Explicitly Correlated Electrons in Molecules

Range Separation of the Interaction Potential in Intermolecular and Intramolecular Symmetry-Adapted Perturbation Theory

Symmetry-adapted perturbation theory (SAPT) is a popular and versatile tool to compute and decompose noncovalent interaction energies between molecules. The intramolecular SAPT (ISAPT) variant provides a similar energy decomposition between two nonbonded fragments of the same molecule, covalently connected by a third fragment. In this work, we explore an alternative approach where the noncovalent interaction is singled out by a range separation of the Coulomb potential. We investigate two common splittings of the 1/r potential into long-range and short-range parts based on the Gaussian and error functions, and approximate either the entire intermolecular/interfragment interaction or only its attractive terms by the long-range contribution. These range separation schemes are tested for a number of intermolecular and intramolecular complexes. We find that the energy corrections from range-separated SAPT or ISAPT are in reasonable agreement with complete SAPT/ISAPT data. This result should be contrasted with the inability of the long-range multipole expansion to describe crucial short-range charge penetration and exchange effects; it shows that the long-range interaction potential does not just recover the asymptotic interaction energy but also provides a useful account of short-range terms. The best consistency is attained for the error-function separation applied to all interaction terms, both attractive and repulsive. This study is the first step toward a fragmentation-free decomposition of intramolecular nonbonded energy.

Basis-Set Limit CCSD(T) Energies for Large Molecules with Local Natural Orbitals and Reduced-Scaling Basis-Set Corrections

The calculation of density-based basis-set correction (DBBSC), which remedies the basis-set incompleteness (BSI) error of the correlation energy, is combined with local approximations. Aiming at large-scale applications, the procedure is implemented in our efficient local natural orbital-based coupled-cluster singles and doubles with perturbative triples [LNO-CCSD(T)] scheme. To this end, the range-separation function, which characterizes the one-electron BSI in space, is decomposed into the sum of contributions from individual localized molecular orbitals (LMOs). A compact domain is constructed around each LMO, and the corresponding contributions are evaluated only within these restricted domains. Furthermore, for the calculation of the complementary auxiliary basis set (CABS) correction, which significantly improves the Hartree-Fock (HF) energy, the local density fitting approximation is utilized. The errors arising from the local approximations are examined in detail, efficient prescreening techniques are introduced to compress the numerical quadrature used for DBBSC, and conservative default thresholds are selected for the truncation parameters. The efficiency of the DBBSC-LNO-CCSD(T) method is demonstrated through representative examples of up to 1000 atoms. Based on the numerical results, we conclude that the corrections drastically reduce the BSI error using double-ζ basis sets, often to below 1 kcal/mol compared to the reliable LNO-CCSD(T) complete basis set references, while significant improvements are also achieved with triple-ζ basis sets. Considering that the calculation of the DBBSC and CABS corrections only moderately increases the wall-clock time required for the post-HF steps in practical applications, the proposed DBBSC-LNO-CCSD(T) method offers a highly efficient and robust tool for large-scale calculations.

Compactification of determinant expansions via transcorrelation

Although selected configuration interaction (SCI) algorithms can tackle much larger Hilbert spaces than the conventional full CI method, the scaling of their computational cost with respect to the system size remains inherently exponential. In addition, inaccuracies in describing the correlation hole at small interelectronic distances lead to the slow convergence of the electronic energy relative to the size of the one-electron basis set. To alleviate these effects, we show that the non-Hermitian, transcorrelated (TC) version of SCI significantly compactifies the determinant space, allowing us to reach a given accuracy with a much smaller number of determinants. Furthermore, we note a significant acceleration in the convergence of the TC-SCI energy as the basis set size increases. The extent of this compression and the energy convergence rate are closely linked to the accuracy of the correlation factor used for the similarity transformation of the Coulombic Hamiltonian. Our systematic investigation of small molecular systems in increasingly large basis sets illustrates the magnitude of these effects.

Energy and spectroscopic parameters of neutral and cations isomers of the CnH2 (n = 2-6) families using high-level ab-initio approaches

Cationic species, previously detected from ion-induced desorption of solid methane by plasma desorption mass spectrometry (PDMS), and neutral species, are investigated using high-level ab-initio approaches. From a set of 25 cationic and 26 neutral structures belonging to CnH2 (n = 2-6) families, it was obtained the energy, rotational constants, harmonic vibrational frequency, charge distribution and excitation energies. The ZPVE-corrected energies, at CCSD(T)-F12; CCSD(T)-F12/RI/(cc-pVTZ-F12, cc-pVTZ-F12-CABS, cc-pVQZ/C) (n = 2-5) and CCSD(T)/cc-pVTZ (n = 6) levels, reveal that the topology of the most stable isomer vary with n and the charge. Out of 674 harmonic frequencies, those with maximum intensity are generally in the 3000-3500 cm-1 range. Analysis of 169 vertical transition energies calculated with the EOM-CCSD approach, suggest three C6H2 species as potential carriers of the diffuse interstellar bands (DIB). Systematic comparison of properties between neutral and cationic species can assist in the structural description of complex matrices.

Striking the right balance of encoding electron correlation in the Hamiltonian and the wavefunction ansatz

Multi-configurational electronic structure theory delivers the most versatile approximations to many-electron wavefunctions, flexible enough to deal with all sorts of transformations, ranging from electronic excitations, to open-shell molecules and chemical reactions. Multi-configurational models are therefore essential to establish universally applicable, predictive ab initio methods for chemistry. Here, we present a discussion of explicit correlation approaches which address the nagging problem of dealing with static and dynamic electron correlation in multi-configurational active-space approaches. We review the latest developments and then point to their key obstacles. Our discussion is supported by new data obtained with tensor network methods. We argue in favor of simple electron-only correlator expressions that may allow one to define transcorrelated models in which the correlator does not bear a dependence on molecular structure.

Accelerated basis-set convergence of coupled-cluster excitation energies using the density-based basis-set correction method

We present the first application to real molecular systems of the recently proposed linear-response theory for the density-based basis-set correction method [J. Chem. Phys., 158, 234107 (2023)]. We apply this approach to accelerate the basis-set convergence of excitation energies in the equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) method. We use an approximate linear-response framework that neglects the second-order derivative of the basis-set correction density functional and consists in simply adding to the usual Hamiltonian the one-electron potential generated by the first-order derivative of the functional. This additional basis-set correction potential is evaluated at the Hartree-Fock density, leading to a very computationally cheap basis-set correction. We tested this approach over a set of about 30 excitation energies computed for five small molecular systems and found that the excitation energies from the ground state to Rydberg states are the main source of basis-set error. These excitation energies systematically increase when the size of the basis set is increased, suggesting a biased description in favour of the excited state. Despite the simplicity of the present approach, the results obtained with the basis-set-corrected EOM-CCSD method are encouraging as they yield a mean absolute deviation of 0.02 eV for the aug-cc-pVTZ basis set, while it is 0.04 eV using the standard EOM-CCSD method. This might open the path to an alternative to explicitly correlated approaches to accelerate the basis-set convergence of excitation energies.

Orbital optimisation in xTC transcorrelated methods

We present a combination of the bi-orthogonal orbital optimisation framework with the recently introduced xTC version of transcorrelation. This allows us to implement non-iterative perturbation based methods on top of the transcorrelated Hamiltonian. Additionally, the orbital optimisation influences results of other truncated methods, such as the distinguishable cluster with singles and doubles. The accuracy of these methods in comparison to standard xTC methods is demonstrated, and the advantages and disadvantages of the orbital optimisation are discussed.

Dynamical Electron Correlation and the Chemical Bond. IV. Covalent Bonds in A2 Molecules (A = N-As and F-Br)

In a series of recent papers, we investigated the effect of dynamical electron correlation on the potential energy curves and spectroscopic constants of several diatomic molecules, including the simple diatomic hydrides (AH) and the more complex diatomic fluorides (AF) and homonuclear diatomic molecules (A2) with A = B-F (AF) or A = C-F (A2), respectively. Our goal was to understand the dependence of the dynamical electron correlation energy, EDEC, on the internuclear distance, R, and quantify how dynamical electron correlation influences the spectroscopic constants (De, Re, and ωe) of these molecules. At large R, we found that the magnitude of EDEC(R) had a simple dependence on R, with EDEC(R) increasing nearly exponentially with decreasing R. However, as R continued to decrease, there were significant variations in EDEC(R). These variations led to differing changes in the predicted spectroscopic constants of the molecules. In many molecules, the changes in EDEC(R) could be correlated with changes in the underlying spin-coupled generalized valence bond wave function, either in the orbitals or the spin-coupling coefficients. In the current paper, we extend these studies to higher main group elements, comparing the effects of EDEC(R) on P2 and As2 versus N2, and on Cl2 and Br2 versus F2. We find that there are significant differences between the effects of dynamical electron correlation on the molecules in the first and subsequent rows of the periodic table.

Investigating the Basis Set Convergence of Diagrammatically Decomposed Coupled-Cluster Correlation Energy Contributions for the Uniform Electron Gas

We investigate the convergence of coupled-cluster (CC) correlation energies and related quantities with respect to the employed basis set size for the uniform electron gas (UEG) to gain a better understanding of the basis set incompleteness error (BSIE). To this end, coupled-cluster doubles (CCD) theory is applied to the three-dimensional UEG for a range of densities, basis set sizes, and electron numbers. We present a detailed analysis of individual diagrammatically decomposed contributions to the amplitudes at the level of CCD theory. In particular, we show that only two terms from the amplitude equations contribute to the asymptotic large-momentum behavior of the transition structure factor, corresponding to the cusp region at short interelectronic distances. However, due to the coupling present in the amplitude equations, all decomposed correlation energy contributions show the same asymptotic convergence behavior to the complete basis set limit. These findings provide an additional rationale for the success of a recently proposed correction to the BSIE of CC theory. Lastly, we examine the BSIE in the CCD plus perturbative triples [CCD(T)] method, as well as in the newly proposed CCD plus complete perturbative triples [CCD(cT)] method.

Ion molecule reactions in the HBr+ + CH4 system: a combined experimental and theoretical study

Reactions in the system HBr+ + CH4 have been investigated inside a guided ion-beam apparatus under single-collision conditions. The HBr+ is vibrational and rotational state selected in the electronic X2Π1/2 state created by (2+1)-REMPI. Due to the exitation scheme employed different rotational states of the HBr+ are accessible. Four reaction channels have been observed. The cross section, σ, for the exothermic proton transfer channel (PT) decreases with increasing collision energy, steeper than predicted by the Langevin model. The cross section also decreases with increasing rotational energy in the HBr+, with the effect of the rotational energy being stronger than that of translational energy. The cross section for the endothermic charge transfer (CT) increased with increasing collision energy. The energy dependence is well reproduced by a simple line of center (loc) model. Although the bromine transfer (BT) is exothermic the observed cross section increased with increasing collision energy due to an activation barrier on the potential energy surface (PES). Analysis by a modified loc model suggest the relevance of an angle dependence of σ. The cross section for the endothermic hydrogen atom abstraction (HA) exhibits a maximum at 2 eV Ecm. The measured cross sections are rationalized by means of reaction dynamics simulations which show good agreement with the experimental cross sections. The dynamics simulations are carried out with a machine learning potential that is developed and benchmarked with ab initio molecular dynamics simulation. The absolute cross sections predicted by reaction dynamics simulations are well within the same order of magnitude while reproducing the trends over three different collision energies for all four reaction channels. Furthermore, the simulations demonstrate various reaction mechanisms for these reaction channels, including a very interesting HBr+ orientation selectivity for the BT reaction channel.

Monohydrides of the Group 13 Elements M=B, Al and Ga: Axial Bi-Nucleophilicity and the Propensity to Form Both H-M⋅⋅⋅HX and M-H⋅⋅⋅HX Hydrogen Bonds (X=F, Cl, Br, I, CN, CCH, CP)

Equilibrium dissociation energies De of the hydrogen-bonded complexes HAl⋅⋅⋅HX and HGa⋅⋅⋅HX (X=F, Cl, Br, I, CN, CCH, and CP) were calculated ab initio at the CCSD(T)-(F12c)/cc-pVDZ-F12 level of theory. The gradients of graphs of De versus the electrophilicity EHX of the Lewis acids HX yielded the nucleophilicities NM-X of the Group 13 atoms M in these diatomic molecules. Molecular electrostatic surfaces potentials reveal that H-Al and H-Ga are bi-nucleophilic and that the H ends of these H-M molecules are more nucleophilic than the M ends for M=Al and Ga, but not when M=boron. Therefore, the complexes M-H⋅⋅⋅HX were investigated using the same approach. It was concluded for M=Al and Ga that, for a given X, the M-H⋅⋅⋅HX complexes were more strongly bound than the corresponding H-M⋅⋅⋅HX complexes for both M=Al and Ga but the reverse order applies for M = boron. The effects of substituting the H atoms in the M-H molecules by F atoms and by methyl groups were investigated to measure the -I and +I inductive effects relative to H, respectively, on the nucleophilicities of the molecules M-H when M is acting as hydrogen-bond acceptor in complexes H-M⋅⋅⋅H-X.

A density-fitting implementation of the density-based basis-set correction method

This work reports an efficient density-fitting implementation of the density-based basis-set correction (DBBSC) method in the MOLPRO software. This method consists in correcting the energy calculated by a wave-function method with a given basis set by an adapted basis-set correction density functional incorporating the short-range electron correlation effects missing in the basis set, resulting in an accelerated convergence to the complete-basis-set limit. Different basis-set correction density-functional approximations are explored and the complementary-auxiliary-basis-set single-excitation correction is added. The method is tested on a benchmark set of reaction energies at the second-order Møller-Plesset (MP2) level and a comparison with the explicitly correlated MP2-F12 method is provided. The results show that the DBBSC method greatly accelerates the basis convergence of MP2 reaction energies, without reaching the accuracy of the MP2-F12 method but with a lower computational cost.

Toward Real Chemical Accuracy on Current Quantum Hardware Through the Transcorrelated Method

Quantum computing is emerging as a new computational paradigm with the potential to transform several research fields including quantum chemistry. However, current hardware limitations (including limited coherence times, gate infidelities, and connectivity) hamper the implementation of most quantum algorithms and call for more noise-resilient solutions. We propose an explicitly correlated Ansatz based on the transcorrelated (TC) approach to target these major roadblocks directly. This method transfers, without any approximation, correlations from the wave function directly into the Hamiltonian, thus reducing the resources needed to achieve accurate results with noisy quantum devices. We show that the TC approach allows for shallower circuits and improves the convergence toward the complete basis set limit, providing energies within chemical accuracy to experiment with smaller basis sets and, thus, fewer qubits. We demonstrate our method by computing bond lengths, dissociation energies, and vibrational frequencies close to experimental results for the hydrogen dimer and lithium hydride using two and four qubits, respectively. To demonstrate our approach's current and near-term potential, we perform hardware experiments, where our results confirm that the TC method paves the way toward accurate quantum chemistry calculations already on today's quantum hardware.

Approaching the Complete Basis Set Limit for Spin-State Energetics of Mononuclear First-Row Transition Metal Complexes

Convergence to the complete basis set (CBS) limit is analyzed for the problem of spin-state energetics in mononuclear first-row transition metal (TM) complexes by taking under scrutiny a benchmark set of 18 energy differences between spin states for 13 chemically diverse TM complexes. The performance of conventional CCSD(T) and explicitly correlated CCSD(T)-F12a/b calculations in approaching the CCSD(T)/CBS limits is systematically studied. An economic computational protocol is developed based on the CCSD-F12a approximation and (here proposed) modified scaling of the perturbative triples term (T#). This computational protocol recovers the relative spin-state energetics of the benchmark set in excellent agreement with the reference CCSD(T)/CBS limits (mean absolute deviation of 0.4, mean signed deviation of 0.2, and maximum deviation of 0.8 kcal/mol) and enables performing canonical CCSD(T) calculations for mononuclear TM complexes sized up to ca. 50 atoms, which is illustrated by application to heme-related metalloporphyrins. Furthermore, a good transferability of the basis set incompleteness error (BSIE) is demonstrated for spin-state energetics computed using CCSD(T) and other wave function methods (MP2, CASPT2, CASPT2/CC, NEVPT2, and MRCI + Q), which justifies efficient focal-point approximations and simplifies the construction of multimethod benchmark studies.

Improved CPS and CBS Extrapolation of PNO-CCSD(T) Energies: The MOBH35 and ISOL24 Data Sets

Computation of heats of reaction of large molecules is now feasible using the domain-based pair natural orbital (PNO)-coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] theory. However, to obtain agreement within 1 kcal/mol of experiment, it is necessary to eliminate basis set incompleteness error, which comprises both the AO basis set error and the PNO truncation error. Our investigation into the convergence to the canonical limit of PNO-CCSD(T) energies with the PNO truncation threshold T shows that errors follow the model E ( T ) = E + A T 1 / 2 . Therefore, PNO truncation errors can be eliminated using a simple two-point CPS extrapolation to the canonical limit so that subsequent CBS extrapolation is not limited by the residual PNO truncation error. Using the ISOL24 and MOBH35 data sets, we find that PNO truncation errors are larger for molecules with significant static correlation and that it is necessary to use very tight thresholds of T = 10 - 8 to ensure that errors do not exceed 1 kcal/mol. We present a lower-cost extrapolation scheme that uses information from small basis sets to estimate the PNO truncation errors for larger basis sets. In this way, the canonical limit of CCSD(T) calculations on sizable molecules with large basis sets can be reliably estimated in a practical way. Using this approach, we report near complete basis set (CBS)-CCSD(T) reaction energies for the full ISOL24 and MOBH35 data sets.

Exploration of the Two-Electron Excitation Space with Data-Driven Coupled Cluster

Computational cost limits the applicability of post-Hartree-Fock methods such as coupled-cluster on larger molecular systems. The data-driven coupled-cluster (DDCC) method applies machine learning to predict the coupled-cluster two-electron amplitudes (t2) using data from second-order perturbation theory (MP2). One major limitation of the DDCC models is the size of training sets that increases exponentially with the system size. Effective sampling of the amplitude space can resolve this issue. Five different amplitude selection techniques that reduce the amount of data used for training were evaluated, an approach that also prevents model overfitting and increases the portability of data-driven coupled-cluster singles and doubles to more complex molecules or larger basis sets. In combination with a localized orbital formalism to predict the CCSD t2 amplitudes, we have achieved a 10-fold error reduction for energy calculations.

Constraints upon Functionals of the 1-Matrix, Universal Properties of Natural Orbitals, and the Fallacy of the Collins "Conjecture"

Reliability of quantum-chemical calculations based upon the density functional theory and its 1-matrix counterpart hinges upon minimizing the extent of empirical parameterization in the approximate energy expressions of these formalisms while imposing as many rigorous constraints upon them as possible. The recently uncovered universal properties of the natural orbitals facilitate the construction of such constraints for the 1-matrix functionals. The benefits of their employment in the validation of these functionals are vividly demonstrated by a critical review of the three incarnations of the so-called Collins conjecture. Although the incorporation of rigorous definitions of the correlation energy and entropy, and the identification of individual potential energy hypersurfaces as probable domains of its applicability turn the originally published unsubstantiated claim into a proper conjecture, the resulting formalism is found to be merely a conduit for incorporation of static correlation effects in electronic structure calculations that is unlikely to allow attaining chemical accuracy.

Developing correlation-consistent numeric atom-centered orbital basis sets for krypton: Applications in RPA-based correlated calculations

Localized atomic orbitals are the preferred basis set choice for large-scale explicit correlated calculations, and high-quality hierarchical correlation-consistent basis sets are a prerequisite for correlated methods to deliver numerically reliable results. At present, numeric atom-centered orbital (NAO) basis sets with valence correlation consistency (VCC), designated as NAO-VCC-nZ, are only available for light elements from hydrogen (H) to argon (Ar) [Zhang et al., New J. Phys. 15, 123033 (2013)]. In this work, we extend this series by developing NAO-VCC-nZ basis sets for krypton (Kr), a prototypical element in the fourth row of the periodic table. We demonstrate that NAO-VCC-nZ basis sets facilitate the convergence of electronic total-energy calculations using the Random Phase Approximation (RPA), which can be used together with a two-point extrapolation scheme to approach the complete basis set limit. Notably, the Basis Set Superposition Error (BSSE) associated with the newly generated NAO basis sets is minimal, making them suitable for applications where BSSE correction is either cumbersome or impractical to do. After confirming the reliability of NAO basis sets for Kr, we proceed to calculate the Helmholtz free energy for Kr crystal at the theoretical level of RPA plus renormalized single excitation correction. From this, we derive the pressure-volume (P-V) diagram, which shows excellent agreement with the latest experimental data. Our work demonstrates the capability of correlation-consistent NAO basis sets for heavy elements, paving the way toward numerically reliable correlated calculations for bulk materials.

Basis Set Limit of CCSD(T) Energies: Explicit Correlation Versus Density-Based Basis-Set Correction

A thorough comparison is carried out for explicitly correlated and density-based basis-set correction approaches, which were primarily developed to mitigate the basis-set incompleteness error of wave function methods. An efficient implementation of the density-based scheme is also presented, utilizing the density-fitting approximation. The performance of these approaches is comprehensively tested for the second-order Møller-Plesset (MP2), coupled-cluster singles and doubles (CCSD), and CCSD with perturbative triples [CCSD(T)] methods with respect to the corresponding complete basis set references. It is demonstrated that the density-based correction together with complementary auxiliary basis set (CABS)-corrected Hartree-Fock energies is highly robust and effectively reduces the error of the standard approaches; however, it does not outperform the corresponding explicitly correlated methods. Nevertheless, what still makes the density-corrected CCSD and CCSD(T) methods competitive is that their computational costs are roughly half of those of the corresponding explicitly correlated variants. Additionally, an incremental approach for standard CCSD and CCSD(T) is introduced. In this simple scheme, the total energies are corrected with the CABS correction and explicitly correlated MP2 contributions. As demonstrated, the resulting methods yield surprisingly good results, below 1 kcal/mol for thermochemical properties even with a double-ζ basis, while their computational expenses are practically identical to those of the density-based basis-set correction approaches.

Linear-Scaling Local Natural Orbital CCSD(T) Approach for Open-Shell Systems: Algorithms, Benchmarks, and Large-Scale Applications

The extension of the highly optimized local natural orbital (LNO) coupled cluster (CC) with single-, double-, and perturbative triple excitations [LNO-CCSD(T)] method is presented for high-spin open-shell molecules based on restricted open-shell references. The techniques enabling the outstanding efficiency of the closed-shell LNO-CCSD(T) variant are adopted, including the iteration- and redundancy-free second-order Møller-Plesset and (T) formulations as well as the integral-direct, memory- and disk use-economic, and OpenMP-parallel algorithms. For large molecules, the efficiency of our open-shell LNO-CCSD(T) method approaches that of its closed-shell parent method due to the application of restricted orbital sets for demanding integral transformations and a novel approximation for higher-order long-range spin-polarization effects. The accuracy of open-shell LNO-CCSD(T) is extensively tested for radicals and reactions thereof, ionization processes, as well as spin-state splittings, and transition-metal compounds. At the size range where the canonical CCSD(T) reference is accessible (up to 20-30 atoms), the average open-shell LNO-CCSD(T) correlation energies are found to be 99.9 to 99.95% accurate, which translates into average absolute deviations of a few tenths of kcal/mol in the investigated energy differences already with the default settings. For more extensive molecules, the local errors may grow, but they can be estimated and decreased via affordable systematic convergence studies. This enables the accurate modeling of large systems with complex electronic structures, as illustrated on open-shell organic radicals and transition-metal complexes of up to 179 atoms as well as on challenging biochemical systems, including up to 601 atoms and 11,000 basis functions. While the protein models involve difficulties for local approximations, such as the spin states of a bounded iron ion or an extremely delocalized singly occupied orbital, the corresponding single-node LNO-CCSD(T) computations were feasible in a matter of days with 10s to 100 GB of memory use. Therefore, the new LNO-CCSD(T) implementation enables highly accurate computations for open-shell systems of unprecedented size and complexity with widely accessible hardware.

Cluster-in-Molecule Approach with Explicitly Correlated Methods for Large Molecules

In this article, we present a series of explicitly correlated local correlation methods developed under the cluster-in-molecule (CIM) framework, including explicitly correlated second-order Møller-Plesset perturbation (MP2), coupled-cluster singles and doubles (CCSD), domain-based local pair natural orbital CCSD (DLPNO-CCSD), and DLPNO-CCSD with perturbative triples (DLPNO-CCSD(T)). In these methods, F12 correction is decomposed into contributions from each occupied local molecular orbital and then evaluated independently in a given cluster, which consists of a subset of localized orbitals. These newly developed methods allow F12 calculations of large molecules (up to 145 atoms for quasi-one-dimensional systems) on a single node. We use these methods to investigate the relative stability between extended and folded alkane C30H62, the relative stability of four secondary structures of a polyglycine Ace(Gly)10NH2, and the binding energies of two host-guest complexes. The results demonstrate that the combination of CIM with F12 methods is a promising way to investigate large molecules with small basis set errors.

Concerted versus stepwise proton transfer reactions in the [2, 2'-bipyridyl]-3-3'-diol molecule: A static and dynamic ab-initio investigation

The double proton transfer (PT) reaction has been investigated in the [2,2'-bipyridyl]-3-3'-diol, a complex molecule where the proton movements is coupled to significant rearrangement of the electronic structure. Moreover, the reaction could be concerted, that is the two protons are exchanged simultaneously, or stepwise, where the two protons are transferred sequentially. To this end, a static exploration of the potential energy surface (PES) was carried together with the analysis of the free-energy surface (FES), both surfaces being evaluated at density functional theory level and different exchange-correlation functionals. While the concerted mechanism has been clearly discharged, the characteristics of the stepwise PT significantly depends on the chosen functionals, some suggesting a clear stepwise mechanism characterized by a stable reaction intermediates and two transitions states, whereas other approaches propend for a asynchronous PT, with a single TS. These features appear on both PES and FES, albeit some differences appears due to their different nature.

Nonunitary projective transcorrelation theory inspired by the F12 ansatz

An alternative nonunitary transcorrelation, inspired by the F12 ansatz, is investigated. In contrast to the Jastrow transcorrelation of Boys-Handy, the effective Hamiltonian of this projective transcorrelation features: 1. a series terminating formally at four-body interactions. 2. no spin-contamination within the non-relativistic framework. 3. simultaneous satisfaction of the singlet and triplet first-order cusp conditions. 4. arbitrary choices of pairs for correlation including frozen-core approximations. We discuss the connection between the projective transcorrelation and F12 theory with applications to small molecules, to show that the cusp conditions play an important role to reduce the uncertainty arising from the nonunitary transformation.

Direct Determination of Optimal Real-Space Orbitals for Correlated Electronic Structure of Molecules

We demonstrate how to determine numerically nearly exact orthonormal orbitals that are optimal for the evaluation of the energy of arbitrary (correlated) states of atoms and molecules by minimization of the energy Lagrangian. Orbitals are expressed in real space using a multiresolution spectral element basis that is refined adaptively to achieve the user-specified target precision while avoiding the ill-conditioning issues that plague AO basis set expansions traditionally used for correlated models of molecular electronic structure. For light atoms, the orbital solver, in conjunction with a variational electronic structure model [selected Configuration Interaction (CI)] provides energies of comparable precision to a state-of-the-art atomic CI solver. The computed electronic energies of atoms and molecules are significantly more accurate than the counterparts obtained with the orbital sets of the same rank expanded in Gaussian AO bases, and can be determined even when linear dependence issues preclude the use of the AO bases. It is feasible to optimize more than 100 fully correlated numerical orbitals on a single computer node, and significant room exists for additional improvement. These findings suggest that real-space orbital representations might be the preferred alternative to AO representations for high-end models of correlated electronic states of molecules and materials.

A Perspective on Sustainable Computational Chemistry Software Development and Integration

The power of quantum chemistry to predict the ground and excited state properties of complex chemical systems has driven the development of computational quantum chemistry software, integrating advances in theory, applied mathematics, and computer science. The emergence of new computational paradigms associated with exascale technologies also poses significant challenges that require a flexible forward strategy to take full advantage of existing and forthcoming computational resources. In this context, the sustainability and interoperability of computational chemistry software development are among the most pressing issues. In this perspective, we discuss software infrastructure needs and investments with an eye to fully utilize exascale resources and provide unique computational tools for next-generation science problems and scientific discoveries.

Basis-Set-Error-Free Random-Phase Approximation Correlation Energies for Atoms Based on the Sternheimer Equation

The finite basis set errors for all-electron random-phase approximation (RPA) correlation energy calculations are analyzed for isolated atomic systems. We show that, within the resolution-of-identity (RI) RPA framework, the major source of the basis set errors is the incompleteness of the single-particle atomic orbitals used to expand the Kohn-Sham eigenstates, instead of the auxiliary basis set (ABS) to represent the density response function χ0 and the bare Coulomb operator v. By solving the Sternheimer equation for the first-order wave function on a dense radial grid, we are able to eliminate the major error─the incompleteness error of the single-particle atomic basis set─for atomic RPA calculations. The error stemming from a finite ABS can be readily rendered vanishingly small by increasing the size of the ABS, or by iteratively determining the eigenmodes of the χ0v operator. The variational property of the RI-RPA correlation energy can be further exploited to optimize the ABS in order to achieve fast convergence of the RI-RPA correlation energy. These numerical techniques enable us to obtain basis-set-error-free RPA correlation energies for atoms, and in this work, such energies for atoms from H to Kr are presented. The implications of the numerical techniques developed in the present work for addressing the basis set issue for molecules and solids are discussed.

Correlation Consistent Basis Sets for Explicitly Correlated Theory: The Transition Metals

We present correlation consistent basis sets for explicitly correlated (F12) calculations, denoted VnZ(-PP)-F12-wis (n = D,T), for the d-block elements. The cc-pVDZ-F12-wis basis set is contracted to [8s7p5d2f] for the 3d-block, while its ECP counterpart for the 4d and 5d-blocks, cc-pVDZ-PP-F12-wis, is contracted to [6s6p5d2f]. The corresponding contracted sizes for cc-pVTZ(-PP)-F12-wis are [9s8p6d3f2g] for the 3d-block elements and [7s7p6d3f2g] for the 4d and 5d-block elements. Our VnZ(-PP)-F12-wis basis sets are evaluated on challenging test sets for metal-organic barrier heights (MOBH35) and group-11 metal clusters (CUAGAU-2). In F12 calculations, they are found to be about as close to the complete basis set limit as the combination of standard cc-pVnZ-F12 on main-group elements with the standard aug-cc-pV(n+1)Z(-PP) basis sets on the transition metal(s). While our basis sets are somewhat more compact than aug-cc-pV(n+1)Z(-PP), the CPU time benefit is negligible for catalytic complexes that contain only one or two transition metals among dozens of main-group elements; however, it is somewhat more significant for metal clusters.

Correlation consistent auxiliary basis sets in density fitting Hartree-Fock: The atoms sodium through argon revisited

We present a series of auxiliary basis sets, for the elements Na to Ar, for use in density-fitted Hartree-Fock calculations with the correlation consistent cc-pV(n + d)Z orbital basis sets. Benchmarking on total molecular energies, reaction energies and the spectroscopic constants of the SO molecule demonstrate that the new sets address the deficiencies of using existing auxiliary sets in combination with these orbital basis sets. We also report auxiliary basis sets for Na and Mg matched to cc-pVnZ, along with recommendations for pairing auxiliary basis sets to the cc-pVnZ-F12 basis sets for Hartree-Fock calculations.

Energy natural orbital characterization of nonadiabatic electron wavepackets in the densely quasi-degenerate electronic state manifold

Dynamics and energetic structure of largely fluctuating nonadiabatic electron wavepackets are studied in terms of Energy Natural Orbitals (ENOs) [K. Takatsuka and Y. Arasaki, J. Chem. Phys. 154, 094103 (2021)]. Such huge fluctuating states are sampled from the highly excited states of clusters of 12 boron atoms (B₁₂), which have densely quasi-degenerate electronic excited-state manifold, each adiabatic state of which gets promptly mixed with other states through the frequent and enduring nonadiabatic interactions within the manifold. Yet, the wavepacket states are expected to be of very long lifetimes. This excited-state electronic wavepacket dynamics is extremely interesting but very hard to analyze since they are usually represented in large time-dependent configuration interaction wavefunctions and/or in some other complicated forms. We have found that ENO gives an invariant energy orbital picture to characterize not only the static highly correlated electronic wavefunctions but also those time-dependent electronic wavefunctions. Hence, we first demonstrate how the ENO representation works for some general cases, choosing proton transfer in water dimer and electron-deficient multicenter chemical bonding in diborane in the ground state. We then penetrate with ENO deep into the analysis of the essential nature of nonadiabatic electron wavepacket dynamics in the excited states and show the mechanism of the coexistence of huge electronic fluctuation and rather strong chemical bonds under very random electron flows within the molecule. To quantify the intra-molecular energy flow associated with the huge electronic-state fluctuation, we define and numerically demonstrate what we call the electronic energy flux.

Kylin 1.0: An ab-initio density matrix renormalization group quantum chemistry program

The accurate evaluation of electron correlations is highly necessary for the proper descriptions of the electronic structures in strongly correlated molecules, ranging from bond-dissociating molecules, polyradicals, to large conjugated molecules and transition metal complexes. For this purpose, in this paper, a new ab-initio quantum chemistry program Kylin 1.0 for electron correlation calculations at various quantum many-body levels, including configuration interaction (CI), perturbation theory (PT), and density matrix renormalization group (DMRG), is presented. Furthermore, fundamental quantum chemistry methods such as Hartree-Fock self-consistent field (HF-SCF) and the complete active space SCF (CASSCF) are also implemented. The Kylin 1.0 program possesses the following features: (1) a matrix product operator (MPO) formulation-based efficient DMRG implementation for describing static electron correlation within a large active space composed of more than 100 orbitals, supporting both U 1 n × U 1 S z $$ \mathrm{U}{(1)}_{\mathrm{n}}\times \mathrm{U}{(1)}_{{\mathrm{S}}_{\mathrm{z}}} $$ and U 1 n × SU 2 S $$ \mathrm{U}{(1)}_{\mathrm{n}}\times \mathrm{SU}{(2)}_{\mathrm{S}} $$ symmetries; (2) an efficient second-order DMRG-self-consistent field (SCF) implementation; (3) an externally contracted multi-reference CI (MRCI) and Epstein-Nesbet PT with DMRG reference wave functions for including the remaining dynamic electron correlation outside the large active spaces. In this paper, we introduce the capabilities and numerical benchmark examples of the Kylin 1.0 program.

Influence of the complete basis set approximation, tight weighted-core, and diffuse functions on the DLPNO-CCSD(T1) atomization energies of neutral H,C,O-compounds

The impact of complete basis set extrapolation schemes (CBS), diffuse functions, and tight weighted-core functions on enthalpies of formation predicted via the DLPNO-CCSD(T1) reduced Feller-Peterson-Dixon approach has been examined for neutral H,C,O-compounds. All tested three-point (TZ/QZ/5Z) extrapolation schemes result in mean unsigned deviation (MUD) below 2 kJ mol^-1 relative to the experiment. The two-point QZ/5Z and TZ/QZ CBS 1 / l max 3 extrapolation schemes are inferior to their inverse power counterpart ( 1 / l max + 1 / 2 4 ) by 1.3 and 4.3 kJ mol^-1 . The CBS extrapolated frozen core atomization energies are insensitive (within 1 kJ mol^-1 ) to augmentation of the basis set with tight weighted core functions. The core-valence correlation effects converge already at triple-ζ, although double-ζ/triple-ζ CBS extrapolation performs better and is recommended. The effect of diffuse function augmentation converges slowly, and cannot be reproduced with double- ζ or triple- ζ calculations as these are plagued with basis set superposition and incompleteness errors.

Reduced Nucleophilicities ИB of Lewis Bases B: Is ИB Independent of Whether B is Involved in a Hydrogen Bond or a Halogen Bond?

Reduced nucleophilicities И_B of axially symmetric molecules B were determined from , where D_e is the equilibrium dissociation energy of the complexes B⋅⋅⋅XY, N_B is the nucleophilicity of B, E_XY is the electrophilicity of the halogen-bond donor XY and σ min ${{\sigma }_{{\rm { min}}}{\rm \ }}$ is the minimum electrostatic surface potential of B. The series B⋅⋅⋅ClY, B⋅⋅⋅BrY, B⋅⋅⋅IY (Y=F, Cl, Br, I, CN, and CCH) as well as (B⋅⋅⋅XY, XY=F₂ , Cl₂ , Br₂ ,and BrCl) of complexes were investigated. Molecules B were grouped so that the terminal atom involved in the halogen bond was fixed within the group. Groups having N as the terminal atom were RCN (R=CH₃ , H, and F) or RN (R=N and P), those with C as the terminal atom were RNC (R=H and F) and RC (R=O, S and Se), and those with a terminal O atom were R=C=O (R=O or S). Graphs of D e ${{D}_{{\rm { e}}}}$ versus E_XY for each group were straight lines through the origin, with generally different gradients, hence implying different N_B . By contrast, when D e / σ min ${{D}_{{\rm { e}}}/{\sigma }_{{\rm { min}}}}$ was the ordinate the lines conflated to give a single straight line, which then defines a common (reduced) nucleophilicity И_B for that group of B. Hence it was concluded that И_B is an intrinsic property of the terminal atom, independent of the remainder of B, and only weakly dependent on the type (C, N or O) of the terminal atom. Moreover, И_B for each B was the same as determined previously from the hydrogen-bonded series B⋅⋅⋅HX, (X=F, Cl, Br, I, CN, CCH, and CP).

A comprehensive benchmark investigation of quantum chemical methods for carbocations

The application of various density functional approximations (DFAs) and an emphasis on popular methods without any consensus have prevailed in computational studies dedicated to carbocations. More importantly, an extensive and rigorous benchmark investigation on density functionals for the class is still lacking. To close this gap, we present a comprehensive benchmark study of quantum chemical methods on a series of classical and nonclassical carbocations, the CARBO33 dataset. We evaluate a total of 107 DFT methods from all rungs giving particular attention to double hybrid density functionals as the potential of the class has been largely undermined in the context of carbocations. To support our findings, DLPNO-CCSD(T) at the complete basis set (CBS) limit and W1-F12 are used as reference methods. Our results indicate that the composite CBS-QB3 method performs poorly and should not be adopted for target energies. Oftentimes, the tested DFAs of a lower rung perform better than several DFAs in a higher rung of Perdew's "Jacob's ladder". Nonetheless, double hybrids DSD-PBEP86-NL and ωB97X-2-D3(BJ) stand out by showing the overall best performance. Among the hybrids evaluated, about half of them show mean absolute deviation (MAD) below 1.1 kcal mol^-1, including the popular hybrids M06-2X and mPW1PW91. In this family, MN15-D3(BJ) performs particularly well (MAD = 0.77 kcal mol^-1) displaying reliable results across various tests. Highly popular B3LYP exhibited one of the worst performances (MAD = 4.74 kcal mol^-1), and we do not recommend its application to carbocations. We also assess the 24 general-purpose basis sets of single- up to quadruple-ζ quality. The best compromise between accuracy and computational cost is achieved with cc-pVTZ followed by def2-TZVP. Computations on larger structures of general interest, including terpene carbocations, are also presented for selected DFT methods confirming general trends in the results.

Basis Set Limit CCSD(T) Energies for Extended Molecules via a Reduced-Cost Explicitly Correlated Approach

Several approximations are introduced and tested to reduce the computational expenses of the explicitly correlated coupled-cluster singles and doubles with perturbative triples [CCSD(T)] method for both closed and open-shell species. First, the well-established frozen natural orbital (FNO) technique is adapted to explicitly correlated CC approaches. Second, our natural auxiliary function (NAF) scheme is employed to reduce the size of the auxiliary basis required for the density fitting approximation regularly used in explicitly correlated calculations. Third, a new approach, termed the natural auxiliary basis (NAB) approximation, is proposed to decrease the size of the auxiliary basis needed for the expansion of the explicitly correlated geminals. The performance of the above approximations and that of the combined FNO-NAF-NAB approach are tested for atomization and reaction energies. Our results show that overall speedups of 7-, 5-, and 3-times can be achieved with double-, triple-, and quadruple-ζ basis sets, respectively, without any loss in accuracy. The new method can provide, e.g., reaction energies and barrier heights well within chemical accuracy for molecules with more than 40 atoms within a few days using a few dozen processor cores, and calculations with 50+ atoms are still feasible. These routinely affordable computations considerably extend the reach of explicitly correlated CCSD(T).

Reduced-Scaling Double Hybrid Density Functional Theory with Rapid Basis Set Convergence through Localized Pair Natural Orbital F12

Following earlier work [Mehta, N.; Martin, J. M. L. J. Chem. Theory Comput.2022, 10.1021/acs.jctc.2c00426] that showed how the slow basis set convergence of the double hybrid density functional theory can be obviated by the use of F12 explicit correlation in the GLPT2 step (second order Görling-Levy perturbation theory), we demonstrate here for the very large and chemically diverse GMTKN55 benchmark suite that the CPU time scaling of this step can be reduced (asymptotically linearized) using the localized pair natural orbital (PNO-L) approximation at negligible cost in accuracy.

Explicitly Correlated Double-Hybrid DFT: A Comprehensive Analysis of the Basis Set Convergence on the GMTKN55 Database

Double-hybrid density functional theory (DHDFT) offers a pathway to accuracy approaching composite wavefunction approaches such as G4 theory. However, the Görling-Levy second-order perturbation theory (GLPT2) term causes them to partially inherit the slow ∝L-3 (with L the maximum angular momentum) basis set convergence of correlated wavefunction methods. This could potentially be remedied by introducing F12 explicit correlation: we investigate the basis set convergence of both DHDFT and DHDFT-F12 (where GLPT2 is replaced by GLPT2-F12) for the large and chemically diverse general main-group thermochemistry, kinetics, and noncovalent interactions (GMTKN55) benchmark suite. The B2GP-PLYP-D3(BJ) and revDSD-PBEP86-D4 DHDFs are investigated as test cases, together with orbital basis sets as large as aug-cc-pV5Z and F12 basis sets as large as cc-pVQZ-F12. We show that F12 greatly accelerates basis set convergence of DHDFs, to the point that even the modest cc-pVDZ-F12 basis set is closer to the basis set limit than cc-pV(Q+d)Z or def2-QZVPPD in orbital-based approaches, and in fact comparable in quality to cc-pV(5+d)Z. Somewhat surprisingly, aug-cc-pVDZ-F12 is not required even for the anionic subsets. In conclusion, DHDF-F12/VDZ-F12 eliminates concerns about basis set convergence in both the development and applications of double-hybrid functionals. Mass storage and I/O bottlenecks for larger systems can be circumvented by localized pair natural orbital approximations, which also exhibit much gentler system size scaling.

Automatic generation of complementary auxiliary basis sets for explicitly correlated methods

Explicitly correlated calculations, aside from the orbital basis set, typically require three auxiliary basis sets: Coulomb-exchange fitting (JK), resolution of the identity MP2 (RI-MP2), and complementary auxiliary basis set (CABS). If unavailable for the orbital basis set and chemical elements of interest, the first two can be auto-generated on the fly using existing algorithms, but not the third. In this paper, we present a quite simple algorithm named autoCABS; a Python implementation under a free software license is offered at Github. For the cc-pVnZ-F12 (n = D,T,Q,5), the W4-08 thermochemical benchmark, and the HFREQ2014 set of harmonic frequencies, we demonstrate that autoCABS-generated CABS basis sets are comparable in quality to purpose-optimized OptRI basis sets from the literature, and that the quality difference becomes entirely negligible as n increases.

A Universal Power Law Governing the Accuracy of Wave Function-Based Electronic Structure Calculations

A universal power law governing the accuracy of wave function-based electronic structure calculations is derived from first principles. The resulting expression ΔE(N,N)N≳19π2gNN, where g is a system-specific factor assuming values between zero and one and ≳ stands for asymptotic inequality at the limit of N→∞, allows facile estimation of the error ΔE(N,N) in the electronic energy of a singlet state of an N-electron system computed with a basis set of N one-electron functions. Several approaches to the estimation of the factor g, which depends on the on-top two-electron density, are presented.

Prediction of Correlation Energies using Variational Subspace Valence Bond

In the variational subspace valence bond (VSVB)1 method the electronic orbitals comprising the wave function correspond to chemically meaningful objects such as bonds, lone pairs, atomic cores, and so on. Selected regions of a molecule (for example, a single chemical bond, as opposed to all the bonds) can be modeled with different levels of basis set and possible methods for modeling correlation from the other regions. The interactions between the components of a molecule (say, a bond and a neighboring orbital) can then be studied in detail for their impact on a chemical phenomenon while avoiding the expense of necessarily applying the higher levels and methods to the entire molecule. This work presents the theoretical basis for modeling correlation effects between specific electron pairs by incorporating terms in the inter-electronic coordinates (') into VSVB. The approach is validated with calculations on small systems using single-reference wave functions.

Quantum Simulation of Molecular Electronic States with a Transcorrelated Hamiltonian: Higher Accuracy with Fewer Qubits

Simulation of electronic structure is one of the most promising applications on noisy intermediate-scale quantum (NISQ) era devices. However, NISQ devices suffer from a number of challenges like limited qubit connectivity, short coherence times, and sizable gate error rates. Thus, desired quantum algorithms should require shallow circuit depths and low qubit counts to take advantage of these devices. Here, we attempt to reduce quantum resource requirements for molecular simulations on a quantum computer while maintaining the desired accuracy with the help of classical quantum chemical theories of canonical transformation and explicit correlation. In this work, compact ab initio Hamiltonians are generated classically, in the second quantized form, through an approximate similarity transformation of the Hamiltonian with (a) an explicitly correlated two-body unitary operator with generalized pair excitations that remove the Coulombic electron-electron singularities from the Hamiltonian and (b) a unitary one-body operator to efficiently capture the orbital relaxation effects required for accurate description of the excited states. The resulting transcorrelated Hamiltonians are able to describe both the ground and the excited states of molecular systems in a balanced manner. Using the variational quantum eigensolver (VQE) method based on the unitary coupled cluster with singles and doubles (UCCSD) ansatz and only a minimal basis set (ANO-RCC-MB), we demonstrate that the transcorrelated Hamiltonians can produce ground state energies comparable to the reference CCSD energies with the much larger cc-pVTZ basis set. This leads to a reduction in the number of required CNOT gates by more than 3 orders of magnitude for the chemical species studied in this work. Furthermore, using the quantum equation of motion (qEOM) formalism in conjunction with the transcorrelated Hamiltonian, we are able to reduce the deviations in the excitation energies from the reference EOM-CCSD/cc-pVTZ values by an order of magnitude. The transcorrelated Hamiltonians developed here are Hermitian and contain only one- and two-body interaction terms and thus can be easily combined with any quantum algorithm for accurate electronic structure simulations.

A litmus test for the balanced description of dispersion interactions and coordination chemistry of lanthanoids

The influence of long-range interactions on the structure of complexes of Eu(III) with four 9-hydroxy-phenalen-1-one ligands (HPLN) and one alkaline earth metal dication [Eu(PLN)₄AE]⁺ (AE: Mg, Ca, Sr, and Ba) is analyzed. Through the [Eu(PLN)₄Ca]⁺ complex, which is a charged complex with two metals-one of them a lanthanoid-and with four relatively fluxional π-ligands, the difficulties of describing such systems are identified. The inclusion of the D3(BJ) or D4 corrections to different density functionals introduces significant changes in the structure, which are shown to stem from the interaction between pairs of PLN ligands. This interaction is studied further with a variety of density functionals, wave-function based methods, and by means of the random phase approximation. By comparing the computed results with those from experimental evidence of gas-phase photoluminescence and ion mobility measurements it is concluded that the inclusion of dispersion corrections does not always yield structures that are in agreement with the experimental findings.

MP2-F12 Basis Set Convergence near the Complete Basis Set Limit: Are h Functions Sufficient?

We have investigated the title question for the W4-08 thermochemical benchmark using l-saturated truncations of a large reference (REF) basis set, as well as for standard F12-optimized basis sets. With the REF basis set, the root-mean-square (RMS) contribution of i functions to the MP2-F12 total atomization energies (TAEs) is about 0.01 kcal/mol, the largest individual contributions being 0.04 kcal/mol for P₂ and P₄. However, even for these cases, basis set extrapolation from {g,h} basis sets adequately addresses the problem. Using basis sets insufficiently saturated in the spdfgh angular momenta may lead to exaggerated i function contributions. For extrapolation from spdfg and spdfgh basis sets, basis set convergence appears to be quite close to the theoretical asymptotic ∝ L^–7 behavior. We hence conclude that h functions are sufficient even for highly demanding F12 applications. With one-parameter extrapolation, spdf and spdfg basis sets are adequate, aug-cc-pV{T,Q}Z-F12 yielding a RMSD = 0.03 kcal/mol. A limited exploration of CCSD(F12*) and CCSD-F12b suggests our conclusions are applicable to higher-level F12 methods as well.

Nature of chemical bond and potential barrier in an invariant energy-orbital picture

Physical nature of the chemical bond and potential barrier is studied in terms of energy natural orbitals (ENOs), which are extracted from highly correlated electronic wavefunctions. ENO provides an objective one-electron picture about energy distribution in a molecule, just as the natural orbitals (NOs) represent one electron view about electronic charge distribution. ENO is invariant in the same sense as NO is, that is, ENOs converge to the exact ones as a series of approximate wavefunctions approach the exact one, no matter how the methods of approximation are adopted. Energy distribution analysis based on ENO can give novel insights about the nature of chemical bonding and formation of potential barriers, besides information based on the charge distribution alone. With ENOs extracted from full configuration interaction wavefunctions in a finite yet large enough basis set, we analyze the nature of chemical bonding of three low-lying electronic states of a hydrogen molecule, all being in different classes of the so-called covalent bond. The mechanism of energy lowering in bond formation, which gives a binding energy, is important, yet not the only concern for this small molecule. Another key notion in chemical bonding is whether a potential basin is well generated stiff enough to support a vibrational state(s) on it. Based on the virial theorem in the adiabatic approximation and in terms of the energy and force analyses with ENOs, we study the roles of the electronic kinetic energy and its nuclear derivative(s) on how they determine the curvature (or the force constant) of the potential basins. The same idea is applied to the potential barrier and, thereby, the transition states. The rate constant within the transition-state theory is formally shown to be described in terms of the electronic kinetic energy and the nuclear derivatives only. Thus, the chemical bonding and rate process are interconnected behind the scenes. Besides this aspect, we pay attention to (1) the effects of electron correlation that manifests itself not only in the orbital energy but also in the population of ENOs and (2) the role of nonadiabaticity (diabatic state mixing), resulting in double basins and a barrier on a single potential curve in bond formation. These factors differentiate a covalent bond into subclasses.

Performance of a one-parameter correlation factor for transcorrelation: Study on a series of second row atomic and molecular systems

In this work, we investigate the performance of a recently proposed transcorrelated (TC) approach based on a single-parameter correlation factor [E. Giner, J. Chem. Phys. 154, 084119 (2021)] for systems involving more than two electrons. The benefit of such an approach relies on its simplicity as efficient numerical-analytical schemes can be set up to compute the two- and three-body integrals occurring in the effective TC Hamiltonian. To obtain accurate ground state energies within a given basis set, the present TC scheme is coupled to the recently proposed TC-full configuration interaction quantum Monte Carlo method [Cohen et al., J. Chem. Phys. 151, 061101 (2019)]. We report ground state total energies on the Li-Ne series, together with their first cations, computed with increasingly large basis sets and compare to more elaborate correlation factors involving electron-electron-nucleus coordinates. Numerical results on the Li-Ne ionization potentials show that the use of the single-parameter correlation factor brings on average only a slightly lower accuracy (1.2 mH) in a triple-zeta quality basis set with respect to a more sophisticated correlation factor. However, already using a quadruple-zeta quality basis set yields results within chemical accuracy to complete basis set limit results when using this novel single-parameter correlation factor. Calculations on the H₂O, CH₂, and FH molecules show that a similar precision can be obtained within a triple-zeta quality basis set for the atomization energies of molecular systems.

Dynamical electron correlation and the chemical bond. I. Covalent bonds in AH and AF (A = B-F)

Dynamical electron correlation has a major impact on the computed values of molecular properties and the energetics of molecular processes. The present study focused on the effect of dynamical electron correlation on the spectroscopic constants, ( Re , ωe , De ) and potential energy curves, Δ E( R), of the covalently-bound AH and AF molecules, A = B-F. The changes in the spectroscopic constants, (Δ Re , Δωe , Δ De ) caused by dynamical correlation are erratic and, at times, even surprising. These changes could be understood based on the dependence of the dynamical electron correlation energies of the AH and AF molecules as a function of the bond distance, i.e., Δ EDEC( R). At large R, the magnitude of Δ EDEC( R) increases nearly exponentially with decreasing R, but this increase slows as R continues to decrease and, in many cases, even reverses at very short R. The changes in Δ EDEC( R) in the region around Re were as unexpected as they were surprising, e.g., distinct minima and maxima were found in the curves of Δ EDEC( R) for the most polar molecules. The variations in Δ EDEC( R) for R ≲ Re are directly correlated with major changes in the electronic structure of the molecules as revealed by a detailed analysis of the SCGVB wave function. The results reported here indicate that we have much to learn about the nature of dynamical electron correlation and its effect on the chemical bond and molecular properties and processes.

Explicitly Correlated Electronic Structure Calculations with Transcorrelated Matrix Product Operators

In this work, we present the first implementation of the transcorrelated electronic Hamiltonian in an optimization procedure for matrix product states by the density matrix renormalization group (DMRG) algorithm. In the transcorrelation ansatz, the electronic Hamiltonian is similarity-transformed with a Jastrow factor to describe the cusp in the wave function at electron-electron coalescence. As a result, the wave function is easier to approximate accurately with the conventional expansion in terms of one-particle basis functions and Slater determinants. The transcorrelated Hamiltonian in first quantization comprises up to three-body interactions, which we deal with in the standard way by applying robust density fitting to two- and three-body integrals entering the second-quantized representation of this Hamiltonian. The lack of hermiticity of the transcorrelated Hamiltonian is taken care of along the lines of the first work on transcorrelated DMRG [ J. Chem. Phys. 2020, 153, 164115] by encoding it as a matrix product operator and optimizing the corresponding ground state wave function with imaginary-time time-dependent DMRG. We demonstrate our quantum chemical transcorrelated DMRG approach at the example of several atoms and first-row diatomic molecules. We show that transcorrelation improves the convergence rate to the complete basis set limit in comparison to conventional DMRG. Moreover, we study extensions of our approach that aim at reducing the cost of handling the matrix product operator representation of the transcorrelated Hamiltonian.

Nucleophilicity of the boron atom in compounds R-B, (R = F, Cl, Br, I, CN, NC, CH3, SiH3, CF3, H): a new look at the inductive effects of the group R

Nucleophilicities N_R-B of molecules R-B (R = F, Cl, Br, I, CN, NC, CH₃, SiH₃, CF₃, H) are determined from the equilibrium dissociation energies D_e of 70 hydrogen-bonded complexes R-B⋯HX (X = F, Cl, Br, I, HCN. HCCH, HCP). The change in N_R-B relative to N_H-B of H-B allows a quantitative measure of the inductive effect I_R of each group R because only the group R affects the electron density associated with the axial non-bonding electron pair carried by the boron in R-B. An alternative definition of I_R, suggested by the strong correlation of the N_R-B values with the minimum value σ_min of the molecular electrostatic surface potential on the 0.001 e Bohr^-3 iso-surface along the R-B axis leads to excellent agreement between the two definitions.