Softmax parameterization of the occupation numbers for natural orbital functionals based on electron pairing approaches.

Within the framework of natural orbital functional theory, having a convenient representation of the occupation numbers and orbitals becomes critical for the computational performance of the calculations. Recognizing this, we propose an innovative parametrization of the occupation numbers that takes advantage of the electron-pairing approach used in Piris natural orbital functionals through the adoption of the softmax function, a pivotal component in modern deep-learning models. Our approach not only ensures adherence to the N-representability of the first-order reduced density matrix (1RDM) but also significantly enhances the computational efficiency of 1RDM functional theory calculations. The effectiveness of this alternative parameterization approach was assessed using the W4-17-MR molecular set, which demonstrated faster and more robust convergence compared to previous implementations.

Excited States by Coupling Piris Natural Orbital Functionals with the Extended Random-Phase Approximation.

In this work, we explore the use of Piris natural orbital functionals (PNOFs) to calculate excited-state energies by coupling their reconstructed second-order reduced density matrix with the extended random-phase approximation (ERPA). We have named the general method PNOF-ERPA, and specific approaches are referred to as PNOF-ERPA0, PNOF-ERPA1, and PNOF-ERPA2, according to the way the excitation operator is built. The implementation has been tested in the first excited states of H2, HeH+, LiH, Li2, and N2 showing good results compared to the configuration interaction (CI) method. As expected, an increase in accuracy is observed on going from ERPA0 to ERPA1 and ERPA2. We also studied the effect of electron correlation included by PNOF5, PNOF7, and the recently proposed global NOF (GNOF) on the predicted excited states. PNOF5 appears to be good and may even provide better results in very small systems, but including more electron correlation becomes important as the system size increases, where GNOF achieves better results. Overall, the extension of PNOF to excited states has been successful, making it a promising method for further applications.

Charge delocalization error in Piris natural orbital functionals.

Piris Natural Orbital Functionals (PNOFs) have been recognized as a low-scaling alternative to study strong correlated systems. In this work, we address the performance of the fifth functional (PNOF5) and the seventh functional (PNOF7) to deal with another common problem, the charge delocalization error. The effects of this problem can be observed in charged systems of repeated well-separated fragments, where the energy should be the sum of the charged and neutral fragments, regardless of how the charge is distributed. In practice, an energetic overstabilization of fractional charged fragments leads to a preference for having the charge delocalized throughout the system. To establish the performance of PNOFs regarding charge delocalization error, charged chains of helium atoms and the W4-17-MR set molecules were used as base fragments, and their energy, charge distribution, and correlation regime were studied. It was found that PNOF5 prefers localized charge distributions, while PNOF7 improves the treatment of interpair static correlation and tends to the correct energetic limit for several cases, although a preference for delocalized charge distributions may arise in highly strong correlation regimes. Overall, it is concluded that PNOFs can simultaneously deal with static correlation and charge delocalization errors, resulting in a promising choice to study charge-related problems.

Understanding the unusual stiffness of hydrophobic dipeptide crystals.

The hydrophobic diphenylalanine peptide crystal is known to be unusually stiff, with an experimental Young's modulus in the range of 19-27 GPa. Here it is shown by means of density functional theory calculations that phenylalanine-leucine, leucine-phenylalanine, alanine-valine, valine-alanine and valine-valine hydrophobic dipeptide crystals are also unusually stiff, with Young's moduli in the range of 19.7-33.3 GPa. To further our understanding of the origin of that unusual stiffness, a linear correlation is established between Young's modulus and the strength and orientation of the hydrogen bond network developed along the crystals, showing that stiffness in these materials is primarily dictated by hydrogen bonding.

Resolution of the identity approximation applied to PNOF correlation calculations.

In this work, the required algebra to employ the resolution of the identity approximation within the Piris Natural Orbital Functional (PNOF) is developed, leading to an implementation named DoNOF-RI. The arithmetic scaling is reduced from fifth-order to fourth-order, and the memory scaling is reduced from fourth-order to third-order, allowing significant computational time savings. After the DoNOF-RI calculation has fully converged, a restart with four-center electron repulsion integrals can be performed to remove the effect of the auxiliary basis set incompleteness, quickly converging to the exact result. The proposed approach has been tested on cycloalkanes and other molecules of general interest to study the numerical results, as well as the speed-ups achieved by PNOF7-RI when compared with PNOF7.

Asymmetric Density Fitting with Modified Cholesky Decomposition Applied to Second-Order Electron Propagator.

Computation of molecular orbital electron repulsion integrals (MO-ERIs) as a transformation from atomic orbital ERIs (AO-ERIs) is the bottleneck of second-order electron propagator calculations when a single orbital is studied. In this contribution, asymmetric density fitting is combined with modified Cholesky decomposition to generate efficiently the required MO-ERIs. The key point of the presented algorithms is to keep track of integrals through partial contractions performed on three-center AO-ERIs; these contractions are stored in RAM instead of the AO-ERIs. Two implementations are provided, an in-core, which reduces the arithmetic and memory scaling factors as compared to the four-center AO-ERIs contraction method, and a semidirect, which overcomes memory limitations by evaluating antisymmetrized MO-ERIs in batches. On the numerical side, the proposed approach is fast and stable. The bad effects due to ill conditioning, namely, several negative and close to zero eigenvalues due to machine round off errors of the matrix associated with the density fitting process, are effectively controlled by means of a modified Cholesky factorization that avoids the matrix inversion needed to perform the asymmetrical density fitting implementation. The numerical experience presented shows that the in-core implementation is highly competitive to perform calculations on medium and large basis sets, while the semidirect implementation has small variations in time by changes in the available memory. The general applicability is illustrated on a set of selected relatively large-size molecules.

Substrate Specificity and Leaving Group Effect in Ester Cleavage by Metal Complexes of an Oximate Nucleophile.

Deprotonated zinc(II) and cadmium(II) complexes of a tridentate oxime nucleophile (1, OxH) show a very high reactivity, breaking by 2-3 orders of magnitude the previously established limiting reactivity of oximate nucleophiles in the cleavage of substituted phenyl acetates and phosphate triesters, but are unreactive with p-nitrophenyl phosphate di- and monoesters. With reactive substrates, these complexes operate as true catalysts through an acylation-deacylation mechanism. Detailed speciation and kinetic studies in a wide pH interval allowed us to establish as catalytically active forms [Cd(Ox)]+, [Zn(Ox)(OH)], and [Zn(Ox)(OH)2]- complexes. The formation of an unusual and most reactive zinc(II) oximatodihydroxo complex was confirmed by electrospray ionization mass spectrometry data and supported by density functional theory calculations, which also supported the previously noticed fact that the coordinated water in [Zn(OxH)(H2O)2]2+ deprotonates before the oxime. Analysis of the leaving group effect on the cleavage of phenyl acetates shows that the rate-determining step in the reaction with the free oximate anion is the nucleophilic attack, while with both zinc(II) and cadmium(II) oximate complexes, it changes to the expulsion of the leaving phenolate anion. The major new features of these complexes are (1) a very high esterolytic activity surpassing that of enzyme hydrolysis of aryl acetate esters and (2) an increased reactivity of coordinated oxime compared to free oxime in phosphate triester cleavage, contrary to the previously observed inhibitory effect of oxime coordination with these substrates.