How to calculate angle force in MD simulation

In a molecular dynamics simulation, the angle force is calculated using the following equation:

F_i = -dE/dx_i = -k_theta * (cos(theta) - cos(theta_0)) * dtheta/dx_i

where F_i is the force on atom i, E is the angle potential energy, k_theta is the force constant for the angle potential, theta is the angle between three connected atoms, theta_0 is the equilibrium angle, and dtheta/dx_i is the derivative of the angle with respect to the position of atom i.

To calculate the angle force, you need to first calculate the angle between the three connected atoms. You can then use the above equation to calculate the force on each atom. The total force on each atom is the sum of the forces due to all the angles that involve that atom.

Here is the general algorithm for calculating the angle force in a molecular dynamics simulation:

For each angle in the system:
1. Calculate the angle between the three connected atoms.
2. Calculate the force on each atom using the above equation.
3. Add the force to the total force on each atom.

Update the positions of the atoms using the total force and a numerical integration method, such as Verlet integration.

Note that this is a simplified overview of the angle force calculation and there may be additional considerations, such as using cutoffs or scaling factors to improve the accuracy and efficiency of the calculation.

 

#include <iostream>
#include <cmath>

// Define the force constants for the angle potential
const double K_ANGLE = 100.0;
const double THETA_0 = M_PI / 3.0;

// Define a function to calculate the angle force
void calculate_angle_force(double x1, double y1, double z1,
                           double x2, double y2, double z2,
                           double x3, double y3, double z3,
                           double &fx1, double &fy1, double &fz1,
                           double &fx2, double &fy2, double &fz2,
                           double &fx3, double &fy3, double &fz3) {
    // Calculate the distances between the atoms
    double dx12 = x1 - x2;
    double dy12 = y1 - y2;
    double dz12 = z1 - z2;
    double r12 = std::sqrt(dx12*dx12 + dy12*dy12 + dz12*dz12);

    double dx32 = x3 - x2;
    double dy32 = y3 - y2;
    double dz32 = z3 - z2;
    double r32 = std::sqrt(dx32*dx32 + dy32*dy32 + dz32*dz32);

    // Calculate the angle between the bonds
    double cos_theta = (dx12*dx32 + dy12*dy32 + dz12*dz32) / (r12*r32);
    double sin_theta = std::sqrt(1.0 - cos_theta*cos_theta);
    double theta = std::acos(cos_theta);

    // Calculate the angle force
    double k_theta = 2.0 * K_ANGLE * (theta - THETA_0) / sin_theta;
    double fx12 = k_theta * (dy12*dz32 - dz12*dy32) / (r12*r32);
    double fy12 = k_theta * (dz12*dx32 - dx12*dz32) / (r12*r32);
    double fz12 = k_theta * (dx12*dy32 - dy12*dx32) / (r12*r32);

    double fx32 = k_theta * (dy32*dz12 - dz32*dy12) / (r32*r12);
    double fy32 = k_theta * (dz32*dx12 - dx32*dz12) / (r32*r12);
    double fz32 = k_theta * (dx32*dy12 - dy32*dx12) / (r32*r12);

    double fx2 = -fx12 - fx32;
    double fy2 = -fy12 - fy32;
    double fz2 = -fz12 - fz32;

    fx1 = fx12;
    fy1 = fy12;
    fz1 = fz12;

    fx3 = fx32;
    fy3 = fy32;
    fz3 = fz32;
}

// Define a function to calculate the angle potential energy
double calculate_angle_potential(double x1, double y1, double z1,
                                 double x2, double y2, double z2,
                                 double x3, double y3, double z3) {
    // Calculate the distances between the atoms
    double dx12 = x1 - x2;
    double dy12 = y1 - y2;
    double dz12 = z1 - z2;
    double r12 = std::sqrt(dx12*dx12 + dy12*dy12 + dz12*dz12);

    double dx32 = x3 -