Using atomic clustering based on structural and electronic descriptors that consider surrounding environment to evaluate local properties of DFT functionals.

We developed a method for evaluating the accuracies of the local properties of DFT functionals in detail using a clustering method based on machine learning and structural/electronic descriptors. We generated 36 clusters consistent with human intuition using 30,436 carbon atoms from the QM9 dataset. The results were used to evaluate 13C NMR chemical shifts calculated using 84 DFT functionals. Carbon atoms were grouped based on their similar environments, reducing errors within these groups. This enables more accurate assessment of the accuracy using a specific DFT functional. Therefore, the present atomic clustering provides more detailed insight into accuracy verification.

Divide-and-Conquer Linear-Scaling Quantum Chemical Computations.

Fragmentation and embedding schemes are of great importance when applying quantum-chemical calculations to more complex and attractive targets. The divide-and-conquer (DC)-based quantum-chemical model is a fragmentation scheme that can be connected to embedding schemes. This feature article explains several DC-based schemes developed by the authors over the last two decades, which was inspired by the pioneering study of DC self-consistent field (SCF) method by Yang and Lee (J. Chem. Phys. 1995, 103, 5674-5678). First, the theoretical aspects of the DC-based SCF, electron correlation, excited-state, and nuclear orbital methods are described, followed by the two-component relativistic theory, quantum-mechanical molecular dynamics simulation, and the introduction of three programs, including DC-based schemes. Illustrative applications confirmed the accuracy and feasibility of the DC-based schemes.

Machine-learned electron correlation model based on frozen core approximation.

The machine-learned electron correlation (ML-EC) model is a regression model in the form of a density functional that reproduces the correlation energy density based on wavefunction theory. In a previous study [T. Nudejima et al., J. Chem. Phys. 151, 024104 (2019)], the ML-EC model was constructed using the correlation energy density from all-electron calculations with basis sets including core polarization functions. In this study, we applied the frozen core approximation (FCA) to the correlation energy density to reduce the computational cost of the response variable used in machine learning. The coupled cluster singles, doubles, and perturbative triples [CCSD(T)] correlation energy density obtained from a grid-based energy density analysis was analyzed within FCA and correlation-consistent basis sets without core polarization functions. The complete basis set (CBS) limit of the correlation energy density was obtained using the extrapolation and composite schemes. The CCSD(T)/CBS correlation energy densities based on these schemes showed reasonable behavior, indicating its appropriateness as a response variable. As expected, the computational time was significantly reduced, especially for systems containing elements with a large number of inner-shell electrons. Based on the density-to-density relationship, a large number of data (5 662 500 points), which were accumulated from 30 molecules, were sufficient to construct the ML-EC model. The valence-electron correlation energies and reaction energies calculated using the constructed model were in good agreement with the reference values, the latter of which were superior in accuracy to density functional calculations using 71 exchange-correlation functionals. The numerical results indicate that the FCA is useful for constructing a versatile model.

Bond Energy Density Analysis Combined with Informatics Technique.

Bond energy density analysis, a two-body energy decomposition scheme, was extended by revisiting the constraint conditions and using the informatics technique. The present scheme can evaluate the bond energies (BEs) for all interatomic pairs including both strong chemical bonds and weak through-space/bond interactions, and bond dissociation energies (BDEs) constructed from BEs. The newly derived formula, presented in the form of the system of linear equations, tends to result in the overfitting problem owing to the small components originating from the weak through-space/bond interactions. Hence, we adopt the least absolute shrinkage and selection operator technique. Numerical assessments of the present scheme were performed for C-C and C-H bonds in typical hydrocarbons as well as 44 chemical bonds, i.e., covalent and ionic bonds, in 33 small molecules involving second- and third-row atoms. The statistics for the BDE estimation confirms the accuracy of the present scheme.

Machine-learned electron correlation model based on correlation energy density at complete basis set limit.

We propose a machine-learned correlation model that is built using the regression between density variables such as electron density and correlation energy density. The correlation energy density of coupled cluster singles, doubles, and perturbative triples [CCSD(T)] is derived based on grid-based energy density analysis. The complete basis set (CBS) limit is estimated using the composite method, which has been reported to calculate the total correlation energy. The numerical examination revealed that the correlation energy density of the CCSD(T)/CBS level is appropriate for the response variable of machine learning. In addition to the density variables used in the exchange-correlation functionals of the density functional theory, the Hartree-Fock (HF) exchange energy density and electron density based on the fractional occupation number of molecular orbitals were employed as explanatory variables. Numerical assessments confirmed the accuracy and efficiency of the present correlation model. Consequently, the present protocol, namely, learning the CCSD(T)/CBS correlation energy density using density variables obtained by the HF calculation with a small basis set, yields an efficient correlation model.

Extension and acceleration of relativistic density functional theory based on transformed density operator.

We report an extension of relativistic density functional theory (RDFT) within one-component or two-component expressions that relies on a unitary-transformed density operator as well as a unitary-transformed Hamiltonian [Oyama et al., Chem. Phys. Lett. 680, 37 (2017)]. The transformed density operator is introduced to avoid the picture-change effect in the electron density, density gradient, kinetic energy density, and exchange-correlation potential. We confirmed that the implementation based on the spin-free infinite-order Douglas-Kroll-Hess method gives total, orbital, and excitation energies close to the reference values given by four-component RDFT calculations. To reduce the computational cost due to the transformed density operator, the local unitary transformation was also implemented. Numerical assessments revealed that the present scheme enabled the RDFT calculation of polyatomic systems with negligibly small picture-change effect.

RAQET: Large-scale two-component relativistic quantum chemistry program package.

The Relativistic And Quantum Electronic Theory (RAQET) program is a new software package, which is designed for large-scale two-component relativistic quantum chemical (QC) calculations. The package includes several efficient schemes and algorithms for calculations involving large molecules which contain heavy elements in accurate relativistic formalisms. These calculations can be carried out in terms of the two-component relativistic Hamiltonian, wavefunction theory, density functional theory, core potential scheme, and evaluation of electron repulsion integrals. Furthermore, several techniques, which have frequently been used in non-relativistic QC calculations, have been customized for relativistic calculations. This article introduces the brief theories and capabilities of RAQET with several calculation examples. © 2018 Wiley Periodicals, Inc.

Semi-local machine-learned kinetic energy density functional with third-order gradients of electron density.

A semi-local kinetic energy density functional (KEDF) was constructed based on machine learning (ML). The present scheme adopts electron densities and their gradients up to third-order as the explanatory variables for ML and the Kohn-Sham (KS) kinetic energy density as the response variable in atoms and molecules. Numerical assessments of the present scheme were performed in atomic and molecular systems, including first- and second-period elements. The results of 37 conventional KEDFs with explicit formulae were also compared with those of the ML KEDF with an implicit formula. The inclusion of the higher order gradients reduces the deviation of the total kinetic energies from the KS calculations in a stepwise manner. Furthermore, our scheme with the third-order gradient resulted in the closest kinetic energies to the KS calculations out of the presented functionals.

Gauge-origin independent formalism of two-component relativistic framework based on unitary transformation in nuclear magnetic shielding constant.

This article proposes a gauge-origin independent formalism of the nuclear magnetic shielding constant in the two-component relativistic framework based on the unitary transformation. The proposed scheme introduces the gauge factor and the unitary transformation into the atomic orbitals. The two-component relativistic equation is formulated by block-diagonalizing the Dirac Hamiltonian together with gauge factors. This formulation is available for arbitrary relativistic unitary transformations. Then, the infinite-order Douglas-Kroll-Hess (IODKH) transformation is applied to the present formulation. Next, the analytical derivatives of the IODKH Hamiltonian for the evaluation of the nuclear magnetic shielding constant are derived. Results obtained from the numerical assessments demonstrate that the present formulation removes the gauge-origin dependence completely. Furthermore, the formulation with the IODKH transformation gives results that are close to those in four-component and other two-component relativistic schemes.

Computerized implementation of higher-order electron-correlation methods and their linear-scaling divide-and-conquer extensions.

We have implemented a linear-scaling divide-and-conquer (DC)-based higher-order coupled-cluster (CC) and Møller-Plesset perturbation theories (MPPT) as well as their combinations automatically by means of the tensor contraction engine, which is a computerized symbolic algebra system. The DC-based energy expressions of the standard CC and MPPT methods and the CC methods augmented with a perturbation correction were proposed for up to high excitation orders [e.g., CCSDTQ, MP4, and CCSD(2)TQ ]. The numerical assessment for hydrogen halide chains, polyene chains, and first coordination sphere (C1) model of photoactive yellow protein has revealed that the DC-based correlation methods provide reliable correlation energies with significantly less computational cost than that of the conventional implementations. © 2017 Wiley Periodicals, Inc.

Unveiling a New Aspect of Simple Arylboronic Esters: Long-Lived Room-Temperature Phosphorescence from Heavy-Atom-Free Molecules.

Arylboronic esters can be used as versatile reagents in organic synthesis, as represented by Suzuki-Miyaura cross-coupling. Here we report a serendipitous finding that simple arylboronic esters are phosphorescent in the solid state at room temperature with a lifetime on the order of several seconds. The phosphorescence properties of arylboronic esters are remarkable in light of the general notion that phosphorescent organic molecules require heavy atoms and/or carbonyl groups for the efficient generation of a triplet excited state. Theoretical calculations on phenylboronic acid pinacol ester indicated that this molecule undergoes an out-of-plane distortion at the (pinacol)B-Cipso moiety in the T1 excited state, which is responsible for its phosphorescence. A compound survey with 19 arylboron compounds suggested that the phosphorescence properties might be determined by solid-state molecular packing rather than by the patterns and numbers of boron substituents on the aryl units. The present finding may update the general notion of phosphorescent organic molecules.

Informatics-Based Energy Fitting Scheme for Correlation Energy at Complete Basis Set Limit.

Energy fitting schemes based on informatics techniques using hierarchical basis sets with small cardinal numbers were numerically investigated to estimate correlation energies at the complete basis set limits. Numerical validations confirmed that the conventional two-point extrapolation models can be unified into a simple formula with optimal parameters obtained by the same test sets. The extrapolation model was extended to two-point fitting models by a relaxation of the relationship between the extrapolation coefficients or a change of the fitting formula. Furthermore, n-scheme fitting models were developed by the combinations of results calculated at several theory levels and basis sets to compensate for the deficiencies in the fitting model at one level of theory. Systematic assessments on the Gaussian-3X and Gaussian-2 sets revealed that the fitting models drastically reduced errors with equal or smaller computational effort. © 2016 Wiley Periodicals, Inc.

Implementation of Analytical Energy Gradient of Spin-Dependent General Hartree-Fock Method Based on the Infinite-Order Douglas-Kroll-Hess Relativistic Hamiltonian with Local Unitary Transformation.

An analytical energy gradient for the spin-dependent general Hartree-Fock method based on the infinite-order Douglas-Kroll-Hess (IODKH) method was developed. To treat realistic systems, the local unitary transformation (LUT) scheme was employed both in energy and energy gradient calculations. The present energy gradient method was numerically assessed to investigate the accuracy in several diatomic molecules containing fifth- and sixth-period elements and to examine the efficiency in one-, two-, and three-dimensional silver clusters. To arrive at a practical calculation, we also determined the geometrical parameters of fac-tris(2-phenylpyridine)iridium and investigated the efficiency. The numerical results confirmed that the present method describes a highly accurate relativistic effect with high efficiency. The present method can be a powerful scheme for determining geometries of large molecules, including heavy-element atoms.

Accompanying coordinate expansion and recurrence relation method using a transfer relation scheme for electron repulsion integrals with high angular momenta and long contractions.

An efficient algorithm for the rapid evaluation of electron repulsion integrals is proposed. The present method, denoted by accompanying coordinate expansion and transferred recurrence relation (ACE-TRR), is constructed using a transfer relation scheme based on the accompanying coordinate expansion and recurrence relation method. Furthermore, the ACE-TRR algorithm is extended for the general-contraction basis sets. Numerical assessments clarify the efficiency of the ACE-TRR method for the systems including heavy elements, whose orbitals have long contractions and high angular momenta, such as f- and g-orbitals.

An ab initio study of nuclear volume effects for isotope fractionations using two-component relativistic methods.

We investigate the accuracy of two-component Douglas-Kroll-Hess (DKH) methods in calculations of the nuclear volume term (≡ lnK(nv)) in the isotope fractionation coefficient. lnK(nv) is a main term in the chemical equilibrium constant for isotope exchange reactions in heavy element. Previous work based on the four-component method reasonably reproduced experimental lnK(nv) values of uranium isotope exchange. In this work, we compared uranium reaction lnK(nv) values obtained from the two-component and four-component methods. We find that both higher-order relativistic interactions and spin-orbit interactions are essential for quantitative description of lnK(nv). The best alternative is the infinite-order Douglas-Kroll-Hess method with infinite-order spin-orbit interactions for the one-electron term and atomic-mean-field spin-same-orbit interaction for the two-electron term (IODKH-IOSO-MFSO). This approach provides almost equivalent results for the four-component method, while being 30 times faster. The IODKH-IOSO-MFSO methodology should pave the way toward computing larger and more general molecules beyond the four-component method limits.

Extension of accompanying coordinate expansion and recurrence relation method for general-contraction basis sets.

An algorithm of the accompanying coordinate expansion and recurrence relation (ACE-RR), which is used for the rapid evaluation of the electron repulsion integral (ERI), has been extended to the general-contraction (GC) scheme. The present algorithm, denoted by GC-ACE-RR, is designed for molecular calculations including heavy elements, whose orbitals consist of many primitive functions with and without higher angular momentum such as d- and f-orbitals. The performance of GC-ACE-RR was assessed for (ss|ss)-, (pp|pp)-, (dd|dd)-, and (ff|ff)-type ERIs in terms of contraction length and the number of GC orbitals. The present algorithm was found to reduce the central processing unit time compared with the ACE-RR algorithm, especially for higher angular momentum and highly contracted orbitals. Compared with HONDOPLUS and GAMESS program packages, GC-ACE-RR computations for ERIs of three-dimensional gold clusters Aun (n = 1, 2, …, 10, 15, 20, and 25) are more than 10 times faster.

Local unitary transformation method toward practical electron correlation calculations with scalar relativistic effect in large-scale molecules.

In order to perform practical electron correlation calculations, the local unitary transformation (LUT) scheme at the spin-free infinite-order Douglas-Kroll-Hess (IODKH) level [J. Seino and H. Nakai, J. Chem. Phys. 136, 244102 (2012); and ibid. 137, 144101 (2012)], which is based on the locality of relativistic effects, has been combined with the linear-scaling divide-and-conquer (DC)-based Hartree-Fock (HF) and electron correlation methods, such as the second-order Mo̸ller-Plesset (MP2) and the coupled cluster theories with single and double excitations (CCSD). Numerical applications in hydrogen halide molecules, (HX)n (X = F, Cl, Br, and I), coinage metal chain systems, Mn (M = Cu and Ag), and platinum-terminated polyynediyl chain, trans,trans-{(p-CH3C6H4)3P}2(C6H5)Pt(C≡C)4Pt(C6H5){(p-CH3C6H4)3P}2, clarified that the present methods, namely DC-HF, MP2, and CCSD with the LUT-IODKH Hamiltonian, reproduce the results obtained using conventional methods with small computational costs. The combination of both LUT and DC techniques could be the first approach that achieves overall quasi-linear-scaling with a small prefactor for relativistic electron correlation calculations.

Analytical energy gradient based on spin-free infinite-order Douglas-Kroll-Hess method with local unitary transformation.

In this study, the analytical energy gradient for the spin-free infinite-order Douglas-Kroll-Hess (IODKH) method at the levels of the Hartree-Fock (HF), density functional theory (DFT), and second-order Møller-Plesset perturbation theory (MP2) is developed. Furthermore, adopting the local unitary transformation (LUT) scheme for the IODKH method improves the efficiency in computation of the analytical energy gradient. Numerical assessments of the present gradient method are performed at the HF, DFT, and MP2 levels for the IODKH with and without the LUT scheme. The accuracies are examined for diatomic molecules such as hydrogen halides, halogen dimers, coinage metal (Cu, Ag, and Au) halides, and coinage metal dimers, and 20 metal complexes, including the fourth-sixth row transition metals. In addition, the efficiencies are investigated for one-, two-, and three-dimensional silver clusters. The numerical results confirm the accuracy and efficiency of the present method.

Local unitary transformation method for large-scale two-component relativistic calculations. II. Extension to two-electron Coulomb interaction.

The local unitary transformation (LUT) scheme at the spin-free infinite-order Douglas-Kroll-Hess (IODKH) level [J. Seino and H. Nakai, J. Chem. Phys. 136, 244102 (2012)], which is based on the locality of relativistic effects, has been extended to a four-component Dirac-Coulomb Hamiltonian. In the previous study, the LUT scheme was applied only to a one-particle IODKH Hamiltonian with non-relativistic two-electron Coulomb interaction, termed IODKH/C. The current study extends the LUT scheme to a two-particle IODKH Hamiltonian as well as one-particle one, termed IODKH/IODKH, which has been a real bottleneck in numerical calculation. The LUT scheme with the IODKH/IODKH Hamiltonian was numerically assessed in the diatomic molecules HX and X(2) and hydrogen halide molecules, (HX)(n) (X = F, Cl, Br, and I). The total Hartree-Fock energies calculated by the LUT method agree well with conventional IODKH/IODKH results. The computational cost of the LUT method is reduced drastically compared with that of the conventional method. In addition, the LUT method achieves linear-scaling with respect to the system size and a small prefactor.