Conventions

This page documents the project-wide sign conventions used across all interaction modules (electrostatics, dispersion, Lennard-Jones).

Virial

All modules return the virial tensor defined as the negative derivative of the energy with respect to the row-vector affine displacement tensor:

\[W_{ab} = -\frac{\partial E}{\partial u_{ab}}\]

where deformed coordinates and cells are built as:

\[R' = R (I + u), \qquad C' = C (I + u).\]

This matches the nvalchemi-toolkit prepare_strain / autograd_stresses convention. The displacement tensor is not symmetrized by the energy-derivative recipe.

For pairwise real-space interactions this is equivalent to:

\[W = -\sum_{i < j} \mathbf{r}_{ij} \otimes \mathbf{F}_{ij}\]

where \(\mathbf{r}_{ij} = \mathbf{r}_j - \mathbf{r}_i\) and \(\mathbf{F}_{ij}\) is the force on atom \(i\) due to atom \(j\). Individual kernel implementations may use the reversed separation vector or the reaction force internally, but returned virials always follow this convention.

Stress

The tensile-positive Cauchy stress is obtained from the virial as:

\[\sigma = -\frac{W}{V}\]

where \(V = |\det(\mathbf{C})|\) is the cell volume.

Equivalently, when using the displacement recipe above:

\[\sigma = \frac{1}{V}\frac{\partial E}{\partial u}.\]

Note

Some molecular-dynamics codes use the opposite (compression-positive or “pressure”) convention \(\sigma = W / V\). When comparing against external references, check which convention they follow.

Separation Vector

The canonical separation vector points from atom \(i\) to atom \(j\):

\[\mathbf{r}_{ij} = \mathbf{r}_j - \mathbf{r}_i\]

Individual kernel implementations may use either direction internally, but the returned virial always follows the convention above.