Ab initiocalculation of force constants and equilibrium geometries in polyatomic molecules

OpenQP: A Quantum Chemical Platform Featuring MRSF-TDDFT with an Emphasis on Open-Source Ecosystem

The OpenQP (Open Quantum Platform) is a new open-source quantum chemistry library developed to tackle sustainability and interoperability challenges in the field of computational chemistry. OpenQP provides various popular quantum chemical theories as autonomous modules such as energy and gradient calculations of HF, DFT, TDDFT, SF-TDDFT, and MRSF-TDDFT, thereby allowing easy interconnection with third-party software. A scientifically notable feature is the innovative mixed-reference spin-flip time-dependent density functional theory (MRSF-TDDFT) and its customized exchange-correlation functionals such as the DTCAM series of VAEE, XI, XIV, AEE, and VEE, which significantly expand the applicability scope of DFT and TDDFT. OpenQP also supports parallel execution and is optimized with BLAS and LAPACK for high performance. Future enhancements such as extended Koopman's theorem (EKT)-MRSF-TDDFT and spin-orbit coupling (SOC)-MRSF-TDDFT will further expand OpenQP's capabilities. Additionally, a Python wrapper PyOQP is provided that performs tasks such as geometry optimization, conical intersection searches, and nonadiabatic coupling calculations, among others, by prototyping the modules of the OpenQP library in combination with third-party libraries. Overall, OpenQP aligns with modern trends in high-performance scientific software development by offering flexible prototyping and operation while retaining the performance benefits of compiled languages like Fortran and C. They enhance the sustainability and interoperability of quantum chemical software, making OpenQP a crucial platform for accelerating the development of advanced quantum theories like MRSF-TDDFT.

Observation and enhancement through alkali metal doping of p-type conductivity in the layered oxyselenides Sr2ZnO2Cu2Se2 and Ba2Zn1-x O2-x Cu2Se2

The optoelectronic properties of two layered copper oxyselenide compounds, with nominal composition Sr2ZnO2Cu2Se2 and Ba2ZnO2Cu2Se2, have been investigated to determine their suitability as p-type conductors. The structure, band gaps and electrical conductivity of pristine and alkali-metal-doped samples have been determined. We find that the strontium-containing compound, Sr2ZnO2Cu2Se2, adopts the expected tetragonal Sr 2 Mn 3 SbO 2 structure with I4/mmm symmetry, and has a band gap of 2.16 eV, and a room temperature conductivity of 4.8 × 10-1 S cm-1. The conductivity of the compound could be increased to 4.2 S cm-1 when sodium doped to a nominal composition of Na0.1Sr1.9ZnO2Cu2Se2. In contrast, the barium containing material was found to have a small zinc oxide deficiency, with a sample dependent compositional range of Ba2Zn1-x O2-x Cu2Se2 where 0.01 < x < 0.06, as determined by single crystal X-ray diffraction and powder neutron diffraction. The barium-containing structure could also be modelled using the tetragonal I4/mmm structure, but significant elongation of the oxygen displacement ellipsoid along the Zn-O bonds in the average structure was observed. This indicated that the oxide ion position was better modelled as a disordered split site with a displacement to change the local zinc coordination from square planar to linear. Electron diffraction data confirmed that the oxide site in Ba2Zn1-x O2-x Cu2Se2 does not adopt a long range ordered arrangement, but also that the idealised I4/mmm structure with an unsplit oxide site was not consistent with the extra reflections observed in the electron diffractograms. The band gap and conductivity of Ba2Zn1-x O2-x Cu2Se2 were determined to be 2.22 eV and 2.0 × 10-3 S cm-1 respectively. The conductivity could be increased to 1.5 × 10-1 S cm-1 with potassium doping in K0.1Ba1.9Zn1-x O2-x Cu2Se2. Hall measurements confirmed that both materials were p-type conductors with holes as the dominant charge carriers.

A Constraint-Based Orbital-Optimized Excited State Method (COOX)

In this work, we present a novel method to directly calculate targeted electronic excited states within a self-consistent field calculation based on constrained density functional theory (cDFT). The constraint is constructed from the static occupied-occupied and virtual-virtual parts of the excited state difference density from (simplified) linear-response time-dependent density functional theory calculations (LR-TDDFT). Our new method shows a stable convergence behavior, provides an accurate excited state density adhering to the Aufbau principle, and can be solved within a restricted SCF for singlet excitations to avoid spin contamination. This also allows the straightforward application of post-SCF electron-correlation methods like MP2 or direct RPA methods. We present the details of our constraint-based orbital-optimized excited state method (COOX) and compare it to similar schemes. The accuracy of excitation energies will be analyzed for a benchmark of systems, while the quality of the resulting excited state densities is investigated by evaluating excited state nuclear forces and excited state structure optimizations. We also investigate the performance of the proposed COOX method for long-range charge transfer excitations and conical intersections with the ground-state.

Low-Scaling, Efficient and Memory Optimized Computation of Nuclear Magnetic Resonance Shieldings within the Random Phase Approximation Using Cholesky-Decomposed Densities and an Attenuated Coulomb Metric

An efficient method for the computation of nuclear magnetic resonance (NMR) shielding tensors within the random phase approximation (RPA) is presented based on our recently introduced resolution-of-the-identity (RI) atomic orbital RPA NMR method [Drontschenko, V. J. Chem. Theory Comput. 2023, 19, 7542-7554] utilizing Cholesky decomposed density type matrices and employing an attenuated Coulomb RI metric. The introduced sparsity is efficiently exploited using sparse matrix algebra. This allows for an efficient and low-scaling computation of RPA NMR shielding tensors. Furthermore, we introduce a batching method for the computation of memory demanding intermediates that accounts for their sparsity. This extends the applicability of our method to even larger systems that would have been out of reach before, such as, e.g., a DNA strand with 260 atoms and 3408 atomic orbital basis functions.

Enabling Ab Initio Molecular Dynamics under Bias: The CP2K+SMEAGOL Interface for Integrating Density Functional Theory and Non-Equilibrium Green Functions

Density functional theory (DFT) combined with non-equilibrium Green's functions (NEGF) is a powerful approach to model quantum transport under external bias potentials at reasonable computational cost. In this work, we present a new interface between the popular mixed Gaussian/plane waves electronic structure package, CP2K, and the NEGF, code SMEAGOL, the most feature-rich implementation of DFT-NEGF available for CP2K to date. The CP2K+SMEAGOL interface includes the implementation of current induced forces. We verify this implementation for a variety of systems: an infinite 1D Au wire, a parallel-plate capacitor, and a Au-H2-Au junction. We find good agreement with SMEAGOL calculations performed with SIESTA for the same systems and with the example of a solvated Au wire demonstrating for the first time that DFT-NEGF can be used to perform molecular dynamics simulations under bias of large-scale condensed phase systems under realistic operating conditions.

Exchange-correlation kernel for perturbation dependent auxiliary functions in auxiliary density perturbation theory

CONTEXT

Analytic exchange-correlation kernel formulations are of the outermost importance for density functional theory (DFT) perturbation calculations. In this paper, the working equation for the exchange-correlation kernel of the generalized gradient approximation (GGA) for perturbation dependent auxiliary functions is derived and discussed in the framework of auxiliary density functional theory (ADFT). The presented new formulation is extended to the unrestricted approach, too. A comprehensive discussion of the implementation of the GGA ADFT kernel, using either the native exchange-correlation functional implementations in deMon2k or the ones from the LibXC library, is given. Calculations with analytic exchange-correlation kernels are compared to their finite difference counterparts. The obtained results are in quantitative agreement. Nevertheless, analytic GGA ADFT kernel implementations show substantial improvement in the computational performance. Similar results are reported for analytic second derivatives of effective core potential (ECP) and model core potential (MCP) matrix elements when compared to their finite difference counterparts in molecular frequency analyses.

METHOD

All calculations are performed in the framework of ADFT as implemented in deMon2k. In the ADFT analytic frequency calculations, auxiliary density perturbation theory was used. The underlying two-center exchange-correlation kernel matrix elements are calculated by numerical integration either with analytic or finite difference kernel expressions. Validation calculations are performed with the VWN and PBE functionals employing DFT-optimized DZVP basis sets in conjunction with automatically generated GEN-A2 auxiliary density function sets. In the (Pt3Cu)n cluster benchmark calculations, the RPBE functional was used. For Pt atoms, the quasi-relativistic LANL2DZ effective core potential with the corresponding valence basis set was employed, whereas for Cu atoms, the all-electron DFT-optimized TZVP basis was applied. The auxiliary density was expanded by the automatically generated GEN-A2* auxiliary function set. We run all benchmark calculations in parallel on 24 cores.

A Simple Model for the Pauli Repulsion with Possible Utility in QM, MM and Chemical Education

The Pauli repulsion is the intermolecular force responsible for the volume and low compressibility of condensed-phase matter at normal conditions. A simple model for this force is presented, wherein per-atom electron densities are represented as spherical charge distributions that are prevented from significantly overlapping. In the example of two noble gas atoms approaching one another beyond their van der Waals radii, the distance between the centers of the electronic charge distributions becomes larger than the distance between the nuclei, giving rise to an unfavorable electrostatic interaction. For the purpose of calculating this interaction, the model is further simplified by representing the per-atom electron density as a negative point charge, loosely inspired by the classical Drude oscillator. The dispersion interaction is simplified to an R-6 term, centered on the aforementioned point charges. Despite the gross simplicity of the resulting formalism, near-quantitative agreement with high-level QM interaction energies of noble gas dimers is achieved. Accordingly, the present model is thought to have utility in force fields, in post-HF and post-DFT dispersion corrections, and in chemical education.

Optimizing Conical Intersections without Explicit Use of Non-Adiabatic Couplings

We present two alternative methods for optimizing minimum energy conical intersection (MECI) molecular geometries without knowledge of the derivative coupling (DC). These methods are based on the utilization of Lagrange multipliers: (i) one method uses an approximate calculation of the DC, while the other (ii) do not require the DC. Both methods use the fact that information on the DC is contained in the Hessian of the squared energy difference. Tests done on a set of small molecular systems, in comparison with other methods, show the ability of the proposed methods to optimize MECIs. Finally, we apply the methods to the furimamide molecule, to optimize and characterize its S1/S2 MECI, and to optimizing the S0/S1 MECI of the silver trimer.

Formulation of transition dipole gradients for non-adiabatic dynamics with polaritonic states

A general formulation of the strong coupling between photons confined in a cavity and molecular electronic states is developed for the state-interaction state-average spin-restricted ensemble-referenced Kohn-Sham method. The light-matter interaction is included in the Jaynes-Cummings model, which requires the derivation and implementation of the analytical derivatives of the transition dipole moments between the molecular electronic states. The developed formalism is tested in the simulations of the nonadiabatic dynamics in the polaritonic states resulting from the strong coupling between the cavity photon mode and the ground and excited states of the penta-2,4-dieniminium cation, also known as PSB3. Comparison with the field-free simulations of the excited-state decay dynamics in PSB3 reveals that the light-matter coupling can considerably alter the decay dynamics by increasing the excited state lifetime and hindering photochemically induced torsion about the C=C double bonds of PSB3. The necessity of obtaining analytical transition dipole gradients for the accurate propagation of the dynamics is underlined.

Ab Initio Vibro-Polaritonic Spectra in Strongly Coupled Cavity-Molecule Systems

Recent experiments have revealed the profound effect of strong light-matter interactions in optical cavities on the electronic ground state of molecular systems. This phenomenon, known as vibrational strong coupling, can modify reaction rates and induce the formation of molecular vibrational polaritons, hybrid states involving both photon modes, and vibrational modes of molecules. We present an ab initio methodology based on the cavity Born-Oppenheimer Hartree-Fock ansatz, which is specifically powerful for ensembles of molecules, to calculate vibro-polaritonic IR spectra. This method allows for a comprehensive analysis of these hybrid states. Our semiclassical approach, validated against full quantum simulations, reproduces key features of the vibro-polaritonic spectra. The underlying analytic gradients also allow for optimization of cavity-coupled molecular systems and performing semiclassical dynamics simulations.

Cartesian constraints in QM/MM optimizations

With the rise of quantum mechanical/molecular mechanical (QM/MM) methods, the interest in the calculation of molecular assemblies has increased considerably. The structures and dynamics of such assemblies are usually governed to a large extend by intermolecular interactions. As a result, the corresponding potential energy surfaces are topological rich and possess many shallow minima. Therefore, local structure optimizations of QM/MM molecular assemblies can be challenging, in particular if optimization constraints are imposed. To overcome this problem, structure optimization in normal coordinate space is advocated. To do so, the external degrees of freedom of a molecule are separated from the internal ones by a projector matrix in the space of the Cartesian coordinates. Here we extend this approach to Cartesian constraints. To this end, we devise an algorithm that adds the Cartesian constraints directly to the projector matrix and in this way eliminates them from the reduced coordinate space in which the molecule is optimized. To analyze the performance and stability of the constrained optimization algorithm in normal coordinate space, we present constrained minimizations of small molecular systems and amino acids in gas phase as well as water employing QM/MM constrained optimizations. All calculations are performed in the framework of auxiliary density functional theory as implemented in the program deMon2k.

Analytical Second-Order Properties for the Random Phase Approximation: Nuclear Magnetic Resonance Shieldings

A method for the analytical computation of nuclear magnetic resonance (NMR) shieldings within the direct random phase approximation (RPA) is presented. As a starting point, we use the RPA ground-state energy expression within the resolution-of-the-identity approximation in the atomic-orbital formalism. As has been shown in a recent benchmark study using numerical second derivatives [Glasbrenner, M. J. Chem. Theory Comput. 2022, 18, 192], RPA based on a Hartree-Fock reference shows accuracies comparable to coupled cluster singles and doubles (CCSD) for NMR chemical shieldings. Together with the much lower computational cost of RPA, it has emerged as an accurate method for the computation of NMR shieldings. Therefore, we aim to extend the applicability of RPA NMR to larger systems by introducing analytical second-order derivatives, making it a viable method for the accurate and efficient computation of NMR chemical shieldings.

Response properties in phaseless auxiliary field quantum Monte Carlo

We present a method for calculating first-order response properties in phaseless auxiliary field quantum Monte Carlo by applying automatic differentiation (AD). Biases and statistical efficiency of the resulting estimators are discussed. Our approach demonstrates that AD enables the calculation of reduced density matrices with the same computational cost scaling per sample as energy calculations, accompanied by a cost prefactor of less than four in our numerical calculations. We investigate the role of self-consistency and trial orbital choice in property calculations. We find that orbitals obtained using density functional theory perform well for the dipole moments of selected molecules compared to those optimized self-consistently.

Orbital-Free Density Functional Theory: An Attractive Electronic Structure Method for Large-Scale First-Principles Simulations

Kohn-Sham Density Functional Theory (KSDFT) is the most widely used electronic structure method in chemistry, physics, and materials science, with thousands of calculations cited annually. This ubiquity is rooted in the favorable accuracy vs cost balance of KSDFT. Nonetheless, the ambitions and expectations of researchers for use of KSDFT in predictive simulations of large, complicated molecular systems are confronted with an intrinsic computational cost-scaling challenge. Particularly evident in the context of first-principles molecular dynamics, the challenge is the high cost-scaling associated with the computation of the Kohn-Sham orbitals. Orbital-free DFT (OFDFT), as the name suggests, circumvents entirely the explicit use of those orbitals. Without them, the structural and algorithmic complexity of KSDFT simplifies dramatically and near-linear scaling with system size irrespective of system state is achievable. Thus, much larger system sizes and longer simulation time scales (compared to conventional KSDFT) become accessible; hence, new chemical phenomena and new materials can be explored. In this review, we introduce the historical contexts of OFDFT, its theoretical basis, and the challenge of realizing its promise via approximate kinetic energy density functionals (KEDFs). We review recent progress on that challenge for an array of KEDFs, such as one-point, two-point, and machine-learnt, as well as some less explored forms. We emphasize use of exact constraints and the inevitability of design choices. Then, we survey the associated numerical techniques and implemented algorithms specific to OFDFT. We conclude with an illustrative sample of applications to showcase the power of OFDFT in materials science, chemistry, and physics.

Implementation of the meta-GGA exchange-correlation functional in numerical atomic orbital basis: With systematic testing on SCAN, rSCAN, and r2SCAN functionals

Kohn-Sham density functional theory (DFT) is nowadays widely used for electronic structure theory simulations, and the accuracy and efficiency of DFT rely on approximations of the exchange-correlation functional. By including the kinetic energy density τ, the meta-generalized-gradient approximation (meta-GGA) family of functionals achieves better accuracy and flexibility while retaining the efficiency of semi-local functionals. For example, the strongly constrained and appropriately normed (SCAN) meta-GGA functional has been proven to yield accurate results for solid and molecular systems. We implement meta-GGA functionals with both numerical atomic orbitals and plane wave bases in the ABACUS package. Apart from the exchange-correlation potential, we also discuss the evaluation of force and stress. To validate our implementation, we perform finite-difference tests and convergence tests with the SCAN, rSCAN, and r2SCAN meta-GGA functionals. We further test water hexamers, weakly interacting molecules from the S22 dataset, as well as 13 semiconductors using the three functionals. The results show satisfactory agreement with previous calculations and available experimental values.

Understanding the electronic structure of Y2Ti2O5S2 for green hydrogen production: a hybrid-DFT and GW study

Utilising photocatalytic water splitting to produce green hydrogen is the key to reducing the carbon footprint of this crucial chemical feedstock. In this study, density functional theory (DFT) is employed to gain insights into the photocatalytic performance of an up-and-coming photocatalyst Y2Ti2O5S2 from first principles. Eleven non-polar clean surfaces are evaluated at the generalised gradient approximation level to obtain a plate-like Wulff shape that agrees well with the experimental data. The (001), (101) and (211) surfaces are considered further at hybrid-DFT level to determine their band alignments with respect to vacuum. The large band offset between the basal (001) and side (101) and (211) surfaces confirms experimentally observed spatial separation of hydrogen and oxygen evolution facets. Furthermore, relevant optoelectronic bulk properties were established using a combination of hybrid-DFT and many-body perturbation theory. The optical absorption of Y2Ti2O5S2 weakly onsets due to dipole-forbidden transitions, and hybrid Wannier-Mott/Frenkel excitonic behaviour is predicted to occur due to the two-dimensional electronic structure, with an exciton binding energy of 0.4 eV.

Combining fragmentation method and high-performance computing: Geometry optimization and vibrational spectra of proteins

Exploring the structures and spectral features of proteins with advanced quantum chemical methods is an uphill task. In this work, a fragment-based molecular tailoring approach (MTA) is appraised for the CAM-B3LYP/aug-cc-pVDZ-level geometry optimization and vibrational infrared (IR) spectra calculation of ten real proteins containing up to 407 atoms and 6617 basis functions. The use of MTA and the inherently parallel nature of the fragment calculations enables a rapid and accurate calculation of the IR spectrum. The applicability of MTA to optimize the protein geometry and evaluate its IR spectrum employing a polarizable continuum model with water as a solvent is also showcased. The typical errors in the total energy and IR frequencies computed by MTA vis-à-vis their full calculation (FC) counterparts for the studied protein are 5-10 millihartrees and 5 cm-1, respectively. Moreover, due to the independent execution of the fragments, large-scale parallelization can also be achieved. With increasing size and level of theory, MTA shows an appreciable advantage in computer time as well as memory and disk space requirement over the corresponding FCs. The present study suggests that the geometry optimization and IR computations on the biomolecules containing ∼1000 atoms and/or ∼15 000 basis functions using MTA and HPC facility can be clearly envisioned in the near future.

Efficient implementation of analytical gradients for periodic hybrid functional calculations within fitted numerical atomic orbitals from NAO2GTO

The NAO2GTO scheme provides an efficient way to evaluate the electron repulsion integrals (ERIs) over numerical atomic orbitals (NAOs) with auxiliary Gaussian-type orbitals (GTOs). However, the NAO2GTO fitting will significantly impact the accuracy and convergence of hybrid functional calculations. To address this issue, here we propose to use the fitted orbitals as a new numerical basis to properly handle the mismatch between NAOs and fitted GTOs. We present an efficient and linear-scaling implementation of analytical gradients of Hartree-Fock exchange (HFX) energy for periodic HSE06 calculations with fitted NAOs in the HONPAS package. In our implementation, the ERIs and their derivatives for HFX matrix and forces are evaluated analytically with the auxiliary GTOs, while other terms are calculated using numerically discretized GTOs. Several integral screening techniques are employed to reduce the number of required ERI derivatives. We benchmark the accuracy and efficiency of our implementation and demonstrate that our results of lattice constants, bulk moduli, and band gaps of several typical semiconductors are in good agreement with the experimental values. We also show that the calculation of HFX forces based on a master-worker dynamic parallel scheme has a very high efficiency and scales linearly with respect to system size. Finally, we study the geometry optimization and polaron formation due to an excess electron in rutile TiO2 by means of HSE06 calculations to further validate the applicability of our implementation.

Semi-Empirical Shadow Molecular Dynamics: A PyTorch Implementation

Extended Lagrangian Born-Oppenheimer molecular dynamics (XL-BOMD) in its most recent shadow potential energy version has been implemented in the semiempirical PyTorch-based software PySeQM. The implementation includes finite electronic temperatures, canonical density matrix perturbation theory, and an adaptive Krylov subspace approximation for the integration of the electronic equations of motion within the XL-BOMB approach (KSA-XL-BOMD). The PyTorch implementation leverages the use of GPU and machine learning hardware accelerators for the simulations. The new XL-BOMD formulation allows studying more challenging chemical systems with charge instabilities and low electronic energy gaps. The current public release of PySeQM continues our development of modular architecture for large-scale simulations employing semi-empirical quantum-mechanical treatment. Applied to molecular dynamics, simulation of 840 carbon atoms, one integration time step executes in 4 s on a single Nvidia RTX A6000 GPU.

Investigation of Polymer Aging Mechanisms Using Molecular Simulations: A Review

Aging has a serious impact on the properties of functional polymers. Therefore, it is necessary to study the aging mechanism to prolong the service and storage life of polymer-based devices and materials. Due to the limitations of traditional experimental methods, more and more studies have adopted molecular simulations to analyze the intrinsic mechanisms of aging. In this paper, recent advances in molecular simulations of the aging of polymers and their composites are reviewed. The characteristics and applications of commonly used simulation methods in the study of the aging mechanisms (traditional molecular dynamics simulation, quantum mechanics, and reactive molecular dynamics simulation) are outlined. The current simulation research progress of physical aging, aging under mechanical stress, thermal aging, hydrothermal aging, thermo-oxidative aging, electric aging, aging under high-energy particle impact, and radiation aging is introduced in detail. Finally, the current research status of the aging simulations of polymers and their composites is summarized, and the future development trend has been prospected.

An improved parameterization procedure for NDDO-descendant semi-empirical methods

CONCEPT

MNDO-based semi-empirical methods in quantum chemistry have found widespread application in the modelling of large and complex systems. A method for the analytic evaluation of first and second derivatives of molecular properties against semi-empirical parameters in MNDO-based NDDO-descendant models is presented, and the resultant parameter Hessian is compared against the approximant currently used in parameterization for the PMx models.

METHODS

As a proof of concept, the exact parameter Hessian is employed in a limited reparameterization of MNDO for the elements C, H, N, O and F using 1206 molecules for reference data (heats of formation, ionization energies, dipole moments and reference geometries). The correctness of our MNDO implementation was verified by comparing the calculated molecular properties with the MOPAC program.

Efficient quantum analytic nuclear gradients with double factorization

Efficient representations of the Hamiltonian, such as double factorization, drastically reduce the circuit depth or the number of repetitions in error corrected and noisy intermediate-scale quantum (NISQ) algorithms for chemistry. We report a Lagrangian-based approach for evaluating relaxed one- and two-particle reduced density matrices from double factorized Hamiltonians, unlocking efficiency improvements in computing the nuclear gradient and related derivative properties. We demonstrate the accuracy and feasibility of our Lagrangian-based approach to recover all off-diagonal density matrix elements in classically simulated examples with up to 327 quantum and 18 470 total atoms in QM/MM simulations with modest-sized quantum active spaces. We show this in the context of the variational quantum eigensolver in case studies, such as transition state optimization, ab initio molecular dynamics simulation, and energy minimization of large molecular systems.

Automated handling of complex chemical structures in Z-matrix coordinates-The chemcoord library

In this work, we present a fully automated method for the construction of chemically meaningful sets of hierarchical nonredundant internal coordinates (ICs; also commonly denoted as Z-matrices) from the Cartesian coordinates of a molecular system. Particular focus is placed on avoiding ill-definitions of angles and dihedrals due to linear arrangements of atoms, to consistently guarantee a well-defined transformation to Cartesian coordinates, even after structural changes. The representations thus obtained are particularly well suited for pathway construction in double-ended methods for transition state search and optimizations with nonlinear constraints. Analytical gradients for the transformation between the coordinate systems were derived for analytical geometry optimizations purely in Z-matrix coordinates. The geometry optimization was coupled with a Symbolic Algebra package to support arbitrary nonlinear constraints in Z-matrix coordinates, while retaining analytical energy gradient conversion. The difference to the commonly used nonhierarchical IC transformations is discussed. Sample applications are provided for a number of common chemical reactions and illustrative examples.

Interpolative Separable Density Fitting for Accelerating Two-Electron Integrals: A Theoretical Perspective

Low-rank approximations have long been considered an efficient way to accelerate electronic structure calculations associated with the evaluation of electron repulsion integrals (ERIs). As an accurate and efficient algorithm for compressing the ERI tensor, the interpolative separable density fitting (ISDF) decomposition has recently attracted great attention in this context. In this perspective, we introduce the ISDF decomposition from the theoretical aspects and technique details. The ISDF decomposition can construct a fully separable low-rank approximation (tensor hypercontraction factorization) of ERIs in real space with a cubic cost, offering great flexibility for accelerating high-scaling electronic structure calculations. We review the typical applications of ISDF in hybrid functionals, time-dependent density functional theory, and GW approximation. Finally, we discuss the promising directions for future development of ISDF.

Ground- and Excited-State Dipole Moments and Oscillator Strengths of Full Configuration Interaction Quality

We report ground- and excited-state dipole moments and oscillator strengths (computed in different "gauges" or representations) of full configuration interaction (FCI) quality using the selected configuration interaction method known as Configuration Interaction using a Perturbative Selection made Iteratively (CIPSI). Thanks to a set encompassing 35 ground- and excited-state properties computed in 11 small molecules, the present near-FCI estimates allow us to assess the accuracy of high-order coupled-cluster (CC) calculations including up to quadruple excitations. In particular, we show that incrementing the excitation degree of the CC expansion (from CC with singles and doubles (CCSD) to CC with singles, doubles, and triples (CCSDT) or from CCSDT to CC with singles, doubles, triples, and quadruples (CCSDTQ)) reduces the average error with respect to the near-FCI reference values by approximately 1 order of magnitude.

Accurate Hellmann-Feynman forces from density functional calculations with augmented Gaussian basis sets

The Hellmann-Feynman (HF) theorem provides a way to compute forces directly from the electron density, enabling efficient force calculations for large systems through machine learning (ML) models for the electron density. The main issue holding back the general acceptance of the HF approach for atom-centered basis sets is the well-known Pulay force which, if naively discarded, typically constitutes an error upward of 10 eV/Å in forces. In this work, we demonstrate that if a suitably augmented Gaussian basis set is used for density functional calculations, the Pulay force can be suppressed, and HF forces can be computed as accurately as analytical forces with state-of-the-art basis sets, allowing geometry optimization and molecular dynamics to be reliably performed with HF forces. Our results pave a clear path forward for the accurate and efficient simulation of large systems using ML densities and the HF theorem.

Development of nonlocal kinetic-energy density functional for the hybrid QM/MM interaction

Development of the electronic kinetic-energy density functional is a subject of major interest in theoretical physics and chemistry. In this work, the nonlocal kinetic-energy functional is developed in terms of the response function for the molecular system to realize the orbital free density-functional theory (OF-DFT) to be utilized in the hybrid QM/MM (quantum mechanical/molecular mechanical) method. The present approach shows a clear contrast to the previous functionals where the homogeneous electron gas serves as a reference to build the response function. As a benchmark test, we apply the method to a QM water molecule in a dimer system and that embedded in a condensed environment to make comparisons with the results given by the QM/MM calculations employing the Kohn-Sham DFT. It was found that the energetics and the polarization density of the QM solute under the influence of the MM environment can be adequately reproduced with our approach. This work suggests the potential ability of the kinetic-energy functional based on the response functions for the molecular reference systems.

Novel asymmetrical azines appending 1,3,4-thiadiazole sulfonamide: synthesis, molecular structure analyses, in silico ADME, and cytotoxic effect†

Toward finding potential and novel anticancer agents, we designed and prepared novel differently substituted unsymmetrical azine-modified thiadiazole sulfonamide derivatives using the “combi-targeting approach”. An efficient procedure for synthesizing the designed compounds starts with 5-acetyl-3-N-(4-sulfamoylphenyl)-2-imino-1,3,4-thiadi-azoline 4. The E/Z configuration for compound 5 was investigated based on spectral analysis combined with quantum mechanical calculation applying the DFT-B3LYP method and 6-31G(d) basis set. The computational results found that the E isomer was energetically more favorable than the Z isomer by 2.21 kcal mol⁻¹. Moreover, ¹H and ¹³C chemical shifts for the E and Z isomers in DMSO were predicted using the GIAO-B3LYP/6-31G(d) computations and IEF-PCM solvation model. The computed chemical shifts for both isomers are consistent with those observed experimentally, indicating that they exist in the solution phase. Moreover, the E/Z configuration for the synthesized azines 7a–c, 9, 11, 13, 15a and 15b was also studied theoretically using the DFT-B3LYP/6-31G(d) calculations. In silico prediction for the biological activities was reported regarding the HOMO–LUMO energy gaps and molecular reactivity descriptors besides the ADMT/drug-likeness properties. The cytotoxic effect of the synthesized compounds has been assayed via the determination of their IC₅₀.

Toward finding potential and novel anticancer agents, we designed and prepared novel differently substituted unsymmetrical azine-modified thiadiazole sulfonamide derivatives using the “combi-targeting approach”.

Efficient Method for the Computation of Frozen-Core Nuclear Gradients within the Random Phase Approximation

A method for the evaluation of analytical frozen-core gradients within the random phase approximation is presented. We outline an efficient way to evaluate the response of the density of active electrons arising only when introducing the frozen-core approximation and constituting the main difficulty, together with the response of the standard Kohn-Sham density. The general framework allows to extend the outlined procedure to related electron correlation methods in the atomic orbital basis that require the evaluation of density responses, such as second-order Møller-Plesset perturbation theory or coupled cluster variants. By using Cholesky decomposed densities─which reintroduce the occupied index in the time-determining steps─we are able to achieve speedups of 20-30% (depending on the size of the basis set) by using the frozen-core approximation, which is of similar magnitude as for molecular orbital formulations. We further show that the errors introduced by the frozen-core approximation are practically insignificant for molecular geometries.

Y2Ti2O5S2 - a promising n-type oxysulphide for thermoelectric applications

Thermoelectric materials offer an unambiguous solution to the ever-increasing global demand for energy by harnessing the Seebeck effect to convert waste heat to electrical energy. Mixed-anion materials are ideal candidate thermoelectric materials due to their thermal stability and potential for "phonon-glass, electron-crystal" behaviour. In this study, we use density-functional theory (DFT) calculations to investigate Y₂Ti₂O₅S₂, a cation-deficient Ruddlesden-Popper system, as a potential thermoelectric. We use hybrid DFT to calculate the electronic structure and band alignment, which indicate a preference for n-type doping with highly anisotropic in-plane and the out-of-plane charge-carrier mobilities as a result of the anisotropy in the crystal structure. We compute phonon spectra and calculate the lattice thermal conductivity within the single-mode relaxation-time approximation using lifetimes obtained by considering three-phonon interactions. We also calculate the transport properties using the momentum relaxation-time approximation to solve the electronic Boltzmann transport equations. The predicted transport properties and lattice thermal conductivity suggest a maximum in-plane ZT of 1.18 at 1000 K with a carrier concentration of 2.37 × 10²⁰ cm^-3. Finally, we discuss further the origins of the low lattice thermal conductivity, in particular exploring the possibility of nanostructuring to lower the phonon mean free path, reduce the thermal conductivity, and further enhance the ZT. Given the experimentally-evidenced high thermal stability and the favourable band alignment found in this work, Y₂Ti₂O₅S₂ has the potential to be a promising high-temperature n-type thermoelectric.

Localized Resolution of Identity Approach to the Analytical Gradients of Random-Phase Approximation Ground-State Energy: Algorithm and Benchmarks

We develop and implement a formalism which enables calculating the analytical gradients of particle-hole random-phase approximation (RPA) ground-state energy with respect to the atomic positions within the atomic orbital basis set framework. Our approach is based on a localized resolution of identity (LRI) approximation for evaluating the two-electron Coulomb integrals and their derivatives, and the density functional perturbation theory for computing the first-order derivatives of the Kohn-Sham (KS) orbitals and orbital energies. Our implementation allows one to relax molecular structures at the RPA level using both Gaussian-type orbitals (GTOs) and numerical atomic orbitals (NAOs). Benchmark calculations against previous implementations show that our approach delivers adequate numerical precision, highlighting the usefulness of LRI in the context of RPA gradient evaluations. A careful assessment of the quality of RPA geometries for small molecules reveals that post-KS RPA systematically overestimates the bond lengths. We furthermore optimized the geometries of the four low-lying water hexamers-cage, prism, cyclic, and book isomers, and determined the energy hierarchy of these four isomers using RPA. The obtained RPA energy ordering is in good agreement with that yielded by the coupled cluster method with single, double and perturbative triple excitations, despite that the dissociation energies themselves are appreciably underestimated. The underestimation of the dissociation energies by RPA is well corrected by the renormalized single excitation correction.

Adiabatic models for the quantum dynamics of surface scattering with lattice effects

In this contribution, we review models for the lattice effects in quantum dynamics calculations on surface scattering, which is important to modeling heterogeneous catalysis for achieving an interpretation of experimental measurements. Unlike dynamics models for reactions in the gas phase, those for heterogeneous reactions have to include the effects of the surface. For manageable computational costs in calculations, the effects of static surface (SS) are firstly modeled as this is simply and easily implemented. Then, the SS model has to be improved to include the effects of the flexible surface, that is the lattice effects. To do this, various surface models have been designed where the coordinates of the surface atoms are introduced in the Hamiltonian operator, especially those of the top surface atom. Based on this model Hamiltonian operator, extensive multi-dimension quantum dynamics calculations can be performed to recover the lattice effects. Here, we first review an overview of the techniques in constructing the Hamiltonian operator, which is a sum of the kinetic energy operator (KEO) and potential energy surface (PES). Since the PES containing the coordinates of the surface atoms in a cell is still expensive, the SS model is often accepted. We consider a mathematical model, called the coupled harmonic oscillator (CHO) model, to introduce the concepts of adiabatic and diabatic representations for separating the molecule and surface. Under the adiabatic model, we further introduce the expansion model where the potential function is Taylor expanded around the optimized geometry of the surface. By an expansion model truncated at the first and second order, various coupling surface models between the molecule and surface are derived. Moreover, by further and deeply understanding the adiabatic representation, an effective Hamiltonian operator is obtained by optimizing the total wave function in factorized form. By this factorized form of wave function and effective Hamiltonian operator, the geometry phase of the surface wave function is theoretically found. This theoretical prediction may be measured by carefully designing experiments. Finally, discussions on the adiabatic representation, the PES construction, and possibility of the classical-dynamics solutions are given. Based on these discussions, a simple outlook on the dynamics of photocatalytics is finally given.

Solving the electronic Schrödinger equation for multiple nuclear geometries with weight-sharing deep neural networks

The Schrödinger equation describes the quantum-mechanical behaviour of particles, making it the most fundamental equation in chemistry. A solution for a given molecule allows computation of any of its properties. Finding accurate solutions for many different molecules and geometries is thus crucial to the discovery of new materials such as drugs or catalysts. Despite its importance, the Schrödinger equation is notoriously difficult to solve even for single molecules, as established methods scale exponentially with the number of particles. Combining Monte Carlo techniques with unsupervised optimization of neural networks was recently discovered as a promising approach to overcome this curse of dimensionality, but the corresponding methods do not exploit synergies that arise when considering multiple geometries. Here we show that sharing the vast majority of weights across neural network models for different geometries substantially accelerates optimization. Furthermore, weight-sharing yields pretrained models that require only a small number of additional optimization steps to obtain high-accuracy solutions for new geometries.

Theory of electronic resonances: fundamental aspects and recent advances

Electronic resonances are states that are unstable towards loss of electrons. They play critical roles in high-energy environments across chemistry, physics, and biology but are also relevant to processes under ambient conditions that involve unbound electrons. This feature article focuses on complex-variable techniques such as complex scaling and complex absorbing potentials that afford a treatment of electronic resonances in terms of discrete square-integrable eigenstates of non-Hermitian Hamiltonians with complex energy. Fundamental aspects of these techniques as well as their integration into molecular electronic-structure theory are discussed and an overview of some recent developments is given: analytic gradient theory for electronic resonances, the application of rank-reduction techniques and quantum embedding to them, as well as approaches for evaluating partial decay widths.

Quasi-Relativistic Calculation of EPR g Tensors with Derivatives of the Decoupling Transformation, Gauge-Including Atomic Orbitals, and Magnetic Balance

We present an exact two-component (X2C) ansatz for the EPR g tensor using gauge-including atomic orbitals (GIAOs) and a magnetically balanced basis set expansion. In contrast to previous X2C and four-component relativistic ansätze for the g tensor, this implementation results in a gauge-origin-invariant formalism. Furthermore, the derivatives of the relativistic decoupling matrix are incorporated to form the complete analytical derivative of the X2C Hamiltonian. To reduce the associated computational costs, we apply the diagonal local approximation to the unitary decoupling transformation (DLU). The quasi-relativistic X2C and DLU-X2C Hamiltonians accurately reproduce the results of the parent four-component relativistic theory when accounting for two-electron picture-change effects with the modified screened nuclear spin-orbit approximation in the respective one-electron integrals and integral derivatives. According to our benchmark studies, the uncontracted Dyall and segmented-contracted Karlsruhe x2c-type basis sets perform well when compared to large even-tempered basis sets. Moreover, (range-separated) hybrid density functional approximations such as LC-ωPBE and ωB97X-D are needed to match the experimental findings. The impact of the GIAOs depends on the distribution of the spin density, and their use may change the Δg shifts by 10-50% as shown for [(C₅Me₅)₂Y(μ-S)₂Mo(μ-S)₂Y(C₅Me₅)₂]^-. Routine calculations of large molecules are possible with widely available and comparably low-cost hardware as demonstrated for [Pt(C₆Cl₅)₄]^- with 3003 basis functions and three spin-(1/2) La(II) and Lu(II) compounds, for which we observe good agreement with the experimental findings.

Investigation of factors affecting the stability of compounds formed by isovalent substitution in layered oxychalcogenides, leading to identification of Ba3Sc2O5Cu2Se2, Ba3Y2O5Cu2S2, Ba3Sc2O5Ag2Se2 and Ba3In2O5Ag2Se2

Four novel compositions containing chalcogenide layers, adopting the Ba₃M₂O₅M'₂Ch₂ layered structure have been identified: Ba₃Sc₂O₅Cu₂Se₂, Ba₃Y₂O₅Cu₂S₂, Ba₃Sc₂O₅Ag₂Se₂ and Ba₃In₂O₅Ag₂Se₂. A comprehensive comparison of experimental and computational results providing the crystallographic and electronic structure of the compounds under investigation has been conducted. Materials were synthesised at 800 °C under vacuum using a conventional ceramic synthesis route with combination of binary oxide and chalcogenide precursors. We report their structures determined by Rietveld refinement of X-ray powder diffraction patterns, and band gaps determined from optical measurements, which range from 1.44 eV to 3.04 eV. Through computational modelling we can also present detailed band structures and propose that, based on their predicted transport properties, Ba₃Sc₂O₅Ag₂Se₂ has potential as a visible light photocatalyst and Ba₃Sc₂O₅Cu₂Se₂ is of interest as a p-type transparent conductor. These four novel compounds are part of a larger series of sixteen compounds adopting the Ba₃M₂O₅M'₂Ch₂ structure that we have considered, of which approximately half are stable and can be synthesized. Analysis of the compounds that cannot be synthesized from this group allows us to identify why compounds containing either M = La, or silver sulfide chalcogenide layers, cannot be formed in this structure type.

Forces from Stochastic Density Functional Theory under Nonorthogonal Atom-Centered Basis Sets

We develop a formalism for calculating forces on the nuclei within the linear-scaling stochastic density functional theory (sDFT) in a nonorthogonal atom-centered basis set representation (Fabian et al. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2019, 9, e1412, 10.1002/wcms.1412) and apply it to the Tryptophan Zipper 2 (Trp-zip2) peptide solvated in water. We use an embedded-fragment approach to reduce the statistical errors (fluctuation and systematic bias), where the entire peptide is the main fragment and the remaining 425 water molecules are grouped into small fragments. We analyze the magnitude of the statistical errors in the forces and find that the systematic bias is of the order of 0.065 eV/Å (∼1.2 × 10-3Eh/a0) when 120 stochastic orbitals are used, independently of system size. This magnitude of bias is sufficiently small to ensure that the bond lengths estimated by stochastic DFT (within a Langevin molecular dynamics simulation) will deviate by less than 1% from those predicted by a deterministic calculation.

Analytic Alchemical Derivatives for the Analysis of Differential Acidity Assisted by the h Function

Analytical calculation of alchemical derivatives based on auxiliary density perturbation theory is described, coded, and validated. For the case where the nucleus is a hydrogen atom and the nuclear charge is changed from 1 to 0, it turns out that a good estimate of the proton binding energies can be obtained very efficiently. First-order results correspond exactly to the molecular electrostatic potential evaluated at the hydrogen nucleus location (removing self-repulsion), in agreement with previously reported extensive studies. Therefore, the second-order results reported here are refinements in accuracy that finally allow a quantitative exploration of differential acidity. Furthermore, the recently reported h function is produced in its analytical form as a byproduct and local descriptor associated with the proton binding energy values found with this approach. In an example application, proton binding energies are computed for a family of imidazolium derivatives to demonstrate the capabilities and the stability of the method with respect to changes in basis set or exchange-correlation functional.

Co(ii), Ni(ii), Cu(ii) and Cd(ii)-thiocarbonohydrazone complexes: spectroscopic, DFT, thermal, and electrical conductivity studies

New and stable coordinated compounds have been isolated in a good yield. The chelates have been prepared by mixing Co(ii), Ni(ii), Cu(ii), and Cd(ii) metal ions with (1E)-1-((6-methyl-4-oxo-4H-chromen-3-yl)methylene)thiocarbonohydrazide (MCMT) in 2 : 1 stoichiometry (MCMT : M²⁺). Various techniques, including elemental microanalyses, molar conductance, thermal studies, FT-IR, ¹H-NMR, UV-Vis, and XRD spectral analyses, magnetic moment measurements, and electrical conductivity, were applied for the structural and spectroscopic elucidation of the coordinating compounds. Further, computational studies using the DFT-B3LYP method were reported for MCMT and its metal complexes. MCMT behaves as a neutral NS bidentate moiety that forms octahedral complexes with general formula [M(MCMT)₂Cl(OH₂)]Cl·XH₂O (M = Cu²⁺; (X = ½), Ni²⁺, Co²⁺; (X = 1)); [Cd(MCMT)₂Cl₂]·½H₂O. There is good confirmation between experimental infrared spectral data and theoretical DFT-B3LYP computational outcomes where MCMT acts as a five-membered chelate bonded to the metal ion through azomethine nitrogen and thiocarbonyl sulphur donors. The thermal analysis is studied to confirm the elucidated structure of the complexes. Also, the kinetic and thermodynamic parameters of the thermal decomposition steps were evaluated. The measured optical band gap values of the prepared compounds exhibited semiconducting nature. AC conductivity and dielectric properties of the ligand and its complexes were examined, which showed that Cu(ii) complex has the highest dielectric constant referring to its high polarization and storage ability.

Electric fields and potentials in condensed phases

The electric fields and potentials inside and at the interface of matter are relevant to many branches of physics, chemistry, and biology. Accurate quantification of these fields and/or potentials is essential to control and exploit chemical and physical transformations. Before we understand the response of matter to external fields, it is first important to understand the intrinsic interior and interfacial fields and potentials, both classically and quantum mechanically, as well as how they are probed experimentally. Here we compare and contrast, beginning with the hydrogen atom in vacuum and ending with concentrated aqueous NaCl electrolyte, both classical and quantum mechanical electric potentials and fields. We make contact with experimental vibrational Stark, electrochemical, X-ray, and electron spectroscopic probes of these potentials and fields, outline relevant conceptual difficulties, and underscore the advantage of electron holography as a basis to better understand electrostatics in matter.

Exploring Dyson's Orbitals and Their Electron Binding Energies for Conceptualizing Excited States from Response Methodology

The molecular orbital (MO) concept is a useful tool, which relates the molecular ground-state energy with the energies (and occupations) of the individual orbitals. However, analysis of the excited states from linear response computations is performed in terms of the initial state MOs or some other forms of orbitals, e.g., natural or natural transition orbitals. Because these orbitals lack the respective energies, they do not allow developing a consistent orbital picture of the excited states. Herein, we argue that Dyson's orbitals enable description of the response states compatible with the concepts of molecular orbital theory. The Dyson orbitals and their energies obtained by mixed-reference spin-flip time-dependent density functional theory (MRSF-TDDFT) for the response ground state are remarkably similar to the canonical MOs obtained by the usual DFT calculation. For excited states, the Dyson orbitals provide a chemically sensible picture of the electronic transitions, thus bridging the chasm between orbital theory and response computations.

Ca4Sb2O and Ca4Bi2O: two promising mixed-anion thermoelectrics

The environmental burden of fossil fuels and the rising impact of global warming have created an urgent need for sustainable clean energy sources. This has led to widespread interest in thermoelectric (TE) materials to recover part of the ∼60% of global energy currently wasted as heat as usable electricity. Oxides are particularly attractive as they are thermally stable, chemically inert, and formed of earth-abundant elements, but despite intensive efforts there have been no reports of oxide TEs matching the performance of flagship chalcogenide materials such as PbTe, Bi2Te3 and SnSe. A number of ternary X4Y2Z mixed-anion systems, including oxides, have predicted band gaps in the useful range for several renewable-energy applications, including as TEs, and some also show the complex crystal structures indicative of low lattice thermal conductivity. In this study, we use ab initio calculations to investigate the TE performance of two structurally-similar mixed-anion oxypnictides, Ca4Sb2O and Ca4Bi2O. Electronic-structure and band-alignment calculations using hybrid density-functional theory (DFT), including spin-orbit coupling, suggest that both materials are likely to be p-type dopable with large charge-carrier mobilities. Lattice-dynamics calculations using third-order perturbation theory predict ultra-low lattice thermal conductivities of ∼0.8 and ∼0.5 W m-1 K-1 above 750 K. Nanostructuring to a crystal grain size of 20 nm is predicted to further reduce the room temperature thermal conductivity by around 40%. Finally, we use the electronic- and thermal-transport calculations to estimate the thermoelectric figure of merit ZT, and show that with p-type doping both oxides could potentially serve as promising earth-abundant oxide TEs for high-temperature applications.

Reinforcement Learning Configuration Interaction

Selected configuration interaction (sCI) methods exploit the sparsity of the full configuration interaction (FCI) wave function, yielding significant computational savings and wave function compression without sacrificing the accuracy. Despite recent advances in sCI methods, the selection of important determinants remains an open problem. We explore the possibility of utilizing reinforcement learning approaches to solve the sCI problem. By mapping the configuration interaction problem onto a sequential decision-making process, the agent learns on-the-fly which determinants to include and which to ignore, yielding a compressed wave function at near-FCI accuracy. This method, which we call reinforcement-learned configuration interaction, adds another weapon to the sCI arsenal and highlights how reinforcement learning approaches can potentially help solve challenging problems in electronic structure theory.