Equivalence Between Total Energy Minimization and Solving Kohn-Sham Equation

In this tutorial, we demonstrate that minimizing the total energy in density functional theory (DFT) is mathematically equivalent to solving the Kohn-Sham equations. This equivalence underpins direct minimization methods, an alternative to conventional diagonalization-based approaches for DFT calculations.

Note

For background on total energy minimization, refer to the tutorial Total Energy Minimization.

Note

Assumptions:

• Occupation numbers are omitted (i.e., all orbitals are singly occupied).

• Spin-spin_restricted formalism is used. For generalization with occupation numbers, see [Li2024].

Total Energy Functional

The total energy functional for a system of non-interacting electrons is expressed as:

\[E_{\text{total}}[\{\psi_i\}] = T_s[\{\psi_i\}] + E_{\text{Hxc}}[n] + \int V_{ext}(r) n(r) dr\]

where

1. Kinetic energy (\(T_s\)): The kinetic energy of non-interacting electrons:

\[T_s[\{\psi_i\}] = \frac{1}{2} \sum_{i} \int \psi_i^*(r) \nabla \psi_i(r) dr\]

2. Hartree-exchange-correlation energy (\(E_{\text{Hxc}}\)): The Hartree-exchange-correlation energy of the system:

\[E_{\text{Hxc}}[n] = \dfrac12 \int \dfrac{n(r) n(r')}{|r - r'|} dr dr' + \int \varepsilon_{\text{xc}}(\boldsymbol{r}) n(\boldsymbol{r}) dr\]

3. External potential (\(V_{ext}\)): Represents interactions with ions or applied fields.

The electron density \(n(\mathbf{r})\) is:

\[n(\mathbf{r}) = \sum_i |\psi_i(\mathbf{r})|^2\]

Constrained Minimization

The orbitals \(\psi_i(\mathbf{r})\) must satisfy orthonormality:

\[\int \psi^*_i(\mathbf{r}) \psi_j(\mathbf{r}) d\mathbf{r} = \delta_{ij}\]

To enforce this, we use Lagrange multipliers \(\lambda_{ij}\) and construct the Lagrangian:

\[L[{\psi_i}, {\lambda_{ij}}] = E_{\text{total}}[{\psi_i}] - \sum_{i,j} \lambda_{ij} \left( \int \psi^*_i(\mathbf{r}) \psi_j(\mathbf{r}) d\mathbf{r} - \delta_{ij} \right)\]

Functional Derivatives and Stationarity

Taking the functional derivative of the Lagrangian \(L\) with respect to \(\psi_i^*(\mathbf{r})\) (conjugate variable) gives:

\[\frac{\delta L}{\delta \psi^*_i (\mathbf{r})} = \frac{\delta T_s}{\delta \psi^*_i(\mathbf{r})} + \frac{\delta E_{\text{Hxc}}}{\delta \psi^*_i(\mathbf{r})} + \frac{\delta\int V_{\text{ext}} n(\mathbf{r})d\mathbf{r} }{\delta \psi^*_i(\mathbf{r})} - \sum_j \lambda_{ij} \psi_j(\mathbf{r}) = 0\]

Let’s break down each term:

• Kinetic Term:

\[\frac{\delta T_s}{\delta \psi_i^*(\mathbf{r})} = -\frac{1}{2} \nabla^2 \psi_i(\mathbf{r})\]

• Hartree-XC Term:

\[\frac{\delta E_{\text{Hxc}}}{\delta \psi_i^*(\mathbf{r})} = \int \frac{n(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|} d\mathbf{r}' + V_{\text{xc}}(\mathbf{r})\]

The first terms of the potential is also known as the Hartree potential:

\[\begin{equation} V_H(\mathbf{r}) := \int \frac{n(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|} d\mathbf{r}' \end{equation}\]

• External Potential Term:

\[\begin{align} \dfrac{\delta E_{\text{ext}}}{\delta \psi_i^*(\mathbf{r})} \int V_{\text{ext}}(\mathbf{r}) n(\mathbf{r}) d\mathbf{r} &= V_{\text{ext}}(\mathbf{r}) \psi_i(\mathbf{r}) \end{align}\]

Kohn-Sham Equation

Combining all terms, the stationary condition becomes:

\[\left[ -\frac{1}{2} \nabla^2 + V_H(\mathbf{r}) + V_{\text{xc}}(\mathbf{r}) + V_{\text{ext}}(\mathbf{r}) \right] \psi_i(\mathbf{r}) = \sum_j \lambda_{ij} \psi_j(\mathbf{r})\]

The matrix \(\lambda_{ij}\) is Hermitian due to orthonormality constraints. By choosing a unitary transformation that diagonalizes \(\lambda_{ij}\), we obtain the Kohn-Sham eigenvalue equation:

\[\hat{H}_{\text{KS}} \psi_i(\mathbf{r}) = \varepsilon_i \psi_i(\mathbf{r})\]

where the Kohn-Sham Hamiltonian is:

\[\hat{H}_{\text{KS}} = -\frac{1}{2} \nabla^2 + V_H(\mathbf{r}) + V_{\text{xc}}(\mathbf{r}) + V_{\text{ext}}(\mathbf{r}).\]

References

[Li2024]

Li, Tianbo, et al. “Diagonalization without Diagonalization: A Direct Optimization Approach for Solid-State Density Functional Theory.” arXiv preprint arXiv:2411.05033 (2024).