DoNOF: An open-source implementation of natural-orbital-functional-based methods for quantum chemistry

The natural orbital functional theory (NOFT) has emerged as an alternative formalism to both density functional (DF) and wavefunction methods. In NOFT, the electronic structure is described in terms of the natural orbitals (NOs) and their occupation numbers (ONs). The approximate NOFs have proven to be more accurate than those of the density for systems with a significant multiconfigurational character, on one side, and scale better with the number of basis functions than correlated wavefunction methods, on the other side. A challenging task in NOFT is to efficiently perform orbital optimization. In this article we present DoNOF, our open source implementation based on diagonalizations that allows to obtain the resulting orbitals automatically orthogonal. The one-particle reduced-density matrix (1RDM) of the ensemble of pure-spin states provides the proper description of spin multiplets. The capabilities of the code are tested on the water molecule, namely, geometry optimization, natural and canonical representations of molecular orbitals, ionization potential, and electric moments. In DoNOF, the electron-pair-based NOFs developed in our group PNOF5, PNOF7 and PNOF7s are implemented. These JKL-only NOFs take into account most non-dynamic effects plus intrapair-dynamic electron correlation, but lack a significant part of interpair-dynamic correlation. Correlation corrections are estimated by the single-reference NOF-MP2 method that simultaneously calculates static and dynamic electron correlations taking as a reference the Slater determinant formed with the NOs of a previous PNOF calculation. The NOF-MP2 method is used to analyze the potential energy surface (PES) and the binding energy for the symmetric dissociation of the water molecule, and compare it with accurate wavefunction-based methods.

自然轨道泛函理论（Natural Orbital Functional Theory, NOFT）已成为密度泛函（density functional, DF）与波函数方法之外的另一套可选形式化理论框架。在自然轨道泛函理论中，电子结构通过自然轨道（natural orbitals, NOs）及其占据数（occupation numbers, ONs）进行描述。一方面，对于具有显著多组态特征的体系，近似自然轨道泛函（NOFs）已被证明比基于密度的泛函更为精准；另一方面，相较于相关波函数方法，其计算复杂度随基函数数目增长的标度性更优。自然轨道泛函理论中的一项挑战性任务是实现高效的轨道优化。本文中，我们提出了开源程序DoNOF：该程序基于对角化方法，可自动生成正交的目标轨道。纯自旋态系综的单粒子约化密度矩阵（one-particle reduced-density matrix, 1RDM）可对自旋多重态进行准确描述。我们以水分子为测试体系，验证了该程序的多项功能，包括几何优化、分子轨道的自然轨道与正则轨道表示、电离势以及电矩的计算。DoNOF程序集成了我们课题组开发的三类基于电子对的自然轨道泛函：PNOF5、PNOF7与PNOF7s。这类仅包含JKL项的自然轨道泛函可涵盖绝大多数非动态关联效应与对内动态电子关联，但仍缺失了相当一部分间对动态电子关联。关联校正可通过单参考态NOF-MP2方法完成：该方法以先前PNOF计算得到的自然轨道构建的斯莱特行列式（Slater determinant）为参考，同时计算静态与动态电子关联。我们采用NOF-MP2方法分析了水分子对称解离过程的势能面（potential energy surface, PES）与结合能，并将结果与高精度基于波函数的方法进行了对比。