Force Field¶
Bonds¶
Cassandra is designed assuming all bond lengths are fixed. If you wish to utilize a force field developed with flexible bond lengths, we recommend that you either use the nominal or “equilibrium” bond lengths of the force field as the fixed bond lengths specified for a Cassandra simulation or carry out an energy minimization of the molecule with a package that treats flexible bond lengths and utilize the bond lengths obtained from the minimization.
| Parameter | Symbol | Units |
|---|---|---|
| Bond length | \(l\) | Å |
Angles¶
Cassandra supports two types of bond angles:
fixed: The angle declared as fixed is not perturbed during the course of the simulation.harmonic: The bond angle energy is calculated as\[E_\theta = K_\theta (\theta - \theta_0)^2 \label{Eq:angle_potential}\]
where the user must specify \(K_\theta\) and \(\theta_0\). Note that a factor of \(1/2\) is not used in the energy calculation of a bond angle. Make sure you know how the force constant is defined in any force field you use.
| Parameter | Symbol | Units |
|---|---|---|
| Nominal bond angle | \(\theta_0\) | degrees |
| Bond angle force constant | \(K_\theta\) | K/rad2 |
Dihedrals¶
Cassandra can handle four different types of dihedral angles:
OPLS: The functional form of the dihedral potential is\[E_\phi = a_0 + a_1\, \left ( 1 + \cos(\phi) \right ) + a_2 \, \left ( 1 - \cos(2\phi)\right ) + a_3 \, \left ( 1 + \cos (3\phi)\right )\]
where \(a_0\), \(a_1\), \(a_2\) and \(a_3\) are specified by the user.
CHARMM: The functional form of the potential is\[E_\phi = a_0 (1 + \cos (a_1\phi - \delta))\]
where \(a_0\), \(a_1\) and \(\delta\) are specified by the user.
harmonic: The dihedral potential is of the form:\[E_\phi = K_\phi (\phi - \phi_0)^2\]
where \(K_\phi\) and \(\phi_0\) are specified by the user.
none: There is no dihedral potential between the given atoms.
| Functional Form | Parameter | Units |
|---|---|---|
| OPLS | \(a_0\), \(a_1\), \(a_2\), \(a_3\) | kJ/mol |
| CHARMM | \(a_0\) | kJ/mol |
| CHARMM | \(a_1\) | dimensionless |
| CHARMM | \(\delta\) | degrees |
| harmonic | \(K_\phi\) | K/rad2 |
| harmonic | \(\phi_0\) | degrees |
Impropers¶
Improper energy calculations can be carried out with the following two options:
none: The improper energy is set to zero for the improper angle.harmonic: The following functional form is used to calculate the energy due to an improper angle\[E_\psi = K_\psi \left ( \psi - \psi_0 \right )^2\]
where \(K_\psi\) and \(\psi_0\) are specified by the user.
| Parameter | Symbol | Units |
|---|---|---|
| Force constant | \(K_\psi\) | K/rad2 |
| Improper | \(\psi_0\) | degrees |
Nonbonded¶
The nonbonded interactions between two atoms \(i\) and \(j\) are due to repulsion-dispersion interactions and electrostatic interactions (if any).
Repulsion-Dispersion Interactions¶
The repulsion-dispersion interactions can take one of the following forms:
Lennard-Jones 12-6 potential (LJ):
\[{\cal V}(r_{ij})= 4 \epsilon_{ij} \left [ \left ( \frac {\sigma_{ij}} { r_{ij} }\right )^{12} - \left ( \frac {\sigma_{ij}} { r_{ij} }\right )^{6}\ \right ]\]
where \(\epsilon_{ij}\) and \(\sigma_{ij}\) are the energy and size parameters set by the user. For unlike interactions, different combining rules can be used, as described elsewhere. Note that this option only evaluates the energy up to a specified cutoff distance. As described below, analytic tail corrections to the pressure and energy can be specified to account for the finite cutoff distance.
Cut and shift potential:
\[{\cal V}(r_{ij})= 4 \epsilon_{ij} \left [ \left ( \frac {\sigma_{ij}} { r_{ij} }\right )^{12} - \left ( \frac {\sigma_{ij}} { r_{ij} }\right )^{6}\ \right ] - 4 \epsilon_{ij} \left [ \left ( \frac {\sigma_{ij}} { r_{cut}}\right )^{12} - \left ( \frac {\sigma_{ij}} { r_{cut} }\right )^{6}\ \right ] \label{Eq:cut_shift}\]
where \(\epsilon_{ij}\) and \(\sigma_{ij}\) are the energy and size parameters set by the user and \(r_{cut}\) is the cutoff distance. This option forces the potential energy to be zero at the cutoff distance. For unlike interactions, different combining rules can be used, as described elsewhere.
Cut and switch potential:
\[{\cal V}(r_{ij})= 4 \epsilon_{ij} \left [ \left ( \frac {\sigma_{ij}} { r_{ij} }\right )^{12} - \left ( \frac {\sigma_{ij}} { r_{ij} }\right )^{6}\ \right ] f \label{Eq:cut_switch}\]
The factor \(f\) takes the following values:
\[ \begin{align}\begin{aligned}\begin{aligned} f = \begin{cases}\\\begin{split} 1.0 \, \, \, & r_ {ij} \le r_{on} \\ \frac { (r_{off}^2 - r_{ij}^2)^2 (r_{off}^2 - 3r_{on}^2 + 2r_{ij}^2)} {\left ( r_{off}^2 - r_{on}^2 \right )^3} \, \, \, & r_{on} < r_{ij} < r_{off}\\ 0.0 \, \, \, & r_{ij} \ge r_{off}\end{split}\\ \end{cases}\end{aligned}\end{aligned}\end{align} \]
where \(\epsilon_{ij}\) and \(\sigma_{ij}\) are the energy and size parameters set by the user. This option smoothly forces the potential to go to zero at a distance \(r_{off}\), and begins altering the potential at a distance of \(r_{on}\). Both of these parameters must be specified by the user. For unlike interactions, different combining rules can be used, as described elsewhere.
Mie potential (generalized form of LJ):
\[{\cal V}(r_{ij})= \left ( \frac{n}{n-m} \right ) \left ( \frac {n}{m} \right )^{\frac{m}{n-m}}\epsilon_{ij} \left [ \left ( \frac {\sigma_{ij}} { r_{ij} }\right )^{n} - \left ( \frac {\sigma_{ij}} { r_{ij} }\right )^{m}\ \right ] \label{Eq:mie}\]
where \(\epsilon_{ij}\) and \(\sigma_{ij}\) are the energy and size parameters and \(n\) and \(m\) are the repulsive and attractive exponents set by the user. This option allows for the use of a generalized LJ potential (for LJ, \(n\) = 12 and \(m\) = 6). Note that this option only evaluates the energy up to a specified cutoff distance. Both n and m can take on separate integer or float values set by the user. For unlike interactions, different combining rules can be used, as described elsewhere.
Mie cut and shift potential:
\[{\cal V}(r_{ij})= \left ( \frac{n}{n-m} \right ) \left ( \frac {n}{m} \right )^{\frac{m}{n-m}}\epsilon_{ij} \left [ \left ( \frac {\sigma_{ij}} { r_{ij} }\right )^{n} - \left ( \frac {\sigma_{ij}} { r_{ij} }\right )^{m}\ \right ] - \left ( \frac{n}{n-m} \right ) \left ( \frac {n}{m} \right )^{\frac{m}{n-m}}\epsilon_{ij} \left [ \left ( \frac {\sigma_{ij}} { r_{cut}}\right )^{n} - \left ( \frac {\sigma_{ij}} { r_{cut} }\right )^{m}\ \right] \label{Eq:mie_cut_shift}\]
where \(\epsilon_{ij}\) and \(\sigma_{ij}\) are the energy and size parameters and \(n\) and \(m\) are the repulsive and attractive exponents set by the user. This option forces the potential energy to be zero at the cutoff distance (i.e. setting \(n\) = 12 and \(m\) = 6 provides the same potential as the LJ cut and shift option). For unlike interactions, different combining rules can be used, as described elsewhere.
- Tail corrections: If the Lennard-Jones potential is used, standard Lennard-Jones tail corrections are used to approximate the long range dispersion interactions
| Parameter | Symbol | Units |
|---|---|---|
| Energy parameter | \(\epsilon/k_B\) | K |
| Collision diameter | \(\sigma\) | Å |
Electrostatics¶
Electrostatic interactions are given by Coulomb’s law
where \(q_i\) and \(q_j\) are the partial charges set by the user, which are placed on atomic positions given by \(r_i\) and \(r_j\). In a simulation, the electrostatic interactions are calculated using either an Ewald summation, the Damped Shifted Force, or a direct summation using the minimum image convention. Note that the total energy that is printed out in the property file is extensive. Consequently, to obtain intensive energies, the printed energies must divided by the total number of molecules in the system.
| Parameter | Symbol | Units |
|---|---|---|
| Charge | \(q\) | e |
Summary of Cassandra units¶
| Item | Parameter | Units |
|---|---|---|
| Bonds | \(l\) | Å |
| Bond angles | \(\theta_0\) | degrees |
| Bond angles | \(K_\theta\) | K/rad2 |
| OPLS dihedrals | \(a_0\), \(a_1\), \(a_2\), \(a_3\) | kJ/mol |
| CHARMM dihedrals | \(a_0\) | kJ/mol |
| CHARMM dihedrals | \(a_1\) | dimensionless |
| CHARMM dihedrals | \(\delta\) | degrees |
| Harmonic dihedrals | \(K_\phi\) | K/rad2 |
| Harmonic dihedrals | \(\phi_0\) | degrees |
| Impropers | \(K_\psi\) | K/rad2 |
| Impropers | \(\psi_0\) | degrees |
| Simulation box length | Å | |
| Distances | Å | |
| Volume | Å3 | |
| Rotational width | degrees | |
| Temperature | K | |
| Pressure | bar | |
| Chemical potential | kJ/mol | |
| Energy | kJ/mol |