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Recent developments in the PySCF program package

Qiming Sun, Xing Zhang, Samragni Banerjee, Peng Bao, Marc Barbry, Nick S. Blunt, Nikolay A. Bogdanov, George H. Booth, Jia Chen, Zhi-Hao Cui, Janus Juul Eriksen, Yang Gao, Sheng Guo, Jan Hermann, Matthew R. Hermes, Kevin Koh, Peter Koval, Susi Lehtola, Zhendong Li, Junzi Liu, Narbe Mardirossian, James D. McClain, Mario Motta, Bastien Mussard, Hung Q. Pham, Artem Pulkin, Wirawan Purwanto, Paul J. Robinson, Enrico Ronca, Elvira Sayfutyarova, Maximilian Scheurer, Henry F. Schurkus, James E. T. Smith, Chong Sun, Shi-Ning Sun, Shiv Upadhyay, Lucas K. Wagner, Xiao Wang, Alec White, James Daniel Whitfield, Mark J. Williamson, Sebastian Wouters, Jun Yang, Jason M. Yu, Tianyu Zhu, Timothy C. Berkelbach, Sandeep Sharma, Alexander Sokolov, Garnet Kin-Lic Chan

arXiv:2002.12531v2physics.chem-phphysics.comp-ph

TL;DR

Electronic-structure software often separates molecular and materials simulations and offers limited support for user customization. This paper presents PYSCF as an extensible development platform with a common framework for both domains, demonstrating broad computational capabilities and real-world scalability.

  • Problem

    Electronic-structure packages often specialize in either molecular or materials simulations, while quantum workflows require programming and customization.

  • Method

    The paper describes PYSCF as an extensible development platform that supports user-implemented methods and common computational workflows for molecules and materials.

  • Results

    PYSCF provides production-ready molecular and solid-state electronic-structure capabilities, including single-k-point crystalline calculations and systems exceeding 10,000 atomic-orbital basis functions.

  • Takeaways & Limitations

    Users can extend PYSCF with their own methods and apply a shared numerical framework across molecular, materials, machine-learning, and quantum-information workflows.

Abstract

from arXiv · show

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.

I. INTRODUCTION · II. THE DESIGN PHILOSOPHY BEHIND PYSCF

PYSCF is an open-source, general-purpose platform designed both for quantum simulations and for developing customized methods. Its philosophy emphasizes modular, accessible programming across molecular and materials applications while preserving competitive performance through optimized compiled components.

  • I. INTRODUCTION: PYSCF evolved from an ab initio quantum embedding tool into a general-purpose, free and open-source development platform.The project began in 2014 and was described at version 1.7.1.
  • I. INTRODUCTION: Its ecosystem includes hundreds of users, over 60 code contributors, and applications spanning chemistry, materials science, data science, machine learning, and quantum computing.PYSCF is distributed through GitHub, pip, conda, and other platforms, with academic and industrial use.
  • I. INTRODUCTION: PYSCF prioritizes functioning as a development platform that lets users modify code, implement methods independently, and reuse modular components.This principle is intended to empower users without assistance from the original developers.
  • I. INTRODUCTION: PYSCF supports molecular and materials calculations equally within the same numerical framework and theoretical approximations.The design is explicitly intended to cover both molecular chemistry and materials science applications.
  • I. INTRODUCTION: PYSCF also enables researchers outside the chemical sciences, including machine-learning and quantum-information researchers, to perform quantum chemistry simulations.The stated guiding principle extends access to quantum chemistry beyond traditional chemical-science users.
  • II. THE DESIGN PHILOSOPHY BEHIND PYSCF: PYSCF is organized as a loosely coupled library of modular components with reusable functions, extensibility hooks, and performance separated into lower-level components.These choices address programming and customization needs ranging from simple workflow automation to embedding customized algorithms.
  • II. THE DESIGN PHILOSOPHY BEHIND PYSCF: Its APIs expose individual algorithmic suboperations, can be called in arbitrary order without side effects, and allow users to focus on the physical theory rather than program state.Examples include reusable operations within Hartree-Fock and density-functional-theory workflows, such as building a Fock matrix from a density matrix.
  • II. THE DESIGN PHILOSOPHY BEHIND PYSCF: Over 90% of PYSCF is written in Python, while optimized NUMPY, SCIPY, custom C, and OpenMP components provide performance and thread-level parallelism.Simulations can spend over 99% of runtime in C libraries, making Python overhead negligible and enabling leading performance in many simulations.

III. A COMMON FRAMEWORK FOR MOLECULES AND CRYSTALLINE MATERIALS

PYSCF unifies molecular and crystalline-material simulations through shared numerical approximations, theoretical models, workflows, and density-fitting intermediates. Periodic calculations additionally require Bloch bases, momentum conservation, complex arithmetic, divergence treatments, periodic-function techniques, and finite-size corrections.

  • Framework design: PYSCF pursues common numerical approximations and theoretical models for molecules and materials, extending its original Gaussian AO molecular code to crystalline Gaussian bases.This design makes molecular and crystalline simulations part of one program framework.
  • Shared representations: The shared framework uses Bloch bases, translational-symmetry momentum conservation, and complex numbers for crystalline calculations, paralleling molecular symmetry and complex-orbital treatments.Bloch functions generally produce complex matrix elements, while momentum conservation exploits translational symmetry.
  • Periodic-specific techniques: Periodic simulations add Coulomb-divergence treatments using density fitting and Ewald summation, FFT and pseudopotential techniques, thermodynamic-limit corrections, and experimental lattice-symmetry support.Finite-size corrections modify exchange at mean-field level, while additional crystal lattice symmetries remain experimentally supported.
  • Density fitting: Three-index density-fitted integrals are PYSCF’s central computational intermediate because they support fast Gaussian implementations, plane-wave FFT equivalence, divergence removal, and compact crystalline Coulomb intermediates.The same density-fitted representation connects molecular and crystalline implementations.
  • Unified implementation: PYSCF exposes a unified density-fitting API and similar molecular-periodic workflows, while supporting Gaussian, plane-wave, crystalline-Gaussian, and mixed auxiliary bases and common driver functions.The common drivers perform generic operations such as solving Hartree–Fock or coupled-cluster amplitude equations.

IV. DEVELOPING WITH PYSCF: CASE STUDIES · A. Case study: modifying the Hamiltonian

PYSCF supports extending functionality by assembling existing APIs without modifying source code. Its simulation models are independent of specific Hamiltonians, allowing users to customize one- and two-electron interactions for subsequent calculations.

  • IV. DEVELOPING WITH PYSCF: CASE STUDIES: Case studies show how users can modify and extend PYSCF functionality without changing the source code.The examples focus on workflows an average user may encounter.
  • IV. DEVELOPING WITH PYSCF: CASE STUDIES: The program workflow distinguishes k-point-independent, k-point-dependent, and generic driver functions for molecular and periodic calculations.Figure 1 uses orange, purple, and blue boxes for these categories, respectively.
  • IV. DEVELOPING WITH PYSCF: CASE STUDIES: Users can assemble different existing PYSCF APIs to implement new functionality.This approach targets users who do not want to modify the source code.
  • A. Case study: modifying the Hamiltonian: PYSCF simulation models are implemented independently of any specific Hamiltonian and support up to two-body interactions.This design separates wavefunction approximations from the Hamiltonian under study.
  • A. Case study: modifying the Hamiltonian: Users can easily customize the Hamiltonian for model problems or interfaces to different numerical basis approximations.The customization follows from the independence of simulation models and Hamiltonians.
  • A. Case study: modifying the Hamiltonian: Figure 2 presents several ways to define one-electron and two-electron interactions before ground- and excited-state calculations with the custom Hamiltonian.The figure links interaction definitions to subsequent calculations.

B. Case study: optimizing orbitals of arbitrary methods · C. Case study: implementing an embedding model

PYSCF supports orbital optimization for arbitrary electronic-structure methods through density-matrix and energy interfaces, including external solvers. Its standard APIs also enable rapid prototyping of customized embedding models with geometry optimization and approximate nuclear gradients.

  • B. Case study: optimizing orbitals of arbitrary methods: PYSCF’s MCSCF module performs quasi-second-order orbital optimization within active subspaces or the complete orbital space.The algorithm is general purpose and operates within orbital subspaces or over the complete orbital space.
  • B. Case study: optimizing orbitals of arbitrary methods: The orbital optimization framework applies to methods that provide energies and one- and two-particle density matrices, beyond built-in CASCI, CASSCF, and multireference solvers.This generality supports active-space orbital optimization for DMRG, Dice, Arrow, and NECI.
  • B. Case study: optimizing orbitals of arbitrary methods: Users can apply orbital optimization to methods without a native orbital-optimization API by writing a short wrapper around density-matrix and kernel functions.The example uses orbital-optimized MP2 and requires wrappers providing make_rdm12 and kernel.
  • C. Case study: implementing an embedding model: A simple script using standard APIs enables geometry optimization for a Hartree-Fock wavefunction-in-Hartree-Fock embedding model with a CISD solver.The system Hamiltonian is expressed through fragment and environment Hamiltonians before defining the embedding Hamiltonian.
  • C. Case study: implementing an embedding model: Geometry optimization in the embedding model uses approximate nuclear gradients, with fragment Hartree-Fock and configuration-interaction wavefunctions obtained from the embedding Hamiltonian.The fragment wavefunctions are denoted Ψfrag,HF and Ψfrag,CI, and the embedding Hamiltonian is Hemb.
  • C. Case study: implementing an embedding model: The embedding demonstration combines APIs for energies, analytical nuclear gradients, Hartree-Fock potentials, Hamiltonian customization, and nuclear-gradient solver customization.These components illustrate rapid prototyping through PYSCF APIs.

V. SUMMARY OF EXISTING METHODS AND RECENT ADDITIONS · A. Hartree-Fock and density functional theory methods

This section summarizes PYSCF’s current capabilities, emphasizing its Hartree-Fock and density functional theory implementations for molecular and crystalline calculations. These methods support diverse references, exchange-correlation functionals, basis treatments, pseudopotentials, and acceleration strategies.

  • V. SUMMARY OF EXISTING METHODS AND RECENT ADDITIONS: PYSCF summarizes its major current capabilities in Table I and the following subsections.
  • A. Hartree-Fock and density functional theory methods: PYSCF implements Hartree-Fock and density functional theory for molecular and crystalline calculations using restricted, unrestricted, and generalized Slater determinant references.Supported references include restricted closed-shell, restricted open-shell, unrestricted, and generalized noncollinear-spin forms.
  • A. Hartree-Fock and density functional theory methods: PYSCF supports predefined LDA, GGA, hybrid, meta-GGA, nonlocal correlation, and range-separated hybrid exchange-correlation functionals through LIBXC and XCFUN.Users can also create customized exchange-correlation functionals in DFT calculations.
  • A. Hartree-Fock and density functional theory methods: The molecular SCF module can use more than 10,000 basis functions on an SMP machine without integral approximations, using the LIBCINT Gaussian integral engine.Gaussian density fitting and the pseudo-spectral SGX approach provide additional speed-up.
  • A. Hartree-Fock and density functional theory methods: Crystalline HF and DFT calculations support either a single Brillouin-zone point or a k-point mesh, with cost dependent on the Gaussian basis and density fitting.PYSCF also supports GTH pseudopotentials with associated CP2K-developed basis sets.
  • A. Hartree-Fock and density functional theory methods: Gaussian density fitting is the algorithm of choice for calculations with HF exchange in crystalline systems.Figure 6 compares silicon band structures from a GTH-LDA pseudopotential calculation using FFTDF with an all-electron calculation using GDF.

B. Many-body methods · 1. Molecular implementations

PYSCF provides a broad suite of many-body methods for molecules and crystalline materials. Its molecular implementations span CI, coupled cluster, ADC, multireference methods, and scalable workflows for large basis sets and active spaces.

  • B. Many-body methods: PYSCF offers MP2, MRPT, CI, CC, MCSCF, ADC, and G0W0 methods, with most capabilities available for molecules and crystalline materials.
  • 1. Molecular implementations: The CI module includes CISD and a multithreaded FCI solver supporting fermion, boson, and coupled fermion-boson Hamiltonians.The FCI solver efficiently handles active spaces with up to 18 electrons in 18 orbitals.
  • 1. Molecular implementations: The CC module supports CCSD, CCSD(T), Λ-equation density matrices, analytic nuclear gradients, and equation-of-motion variants for EA, IP, and neutral excitation energies.Experimental beyond-doubles corrections are also available for IP and EA through IP-EOM-CCSD* and EA-EOM-CCSD*.
  • 1. Molecular implementations: An AO-driven pathway enables calculations with more than 1500 basis functions, including a CCSD(T) calculation with 50 electrons and 1500 basis functions.
  • 1. Molecular implementations: EA/IP-ADC(n) methods with n = 2, 3 provide lower-cost molecular electron-affinity and ionization-potential calculations than EA/IP-EOM-CCSD.Their one-iteration amplitude equations and Hermitian eigenvalue problem reduce the cost of computing energies and transition intensities.
  • 1. Molecular implementations: The MCSCF module provides CASCI and CASSCF methods, a general second-order orbital optimizer, and orbital optimization for DMRG, FCIQMC, and selected configuration interaction wavefunctions.
  • 1. Molecular implementations: SC-NEVPT2 extends CASCI or CASSCF wavefunctions with dynamic correlation and, with external active-space solvers, enables relatively large active spaces.

2. Crystalline implementations · C. Efficiency

PYSCF’s crystalline implementations closely parallel its molecular modules, enabling many-body calculations at Γ and, for supported modules, at arbitrary single k-points. Its efficiency spans large molecular mean-field systems and many-body calculations, with achievable problem sizes depending on available computational resources.

  • 2. Crystalline implementations: PYSCF’s molecular modules can perform crystalline calculations at the Γ-point, while complex-integral modules also support arbitrary single k-points.Single k-point calculations require supplying appropriate periodic integrals to the many-body solver.
  • 2. Crystalline implementations: The crystalline many-body implementations closely parallel their molecular counterparts.This parallel design allows existing molecular modules to be reused for solid-state calculations.
  • C. Efficiency: Estimated PYSCF problem sizes depend on computational resources, with Table I assuming a cluster node or a few dozen cores.The estimates are rough and illustrate representative rather than absolute limits.
  • C. Efficiency: More than 10,000 AO basis functions can be treated without difficulty in molecular mean-field calculations.This capacity is demonstrated for large water clusters and reflects PYSCF’s molecular mean-field efficiency.
  • C. Efficiency: Around 7 hours are required to build the HF Fock matrix for a water cluster with more than 12,000 basis functions on one node with 32 CPU cores.The calculation uses an integral screening threshold of 10−13 a.u.; MPI parallelization further reduces the Fock-build time.
  • C. Efficiency: Systems with more than 1,500 basis functions can be easily treated at the CCSD level.Tables III and IV provide timing data for exemplary CCSD and FCI calculations.
  • C. Efficiency: The PYSCF FCI implementation is reported to be very efficient, alongside estimated performance data for other implemented many-body methods.The estimates for other methods are summarized in Table I.

D. Properties · E. Orbital localization

PYSCF supports a broad range of molecular and crystalline property calculations, including relativistic treatments, correlated densities, analytic gradients, and gauge-origin-invariant magnetic properties. Its orbital-localization tools provide AO-based orbitals for population analysis, optimized localized orbitals and bond-related representations, and crystalline MLWFs through WANNIER90.

  • D. Properties: PYSCF computes diverse nonrelativistic and four-component relativistic molecular properties, including magnetic, spin, polarizability, and hyperfine-related tensors.Supported properties include NMR shielding, spin-spin coupling, electronic g-tensors, hyperfine coupling, ESR, magnetizability, zero-field splitting, and static and dynamic polarizability.
  • D. Properties: Gauge including atomic orbitals provide approximate gauge-origin invariance for NMR shielding, g-tensors, and magnetizability tensors.This treatment is specifically described for magnetic property calculations.
  • D. Properties: Electric field gradients and Mössbauer parameters can be computed from mean-field or correlated densities using nonrelativistic, X2C, or four-component methods in molecules and crystals.The correlated-density options include spin-free exact-two-component relativistic Hamiltonians.
  • D. Properties: Analytic nuclear gradients are available for mean-field calculations and several correlated methods, with CASCI gradients supporting external solvers such as DMRG.Implemented methods include MP2, CCSD, CISD, CASCI, and CASSCF; TDDFT gradients are also implemented with or without TDA.
  • E. Orbital localization: AO-based localization generates intrinsic atomic orbitals, natural atomic orbitals, and meta-Löwdin orbitals that enable reliable population analysis in arbitrary basis sets.These orbitals are localized according to the atomic character of the basis functions.
  • E. Orbital localization: Cost-function localization includes Boys, Edmiston-Ruedenberg, and Pipek–Mezey procedures, while IAO-based Pipek-Mezey localization constructs intrinsic bond orbitals.Localized intrinsic valence virtual orbitals can also be used to assign core-excited states, and optimization uses the CIAH algorithm for rapid convergence.
  • E. Orbital localization: For crystalline k-point calculations, PYSCF provides maximally-localised Wannier functions through a native WANNIER90 interface.Initial guesses can come from WANNIER90 atomic orbitals, meta-Löwdin orbitals, or localized SCDM orbitals; Figure 12 illustrates IBOs and MLWFs of diamond.

F. QM/MM and solvent · G. Relativistic treatments · H. MPI implementations

PYSCF supports continuum solvation, limited point-charge QM/MM, multiple relativistic treatments, and MPI-based distributed parallelism. Its MPI4PYSCF extension preserves serial APIs while using a server-client architecture for parallel quantum-chemistry and crystalline-material computations.

  • F. QM/MM and solvent: PYSCF implements COSMO and IEF-PCM continuum solvation models efficiently through domain decomposition.Both models are compatible with most electronic-structure methods in PYSCF.
  • F. QM/MM and solvent: Non-equilibrium solvation is available within TDDFT to describe fast solvent response after abrupt changes in solute charge density.Equilibrium solvation instead uses the static electric susceptibility.
  • F. QM/MM and solvent: Limited QM/MM functionality adds classical point charges to the QM region and supports all molecular electronic-structure methods by decorating SCF methods.MM charges can also be combined with X2C and implicit-solvent treatments.
  • G. Relativistic treatments: PYSCF provides scalar relativistic treatments including spin-free X2C, scalar ECP, and relativistic pseudo-potentials for energies, nuclear gradients, and nuclear Hessians.These options are available for all methods.
  • H. MPI implementations: MPI4PYSCF extends the main library with distributed parallelism for common quantum-chemistry and crystalline-material methods while preserving nearly identical serial APIs.For solids, MPI support is currently provided only through parallelization over k-points.
  • H. MPI implementations: MPI efficiency is demonstrated for Fock builds on a system with 12,288 AOs using up to 64 MPI processes and 32 OpenMP threads per process.The evaluation reports wall time and speedup.
  • H. MPI implementations: MPI4PYSCF uses a server-client mechanism in which a master starts Python daemons that receive functions and data on remote nodes.The design retains the simplicity of the PYSCF package structure.
  • H. MPI implementations: For Fock builds, clients receive distributed input variables, evaluate balanced subsets of four-center two-electron integrals, and construct partial Fock matrices.The master process gathers the partial matrices from clients.

VI. THE PYSCF SIMULATION ECOSYSTEM · A. External projects with native interfaces

PYSCF supports a growing ecosystem of external projects through maintained native interfaces and tightly coupled integrations. These interfaces extend geometry optimization, multireference methods, tensor contractions, density-functional calculations, and solvent modeling.

  • VI. THE PYSCF SIMULATION ECOSYSTEM: PYSCF is widely used as a development tool, with external projects interfacing to it or coupling tightly to access greater functionality.The ecosystem is organized into projects with native PYSCF interfaces and projects that build on PYSCF.
  • A. External projects with native interfaces: PYSCF maintains native interfaces to several external projects, extending its capabilities through coordinated software integrations.The paper introduces these interfaces as a distinct part of the PYSCF simulation ecosystem.
  • A. External projects with native interfaces: GEOMETRIC and PYBERNY integrate with PYSCF’s GEOMOPT module for geometry optimization when methods provide energies and nuclear gradients.The module generates an object that can be used by the external optimization libraries.
  • A. External projects with native interfaces: BLOCK, CHEMPS2, DICE, ARROW, and NECI can replace the FCI solver in CASCI and CASSCF calculations for large active-space multireference problems.Their interfaces follow PYSCF’s FCI module conventions.
  • A. External projects with native interfaces: LIBXC and XCFUN are tightly integrated into PYSCF, evaluating individual exchange-correlation functionals and their derivatives for customizable DFT calculations.PYSCF’s DFT module allows users to customize exchange-correlation functionals by linearly combining different functionals.
  • A. External projects with native interfaces: TBLIS provides a PYSCF einsum interface with the numpy.einsum API while avoiding the high memory footprint of transposed tensor intermediates.TBLIS delivers tensor-contraction speedups without the memory overhead associated with the BLAS-based transpose-GEMM-transpose scheme.
  • A. External projects with native interfaces: CPPE integrates a polarizable embedding solvent model into PYSCF ground-state mean-field and post-SCF calculations and supports TDA for excited states.The interface enables solvent modeling across ground-state and supported excited-state workflows.

B. External projects that build on PySCF

External projects use PYSCF as infrastructure across quantum Monte Carlo, quantum embedding, and general quantum chemistry. These projects use PYSCF to exchange wavefunctions and Hamiltonians, perform electronic structure calculations, and support specialized methods and simulations.

  • Quantum Monte Carlo: Quantum Monte Carlo programs including QMCPACK, PYQMC, QWALK, and HANDE read wavefunctions and/or Hamiltonians generated by PYSCF.PYSCF is integrated as a dependent module in PYQMC.
  • Quantum embedding packages: Quantum embedding packages implement density matrix embedding and dynamical mean-field theory on top of PYSCF.QSOME, PDMET, PYDMFET, POTATO, and OPEN-QEMIST use PYSCF to manipulate wavefunctions and embedding Hamiltonians and provide many-electron solvers.
  • General quantum chemistry: General quantum chemistry projects use PYSCF for localized active-space, multiconfiguration, pair-density functional, localized-orbital, and many-body expanded full CI methods.Examples include LASSCF, MC-PDFT, state-averaged CASSCF, PYWANNIER90, and PYMBE.
  • General quantum chemistry: PYSCF supports Green’s functions, linear-scaling, alchemical perturbation, and numerical-atomic-orbital projects as ab initio or quantum-mechanical infrastructure.It underlies GF2, self-consistent GW, LSQC, and APDFT, while PYSCF-NAO adds numerical atomic orbital support for large-scale ground-state and excited-state methods.
  • General quantum chemistry: Other projects combine PYSCF modules with specialized workflows for self-interaction corrections, basis-function development, and spectroscopic or excited-state simulations.PYFLOSIC uses the PYSCF DFT module, MOLSTURM uses PYSCF FCI capabilities, and VELOXCHEM and ADCC use PYSCF post-HF methods.

VII. BEYOND ELECTRONIC STRUCTURE … VIII. OUTLOOK

PYSCF extends beyond electronic-structure simulations into machine learning and quantum information science, where its flexible Python-based infrastructure supports data generation, method development, and molecular simulations. After five years, the project is described as mature, with version 2 planned to expand performance and interdisciplinary capabilities.

  • A. PySCF in the materials genome initiative and machine learning: PYSCF is used with machine learning in molecular and materials simulations, including testing machine-learning-generated exchange-correlation functionals.The DFT module’s flexibility enables these applications.
  • A. PySCF in the materials genome initiative and machine learning: PYSCF serves as a large-scale engine for quantum-chemistry data generation and supports machine-learning approaches to SCF initial guesses and HF orbitals.It also provides Hamiltonian integrals for fermionic neural networks in NETKET.
  • B. PySCF in quantum information science: PYSCF has been rapidly adopted in quantum information science and quantum computing, partly because Python is the community’s de-facto standard language.The passage attributes adoption partly to Python’s status in the quantum-computing community.
  • B. PySCF in quantum information science: PYSCF is a standard prerequisite for molecular simulations in OPENFERMION, QISKIT-AQUA, and OPENQEMIST.These packages connect PYSCF to quantum-computing workflows.
  • VIII. OUTLOOK: After five years of development, PYSCF is described as feature complete and mature while supporting both simulations and customization of new electronic-structure methods.The authors note that no single package can be optimal for all tasks.
  • VIII. OUTLOOK: Version 2 is expected to add faster methods for very large systems, stronger machine-learning and quantum-computing integration, improved HPC-library integration, and more parallel implementations.The authors also mention possible further areas of development.
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