Conventions

This page documents the project-wide sign conventions used by nvalchemi.

Virial

The virial tensor is defined as the negative strain derivative of the energy:

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

where \(\varepsilon\) is the symmetric infinitesimal strain tensor.

Low-level interaction kernels in nvalchemiops return virials using this convention.

Stress

The public stress tensor in nvalchemi follows the tensile-positive Cauchy stress convention:

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

where \(V = |\det(\mathbf{C})|\) is the cell volume. Cauchy stress is the true stress tensor for the current configuration: it maps a surface normal to the traction acting on that surface.

Pressure

Scalar and tensor pressure are positive for compression. From the tensile-positive stress tensor:

\[p = -\frac{1}{3}\operatorname{tr}(\sigma)\]

The NPT/NPH pressure tensor uses the same sign convention:

\[\mathbf{P} = \frac{\mathbf{K} + W}{V}\]

where \(\mathbf{K}\) is the kinetic tensor and \(W\) is the virial. For a static system with \(\sigma = -p\mathbf{I}\), this gives \(\mathbf{P} = p\mathbf{I}\).

References

• Malvern, L. E. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, 1969.

• Thompson, A. P.; Plimpton, S. J.; Mattson, W. “General formulation of pressure and stress tensor for arbitrary many-body interaction potentials under periodic boundary conditions.” J. Chem. Phys. 131, 154107 (2009). doi:10.1063/1.3245303