Please review this C++ listing of an implementation of Leapfrog integration.

Was the algorithm implemented correctly?

#include "stdafx.h"
#include <iostream>
#include <vector>
#include <cmath>
#include <random>

// Constants for Argon
constexpr double epsilon = 119.8;   // Depth of the potential well (in K)
constexpr double sigma = 3.405;     // Distance for zero potential (in Angstrom)
constexpr double mass = 39.948;     // Mass of Argon (in amu)

struct Vector3D {
    double x, y, z;
};

// Lennard-Jones potential function
double lj_potential(const Vector3D& r)
{
    double r_mag = std::sqrt(r.x * r.x + r.y * r.y + r.z * r.z);
    double s_over_r = sigma / r_mag;
    double s_over_r6 = pow(s_over_r, 6);
    return 4.0 * epsilon * (s_over_r6 * s_over_r6 - s_over_r6);
}

// Derivative of the Lennard-Jones potential
Vector3D lj_force(const Vector3D& r)
{
    // Define a small distance for the derivative approximation
    double dr = 1e-6;
    Vector3D force;
    std::vector<Vector3D> r_plus_dr = { r, r, r };
    r_plus_dr[0].x += dr;
    r_plus_dr[1].y += dr;
    r_plus_dr[2].z += dr;
    // The force is the negative derivative of the potential energy
    force.x = -(lj_potential(r_plus_dr[0]) - lj_potential(r)) / dr;
    force.y = -(lj_potential(r_plus_dr[1]) - lj_potential(r)) / dr;
    force.z = -(lj_potential(r_plus_dr[2]) - lj_potential(r)) / dr;
    return force;
}

// Update the 'accel' function
void accel(std::vector<Vector3D>& a, const std::vector<Vector3D>& x)
{
    int n = x.size();   // number of points
    for (int i = 0; i < n; i++)
    {
        auto force = lj_force(x[i]);
        // use Lennard-Jones force law
        a[i].x = -force.x / mass;
        a[i].y = -force.y / mass;
        a[i].z = -force.z / mass;
    }
}

void leapstep(std::vector<Vector3D>& x, std::vector<Vector3D>& v, double dt) {
    int n = x.size();       // number of points
    std::vector<Vector3D> a(n);

    accel(a, x);
    for (int i = 0; i < n; i++)
    {
        v[i].x = v[i].x + 0.5 * dt * a[i].x;    // advance vel by half-step
        v[i].y = v[i].y + 0.5 * dt * a[i].y;    // advance vel by half-step
        v[i].z = v[i].z + 0.5 * dt * a[i].z;    // advance vel by half-step
        x[i].x = x[i].x + dt * v[i].x;      // advance pos by full-step
        x[i].y = x[i].y + dt * v[i].y;      // advance pos by full-step
        x[i].z = x[i].z + dt * v[i].z;      // advance pos by full-step
    }

    accel(a, x);
    for (int i = 0; i < n; i++)
    {
        v[i].x = v[i].x + 0.5 * dt * a[i].x;    // and complete vel. step
        v[i].y = v[i].y + 0.5 * dt * a[i].y;    // and complete vel. step
        v[i].z = v[i].z + 0.5 * dt * a[i].z;    // and complete vel. step
    }
}

// Initialize positions and velocities of particles
void initialize(std::vector<Vector3D>& x, std::vector<Vector3D>& v, int n_particles, double box_size, double max_vel) {
    // Create a random number generator
    std::default_random_engine generator;
    std::uniform_real_distribution<double> distribution(-0.5, 0.5);

    // Resize the vectors to hold the positions and velocities of all particles
    x.resize(n_particles);
    v.resize(n_particles);

    // Initialize positions and velocities
    for (int i = 0; i < n_particles; i++)
    {
        // Assign random initial positions within the box
        x[i].x = box_size * distribution(generator);
        x[i].y = box_size * distribution(generator);
        x[i].z = box_size * distribution(generator);

        // Assign random initial velocities up to max_vel
        v[i].x = max_vel * distribution(generator);
        v[i].y = max_vel * distribution(generator);
        v[i].z = max_vel * distribution(generator);
    }
}

int main() {
    int n_particles = 100;  // number of particles
    double box_size = 10.0; // size of the simulation box
    double max_vel = 0.1;   // maximum initial velocity
    double dt = 0.01;   // time step
    int n_steps = 10000;    // number of time steps

                            // Positions and velocities of the particles
    std::vector<Vector3D> x, v;

    // Initialize the particles
    initialize(x, v, n_particles, box_size, max_vel);

    // Run the simulation
    for (int step = 0; step < n_steps; step++) {
        leapstep(x, v, dt);
    }

    return 0;
}