Kinetic Energy Density (KED)

The kinetic-energy density, or local kinetic energy, is inherently ambiguous, because specifying the expectation value for the kinetic energy of the electrons, \(\tfrac{1}{2m} p^2\), at a specified point in space requires knowing the momentum and location of the electrons simultaneously, which is impossible according to the Heisenberg Uncertainty Principle. This has not stopped scientists from defining a myriad exact, and approximate, definitions for the kinetic energy density. As with other DFT-based tools, these can be defined in either spin-resolved or spin-unresolved manner, depending on how the class is initialized.

While kinetic energy densities are not commonly employed on their own, they are common ingredients in descriptors that are used to elucidate molecular electronic structure and bonding, as well as in topological analysis and partitioning techniques.

Positive-definite Kinetic Energy Density: The manifestly nonnegative kinetic energy density, which is the key ingredient in many descriptors and also in the

\[\tau_\text{PD}(\mathbf{r}) = \tfrac{1}{2}\sum_i^N n_i \rvert\nabla\phi_i(\mathbf{r})\lvert^2\]

where \(\phi_i(\mathbf{r})\) and \(n_i\) are the molecule (spin) orbitals and their occupation numbers.

General(ish) Kinetic Energy Density

\[\tau_\text{G}(\mathbf{r}, \alpha) = \tau_\text{PD}(\mathbf{r}) + \tfrac{1}{4} (a - 1) \nabla^2\rho(\mathbf{r})\]

The integral of this kinetic energy density gives the exact non-interacting kinetic energy. Several of the more common kinetic energy densities in the literature arise as special sub-cases, including the form of Ghosh, Berkowitz, and Parr (GBP, \(a=\tfrac{1}{2}\)) which can be justified by maximizing the entropy of the underlying quasiprobability distribution function and the form of Yang, Liu, and Wang (YLW, \(a=0\)), which can be justified by appealing to the local energy. The form corresponds to the “Schrödinger” kinetic energy that Bader denoted \(K(\mathbf{r})\), while \(a=1\) is the positive-definite kinetic energy that Bader denoted \(G(\mathbf{r})\). While only the form \(a=0\) is is non-negative, but the integral gives the correct global kinetic energy for any value of \(a\).

“Unambiguous” Kohn-Sham Kinetic Energy Density: Using the virial theorem, one can derivate a kinetic-energy density from the Kohn-Sham potential. The usual expression is not invariant to translation/rotation of the electron density, but an alternative expression, which requires solving a Poisson equation, avoids this issue:

\[\tfrac{-1}{4\pi} \nabla^2 \tau_\text{unambiguous}(\mathbf{r}) = \tfrac{3}{8\pi} \nabla \cdot \rho(\mathbf{r}) \nabla v_s(\mathbf{r})\]

Here, \(v_s(\mathbf{r})\) is the Kohn-Sham potential. The integral of this kinetic energy density gives the exact non-interacting kinetic energy.

Thomas-Fermi Kinetic Energy Density \(\tau_\text{TF}\left(\mathbf{r}\right)\) represents non-interacting kinetic energy density that is exact for a uniform electron gas:

\[\begin{split}\tau_\text{TF}(\mathbf{r}) = \begin{cases} \tfrac{3}{10} \left(3\pi\right)^{2/3} \rho(\mathbf{r})^{5/3} \text{ for spin}=\alpha + \beta\\ \tfrac{3}{10} \left(6\pi\right)^{2/3} \rho(\mathbf{r})^{5/3} \text{ for spin}=\alpha, \beta, \alpha - \beta \end{cases}\end{split}\]

Von Weizsacker Kinetic Energy Density \(\tau_\text{W}\left(\mathbf{r}\right)\) The kinetic energy density for a system of bosons with the same density as the electron density. It is exact for one- and two-electron systems (with nondegenerate ground states).

\[\tau_\text{W}(\mathbf{r}) = \frac{\lvert \nabla \rho(\mathbf{r}) \rvert ^2}{8 \rho(\mathbf{r})} = \frac{\nabla\rho(\mathbf{r}) \cdot \nabla\rho(\mathbf{r})}{8 \rho(\mathbf{r})} = \tfrac{1}{2} \nabla\sqrt{\rho(\mathbf{r})} \cdot \nabla\sqrt{\rho(\mathbf{r})}\]

Gradient Expansion Approximation: This expression is exact for the kinetic energy density of a slowly-varying electron gas. It is derived by performing a perturbation expansion about the uniform electron gas limit, and truncating the expansion at second order. The fourth- and sixth-order terms in the expansion are known, but rarely used because they are strongly singular near the nucleus.

\[\tau_\text{GEA}(\mathbf{r}) = \tau_\text{TF}(\mathbf{r}) + \tfrac{1}{9} \tau_\text{W}(\mathbf{r}) + \tfrac{1}{6} \nabla^2\rho(\mathbf{r})\]

Empirical Gradient Expansion Approximation: The gradient expansion approximation with a higher Weizsacker factor, which often gives more accurate results in practice because atoms and molecules are far from the slowly-varying limit.

\[\tau_\text{empGEA}(\mathbf{r}) = \tau_\text{TF} (\mathbf{r}) + \tfrac{1}{5} \tau_\text{W}(\mathbf{r}) + \tfrac{1}{6} \nabla^2 \rho(\mathbf{r})\]

Nuclear-Corrected Kinetic Energy Density: For any kinetic energy expression, this expression makes a correction near the nucleus, based on the fact the Weizsacker kinetic energy density is very accurate there. This is particularly useful for functionals that include Laplacian contributions, which tend to be bad there.

\[w(\mathbf{r}) = \sum_{A=1}^{N_\text{atoms}} \exp\left(- \frac{\left(Z_A \lvert \mathbf{r} - \mathbf{R}_{A} \rvert\right)^4}{\left(\ln2\right)^3}\right)\]

\[\tau_\text{mod}(\mathbf{r}) = w(\mathbf{r}) \tau_\text{W}(\mathbf{r}) + (1 + w(\mathbf{r})) \tau(\mathbf{r})\]

where \(\tau(\mathbf{r})\) can be any kinetic energy density.

While the kinetic energy density from this equation tends to resemble the positive-definite kinetic energy density more strongly than the preceding approximations, its integrated value is not significantly more accurate.

TODO: (Eventually we should also provide some options for correlation-kinetic energies.)

Global Kinetic Energy: We do not recommend usually evaluating the global kinetic energy from the preceding kinetic energy densities, but we support this by allowing one to evaluate the kinetic energy corresponding to any kinetic energy density (by simple integration):

\[T = \int \tau(\mathbf{r}) d\mathbf{r}\]