Force fields and potential energy surfaces are essential tools in molecular dynamics simulations. They help us understand how atoms and molecules interact, allowing us to predict their behavior. These concepts are crucial for studying complex systems like proteins and materials.

Potential energy surfaces map out the energy landscape of molecules, showing stable configurations and transition states. Force fields break down molecular interactions into simpler terms, making it easier to model large systems. Together, they form the backbone of molecular simulations.

Empirical Force Fields

Components of Empirical Force Fields

• Empirical force fields approximate the potential energy of a molecular system as a function of the atomic coordinates
• Consist of two main components: bonded interactions and non-bonded interactions
• Bonded interactions describe the potential energy associated with covalent bonds, angles, and torsions within molecules
  • Include terms for bond stretching (harmonic potential), angle bending (harmonic potential), and torsional rotations (cosine series)
• Non-bonded interactions capture the potential energy between atoms not connected by covalent bonds
  • Include van der Waals interactions (Lennard-Jones potential) and electrostatic interactions (Coulomb potential)

Parameterization of Force Fields

• Parameterization involves determining the numerical values of the force field parameters to accurately reproduce experimental or high-level quantum mechanical data
• Force field parameters include equilibrium bond lengths, angles, force constants, and non-bonded interaction parameters (Lennard-Jones parameters and partial charges)
• Parameterization can be performed by fitting to experimental data such as crystal structures, vibrational spectra, and thermodynamic properties
• High-level quantum mechanical calculations (ab initio or density functional theory) can also be used to derive force field parameters
• Transferability of parameters allows force fields to be applied to a wide range of molecular systems

Intermolecular Potentials

Lennard-Jones Potential

• The Lennard-Jones potential describes the van der Waals interactions between atoms
• Consists of a repulsive term ($r^{-12}$) and an attractive term ($r^{-6}$), where $r$ is the distance between atoms
• The repulsive term dominates at short distances due to the overlap of electron clouds (Pauli repulsion)
• The attractive term captures the dispersion forces (London dispersion) arising from induced dipole-induced dipole interactions
• Lennard-Jones parameters ($\epsilon$ and $\sigma$) determine the depth of the potential well and the distance at which the potential crosses zero

Coulomb Potential

• The Coulomb potential describes the electrostatic interactions between charged particles
• Proportional to the product of the charges divided by the distance between them ($\frac{q_1q_2}{r}$)
• Positive charges repel each other, while opposite charges attract
• Partial charges assigned to atoms in molecules capture the uneven distribution of electron density
• Electrostatic interactions play a crucial role in determining the structure and interactions of polar and charged molecules (water, proteins, DNA)

Potential Energy Surfaces

Concept and Visualization

• A potential energy surface (PES) is a multidimensional surface that represents the potential energy of a molecular system as a function of its atomic coordinates
• The PES is a hypersurface in a high-dimensional space, where each point corresponds to a specific molecular geometry
• Minima on the PES represent stable configurations of the molecule (equilibrium geometries)
• Saddle points on the PES correspond to transition states connecting different minima
• The shape of the PES determines the dynamics and reactivity of the molecular system

Born-Oppenheimer Approximation

• The Born-Oppenheimer approximation allows the separation of nuclear and electronic motion in molecules
• It assumes that the electrons adjust instantly to changes in the positions of the nuclei due to their much smaller mass
• Under this approximation, the PES is calculated by solving the electronic Schrödinger equation for fixed nuclear positions
• The resulting electronic energy, combined with the nuclear repulsion energy, gives the potential energy of the system
• The Born-Oppenheimer approximation simplifies the calculation of PES by treating the nuclei as classical particles moving on the electronic PES