Overview

Dopyqo performs many-body post-processing of Kohn-Sham density functional theory (DFT) calculations. Starting from a Quantum ESPRESSO plane-wave DFT result, it constructs a second-quantized many-body Hamiltonian and solves it with classical (FCI) or quantum (VQE) algorithms, and then can compute Bader charges from the resulting many-body charge density.

Workflow

Quantum ESPRESSO (DFT)
       │
       │  wfc.dat / wfc.hdf5
       │  data-file-schema.xml
       ▼
Kohn-Sham orbitals & crystal structure
       │
       │  active space selection
       ▼
Many-body Hamiltonian (matrix elements and constant energies)
       │
       ├──► FCI (PySCF)   ──► ground-state energy & density
       └──► VQE (TenCirChem / Qiskit) ──► ground-state energy & density
                                                 │
                                                 ▼
                                        Bader charge analysis

Many-body Hamiltonian

The electronic Hamiltonian in second quantization is (see our paper):

\[\hat{H} = \sum_{tu} h_{tu}\, a_t^\dagger a_u + \frac{1}{2} \sum_{tuvw} h_{tuvw}\, a_t^\dagger a_u^\dagger a_v a_w + E_{\text{n-n}} + E_{\text{e-self}}\]

where \(a_t^\dagger\) and \(a_t\) are fermionic creation and annihilation operators for spin-orbital \(t\), \(E_{\text{n-n}}\) is the nuclear repulsion (Ewald) energy, and \(E_{\text{e-self}}\) is the electron self-interaction energy.

Electron Repulsion Integrals

The two-electron integrals are computed efficiently in reciprocal space via pair densities:

\[h_{tuvw} = \frac{4\pi}{V} \sum_{\mathbf{G} \neq 0} \frac{\tilde{\rho}^*_{wt}(\mathbf{G})\,\tilde{\rho}_{uv}(\mathbf{G})}{G^2}\]

where \(V\) is the unit-cell volume, \(\mathbf{G}\) are reciprocal-lattice vectors, and \(\tilde{\rho}_{tu}(\mathbf{G})\) is the Fourier transform of the real-space pair density \(\rho_{tu}(\mathbf{r})=\psi^\ast_t(\mathbf{r}) \psi_u(\mathbf{r})\), where \(\psi_t(\mathbf{r})\) is the real-space representation of the \(t\)-th Kohn-Sham orbital.

The \(\mathbf{G} = 0\) term is omitted, as it cancels with the compensating background charge in a periodic system.

Bader Charge Analysis

After obtaining the many-body ground-state charge density \(\rho(\mathbf{r})\), atoms are assigned charges by integrating over their Bader regions \(\mathcal{B}_I\):

\[Q_I = \int_{\mathcal{B}_I} \rho(\mathbf{r})\, d^3r\]

Each region \(\mathcal{B}_I\) is bounded by zero-flux surfaces running through minima of the charge density. The external Bader charge analysis code is used to determine the region boundaries.

Active Space

Because exactly solving the many-body Hamiltonian with classical algorithms scales exponentially with system size, Dopyqo uses an active space approach: only \(N_e\) electrons in \(N_o\) orbitals are treated at the many-body level. Orbitals outside the active space are either frozen (occupied) or discarded (virtual). The frozen orbitals are treated on a mean-field level using the frozen-core approximation.

The active-space parameters are set via active_electrons and active_orbitals in DopyqoConfig.