Degrees of Freedom

Overview

Teaching: 10 min
Exercises: 0 min

Questions

• How is kinetic energy contained and distributed in a dynamic molecular system

• How to increase efficiency of thermodynamic sampling

• How reduction of degrees of freedom affects dynamics

Objectives

• Explain how kinetic energy is distributed in a simulation system

• Understand how to correctly increase efficiency of thermodynamic sampling

Molecular degrees of freedom

• Refer to the number of unique ways a molecule may move.
• Molecular degrees of freedom describe how kinetic energy is contained and distributed in a molecule.
• Translational, rotational, and vibrational components.

Equipartition theorem

• The energy is shared equally among the degrees of freedom (equipartition theorem).
• Each degree of freedom has an average energy of \(\frac{1}{2}k_BT\) and contributes \(\frac{1}{2}k_B\) to the system’s heat capacity.

Degrees of freedom and thermodynamics properties

• When kinetic energy is applied to simulation systems containing molecules with different degrees of freedom, the temperature increase will vary.
• If energy is spread over many places (degrees of freedom), the temperature will change less.
• The number of degrees of freedom is an essential quantity for estimating various thermodynamic variables for a simulation system (such as heat capacity, entropy, and temperature).

Translational degrees of freedom

• Atoms and molecules have three degrees of freedom associated with the translation of their centers of mass about each coordinate axis.
• Translational degrees of freedom in three dimensions yield \(\frac{3}{2}k_BT\) of energy.

Rotational degrees of freedom

• Atoms have a negligible amount of rotational energy because their mass is concentrated in the nucleus which is very small (about \(10^{-15}\) m).
• A linear molecule, has two rotational degrees of freedom.
• A nonlinear molecule, where the atoms do not lie along a single axis, has three rotational degrees of freedom, because it can rotate around any of three perpendicular axes.
• The rotational degrees of freedom contribute \(k_BT\) to the energy of linear molecules and \(\frac{3}{2}k_BT\) to the energy of non-linear molecules.

Vibrational degrees of freedom

• A diatomic molecule has one molecular vibration mode.
• A linear molecule with N atoms has 3N − 5 vibrational modes.
• A non-linear molecule with N atoms has 3N − 6 vibrational modes.

Angle bend	Symmetric stretch	Asymmetric stretch

• There are two degrees of freedom for each vibrational mode.
• One degree involves the kinetic energy of the moving atoms, and the other involves the potential energy of the springlike chemical bond(s).
• Each vibrational degree of freedom contributes \(k_BT\) to the energy of a molecule. However, this is valid only when \(k_BT\) is much bigger than energy spacing between vibrational modes.
• At low temperature this condition is not satisfied, only a few vibrational states are occupied and the equipartition principle is not typically applicable.

Why do degrees of freedom matter?

• More degrees of freedom provide more ways to store energy.
• Systems with more degrees of freedom generally experience smaller temperature changes for the same energy input.

Degrees of freedom and simulation efficiency.

By reducing the number of degrees of freedom we can increase thermodynamic sampling efficiency.

• Force fields remove the electrons’ degrees of freedom by replacing them with atom centered charges.
• An implicit solvent model eliminates the degrees of freedom associated with the solvent molecules.
• Bond constraints eliminate vibrational degrees of freedom and make possible to use longer time steps.
• Constraints including angles and dihedrals can be also applied.

Coarse-grained models

• Coarse-grained models reduce degrees of freedom by averaging interactions across groups of particles.

The trade-off: efficiency versus realism

Bond and angle constraints can:

• slow down dihedral angle transitions [1]
• shift the frequencies of the normal modes in biomolecules [2]
• perturb the dynamics of polypeptides [3].
• Coarse-grained models produce approximate dynamics by sacrificing atomic details.

Key Points

• Constraints decrease the number of degrees of freedom

• Imposing constraints can affect simulation outcome