Question

Briefly define the following terms and discuss their role in molecular modelling a. Periodic boundary conditions b. Steepest descents minimization c. Force field parameterisation d. Bonded potential e. Cross terms f. Harmonic potential g. Morse potential

Answer 1

Solution-

(a.) Periodic boundary conditions (PBCs) are a set of boundary conditions which are often chosen for approximating a large (infinite) system by using a small part called a unit cell. PBCs are often used in computer simulationsand mathematical models. The topology of two-dimensional PBC is equal to that of a world map of some video games; the geometry of the unit cell satisfies perfect two-dimensional tiling, and when an object passes through one side of the unit cell, it re-appears on the opposite side with the same velocity. In topological terms, the space made by two-dimensional PBCs can be thought of as being mapped onto a torus (compactification). The large systems approximated by PBCs consist of an infinite number of unit cells. In computer simulations, one of these is the original simulation box, and others are copies called images. During the simulation, only the properties of the original simulation box need to be recorded and propagated. The minimum-image convention is a common form of PBC particle bookkeeping in which each individual particle in the simulation interacts with the closest image of the remaining particles in the system.

(b.) The steepest descent method is one of the oldest known methods for minimizing a general nonlinear function. The convergence theory for the method is widely used and is the basis for understanding many of the more sophisticated and well known algorithms.

(c.) Once a particular form for a force field has been chosen, the force parameters have to be determined. Even for a simple force field to be used for modelling a small number of systems this can be a large undertaking. For example to parameterize a simple diagonal force field for alkane molecules there needs to be:

• two parameters (force constant and equilibrium bond length) for the two different bond types (C-C and C-H),
• two for each of the three different angle types (C-C-C, C-C-H, and H-C-H),
• three for each different dihedral type (C-C-C-C, H-C-C-C, H-C-C-H),
• two Van der Waals parameters for the three different combination of atoms (C-C, C-H, H-H)
• up to five different charges.

(d.) Bonded potential :-

To model a covalent bond in a molecular structure, many types of interaction potentials can be used, such as the Morse potential or the finitely extendible nonlinear elastic (FENE) potential. However, the most common potential to be used in any molecular dynamics program is the harmonic bond potential. The equation that describes the potential energy of the harmonic potential is given by

In this equation r0 is the reference bond length, kij the force constant, and rij the distance between atoms i and j, hence the current length of the bond. The figure to the right schematically shows some of these parameters. The harmonic potential basically is a Taylor approximation of more elaborate potentials around the reference bond length. From this potential the force acting on atom i can be deduced (assuming the we are dealing with central and conservative forces) through

Substituting Equation (1) into (2) gives us for the force acting on atom i

The force acting on atom j is, due to Newton's third law, of equal magnitude but opposite in direction to the force acting on atom i.

(E.) Cross terms- Direct or straight might be what you are looking for, as opposed to cross,crossed or mixed (since each resultant term has either one variable to a power or two different variables, a "mixture").

(F.) Harmonic potential -

In classical mechanics, a harmonic potential a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement of x.

where k is a positive constant.

If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).

(G.) Morse potential -

Morse potential convenient interatomic interaction model for the potential energy of a diatomic molecule. It is a better approximation for the vibrational structure of the molecule than the QHO (quantum harmonic oscillator) because it explicitly includes the effects of bond breaking, such as the existence of unbound states. It also accounts for the anharmonicity of real bonds and the non-zero transition probability for overtone and combination bands. The Morse potential can also be used to model other interactions such as the interaction between an atom and a surface. Due to its simplicity (only three fitting parameters), it is not used in modern spectroscopy. However, its mathematical form inspired the MLR (Morse/Long-range) potential, which is the most popular potential energy function used for fitting spectroscopic data.