Quantum chemistry for solids made simple on the Clifford torus

Abstract

We present a general theory to treat periodic solids with quantum-chemistry methods. It relies on two main developments: 1) the modeling of a solid as a Clifford torus which is a torus that is both periodic and flat and 2) the introduction of a periodic gaussian basis set that is compatible with the topology of the Clifford torus. We illustrate our approach by calculating the ground-state energy of a periodic chain of hydrogen atoms within both Hartree-Fock and coupled cluster theory. We demonstrate that our approach yields the correct ground-state energy in the thermodynamic limit by comparing it to the ground-state energy of a ring of hydrogen atoms in the same limit. Since equivalent ring-like calculations for three-dimensional solids are impossible, our approach is an excellent alternative to perform quantum-chemistry calculations of solids. Our Clifford formalism can be seamlessly combined with current implementations of quantum-chemistry methods designed for atoms and molecules to make them applicable to solids.

Figures (4)

• Figure 1: The Euclidean distance between two points in a one-dimensional Clifford supercell of length $L_x$. Left panel: two points in a one-dimensional Clifford supercell. Right Panel : The Euclidean distance between these two points represented on a ring that is topologically equivalent to the Clifford torus.
• Figure 2: Periodic Clifford gaussians $g^{C}_i(x)$ (solid lines) compared to regular non-periodic gaussians $g_i(x)$ (dashed lines) in two neighboring supercells of length $L_x$ for various values of $i$.
• Figure 3: Hartree-Fock ground-state energies per atom $\bar{E}_0(H_N)$ of hydrogen chains with various numbers of atoms and a nearest-neighbor distance of 1.8 a.u. using Clifford periodic boundary conditions and Clifford gaussians (red dots) and a ring configuration with non-periodic gaussians (blue dots). The corresponding solid lines represent the extrapolation to the thermodynamic limit according to Eqs. \ref{['Eq:TDL_torus']} and \ref{['Eq:TDL_ring']}, respectively.
• Figure 4: CCSD(T) ground-state energies per atom $\bar{E}_0(H_N)$ of hydrogen chains with various numbers of atoms and a nearest-neighbor distance of 1.8 a.u. using Clifford periodic boundary conditions and Clifford gaussians (red dots) and a ring configuration with non-periodic gaussians (blue dots). The corresponding solid lines represent the extrapolation to the thermodynamic limit according to Eqs. \ref{['Eq:TDL_torus_CC']} and \ref{['Eq:TDL_ring_CC']}, respectively.