Derk Pieter Kooi

Over the past decade, the Python-based Simulations of Chemistry Framework (PySCF) has developed into a widely used open-source platform for electronic-structure theory and quantum chemical method development. This article reviews the major advances since the previous overview in 2020, covering new modules and methodology, infrastructure changes, and performance benchmarks.

Variational ab initio methods in quantum chemistry stand out among other methods in providing direct access to the wave function. This allows, in principle, straightforward extraction of any other observable of interest, besides the energy, but, in practice, this extraction is often technically difficult and computationally impractical. Here, we consider the electron density as a central observable in quantum chemistry and introduce a novel method to obtain accurate densities from real-space many-electron wave functions by representing the density with a neural network that captures known asymptotic properties and is trained from the wave function by score matching and noise-contrastive estimation. We use variational quantum Monte Carlo with deep-learning Ansätze to obtain highly accurate wave functions free of basis set errors and from them, using our novel method, correspondingly accurate electron densities, which we demonstrate by calculating dipole moments, nuclear forces, contact densities, and other density-based properties.

We numerically study the strong-interaction limit of the exchange–correlation functional for neutral atoms and Bohr atoms as the number of electrons increases. Using a compact representation, we analyze the second-order gradient expansion, comparing it with the one for exchange (weak interaction limit). The two gradient expansions, at strong and weak interaction, turn out to be very similar in magnitude but with opposite signs. We find that the point-charge plus continuum model is surprisingly accurate for the gradient expansion coefficient at strong coupling, while generalized gradient approximations, such as Perdew–Burke–Ernzerhof (PBE) and PBEsol, severely underestimate it. We then use our results to analyze the Lieb–Oxford bound from the point of view of slowly varying densities, clarifying some aspects on the bound at a fixed number of electrons.

Noncovalent interactions (NCIs) play a crucial role in biology, chemistry, material science, and everything in between. To improve pure quantum-chemical simulations of NCIs, we propose a methodology for constructing approximate correlation energies by combining an interpolation along the Møller-Plesset adiabatic connection (MP AC) with a regularization and spin-scaling strategy applied to MP2 correlation energies. This combination yields c_osκ_os-SPL2, which exhibits superior accuracy for NCIs compared to any of the individual strategies. With the N⁴ formal scaling, c_osκ_os-SPL2 is competitive or often outperforms more expensive dispersion-corrected double hybrids for NCIs. The accuracy of c_osκ_os-SPL2 particularly shines for anionic halogen bonded complexes, where it surpasses standard dispersion-corrected DFT by a factor of 3 to 5.

Real materials always contain, to some extent, randomness in the form of defects or irregularities. It is known since the seminal work of Anderson that randomness can drive a metallic phase to an insulating one, and the mechanism responsible for this transition is intrinsically different from the one of the interaction-induced transitions discovered by Mott. Lattice Hamiltonians, with their conceptual and computational advantages, permitted to investigate broadly the interplay of both mechanisms. However, a clear understanding of the differences (or not) with their real-space counterparts is lacking, especially in the presence of long-range Coulomb interactions. This work aims at shedding light on this challenging question by investigating a real-space one-dimensional model of interacting electrons in the presence of a disordered potential. The transition between delocalized and localized phases is characterized using two different indicators, namely, the single-particle occupation entropy and the position-space information entropy. In addition, the performance of density functional approximations to reproduce the exact ground-state densities of this many-body localization model are gauged.

The adiabatic connection that has, as weak-interaction expansion, the Møller–Plesset perturbation series has been recently shown to have a large coupling-strength expansion, in terms of functionals of the Hartree–Fock density with a clear physical meaning. In this work, we accurately evaluate these density functionals and we extract second-order gradient coefficients from the data for neutral atoms, following ideas similar to the ones used in the literature for exchange, with some modifications. These new gradient expansions will be the key ingredient for performing interpolations that have already been shown to reduce dramatically MP2 errors for large noncovalent complexes. As a byproduct, our investigation of neutral atoms with large number of electrons N indicates that the second-order gradient expansion for exchange grows as N log(N) rather than as N, as often reported in the literature.

The “fixed diagonal matrices” (FDM) dispersion formalism [Kooi, D. P.; et al. J. Phys. Chem. Lett. 2019, 10, 1537] is based on a supramolecular wave function constrained to leave the diagonal of the many-body density matrix of each monomer unchanged, reducing dispersion to a balance between kinetic energy and monomer–monomer interaction. The corresponding variational optimization leads to expressions for the dispersion energy in terms of the ground-state pair densities of the isolated monomers only, providing a framework to build new approximations without the need for polarizabilities or virtual orbitals. Despite the underlying microscopic real space mechanism being incorrect, as in the exact case there is density relaxation, the formalism has been shown to give extremely accurate (or even exact) dispersion coefficients for H and He. The question we answer in this work is how accurate the FDM expressions can be for isotropic and anisotropic C₆ dispersion coefficients when monomer pair densities are used from different levels of theory, namely Hartree–Fock, MP2, and CCSD. For closed-shell systems, FDM with CCSD monomer pair densities yield a mean average percent error for isotropic C₆ dispersion coefficients of about 7% and a maximum absolute error within 18%, with a similar accuracy for anisotropies. The performance for open-shell systems is less satisfactory, with CCSD pair densities performing sometimes worse than Hartree–Fock or MP2. In the present implementation, the computational cost on top of the monomer’s ground-state calculations is O(N⁴). The results show little sensitivity to the basis set used in the monomer’s calculations.

We study in detail the first three leading terms of the large coupling-strength limit of the adiabatic connection that has as weak-interaction expansion the Møller–Plesset perturbation theory. We first focus on the H atom, both in the spin-polarized and the spin-unpolarized cases, reporting numerical and analytical results. In particular, we derive an asymptotic equation that turns out to have simple analytical solutions for certain channels. The asymptotic H atom solution for the spin-unpolarized case is then shown to be variationally optimal for the many-electron spin-restricted closed-shell case, providing expressions for the large coupling-strength density functionals up to the third leading order. We also analyze the H₂ molecule and the uniform electron gas.