Tensorized orbitals for computational chemistry

TL;DR

The paper introduces tensorized orbitals built from a quantics representation and Tensor Cross Interpolation (TCI) to overcome the dominant bottleneck of computing the four‑index Coulomb integrals. By encoding orbitals as matrix product states (MPS) on a 3D grid and contracting with MPOs/MPS tools, the authors demonstrate accurate, scalable evaluation of S_{ij}, H_{ij}, and V_{ijkl} for Gaussian, Slater, and real‑space orbitals, including efficient handling of large basis sets. They show substantial energy accuracy gains for small molecules (e.g., H2 and CH4) by moving toward a continuum (CBS) limit using natural orbitals and an enrichment procedure that preserves computational feasibility. Additionally, direct real-space tensorized orbitals optimized with DMRG/DMRG‑like methods yield highly accurate single‑orbital representations that outperform conventional Gaussian expansions. Overall, the approach opens a path to more accurate and compact basis sets and integrates tensor-network techniques with standard quantum chemistry workflows, promising significant improvements in basis‑set convergence and correlation treatment.

Abstract

Choosing a basis set is the first step of a quantum chemistry calculation and it sets its maximum accuracy. This choice of orbitals is limited by strong technical constraints as one must be able to compute a large number of six dimensional Coulomb integrals from these orbitals. Here we use tensor network techniques to construct representations of orbitals that essentially lift these technical constraints. We show that a large class of orbitals can be put into ``tensorized'' form including the Gaussian orbitals, Slater orbitals, linear combination thereof as well as new orbitals beyond the above. Our method provides a path for building more accurate and more compact basis sets beyond what has been accessible with previous technology. As an illustration, we construct optimized tensorized orbitals and obtain a 85% reduction of the error on the energy of the $H_2$ molecules with respect to a reference double zeta calculation (cc-pvDz) of the same size.

Figures (13)

• Figure 1: Tensorisation of Slater orbitals. Error versus bond dimension $\chi$ for the energy of the exact $1s$ orbital of the hydrogen atom (first three panels) and other orbitals ($1s$, $2p_z$, $3d_{xz}$ and $4f_{z(x^2-y^2)}$, last panel). First three panels: error on the kinetic energy $K$, nuclei potential energy $P$ and total energy $E=K+P$ for different grid discretization $n=8,12,16$ and $20$. Last panel: $n=20$ except for the $4f$ orbital for which $n=22$. Energies in Hartree.
• Figure 2: Tensorisation of Gaussian orbitals: LiH molecule in the STO-6G basis set. Error versus bond dimension $\chi$ of the orbitals for the different matrix elements (first three panels, $\sigma$ is the standard deviation) and overall CCSD(T) energy (last panel) for different discretization parameters $n=8,12,16$ and $20$. Distance between the nuclei: $d_{Li-H} = 2.8571 a_0$. Reference calculation: pyscf package sun2018.
• Figure 3: Error on the ground energy of the $H_2$ molecule for different basis sets. The calculation uses $M$ orbitals constructed optimally out of $M_{AO}$ atomic orbitals (colored lines) or using our enrichment algorithm (black circles, $n=20$). Pink shaded area: $\epsilon_{\rm cor}$; Blue shaded area: $\epsilon_{\rm BS}$. The interatomic distance is $d_{H-H} = 1.4013 ~ a_0$, the reference energy is $E_0 = -1.17447498 ~ {\rm Ha}$kolos1968 and the many-body problem is treated exactly with pyscf.
• Figure 4: Results of enrichment process on the $CH_4$ molecule with $M=34$, $M^{\star} = 54$ versus number of enrichment steps. The enrichement is performed from the cc-pvDz ($M_{AO}=34$, blue, single data point), cc-pvTz ($M_{AO}=86$, orange), cc-pvQz ($M_{AO}=175$, green) and the cc-pv5z ($M_{AO}=311$, pink) basis set. Both green and pink curve reach the continuous limit. The inset shows the CCSD(T) calculations of the energy versus the number of optimum orbitals $M$ for $M\le 34$ computed by enrichment (black circles). Also shown are regular cc-pvXz CCSD(T) calculations (color triangles). Parameters of the $CH_4$ molecule: $d_{C-H} = 2.0541a_0$ ; $\theta_{CH_4} = 109.5^{\circ}$. The enrichment was done with $n=20$.
• Figure 5: Errors of the MPS interpolating the 1s orbital of the Hydrogen atom versus its rank $\chi$. a. In sample error ($\epsilon_{\mathrm{IS}}$) in dotted lines and out of sample error ($\epsilon_{\mathrm{OS}}$) in solid lines. Both are converging exponentially to $0$. b. Error on the integral $\epsilon_{\mathrm norm}$. Here, the exact integral is $\int \phi(x) \phi^{\dagger}(x) dx = 1$.