In the context of molecular modeling, a force field refers to the form and parameters of mathematical functions used to describe the potential energy of a system of particles (typically molecules and atoms).The usage of the term force field in chemistry and computational biology differs from the standard usage in physics. In chemistry, it is a system of potential energy functions rather than the gradient of a scalar potential, as defined in physics.
A force field is built up from two distinct components to describe the interaction between particles (typically atoms):
• the set of equations (called the potential functions) used to generate the potential energies and their derivatives, the forces.
• the parameters used in this set of equations.
There are three types of force fields:
• all atom–parameters provided for every single atom within the system.
• united atom–parameters provided for all atoms except non-polar hy-drogens.
• coarse grained–an abstract representation of molecules by grouping several atoms into ”super-atoms”.
Functional Form
The basic functional form of a force field comprises both bonded terms relat-ing to atoms that are linked by covalent bonds, and nonbonded (also called noncovalent) terms describing the long-range electrostatic and van der Waals forces. The specific decomposition of the terms depends on the force field, but a general form for the total energy in an additive force field can be written as
Etotal=Ebonded+Enon−bonded (3.5)
where the components of the covalent and noncovalent contributions are given by the following summations:
Ebonded =Ebond+Eangle+Edihedral+Eω+. . . (3.6) The bond and angle terms are usually modeled as harmonic oscillators in force fields that do not allow bond breaking. Ebond comprises the energy for bond stretching, Es and the energy for bond angle bending, Eb. Eω is the torsional energy due to twisting about bonds; normally added to enforce the planarity of aromatic rings and other conjugated systems, and cross-terms that describe coupling of different internal variables, such as angles and bond lengths.
Enon−bonded =Eelectrostatic+EvanderW aals (3.7) The nonbonded terms are most computationally intensive because they in-clude many more interactions per atom. A popular choice is to limit interac-tions to pairwise energies. The van der Waals term is usually computed with a Lennard-Jones potential and the electrostatic term with Coulomb’s law.
Nevertheless, both can be buffered or scaled by a constant factor to account for electronic polarizability and produce better agreement with experimental observations.
3.4.1 Force Field Development
The design of force fields for molecular mechanics is guided by by the follow-ing principles:
• Nuclei and electrons are lumped into atom-like particles.
• Atom-like particles are spherical (radii obtained from measurements or theory) and have a net charge (obtained from theory).
• Interactions are based on springs and classical potentials.
• Interactions must be preassigned to specific sets of atoms.
• Interactions determine the spatial distribution of atom-like particles and their energies.
Stretching and Bending
Considering the idea of a molecule to be a collection of masses connected by springs, then by applying Hooke’s Law we can evaluate the energy required to stretch and bend bonds from their ideal values. Thus Es and Eb may be expressed as:
Es=
N
X
i=1
kis
2(li−li0)2 (3.8)
Es =
M
X
i<j
kijb
2 (θij −θij0)2 (3.9) where N is the total number of bonds and M is the total number of bond angles in the molecule. ks and kb are the force constants for stretching and bending respectively. li andθij are the actual bond lengths and bond angles.
Finally, l0i and θij0 are ideal bond lengths and bond angle s respectively.
It should however be noted that the formulation above (Eqs. 3.8 and 3.9 ) is only a first approximation. There are various factors which can be taken into account to improve the accuracy for these terms. These include noting that bond stretching requires more energy than bond bending and so for a molecule being deformed most of the distortion should occur in the bond an-gles rather than bond lengths. Another point to consider is that Hooke’s Law overestimates the energy required to achieve large distortions. Another as-pect is that as a bond angle gets compressed the two associated bond lengths become longer.
Torsion
Here, we consider the form of theEω term. The energy due to torsion is usually expressed in terms of a Fourier series:
Eω =X1
2[V1(1 + cosω) +V2(1 + cos 2ω) +V2(1 + cos 3ω) +. . .] (3.10) where the sum is over all unique sequences of bonded atoms, ω is the torsion angle, and V1,V2,V3, are the adjustable parameters. In general, the series is truncated at the third term, with V1, V2 and V3 chosen so that the resultant
conformation agree well with experiment for a given group of molecules.
Non bonded Interactions
The final term contributing to Etotal is the energy from pairwise non bonded interactions. Such interactions are modeled by London dispersion forces (for the attraction) and van der Waals forces (for the repulsion). Some of the common potential functions implementing the above are the Lennard Jones (VLJ and Buckingham (VBuck) potentials [3],
VLJ = A r12 − B
R6 (3.11)
where A = 4σ12 and B = 4σ6. , the well depth and σ, the diameter are parameters in the model which are normally found by experiments.
VBuck=A0exp(B0 r )− C
R6 (3.12)
An important non bonded energy term that is always taken into account is the electrostatic interaction. Typically the electrostatic interaction dom-inates the total energy of a system by a full magnitude. The electrostatic contribution is modeled using a Coulombic potential,
ECoul =X
ij=1
qiqj
4π0rij (3.13)
The electrostatic energy is therefore said to be a function of the charge (q) on the non-bonded atoms, their interatomic distance, (rij) and a molecular dielectric expression that accounts for the decrease of electrostatic interac-tion due to the environment (such as by solvent or the molecule itself). 0 is the permittivity of free space.
The accuracy of the electrostatic term depends on the correct assignment of charges to individual atoms.
Generality is still a problem with force fields, though with the development of the Universal Force Field (UFF) an attempt has been made to develop a gen-eralized force field applicable to a large portion of the periodic table and not be restricted to particular groupings of atoms such as proteins, nucleic acids etc. other force field developed so far are AMBER, OPLS, MMFF, GRO-MOS, CHARMM, MM3, etc. AMBER, OPLS and CHARMM are geared more to larger molecules (proteins, polymers) in condensed phases.
Bibliography
[1] Anna Tomberg, “Gaussian 09W Tutorial:An Introduction to Computa-tional Chemistry Using G09 and Avogadros Software“.
[2] H. Bernhard Schlegel, Perspective on “Ab initio calculation of force con-stants and equilibrium geometries in polyatomic molecules. I. Theory”
Theor. Chem. Acc., 103:294-296, 2000.
[3] Rajarshi Guha (9915607) and Rajesh Sardar (9915610), “Force Fields in Molecular Mechanics”, 2001.
Chapter 4
MD Simulation of
Poly(3-hexylthiophene)
In this chapter, we give an account and details of the computational methods and procedures used in simulating (P3HT)n. Simulation results are also analyzed and discussed in this part.