Grid-based DFT/HF equations: the tensor decomposition approach

We consider Hartree-Fock and DFT equations.

Our approach:

• Fully grid based -- no basis set error.
• Real space orbitals are in the Tucker format $\Longrightarrow$ linear in 1D grid size complexity $\mathcal{O}(n^3) \rightarrow \mathcal{O}(n)$
• Fast 3D convolutions via cross approximation

Some details

• Preconditioned direct minimization procedure -- good for tensor formats
• Derivative free calculations of the Fock matrix
• DIIS procedure
• Aitken's extrapolation

Algorithm

1. Calculate $\widehat \Phi = (\hat \phi_1, \dots, \hat \phi_N)$: $\hat\phi_i =2\, (-\Delta - 2 \lambda_i^{(k)})^{-1} V^{(k)}\phi_i^{(k)}$.
   
   
2. $\widehat \Phi$: $\widetilde \Phi = \widehat \Phi L^{-T}$, where $L$: $LL^T = \int_{\mathbb{R}^3} \widehat \Phi^T \widehat \Phi \, d{\bf r}$.
   
   
3. Calculate the Fock matrix $F = \int_{\mathbb{R}^3} \widetilde\Phi^T H^{(k+1)}\ \widetilde\Phi \, d{\bf r}$ via derivative-free formula:
   
   $$ F = \int_{\mathbb{R}^3} \left( \widetilde\Phi^T V^{(k+1)} \widetilde\Phi - \widetilde\Phi^T V^{(k)} \Phi^{(k)}L^{-T} + L^{-1}\,\widehat\Phi^T\widehat\Phi\Lambda^{(k)} L^{-T} \right) \, d{\bf r}, $$
   
4. Find new orbital energies by diagonalizing $F$: $ \Lambda^{(k+1)} = S^{-1} F S$
   
   
5. Find new orbitals: $\Phi^{(k+1)} = \widetilde \Phi S$

Accuracy and extrapolation

Experiments: Accuracy for He

For Helium $E = -2.861\ 679\ 996$ . We get

Experiments: Molecules

Experiments: Clusters

For clusters we observed that ranks grow sublinearly with the system size:

• H$_{3\times 2 \times 2}$: highest ranks $= 35\times35\times 20$
• H$_{9\times 2 \times 2}$: highest ranks $= 36\times36\times 24$

Experiments: Factors Behavior