### Dallas R. Trinkle

#### Research Group

## Density Functional Theory Energy Density

Density-functional theory (DFT) for electronic structure defines the total energy for a set of atoms and electrons; using this to compute the energy of **crystalline defects** requires that the energy of the system is simply the sum of a single defect and well-defined (simple) reference states. An alternative approach is to define per-atom energies
\[
E_\text{total} = \sum_{i\in\text{atoms}} E_i
\]
in a similar fashion as one would have with classical potentials (atomistics). To do this, we use the density-functional theory **energy density** $\varepsilon(\vec r)$
\[
E_\text{total} = \int_{\Omega} \varepsilon(\vec r)d^3r
\]
To connect these, we need to partition space into unique volumes for each atom, $\Omega_i$. We identify volumes that produce gauge-independent integrals of the energy density, such as the Bader volumes. From a computational efficiency perspective, we also need to ensure that the definition of and integration over the volumes—for functions defined on a discrete grid—is done to minimize error.

#### Publications

- “Application of Energy Density Method on Crystalline Defects.” Min Yu, Dallas R. Trinkle and Richard M. Martin (in preparation)
- “Gauge-independent integration.” Min Yu, Dallas R. Trinkle (in preparation)

This work in collaboration with Richard Martin, UIUC Physics (emeritus).