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Open-source APOST-3D software features a large number of wavefunction analysis tools developed over the past 20 years, aiming at connecting classical chemical concepts with the electronic structure of molecules. APOST-3D relies on the identification of the atom in the molecule (AIM), and several analysis tools are implemented in the most general way so that they can be used in combination with any chosen AIM. Several Hilbert-space and real-space (fuzzy atom) AIM definitions are implemented. In general, global quantities are decomposed into one- and two-center terms, which can also be further grouped into fragment contributions. Real-space AIM methods involve numerical integrations, which are particularly costly for energy decomposition schemes. The current version of APOST-3D features several strategies to minimize numerical error and improve task parallelization. In addition to conventional population analysis of the density and other scalar fields, APOST-3D implements different schemes for oxidation state assignment (effective oxidation state and oxidation states localized orbitals), molecular energy decomposition schemes, and local spin analysis. The APOST-3D platform offers a user-friendly interface and a comprehensive suite of state-of-the-art tools to bridge the gap between theory and experiment, representing a valuable resource for both seasoned computational chemists and researchers with a focus on experimental work. We provide an overview of the code structure and its capabilities, together with illustrative examples. [ABSTRACT FROM AUTHOR]

The method of extracting effective atomic orbitals and effective minimal basis sets from molecular wave function characterizing the state of an atom in a molecule is developed in the framework of the “fuzzy” atoms. In all cases studied, there were as many effective orbitals that have considerable occupation numbers as orbitals in the classical minimal basis. That is considered to be of high conceptual importance. [ABSTRACT FROM AUTHOR]

A conceptually new approach is introduced for the decomposition of the molecular energy calculated at the density functional theory level of theory into sum of one- and two-atomic energy components, and is realized in the “fuzzy atoms” framework. (Fuzzy atoms mean that the three-dimensional physical space is divided into atomic regions having no sharp boundaries but exhibiting a continuous transition from one to another.) The new scheme uses the new concept of “bond order density” to calculate the diatomic exchange energy components and gives them unexpectedly close to the values calculated by the exact (Hartree-Fock) exchange for the same Kohn-Sham orbitals. [ABSTRACT FROM AUTHOR]

The basis set superposition error-free second-order Møller–Plesset perturbation theory of intermolecular interactions, based on the “chemical Hamiltonian approach,” which has been introduced in Part I, is applied here to open-shell systems by using a new, effective computer realization. The results of the numerical examples considered (CH[sub 4]...HO, NO...HF) showed again the perfect performance of the method. Striking agreement has again been found with the results of the a posteriori counterpoise correction (CP) scheme in the case of large, well-balanced basis sets, which is also in agreement with a most recent formal theoretical analysis. The difficulties of the CP correction in open-shell systems are also discussed. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The total energy of a molecule is presented as a sum of one- and two-atomic energy components in terms of “fuzzy” atoms, i.e., such divisions of the three-dimensional physical space into atomic regions in which the regions assigned to the individual atoms have no sharp boundaries but exhibit a continuous transition from one to another. By proper definitions the energy components are on the chemical energy scale. The method is realized by using Becke’s integration scheme and weight function permitting very effective numerical integrations. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The energy decomposition scheme proposed in a recent paper has been realized by performing numerical integrations. The sample calculations carried out for some simple molecules show excellent agreement with the chemical picture of molecules, indicating that such an energy decomposition analysis can be useful from the point of view of connecting quantum mechanics with the genuine chemical concepts. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Geometries, vibrational frequencies, and interaction energies of the CNH...O[sub 3] and HCCH...O[sub 3] complexes are calculated in a counterpoise-corrected (CP-corrected) potential-energy surface (PES) that corrects for the basis set superposition error (BSSE). Ab initio calculations are performed at the Hartree-Fock (HF) and second-order Mo\ller-Plesset (MP2) levels, using the 6-31G(d,p) and D95++(d,p) basis sets. Interaction energies are presented including corrections for zero-point vibrational energy (ZPVE) and thermal correction to enthalpy at 298 K. The CP-corrected and conventional PES are compared; the uncorrected PES obtained using the larger basis set including diffuse functions exhibits a double well shape, whereas use of the 6-31G(d,p) basis set leads to a flat single-well profile. The CP-corrected PES has always a multiple-well shape. In particular, it is shown that the CP-corrected PES using the smaller basis set is qualitatively analogous to that obtained with the larger basis sets, so the CP method becomes useful to correctly describe large systems, where the use of small basis sets may be necessary. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]

In the so-called Interacting Quantum Atoms (IQA) approach, the molecular energy is numerically decomposed as a sum of atomic and diatomic contributions. While proper formulations have been put forward for both Hartree-Fock and post-Hartree-Fock wavefunctions, this is not the case for the Kohn-Sham density functional theory (KS-DFT). In this work, we critically analyze the performance of two fully additive approaches for the IQA decomposition of the KS-DFT energy, namely, the one from Francisco et al., which uses atomic scaling factors, and that from Salvador and Mayer based upon the bond order density (SM-IQA). Atomic and diatomic exchange-correlation (xc) energy components are obtained for a molecular test set comprising different bond types and multiplicities and along the reaction coordinate of a Diels-Alder reaction. Both methodologies behave similarly for all systems considered. In general, the SM-IQA diatomic xc components are less negative than the Hartree-Fock ones, which is in good agreement with the known effect of electron correlation upon (most) covalent bonds. In addition, a new general scheme to minimize the numerical error of the sum of two-electron energy contributions (i.e., Coulomb and exact exchange) in the framework of overlapping atoms is described in detail., (© 2023 Author(s). Published under an exclusive license by AIP Publishing.)

We suggest new strict constraints that the two-particle cumulant matrix should fulfill. The constraints are obtained from the decomposition of ⟨Ŝ(2)⟩, previously developed in our laboratory, and the vanishing number of electrons shared by two non-interacting fragments. The conditions impose stringent constraints into the cumulant structure without any need to perform an orbital optimization procedure thus carrying very small or no computational effort. These constraints are tested on the series of Piris natural orbital functionals (PNOF), which are among the most accurate ones available in the literature. Interestingly, even though all PNOF cumulants ensure correct overall ⟨Ŝ(2)⟩ values, none of them is consistent with the local spin structure of systems that dissociate more than one pair of electrons. A careful analysis of the local spin components reveals the most important missing contributions in the cumulant expression thus suggesting a means to improve PNOF5. The constraints provide an inexpensive tool for the construction and testing of cumulant structures that complement previously known conditions such as the N-representability or the square of the total spin angular momentum, ⟨Ŝ(2)⟩.

A new, more flexible definition of fuzzy Voronoi cells is proposed as a computationally efficient alternative to Bader's Quantum Theory of Atoms in Molecules (QTAIM) partitioning of the physical space for large-scale routine calculations. The new fuzzy scheme provides atomic charges, delocalization indices, and molecular energy components very close to those obtained using QTAIM. The method is flexible enough to either ignore the presence of spurious non-nuclear attractors or to readily incorporate them by introducing additional fuzzy Voronoi cells.

The effective atomic orbitals have been realized in the framework of Bader's atoms in molecules theory for a general wavefunction. This formalism can be used to retrieve from any type of calculation a proper set of orthonormalized numerical atomic orbitals, with occupation numbers that sum up to the respective Quantum Theory of Atoms in Molecules (QTAIM) atomic populations. Experience shows that only a limited number of effective atomic orbitals exhibit significant occupation numbers. These correspond to atomic hybrids that closely resemble the core and valence shells of the atom. The occupation numbers of the remaining effective orbitals are almost negligible, except for atoms with hypervalent character. In addition, the molecular orbitals of a calculation can be exactly expressed as a linear combination of this orthonormalized set of numerical atomic orbitals, and the Mulliken population analysis carried out on this basis set exactly reproduces the original QTAIM atomic populations of the atoms. Approximate expansion of the molecular orbitals over a much reduced set of orthogonal atomic basis functions can also be accomplished to a very good accuracy with a singular value decomposition procedure.

Recently, the surprising result that ab initio calculations on benzene and other planar arenes at correlated MP2, MP3, configuration interaction with singles and doubles (CISD), and coupled cluster with singles and doubles levels of theory using standard Pople's basis sets yield nonplanar minima has been reported. The planar optimized structures turn out to be transition states presenting one or more large imaginary frequencies, whereas single-determinant-based methods lead to the expected planar minima and no imaginary frequencies. It has been suggested that such anomalous behavior can be originated by two-electron basis set incompleteness error. In this work, we show that the reported pitfalls can be interpreted in terms of intramolecular basis set superposition error (BSSE) effects, mostly between the C-H moieties constituting the arenes. We have carried out counterpoise-corrected optimizations and frequency calculations at the Hartree-Fock, B3LYP, MP2, and CISD levels of theory with several basis sets for a number of arenes. In all cases, correcting for intramolecular BSSE fixes the anomalous behavior of the correlated methods, whereas no significant differences are observed in the single-determinant case. Consequently, all systems studied are planar at all levels of theory. The effect of different intramolecular fragment definitions and the particular case of charged species, namely, cyclopentadienyl and indenyl anions, respectively, are also discussed.

The present work provides a generalization of Mayer's energy decomposition for the density-functional theory (DFT) case. It is shown that one- and two-atom Hartree-Fock energy components in Mayer's approach can be represented as an action of a one-atom potential V(A) on a one-atom density rho(A) or rho(B). To treat the exchange-correlation term in the DFT energy expression in a similar way, the exchange-correlation energy density per electron is expanded into a linear combination of basis functions. Calculations carried out for a number of density functionals demonstrate that the DFT and Hartree-Fock two-atom energies agree to a reasonable extent with each other. The two-atom energies for strong covalent bonds are within the range of typical bond dissociation energies and are therefore a convenient computational tool for assessment of individual bond strength in polyatomic molecules. For nonspecific nonbonding interactions, the two-atom energies are low. They can be either repulsive or slightly attractive, but the DFT results more frequently yield small attractive values compared to the Hartree-Fock case. The hydrogen bond in the water dimer is calculated to be between the strong covalent and nonbonding interactions on the energy scale.