
L
L
J
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More generally, a Fourier Series:
Udihedral = dihedrals 0.5 V1[1 + cos()] + dihedrals 0.5 V2[1 - cos(2)] +
dihedrals 0.5 V3[1 + cos(3] .....
= dihedrals 0.5 Vn[1 + cos(n- )]
Additional Covalent Terms
Uimproper: Out-of-plane motions, sp2-type carbons, nitrogens etc. (A,B,C,D;
A-B-C-D not directly bonded);
Uimproper = improper 0.5 Vn[1 + cos(n- )]
Or parabolic in displacement of atom B from plane A-C-D or parabolic in
some angle.
Cross-Terms: Bond Length/Bond Angle cross-term
Ub = Ab,ABC 0.5 kAB,ABC(RAB - RAB,eq)(ABC - ABC,eq)
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Non-Bonding Interactions
Interactions between atoms in different molecules or atoms in the
same molecule which are neither directly bonded to each other nor sharing
an atom. Non-bonding interactions determine overall conformation (e.g.
tertiary structure of proteins) and intermolecular interactions.
Required Terms
 Short-range: repulsive electrostatic and (Pauli) exchange interactions
 Long-range: electrostatic (attractive or repulsive) and dispersion
(attractive) interactions.
Unb = UVdW + Uelec
VdW = Van der Waals interactions: short-range repulsive interactions
(Pauli exclusion interactions) and long-range dispersion interactions. The
latter, in particular, is an inter-molecular concept now being used to also
describe an intra-molecular interaction.
UVdW is typically modeled by a "Lennard-Jones 6 -12 potential"
UVdW = ULJ = AB aAB/RAB12 - AB bAB/RAB6 =
 AB 4AB [(AB /RAB)12 - (AB/RAB)6]
AB = well-depth
AB = distance where interaction energy equals zero
AB = (AA BB)1/2
AB = (AA BB)1/2 or AB = 0.5(AA + BB)
A-B = (1-4), 1-5, 1-6 etc....
R6 term is the "induced dipole-dipole" term
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Or by a "Buckingham-Hill potential"
UVdW = UBH = AB aAB exp(-cABRAB) - AB bAB/RAB6
Uelec: electrostatic interactions between two charge distributions, typically
modeled via atomic point charges or bond-dipoles
Point Charge Model
Uelec = AB QAQB/ABRAB
QA = fractional (net) charge on atom a; AB = dielectric constant
A-B = (1-4), 1-5, 1-6 etc...
Bond Dipole Moment Model
Uelec = AB/CD [AB.CD/RAB/CD3 3(AB.RAB/CD)(CD.RAB/CD)/RAB/CD5]/AB/CD
= AB/CD ABCD/AB/CDRAB/CD3 [cos AB/CD - 3cosABcosCD]
A,B = bonds; AB = bond dipole assigned to bond AB (vector);
AB/CD = dielectric constant
Additional Non-CovalentTerms
Upol: polarization (induced) electrostatic interactions; fluctuating atomic
charges or bond-dipoles
UH-bond: Special terms for hydrogen bonding; DAB/RAB12 - EAB/RAB10
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Force Field Parameterization
Utot = Ucov + Unb = Ubond + Uangle + Udihedral + UVdW + Uelec
+ Uimproper + Upol + Ub...
It appears that we have succeeded in writing a total energy function a force field or a potential energy function - which depends on the nuclear
coordinates and (many) parameters
Utot = Utot(Ri)
Where do the parameters in the force field come from?
Fitting procedure:
(a) Assemble a data bank of test molecules and some of their physical
properties (geometries, heats of formation, vibrational frequencies,
rotational barriers….). The data listed in the data bank may be derived from
experiment and/or high-level computations.
(b) Guess an initial set of values for the parameters in the force field.
Calculate properties for the test set of molecules at optimized (energy
minimized) geometries; compare properties to values in data bank.
(c) Refine the parameters, using a non-linear least-squares fitting program.
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(d) Recalculate properties for the test set of molecules with the improved
force field parameters....and compare to data bank. If agreement is
satisfactory, i.e., a minimum in the
ErrorFunction = weight (reference value - calculated value)2
have been found, go to (e). If not, go back to (c).
(e) A set of “ideal” reference geometrical parameters, force constants,
partial charges, VDW parameters etc. have been established
Can fit to a particular set of molecules (organic, biological, inorganic)
or solids.
Aim to achieve "chemically reasonable" parameters
Parameters mostly transferable, except perhaps partial atomic
charges (Electrostatic Potential (ESP) fitting).
The quality of the parameterization is crucial to the results that will be
obtained on molecules not in the data set.
Different parameters for different atom types [C(alkane), C(carbonyl),
C(alkene), C(alkyne), O(ether), O(aldehyde), O(carbonyl)...]
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Performance
The number of terms in the sums for U(stretch), U(bend), and
U(dihedral) increase roughly as N (N = number of atoms), whereas the
number of terms in the non-bonded interaction sums (VDW and/or
Coulomb) increase roughly as N2. Hence, evaluation of the non-bonded
terms is the time-consuming step in molecular mechanics calculations. The
overall very favorable scaling makes it possible to deal efficiently with large
molecular systems (>1,000,000 atoms).
Assuming that a complete set of parameters are available (a "force
field"), the energy ("steric" or "strain") may be evaluated at a particular
configuration by simply inserting the values for the geometric parameters
into the various expressions and sum up the energy contributions. The
"steric" or "strain" energy obtained is the energy of the real molecule
relative to a hypothetical, "unstrained" molecule where all the structural
values are exactly at their ideal (reference, equilibrium) values. By itself,
this energy has thus no physical meaning and cannot be measured. It may,
in some cases, be converted to an enthalpy of formation (as done, for
example, in Allinger's MM2 and MM3 force fields).
Comparisons may always be made to other molecular conformations
using the same force field. Comparisons may be made to other molecules
using the same force field, if enthalpies of formation are compared.
Evaluation of the steric energy may be done at any geometry, thus
giving rise to a data set of molecular configurations and associated
energies. This provides an immediate means for obtaining energy
differences between configurations as well as finding the "best"
configuration, namely the one of lowest steric energy. Overall, a generous
portion of the potential energy surface may be explored.
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Strengths - Weaknesses of Molecular Mechanics
Strengths
(a) Excellent geometries and energetics for the atom types and situations it
is parameterized for.
(b) Applicable to large systems. Relatively large segments of potential
energy surfaces (many configurations) can be investigated.
(c) Applicable to intra- and inter-molecular interactions in gas, liquid, or
solid phase.
Weaknesses
(a) Not generally applicable to all elements (e.g., transition metals) or
bonding situations (e.g., pi-delocalization).
(b) Large problems in obtaining parameters for new atom types or
combinations; small rings; "unusual" molecules.
(c) Limited set of properties actually computed; different parameters for
different quantities
(d) Interpretational problems may arise from very different force fields
giving rise to the same data (cause <---> effect). Comparisons to be made
within a particular force field method only.
"Use molecular mechanics when you can -- use quantum mechanics
when you must!", Norman Allinger.
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