[Paper Review] Tensorized orbitals for computational chemistry

Nicolas Jolly, Yuriel Núñez Fernández+1 more|arXiv (Cornell University)|Aug 7, 2023

Machine Learning in Materials ScienceMaterials Science3 citations

TL;DR

This paper introduces tensorized orbitals using tensor network techniques—specifically quantics representation and the TCI algorithm—to represent atomic and molecular orbitals with unprecedented accuracy and compactness. By expressing orbitals as tensor trains, the method enables exact computation of electron-electron interaction matrix elements, overcoming the computational bottleneck of traditional Gaussian basis sets, and achieves chemical accuracy with a single orbital where Gaussians require hundreds.

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.

Motivation & Objective

• To overcome the computational bottleneck in quantum chemistry caused by expensive 6D electron-electron integrals in traditional basis sets.
• To develop a new representation of orbitals that enables high-accuracy, compact, and flexible basis sets beyond Gaussian-type orbitals.
• To leverage tensor network techniques—particularly tensor trains and the TCI algorithm—to represent orbitals in a way that simplifies and accelerates matrix element computation.
• To demonstrate that tensorized orbitals can achieve chemical accuracy with fewer orbitals than Gaussian basis sets, especially for challenging cases like chemical bonds.
• To enable iterative basis set improvement via natural orbitals without increasing computational cost, unlike in Gaussian-based methods.

Proposed method

• Discretize spatial coordinates using an exponentially dense grid with 2^n points per dimension, labeling each grid point via n bits for x, y, and z coordinates.
• Represent the orbital as a 3n-index tensor Φ_x1y1z1...xnynzn ≡ φ(x,y,z) using the quantics format, enabling hierarchical, multi-scale representation.
• Express the orbital as a tensor train (TT) decomposition to compress the representation and enable efficient computation of integrals.
• Use the TCI (Tensor Contraction Integrator) algorithm to map the orbital representation into a form compatible with MPO/MPS solvers, enabling direct energy minimization.
• Optimize the tensorized orbital directly via DMRG (Density Matrix Renormalization Group) to minimize the energy of the one-electron Hamiltonian, avoiding explicit functional forms.
• Combine tensorized orbitals with standard quantum chemistry workflows, including iterative natural orbital construction, without increasing computational cost.

Figure 1: Error versus bond dimension $\chi$ for the energy of the $1s$ 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 correspond respectively to the error on the kinetic energy $K$ , nucl

Experimental results

Research questions

• RQ1 Can tensor network techniques be used to represent atomic and molecular orbitals in a way that enables exact and efficient computation of electron-electron interaction matrix elements?
• RQ2 Can tensorized orbitals achieve chemical accuracy with fewer orbitals than traditional Gaussian basis sets, especially for systems with strong electron correlation or bonding nodes?
• RQ3 To what extent can tensorized orbitals represent exact hydrogenoid orbitals, plane waves, and chemical bonds with high fidelity?
• RQ4 How does the computational cost of iterative basis set improvement (e.g., via natural orbitals) scale when using tensorized orbitals versus Gaussian orbitals?
• RQ5 Can the tensorized orbital framework be integrated into existing quantum chemistry software with minimal modification?

Key findings

• Tensorized orbitals achieve chemical accuracy (1.6 mHa) with a single orbital for the H₂⁺ ion, reaching 10⁻⁴ mHa error, while the best Gaussian basis set requires ~100 orbitals to reach 0.01 mHa.
• The method enables exact computation of 6D electron-electron integrals through tensor network structure, eliminating the need for analytical integration formulas.
• The tensorized representation allows for direct optimization of orbitals via DMRG, achieving high accuracy without increasing the number of orbitals or computational cost.
• The electronic density between two protons in H₂⁺ is better captured by the tensorized orbital than by Gaussian orbitals, as shown in the iso-density plot inset.
• Tensorized orbitals can represent a wide class of orbitals—including Gaussians, hydrogenoid orbitals, plane waves, and hybrid functions—within a unified framework.
• Iterative basis set improvement using natural orbitals can be performed indefinitely with tensorized orbitals at no additional computational cost, unlike Gaussian-based methods.

Figure 2: Benchmark of the calculation of the matrix elements for the LiH molecule in the STO-6G basis set. Upper panels: colormap of the different mathematical objects in Hartrees. The worst error for a matrix element is $0.7mHa$ for this calculation ( $n=16$ and $\chi=100$ ). Lower panels: error v