From potential energy to atomic forces

\[V = V_{bonded} + V_{nb}\]

First, let us derive a couple of general expression that should help us later. Consider a system of \(N\) particles. Assume, that the potential energy of the system depends on the coordinates of the particles \(V=V(\{\mathbf{r}_i\})=V(\mathbf{r}_1, \mathbf{r}_2, \ldots \mathbf{r}_N)\). In this case, the force exerted by a particle \(i\) is given by:

\[\begin{split}\mathbf{f}_i = \nabla_i V = \begin{bmatrix}\frac{\partial}{\partial x_i}\\\frac{\partial}{\partial y_i}\\\frac{\partial}{\partial z_i}\end{bmatrix}V\end{split}\]

Potential function usually depends on the relative position of the particles, not their absolute coordinates (e.g. the potential energy of a harmonic spring depends on its extension, not the spacial orientation). Hence, while differentiating and applying the chain rule, we will frequently encounter expressions like \(\nabla_i r_{ij}\), where \(r_{ij}\) is the distance between two particles. Expanding this expression, we get:

\[\begin{split}\begin{split} \nabla_i r_{ij} =& \nabla_i\sqrt{(x_i-x_j)^2 + (y_i-y_j)^2 + (z_i-z_j)^2} = \\ =& \frac{1}{2r_{ij}}\nabla_i\left((x_i-x_j)^2 + (y_i-y_j)^2 + (z_i-z_j)^2\right) = \\ =& \frac{1}{2r_{ij}} \begin{bmatrix} \frac{\partial(x_i-x_j)^2}{\partial x_i}\\ \frac{\partial(y_i-y_j)^2}{\partial y_i}\\ \frac{\partial(z_i-z_j)^2}{\partial z_i} \end{bmatrix} = \frac{1}{2r_{ij}}\begin{bmatrix}-2(x_i-x_j)\\-2(y_i-y_j)\\-2(z_i-z_j)\end{bmatrix} = \frac{\mathbf{r}_{ij}}{r_{ij}} \end{split}\end{split}\]

Note that \(\nabla_j r_{ij}=-\nabla_i r_{ij}\), as suggested by Newtons third law.

Exercise

Compute \(\nabla_i\theta_{ijk}\), \(\nabla_j\theta_{ijk}\) and \(\nabla_k\theta_{ijk}\), where \(\theta_{ijk}\) is the angle between vectors, connecting particles \(j-i\) and \(j-k\) (i.e. angle between \(\mathbf{r_{ji}}\) and \(\mathbf{r_{jk}}\)).

Exercise

Compute \(\nabla_i\phi_{ijkl}\), \(\nabla_j\phi_{ijkl}\), \(\nabla_k\phi_{ijkl}\) and \(\nabla_l\phi_{ijkl}\), where \(\phi_{ijkl}\) is the dihedral (torsion) angle between \(i-j-k\) and \(j-k-l\) planes (i.e. torsion angle for the \(\mathbf{r_{jk}}\) bond).