I would like to find out the Hessian matrix of the following harmonic potential function
$$V=\frac{k}{2}|\vec{r}_i-\vec{r}_j|^2$$ where $r_{ij}^2=|\vec{r}_i-\vec{r}_j|^2=(x_i-x_j)^2+(y_i-y_j)^2+(z_i-z_j)^2$. For this purpose, I visited the page https://physics.stackexchange.com/questions/675733/derivation-of-second-order-derivative-of-harmonic-potential-well and https://web.archive.org/web/20200927012331/https://bougui505.github.io/science/2014/11/02/elastic-network-model-of-proteins-with-python.html but while I am trying to calculate the same, I get the following: $\frac{\partial V}{\partial x_i}=k(x_i-x_j)$, which makes $\frac{\partial^2 V}{\partial x_i\partial y_j}=0$. However, as per the above link, I should get some non zero value of the Hessian matrix but according to my calculation only the diagonal term will survive and all other off diagonal terms will be zero, which seems wrong as per the calculation shown in the link. Would you kindly tell me where my misconception lies? or any other pdf where the details of this calculation has been discussed. Thanking you...