DoNOF 2.0: A modern Open-Source Electronic Structure Program for Natural Orbital Functionals

In this work, we present the second version of the Donostia Natural Orbital Functional Software, an open-source program for natural orbital functional calculations. The new release incorporates improved optimization algorithms, capabilities for excited-state computations, support for ab initio molecular dynamics, and integration with the libcint library. DoNOF 2.0 also extends its property toolbox by enabling the evaluation of nonlinear optical responses, including static polarizabilities and higher-order hyperpolarizabilities via a finite-field Romberg-Richardson scheme. Program Summary [Title: DoNOF; Developer’s repository link: http://github.com/DoNOF/; Program’s Manual link: https://donof.readthedocs.io/; Licensing provisions: GPLv3; Programming language: Fortran; additional implementations available in Python (PyNOF) and Julia (DoNOF.jl); Multinode capability: Support for distributed execution through a hybrid OpenMPI and OpenMP implementation]

💡 Research Summary

This paper presents DoNOF 2.0, the second major version of the Donostia Natural Orbital Functional software, an open-source program dedicated to electronic structure calculations using Natural Orbital Functional Theory (NOFT). The development addresses the practical challenges of implementing one-particle reduced density matrix (1RDM) functional theory, which offers a promising route to capturing static correlation effects at a computational cost comparable to Hartree-Fock or Density Functional Theory.

The core theoretical challenge in NOFT is the N-representability problem of the two-particle reduced density matrix (2RDM). Since the exact functional form is unknown, DoNOF employs a bottom-up strategy, progressively imposing necessary N-representability conditions to construct physically meaningful approximate functionals, leading to the families of Piris NOFs (PNOFs) and Global NOFs (GNOFs). The paper details the electron-pairing models used in these functionals, where the orbital space is divided into subspaces for unpaired electrons (ΩI) and paired electrons (ΩII). Within each pairing subspace ΩII, a sum rule constrains the occupation numbers of one strongly doubly occupied orbital and several weakly doubly occupied orbitals to sum to one.

A significant focus of DoNOF 2.0 is the modernization of its optimization algorithms. For occupation number (ON) optimization, the software introduces two transformation techniques—trigonometric mapping and softmax mapping—that automatically satisfy the pairing constraints by reparameterizing the ONs in terms of unconstrained auxiliary variables. For the more computationally demanding natural orbital (NO) optimization, the program shifts from iterative diagonalization to an orbital-rotation approach. It innovatively adopts the ADAM (Adaptive Moment Estimation) optimization algorithm, commonly used in deep learning, which relies solely on gradient information. This eliminates the need for expensive Hessian evaluations, greatly enhancing computational efficiency and reducing memory overhead.

Implementation-wise, DoNOF is written in Fortran and supports hybrid OpenMP/MPI parallel execution for both single-node and multi-node calculations. It leverages established libraries: LAPACK for linear algebra and the libcint library for evaluating one- and two-electron integrals, enabling support for various Gaussian basis sets. User input is managed through free-format namelists (INPRUN and NOFINP), offering control over calculation types (single-point energy, gradient, Hessian, geometry optimization, dynamics) and optimization parameters.

The new version introduces several major enhancements. It now includes capabilities for excited-state computations and direct ab initio molecular dynamics simulations. A notable expansion is its property toolbox, which now allows for the evaluation of nonlinear optical properties (NLOPs). This is achieved by implementing a finite-field Romberg-Richardson extrapolation scheme to compute static polarizabilities and higher-order hyperpolarizabilities, providing a valuable tool for studying molecular optical responses.

In summary, DoNOF 2.0 represents a substantial advancement in the practical application of NOFT. By integrating modern optimization algorithms (like ADAM), expanding its scope to dynamical and response properties, and improving usability through library integration and documentation, it evolves from a specialized research code into a robust, open-source platform. It positions itself as a competitive alternative for studying complex chemical systems where multireference character is important, such as transition metal complexes, radicals, and extended systems with strong electron correlation.