R12 methods in explicitly correlated molecular electronic structure theory

Structure of free complement wavefunction for the ground and the first excited state of helium atom

We review the fundamental ideas of free complement (FC) method through its application on both ground and first excited states of helium atom. We have found that lower energies can be obtained with fewer number of terms in the FC expansion of the ground state wavefunction. In this case, the optimization of orbital exponents was not necessary for achieving spectroscopic accuracy, especially at higher orders where the structure of the FC wavefunction converges to that of the exact one. We have discovered that permanents naturally appear in the FC expansion of the first triplet excited state wavefunction. Including permanents in the FC expansion is shown to be energetically important for the first triplet excited state of helium atom whereas it is not computationally favorable at higher orders. Finally, considering the group theoretical properties of the symmetric group [Formula: see text] and using immanants, a compact and more elegant form for the FC expansion of the first triplet excited state of the helium atom is achieved.

Prescreening and efficiency in the evaluation of integrals over ab initio effective core potentials

New, efficient schemes for the prescreening and evaluation of integrals over effective core potentials (ECPs) are presented. The screening is shown to give a rigorous, and close bound, to within on average 10% of the true value. A systematic rescaling procedure is given to reduce this error to approximately 0.1%. This is then used to devise a numerically stable recursive integration routine that avoids expensive quadratures. Tests with coupled clusters with single and double excitations and perturbative triple calculations on small silver clusters demonstrate that the new schemes show no loss in accuracy, while reducing both the power and prefactor of the scaling with system size. In particular, speedups of roughly 40 times can be achieved compared to quadrature-based methods.

Laplace-transformed multi-reference second-order perturbation theories in the atomic and active molecular orbital basis

In the present article, we show how to formulate the partially contracted n-electron valence second-order perturbation theory (NEVPT2) energies in the atomic and active molecular orbital basis by employing the Laplace transformation of orbital-energy denominators (OEDs). As atomic-orbital (AO) basis functions are inherently localized and the number of active orbitals is comparatively small, our formulation is particularly suited for a linearly scaling NEVPT2 implementation. In our formulation, there are two kinds of NEVPT2 energy contributions, which differ in the number of active orbitals in the two-electron integrals involved. Those involving integrals with either no or a single active orbital can be formulated completely in the AO basis as single-reference second-order Møller-Plesset perturbation theory and benefit from sparse active pseudo-density matrices-particularly if the active molecular orbitals are localized only in parts of a molecule. Conversely, energy contributions involving integrals with either two or three active orbitals can be obtained from Coulomb and exchange matrices generalized for pairs of active orbitals. Moreover, we demonstrate that Laplace-transformed partially contracted NEVPT2 is nothing less than time-dependent NEVPT2 [A. Y. Sokolov and G. K.-L. Chan, J. Chem. Phys. 144, 064102 (2016)] iff the all-active intermediates are computed with the internal-contraction approximation. Furthermore, we show that for multi-reference perturbation theories it is particularly challenging to find optimal parameters of the numerical Laplace transformation as the fit range may vary among the 8 different OEDs by many orders of magnitude. Selecting the number of quadrature points for each OED separately according to an accuracy-based criterion allows us to control the errors in the NEVPT2 energies reliably.

Corannulene and its complex with water: a tiny cup of water

We report the results of a broadband rotational spectroscopic study of corannulene, C₂₀H₁₀, all of its singly substituted ¹³C isotopologues, and a complex of corannulene with one molecule of water.

SparseMaps--A systematic infrastructure for reduced-scaling electronic structure methods. IV. Linear-scaling second-order explicitly correlated energy with pair natural orbitals

We present a formulation of the explicitly correlated second-order Møller-Plesset (MP2-F12) energy in which all nontrivial post-mean-field steps are formulated with linear computational complexity in system size. The two key ideas are the use of pair-natural orbitals for compact representation of wave function amplitudes and the use of domain approximation to impose the block sparsity. This development utilizes the concepts for sparse representation of tensors described in the context of the domain based local pair-natural orbital-MP2 (DLPNO-MP2) method by us recently [Pinski et al., J. Chem. Phys. 143, 034108 (2015)]. Novel developments reported here include the use of domains not only for the projected atomic orbitals, but also for the complementary auxiliary basis set (CABS) used to approximate the three- and four-electron integrals of the F12 theory, and a simplification of the standard B intermediate of the F12 theory that avoids computation of four-index two-electron integrals that involve two CABS indices. For quasi-1-dimensional systems (n-alkanes), the ON DLPNO-MP2-F12 method becomes less expensive than the conventional ON(5) MP2-F12 for n between 10 and 15, for double- and triple-zeta basis sets; for the largest alkane, C200H402, in def2-TZVP basis, the observed computational complexity is N(∼1.6), largely due to the cubic cost of computing the mean-field operators. The method reproduces the canonical MP2-F12 energy with high precision: 99.9% of the canonical correlation energy is recovered with the default truncation parameters. Although its cost is significantly higher than that of DLPNO-MP2 method, the cost increase is compensated by the great reduction of the basis set error due to explicit correlation.

Molecular Dipole Moments within the Incremental Scheme Using the Domain-Specific Basis-Set Approach

We present the first implementation of the fully automated incremental scheme for CCSD unrelaxed dipole moments using the domain-specific basis-set approach. Truncation parameters are varied, and the accuracy of the method is statistically analyzed for a test set of 20 molecules. The local approximations introduce small errors at second order and negligible ones at third order. For a third-order incremental CCSD expansion with a CC2 error correction, a cc-pVDZ/SV domain-specific basis set (tmain = 3.5 Bohr), and the truncation parameter f = 30 Bohr, we obtain a mean error of 0.00 mau (-0.20 mau) and a standard deviation of 1.95 mau (2.17 mau) for the total dipole moments (Cartesian components of the dipole vectors). By analyzing incremental CCSD energies, we demonstrate that the MP2 and CC2 error correction schemes are an exclusive correction for the domain-specific basis-set error. Our implementation of the incremental scheme provides fully automated computations of highly accurate dipole moments at reduced computational cost and is fully parallelized in terms of the calculation of the increments. Therefore, one can utilize the incremental scheme, on the same hardware, to extend the basis set in comparison to standard CCSD and thus obtain a better total accuracy.

Modeling Polymorphic Molecular Crystals with Electronic Structure Theory

Interest in molecular crystals has grown thanks to their relevance to pharmaceuticals, organic semiconductor materials, foods, and many other applications. Electronic structure methods have become an increasingly important tool for modeling molecular crystals and polymorphism. This article reviews electronic structure techniques used to model molecular crystals, including periodic density functional theory, periodic second-order Møller-Plesset perturbation theory, fragment-based electronic structure methods, and diffusion Monte Carlo. It also discusses the use of these models for predicting a variety of crystal properties that are relevant to the study of polymorphism, including lattice energies, structures, crystal structure prediction, polymorphism, phase diagrams, vibrational spectroscopies, and nuclear magnetic resonance spectroscopy. Finally, tools for analyzing crystal structures and intermolecular interactions are briefly discussed.

Can density cumulant functional theory describe static correlation effects?

We evaluate the performance of density cumulant functional theory (DCT) for capturing static correlation effects. In particular, we examine systems with significant multideterminant character of the electronic wave function, such as the beryllium dimer, diatomic carbon, m-benzyne, 2,6-pyridyne, twisted ethylene, as well as the barrier for double-bond migration in cyclobutadiene. We compute molecular properties of these systems using the ODC-12 and DC-12 variants of DCT and compare these results to multireference configuration interaction and multireference coupled-cluster theories, as well as single-reference coupled-cluster theory with single, double (CCSD), and perturbative triple excitations [CCSD(T)]. For all systems the DCT methods show intermediate performance between that of CCSD and CCSD(T), with significant improvement over the former method. In particular, for the beryllium dimer, m-benzyne, and 2,6-pyridyne, the ODC-12 method along with CCSD(T) correctly predict the global minimum structures, while CCSD predictions fail qualitatively, underestimating the multireference effects. Our results suggest that the DC-12 and ODC-12 methods are capable of describing emerging static correlation effects but should be used cautiously when highly accurate results are required. Conveniently, the appearance of multireference effects in DCT can be diagnosed by analyzing the DCT natural orbital occupations, which are readily available at the end of the energy computation.

Probing the structural and dynamical properties of liquid water with models including non-local electron correlation

Water is a ubiquitous liquid that displays a wide range of anomalous properties and has a delicate structure that challenges experiment and simulation alike. The various intermolecular interactions that play an important role, such as repulsion, polarization, hydrogen bonding, and van der Waals interactions, are often difficult to reproduce faithfully in atomistic models. Here, electronic structure theories including all these interactions at equal footing, which requires the inclusion of non-local electron correlation, are used to describe structure and dynamics of bulk liquid water. Isobaric-isothermal (NpT) ensemble simulations based on the Random Phase Approximation (RPA) yield excellent density (0.994 g/ml) and fair radial distribution functions, while various other density functional approximations produce scattered results (0.8-1.2 g/ml). Molecular dynamics simulation in the microcanonical (NVE) ensemble based on Møller-Plesset perturbation theory (MP2) yields dynamical properties in the condensed phase, namely, the infrared spectrum and diffusion constant. At the MP2 and RPA levels of theory, ice is correctly predicted to float on water, resolving one of the anomalies as resulting from a delicate balance between van der Waals and hydrogen bonding interactions. For several properties, obtaining quantitative agreement with experiment requires correction for nuclear quantum effects (NQEs), highlighting their importance, for structure, dynamics, and electronic properties. A computed NQE shift of 0.6 eV for the band gap and absorption spectrum illustrates the latter. Giving access to both structure and dynamics of condensed phase systems, non-local electron correlation will increasingly be used to study systems where weak interactions are of paramount importance.

Energy benchmarks for methane-water systems from quantum Monte Carlo and second-order Møller-Plesset calculations

The quantum Monte Carlo (QMC) technique is used to generate accurate energy benchmarks for methane-water clusters containing a single methane monomer and up to 20 water monomers. The benchmarks for each type of cluster are computed for a set of geometries drawn from molecular dynamics simulations. The accuracy of QMC is expected to be comparable with that of coupled-cluster calculations, and this is confirmed by comparisons for the CH4-H2O dimer. The benchmarks are used to assess the accuracy of the second-order Møller-Plesset (MP2) approximation close to the complete basis-set limit. A recently developed embedded many-body technique is shown to give an efficient procedure for computing basis-set converged MP2 energies for the large clusters. It is found that MP2 values for the methane binding energies and the cohesive energies of the water clusters without methane are in close agreement with the QMC benchmarks, but the agreement is aided by partial cancelation between 2-body and beyond-2-body errors of MP2. The embedding approach allows MP2 to be applied without loss of accuracy to the methane hydrate crystal, and it is shown that the resulting methane binding energy and the cohesive energy of the water lattice agree almost exactly with recently reported QMC values.

A systematic way for the cost reduction of density fitting methods

We present a simple approach for the reduction of the size of auxiliary basis sets used in methods exploiting the density fitting (resolution of identity) approximation for electron repulsion integrals. Starting out of the singular value decomposition of three-center two-electron integrals, new auxiliary functions are constructed as linear combinations of the original fitting functions. The new functions, which we term natural auxiliary functions (NAFs), are analogous to the natural orbitals widely used for the cost reduction of correlation methods. The use of the NAF basis enables the systematic truncation of the fitting basis, and thereby potentially the reduction of the computational expenses of the methods, though the scaling with the system size is not altered. The performance of the new approach has been tested for several quantum chemical methods. It is demonstrated that the most pronounced gain in computational efficiency can be expected for iterative models which scale quadratically with the size of the fitting basis set, such as the direct random phase approximation. The approach also has the promise of accelerating local correlation methods, for which the processing of three-center Coulomb integrals is a bottleneck.

Ring planarity problem of 2-oxazoline revisited using microwave spectroscopy and quantum chemical calculations

In a previous infrared, Raman, and microwave spectroscopic work,1 it was claimed that 2-oxazoline has a planar ring equilibrium conformation, and the ring-puckering potential function V(z) = 22.2(z(4) + 1.31z(2)) cm(-1), where z is a dimensionless reduced coordinate, was derived. This function poorly reproduces the rotational constants of the lowest and most important puckering states. The microwave spectrum has been reinvestigated and largely extended to include more than 4600 transitions of the ground state and six excited states of the ring-puckering vibration allowing accurate centrifugal distortion constants to be obtained for the first time. A new potential function V(z) = 38.8(z(4) - 0.65z(2)) cm(-1) has been determined. This function yields much better agreement between calculated and observed rotational constants, especially for the lowest puckering states, than the previous function and predicts a nonplanar ring equilibrium conformation. The barrier to ring planarity is determined to be 49(8) J/mol. The ground-state energy level is 35 cm(-1) above the barrier maximum. Theory predicts that three of the five Watson centrifugal distortion constants, ΔJK, ΔK, and δK, should vary with the puckering state, whereas ΔJ and δJ should be unaffected. It was found that ΔJK and ΔK indeed behave in the expected manner, while deviations were seen for the three other centrifugal distortion constants. The ab initio methods HF, MP2, CCSD, CCSD(T), and CCSD(T)-F12 with large basis sets as well as several DFT methods were used in an attempt to reproduce the low experimental barrier to the planar ring. Only the MP2 method yielded a satisfactory prediction of the barrier. The CCSD and the CCSD(T) calculations predict a planar ring, whereas the energy differences between a planar and a nonplanar ring obtained in the CCSD(T)-F12 computations are so small that a definite conclusion cannot be drawn.

Efficient calculation of integrals in mixed ramp-Gaussian basis sets

Algorithms for the efficient calculation of two-electron integrals in the newly developed mixed ramp-Gaussian basis sets are presented, alongside a Fortran90 implementation of these algorithms, RampItUp. These new basis sets have significant potential to (1) give some speed-up (estimated at up to 20% for large molecules in fully optimised code) to general-purpose Hartree-Fock (HF) and density functional theory quantum chemistry calculations, replacing all-Gaussian basis sets, and (2) give very large speed-ups for calculations of core-dependent properties, such as electron density at the nucleus, NMR parameters, relativistic corrections, and total energies, replacing the current use of Slater basis functions or very large specialised all-Gaussian basis sets for these purposes. This initial implementation already demonstrates roughly 10% speed-ups in HF/R-31G calculations compared to HF/6-31G calculations for large linear molecules, demonstrating the promise of this methodology, particularly for the second application. As well as the reduction in the total primitive number in R-31G compared to 6-31G, this timing advantage can be attributed to the significant reduction in the number of mathematically complex intermediate integrals after modelling each ramp-Gaussian basis-function-pair as a sum of ramps on a single atomic centre.

Incremental evaluation of coupled cluster dipole polarizabilities

In this work we present the first implementation of the incremental scheme for coupled cluster linear-response frequency-dependent dipole polarizabilities.

Bracketing subtle conformational energy differences between self-solvated and stretched trifluoropropanol

The interconversion landscape between stretched and folded trifluoropropanol conformations is characterized by Raman jet spectroscopy and theory.