Proof of pressure of system in molecular dynamics

I try to proof this from virial theorem.

$$2langle T rangle = - leftlangle sum_i vec{F}_i cdot vec{r}_i rightrangle $$

I classify the force into (a) interaction force between constituent molecules, and (b) force from the container.

Apparently, the pressure will come from the second category. Let investigate (b) first:

Force from the container occurs for $vec{r}_k$ at the container surface. Where the gas exerts pressure $P$ to the surface perpendicular to the surface $ P dvec{A}$, therefore the surface also renders a same force but in opposite direction $dvec{F} = - P dvec{A}$. This force contributes to the virial term:

$$ sum_{r_k text{ at surface}} vec{F}_k cdot vec{r}_k =- P sum_{r_k text{ at surface}} dvec{A}_k cdot vec{r}_k = -P unicode{x222f}_A dvec{A} cdot vec{r}. $$

We then turn the surface integral into volume integral by Divergent Theorem: $$ unicode{x222f}_A vec{V} cdot dvec{A} = iiint vecnablacdotvec{V} d^3 r$$

We then have: $$ sum_{r_k text{ at surface}} vec{F}_k cdot vec{r}_k = -P iiint vecnablacdotvec{r} d^3r = -3 P iiint d^3r = -3 P V. $$

Then the interaction term:

$$ vec{F}_{ij}cdot vec{r}_i + vec{F}_{ji}cdot vec{r}_i = vec{F}_{ij}cdot left( vec{r}_i - vec{r}_j right) = vec{F}_{ij}cdot vec{r}_{ij}. $$

All together, we have:

$$ - 3 p V + sum_i sum_{jne i} vec{F}_{ij}cdot vec{r}_{ij} + sum_i frac{p_i^2}{m_i} = 0 PV = frac{1}{3} left{ sum_i sum_{jne i} vec{F}_{ij}cdot vec{r}_{ij} + sum_i frac{p_i^2}{m_i} right} $$