PYSCF is a Python-based general-purpose electronic structure platform that both supports first-principles simulations of molecules and solids, as well as accelerates the development of new methodology and complex computational workflows. The present paper explains the design and philosophy behind PYSCF that enables it to meet these twin objectives. With several case studies, we show how users can easily implement their own methods using PYSCF as a development environment. We then summarize the capabilities of PYSCF for molecular and solid-state simulations. Finally, we describe the growing ecosystem of projects that use PYSCF across the domains of quantum chemistry, materials science, machine learning and quantum information science.

We use the recently developed Heat-bath Configuration Interaction (HCI) algorithm as an efficient active space solver to perform multiconfiguration self-consistent field calculations (HCISCF) with large active spaces. We give a detailed derivation of the theory and show that difficulties associated with non-variationality of the HCI procedure can be overcome by making use of the Lagrangian formulation to calculate the HCI relaxed two-body reduced density matrix. HCISCF is then used to study the electronic structure of butadiene, pentacene, and Fe-porphyrin. One of the most striking results of our work is that the converged active space orbitals obtained from HCISCF are relatively insensitive to the accuracy of the HCI calculation. This allows us to obtain nearly converged CASSCF energies with an estimated error of less than 1 mHa using the orbitals obtained from the HCISCF procedure in which the integral transformation is the dominant cost. For example, an HCISCF calculation on the Fe-porphyrin model complex with an active space of (44e, 44o) took only 412 s per iteration on a single node containing 28 cores, out of which 185 s was spent in the HCI calculation and the remaining 227 s was used mainly for integral transformation. Finally, we also show that active space orbitals can be optimized using HCISCF to substantially speed up the convergence of the HCI energy to the Full CI limit because HCI is not invariant to unitary transformations within the active space.

A method to generate electrostatic potential (ESP) derived atomic charges in crystalline solids from periodic quantum mechanical calculations, termed the REPEAT method, is presented. Conventional ESP fitting procedures developed for molecular systems, in general, will not work for crystalline systems because the electrostatic potential in periodic systems is ill-defined up to a constant offset at each spatial position. In this work the problem is circumvented by introducing a new error functional which acts on the relative differences of the potential and not on its absolute values, as it is currently done with molecular ESP charge derivation methods. We formally demonstrate that the new functional reduces to the conventional error functional used in molecular ESP approaches when the simulation box of the periodic calculation becomes infinitely large. Several tests are presented to validate the new technique. For the periodic calculation of isolated molecules, the REPEAT charges are found to be in good agreement with those determined with established molecular ESP charge derivation methods. For siliceous sodalite, it is demonstrated that conventional molecular ESP approaches generate 'unphysical' charges, whereas the REPEAT method produces charges that are both chemically intuitive and consistent between different periodic electronic structure packages. The new approach is employed to generate partial atomic charges of various microporous materials and compared to both experimentally derived and molecular fragment ESP charges. This method can be used to generate partial atomic charges to be used in simulations of microporous and nanoporous materials, such as zeolites and metal organic framework materials.

We explore the computation of high-harmonic generation spectra by means of Gaussian basis sets in approaches propagating the time-dependent Schrödinger equation. We investigate the efficiency of Gaussian functions specifically designed for the description of the continuum proposed by Kaufmann et al. [J. Phys. B 22, 2223(1989]. We assess the range of applicability of this approach by studying the hydrogen atom, i.e. the simplest atom for which "exact" calculations on a grid can be performed. We notably study the effect of increasing the basis set cardinal number, the number of diffuse basis functions, and the number of Gaussian pseudo-continuum basis functions for various laser parameters. Our results show that the latter significantly improve the description of the low-lying continuum states, and provide a satisfactory agreement with grid calculations for laser wavelengths λ0 = 800 and 1064 nm. The Kaufmann continuum functions therefore appear as a promising way of constructing Gaussian basis sets for studying molecular electron dynamics in strong laser fields using time-dependent quantum-chemistry approaches.

A large collaboration carefully benchmarks 20 first principles many-body electronic structure methods on a test set of 7 transition metal atoms, and their ions and monoxides. Good agreement is attained between 3 systematically converged methods, resulting in experiment-free reference values.These reference values are used to assess the accuracy of modern emerging and scalable approaches to the many-electron problem. The most accurate methods obtain energies indistinguishable from experimental results, with the agreement mainly limited by the experimental uncertainties. Comparison between methods enables a unique perspective on calculations of many-body systems of electrons.

In this work we demonstrate that the heat bath configuration interaction (HCI) and its semistochastic extension can be used to treat relativistic effects and electron correlation on an equal footing in large active spaces to calculate the low energy spectrum of several systems including halogen group atoms (F, Cl, Br, I), coinage atoms (Cu, Au), and the neptunyl(VI) dioxide radical. This work demonstrates that despite a significant increase in the size of the Hilbert space due to spin symmetry breaking by the spin-orbit coupling terms, HCI retains the ability to discard large parts of the low importance Hilbert space to deliver converged absolute and relative energies. For instance, by using just over 10 determinants we get converged excitation energies for Au atom in an active space containing (150o,25e) which has over 10 determinants. We also investigate the accuracy of five different two-component relativistic Hamiltonians in which different levels of approximations are made in deriving the one-electron and two-electrons Hamiltonians, ranging from Breit-Pauli (BP) to various flavors of exact two-component (X2C) theory. The relative accuracy of the different Hamiltonians are compared on systems that range in atomic number from first row atoms to actinides.

Range-separated density-functional theory is an alternative approach to Kohn-Sham densityfunctional theory. The strategy of range-separated density-functional theory consists in separating the Coulomb electron-electron interaction into long-range and short-range components, and treating the long-range part by an explicit many-body wave-function method and the short-range part by a density-functional approximation. Among the advantages of using many-body methods for the long-range part of the electron-electron interaction is that they are much less sensitive to the oneelectron atomic basis compared to the case of the standard Coulomb interaction. Here, we provide a detailed study of the basis convergence of range-separated density-functional theory. We study the convergence of the partial-wave expansion of the long-range wave function near the electron-electron coalescence. We show that the rate of convergence is exponential with respect to the maximal angular momentum L for the long-range wave function, whereas it is polynomial for the case of the Coulomb interaction. We also study the convergence of the long-range second-order Møller-Plesset correlation energy of four systems (He, Ne, N2, and H2O) with the cardinal number X of the Dunning basis sets cc-p(C)VXZ, and find that the error in the correlation energy is best fitted by an exponential in X. This leads us to propose a three-point complete-basis-set extrapolation scheme for range-separated density-functional theory based on an exponential formula.

We introduce an orbital-optimized double-hybrid (DH) scheme using the optimized-effectivepotential (OEP) method. The orbitals are optimized using a local potential corresponding to the complete exchange-correlation energy expression including the second-order Møller-Plesset (MP2) correlation contribution. We have implemented a one-parameter version of this OEP-based selfconsistent DH scheme using the BLYP density-functional approximation and compared it to the corresponding non-self-consistent DH scheme for calculations on a few closed-shell atoms and molecules. While the OEP-based self-consistency does not provide any improvement for the calculations of ground-state total energies and ionization potentials, it does improve the accuracy of electron affinities and restores the meaning of the LUMO orbital energy as being connected to a neutral excitation energy. Moreover, the OEP-based self-consistent DH scheme provides reasonably accurate exchangecorrelation potentials and correlated densities.