9.2 Types of interactions in force fields

Force fields in molecular mechanics use mathematical models to describe various interactions within molecules. These models include bonded interactions like bond stretching, angle bending, and torsional rotation, as well as non-bonded interactions such as van der Waals forces and electrostatic interactions.

Understanding these interactions is crucial for accurately simulating molecular behavior. Each type of interaction is represented by specific potential energy functions, allowing researchers to calculate the overall energy of a molecular system and predict its structural and dynamic properties.

Bonded Interactions

Covalent Bond Deformations

• Bond stretching involves changes in the distance between two covalently bonded atoms
  • Modeled using a harmonic oscillator approximation
  • Potential energy increases as bond length deviates from equilibrium
  • $E_{stretch} = \frac{1}{2}k_b(r - r_0)^2$
    • $k_b$ represents the force constant
    • $r$ is the current bond length
    • $r_0$ is the equilibrium bond length
• Angle bending describes the deviation of bond angles from their equilibrium values
  • Also approximated using a harmonic potential
  • $E_{bend} = \frac{1}{2}k_\theta(\theta - \theta_0)^2$
    • $k_\theta$ is the angle bending force constant
    • $\theta$ represents the current angle
    • $\theta_0$ is the equilibrium angle
• Torsional rotation refers to the rotation around single bonds
  • Modeled using a periodic function to account for energy barriers
  • $E_{torsion} = \frac{V_n}{2}[1 + \cos(n\phi - \gamma)]$
    • $V_n$ is the barrier height
    • $n$ determines the periodicity
    • $\phi$ is the torsion angle
    • $\gamma$ represents the phase shift

Non-Bonded Interactions

Intermolecular Forces

• Van der Waals forces encompass weak attractive or repulsive interactions between molecules
  • Include dispersion forces (London forces), dipole-dipole interactions, and induced dipole interactions
  • Arise from temporary fluctuations in electron distribution
  • Strength decreases rapidly with distance (proportional to $1/r^6$)
• Electrostatic interactions occur between charged particles or permanent dipoles
  • Described by Coulomb's law
  • $E_{electrostatic} = \frac{q_1q_2}{4\pi\epsilon_0r}$
    • $q_1$ and $q_2$ are the charges of the interacting particles
    • $\epsilon_0$ is the permittivity of free space
    • $r$ represents the distance between charges
• Hydrogen bonding forms between a hydrogen atom bonded to an electronegative atom and another electronegative atom
  • Stronger than typical van der Waals interactions but weaker than covalent bonds
  • Crucial in determining the structure and properties of water, proteins, and nucleic acids
  • Energy ranges from 4-50 kJ/mol (water hydrogen bonds ~20 kJ/mol)

Potential Energy Functions

• Lennard-Jones potential models both attractive and repulsive components of van der Waals interactions
  • Combines a short-range repulsive term and a long-range attractive term
  • $E_{LJ} = 4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6\right]$
    • $\epsilon$ represents the depth of the potential well
    • $\sigma$ is the distance at which the potential energy is zero
    • $r$ is the distance between particles
  • $r^{-12}$ term models repulsion due to electron cloud overlap
  • $r^{-6}$ term accounts for attractive dispersion forces
• Buckingham potential serves as an alternative to the Lennard-Jones potential
  • $E_{Buckingham} = A e^{-Br} - \frac{C}{r^6}$
    • $A$, $B$, and $C$ are empirically determined parameters
  • Provides a more accurate description of short-range repulsion
  • Computationally more expensive than Lennard-Jones potential