Notes on DFT

revisiting DFT, personal notes meant for a quick brush-up, specific derivation added

• many-body problem
  • complications come from the fact that we cannot separate the coordinates
    • modelling aluminum (13 electrons) requires $(10^6)^{13}$ variables to describe the system
• observables
  • can we rewrite this expression ?
  • operation $O$ is determined by the operator $\hat{O}$ and its Hamiltonian $\hat{H}$
  • $V_{ext}$ contains sufficient info. to determine the observable
    • $O[\Psi]\rightarrow O[V_{ext}]$
    • is there a more compact quantity ? : $O[V_{ext}]\rightarrow O[?]$
  • one-body operator, 2-body operator : density $n_1(x_1), n(x_1, x_2)$ can be used
• Functionals $F$:
  • $F : C^{\infty}(\mathbb{R}) \rightarrow \mathbb{R}$
    • e.g. $F[f] = \int_a^b f(x) = A$
  • functional derivative $\frac{\delta F[f]}{\delta f}$
    • will be useful when dealing with the Kohn-Sham equation derivation
    • obtained from the following expression: \begin{equation} \left . \frac{dF[f+\epsilon\eta(x)]}{d\epsilon} \right|_{\epsilon=0} = \int dx \frac{\delta F[f]}{\delta f} \eta(x) \label{eq:funcderivative} \end{equation}
  • example (to be operative) :
    • $F[f] = \int dx \delta(x-x_0) f(x) = f(x_0)$, compare this to equation (\ref{eq:funcderivative}) \begin{align} \left .\frac{dF[f+\epsilon\eta(x)]}{d\epsilon} \right|_{\epsilon=0} &= \int dx \delta(x-x_0)\eta(x) \Rightarrow \int dx \frac{\delta F[f]}{\delta f} \eta(x) \\ \implies \frac{\delta F[f]}{\delta f} &= \delta(x-x_0) \end{align}
• types of functional
  • local functionals : $E_{ext}[n] = \int \mathbf{r} n(\mathbf{r}) V_{ext}(\mathbf{r})$
  • non-local functions : $ E_{H}[n] = \frac{1}{2} \int d\mathbf{r}d\mathbf{r’} \frac{n(r)n(r’)}{|r-r’|} $
• Hohenberg-Kohn Theorem
  • (1) HK theorem : external potential $V_{ext}$ is determined, within a trivial additive constant, by the electron density ($n(\mathbf{r})$) [2]
    • $n(\mathbf{r})$ determines the ground state wavefunction and other electronic properties of the system
      • hence, it determine all other ground state properties
    • every observable in the ground state is a unique functional of the electron density
    • $O=\bra{\Psi} \hat{O} \ket{\Psi} = O[V_{ext}]$ ??
      • $V_{ext}\rightarrow n(\mathbf{r})$, implies $O[V_{ext}] \rightarrow O[n(\mathbf{r})]$
      • $E[n] = \bra{\Psi} \hat{H} \ket{\Psi} $
  • (2) HK theorem: for a trial density $n(\mathbf{r})>0$ and $\int n(\mathbf{r}) d\mathbf{r}=N$, $E_0 \leq E[n(\mathbf{r})]$, where $E[n(\mathbf{r})]$ is the energy functional
  • How to calculate the density?
• Born-Oppenheimer approximation
• auxilliary system?
  • what could be a good auxilliary system for many electrons be?
• Kohn-Sham theorem:
  • for each system, real system of interacting electrons, there exist an auxiliary system of non-interacting electrons which has the same density
  • what are the best one-particle wavefunctions of the auxialliary system that minimizes the energy functional $E[n]$
    • $\frac{\delta E[n]}{\delta \psi_i^*}=0$, constrained by $\int \psi_i^* \psi_i = 1$
    • use the lagrange multiplier method: $\frac{\delta [E[n]-\lambda (\int d\mathbf{r}\psi_i^* \psi_i - 1)]}{\delta \psi_i^*} = 0$
    • total energy from the Kohn-Sham equation: \begin{align} & \frac{\delta T_s[n]}{\delta \psi_i^*} + \frac{\delta}{\delta \psi_i^*} \int d\mathbf{r} n(\mathbf{r})V_{ext}(\mathbf{r}) + \frac{\delta }{\delta \psi_i^*}\int d\mathbf{r} d\mathbf{r}’ \frac{n(\mathbf{r})n(\mathbf{r’})}{|\mathbf{r}-\mathbf{r}’|}
      \end{align} \begin{align} + \frac{\delta E_{xc}}{\delta \psi^*_i} - \lambda_i \frac{\delta }{\delta \psi^*_i} \int d\mathbf{r} \psi_i^*(\mathbf{r})\psi_i(\mathbf{r}) = 0 \end{align}
      First term functional derivative

      \begin{align} T_s[\psi_i(\mathbf{r})] &= \int d\mathbf{r} \psi_i^*(\mathbf{r})\left( \frac{-\nabla^2}{2}\right) \psi_i(\mathbf{r}) \\ \left. \frac{dT[\psi_i(\mathbf{r}) + \eta(\mathbf{r})\epsilon]}{d\epsilon} \right|_{\epsilon=0} &= \left.\frac{d}{d\epsilon} \int d\mathbf{r} (\psi_i^*(\mathbf{r}) + \eta(\mathbf{r})\epsilon) \left( \frac{-\nabla^2}{2}\right) \psi_i(\mathbf{r}) \right|_{\epsilon=0}\\ &= \int d\mathbf{r} \left[\left( \frac{-\nabla^2}{2}\right) \psi_i(\mathbf{r}) \right] \eta(\mathbf{r}) \\ \implies \frac{\delta T_s[\psi^*_i]}{\delta \psi^*_i} &= \left( \frac{-\nabla^2}{2}\right) \psi_i(\mathbf{r}) \end{align}
      Second term functional derivative

      \begin{align*} F[\psi_i^*] &= \int d\mathbf{r} \sum_j \psi_j^*(\mathbf{r})\psi_j(\mathbf{r}) V_{ext}(\mathbf{r}) \\ \left. \frac{d F[\psi_i^* + \epsilon \eta(\mathbf{r})]}{d\epsilon}\right|_{\epsilon=0} &= \left.\frac{d}{d\epsilon} \int d\mathbf{r} \sum_j (\psi_j^*(\mathbf{r})+ \delta_{ij}\eta(\mathbf{r})\epsilon)\psi_j(\mathbf{r}) V_{ext}(\mathbf{r}) \right|_{\epsilon=0} \\ &= \left. \frac{d}{d\epsilon} \int d\mathbf{r} \sum_j (\psi_j^*(\mathbf{r}) \psi_j(\mathbf{r})) \right|_{\epsilon=0} + \left. \frac{d}{d\epsilon} \int d\mathbf{r}\eta(\mathbf{r})\epsilon \sum_j \delta_{ij}\psi_j(\mathbf{r}) V_{ext}(\mathbf{r}) \right|_{\epsilon=0} \\ &= \left. \frac{d}{d\epsilon} \int d\mathbf{r} \left[ \psi_i(\mathbf{r}) \epsilon V_{ext}(\mathbf{r})\right]\eta(\mathbf{r}) \right|_{\epsilon=0} \\ &= \int d\mathbf{r} [\psi_i(\mathbf{r}) V_{ext}(\mathbf{r})]\eta(\mathbf{r}) \\ \implies \frac{\delta F[\psi^*(\mathbf{r})]}{\delta \psi^*(\mathbf{r})} &=\psi_i(\mathbf{r})V_{ext}(\mathbf{r}) \end{align*}
      Third term functional derivative

      \begin{align*} F[\psi^*(\mathbf{r})] &= \frac{1}{2}\int d\mathbf{r} d\mathbf{r}' \frac{n(\mathbf{r})n(\mathbf{r'})}{|\mathbf{r}-\mathbf{r}'|} \\ \left. \frac{d F[\psi_i^*(\mathbf{r}) + \epsilon \eta(\mathbf{r})]}{d\epsilon}\right|_{\epsilon=0} &= \frac{1}{2}\frac{d}{d\epsilon} \int d\mathbf{r} \int d\mathbf{r'}n(\mathbf{r})\frac{1}{|\mathbf{r}-\mathbf{r}'|}n(\mathbf{r'}) \\ &= \frac{1}{2}\frac{d}{d\epsilon} \int d\mathbf{r} \int d\mathbf{r'} \sum_j (\psi^*_j(\mathbf{r}) + \delta_{ij}\eta(\mathbf{r})\epsilon) \psi_j(\mathbf{r}) \frac{1}{|\mathbf{r}-\mathbf{r}'|} \sum_k (\psi^*_k(\mathbf{r'}) + \delta_{ik}\eta(\mathbf{r'})\epsilon)) \psi_k(\mathbf{r'}) \\ &= \frac{1}{2}\frac{d}{d\epsilon} \int d\mathbf{r} \int d\mathbf{r'} [n(\mathbf{r}) + \eta(\mathbf{r})\epsilon\psi_i(\mathbf{r})] \frac{1}{|\mathbf{r}-\mathbf{r}'|} [ n(\mathbf{r'}) + \eta(\mathbf{r'})\epsilon \psi_i(\mathbf{r'})]\\ \left. \frac{d F}{d\epsilon} \right|_{\epsilon=0} &= \frac{1}{2} \int d\mathbf{r} \int d\mathbf{r}' \frac{\eta(\mathbf{r})\psi_i(\mathbf{r})n(\mathbf{r}') + n(\mathbf{r})\eta(\mathbf{r}')\psi_i(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \\ &= \int d\mathbf{r} \left[ \psi_i(\mathbf{r}) \int d\mathbf{r}' \frac{n(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right] \eta(\mathbf{r}) \\ \implies\frac{\delta F[\psi_i^*(\mathbf{r})]}{\delta \psi^*_i} &= \psi_i(\mathbf{r}) \int d\mathbf{r}' \frac{n(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \end{align*}
      Fourth term functional derivative

      \begin{align} \frac{\delta E_{xc}}{\delta \psi_i^*} &= \frac{\delta E_{xc}}{\delta n(\mathbf{r})} \frac{\delta n(\mathbf{r})}{\delta \psi_i^*} \\ &= \frac{\delta E_{xc}}{\delta n(\mathbf{r})}\frac{\delta \sum_j \psi^*_j(\mathbf{r}) \psi_j(\mathbf{r})}{\delta \psi^*_i(\mathbf{r})} \\ &= \frac{\delta E_{xc}}{\delta n(\mathbf{r})} \delta_{ij} \psi_j(\mathbf{r}) \\ \implies \frac{\delta E_{xc}}{\delta \psi_i^*}&= \frac{\delta E_{xc}}{\delta n(\mathbf{r})}\psi_i(\mathbf{r}) \end{align}
      Constraint term functional derivative

      \begin{align*} F[\psi^*(\mathbf{r})] &= \int d\mathbf{r} \psi_i^*(\mathbf{r})\psi_i(\mathbf{r}) \\ \left. \frac{d F[\psi_i^*(\mathbf{r}) + \epsilon \eta(\mathbf{r})]}{d\epsilon}\right|_{\epsilon=0} &= \left.\frac{d}{d\epsilon} \int d\mathbf{r} (\psi_i^*(\mathbf{r}) + \eta(\mathbf{r})\epsilon)\psi_i(\mathbf{r}) \right|_{\epsilon=0} \\ &= \left.\frac{d}{d\epsilon} \int d\mathbf{r} (\psi_i^*(\mathbf{r}) \psi_i(\mathbf{r}) \right|_{\epsilon=0} + \left.\frac{d}{d\epsilon} \int d\mathbf{r} \eta(\mathbf{r})\epsilon\psi_i(\mathbf{r}) \right|_{\epsilon=0} \\ &= \ \left.\frac{d}{d\epsilon} \int d\mathbf{r} \eta(\mathbf{r})\epsilon\psi_i(\mathbf{r}) \right|_{\epsilon=0} \\ &= \int d\mathbf{r} \eta(\mathbf{r})\psi_i(\mathbf{r}) \\ \implies \frac{\delta F[\psi^*(\mathbf{r})]}{\delta \psi^*(\mathbf{r})} &= \psi_i(\mathbf{r}) \end{align*}
  • because of the auxilliary system idea and the variational principle, we are able to write one particle equations
  • KS gives the exact electron density
  • unknown: $E_{xc}$ or $V_{xc}$
    • Exchange-correlation term
      • strategies for approximation: empirical vs. non-empirical
      • Jacob’s ladder of approximation
      • LDA, GGA, meta GGA, hybrid
  • how to solve the KS equation – self-consistent field method
• Universal Functional
• Material Properties
  • band gap
  • bond dissociation energies
  • bond length, lattice constants
  • cohesive energy
  • bulk moduli
  • dipole moment
• Limitations
  • $E_{xc}, V_{xc}$ are just approximate
  • large variety of xc, no systematic approach
  • problems: barriers of chemical reactions, dissocation energies of ions, van der Waals bonding, phase transition
  • uses of KS equation: band structure, effective masses, optical, photo emission
• ML/DL in DFT
  • machine-learned density functional
  • electron density learning

1. Sottile, F., & Reining, L. (2021). Density Functional Theory. Coursera.
2. Parr, R. G. (1989). Density functional theory of atoms and molecules. Springer.
3. Burke, K., & Wagner, L. (2014). ABC of ground-state DFT. University lecture.