Avoiding Negligible Shell Pairs and Quartets in Electronic Structure Calculations.

We define a significant shell pair in an electronic structure calculation as one that generates at least one two-electron integral larger than a preset threshold. We define a significant shell quartet similarly. We then explore several methods for identifying nonsignificant pairs and quartets so that they can be avoided and computational efficiency improved. We find that the widely used Cauchy-Schwarz bound identifies most nonsignificant quartets but that the Hölder bound is slightly more powerful for identifying nonsignificant pairs.

Software for the frontiers of quantum chemistry: An overview of developments in the Q-Chem 5 package.

This article summarizes technical advances contained in the fifth major release of the Q-Chem quantum chemistry program package, covering developments since 2015. A comprehensive library of exchange-correlation functionals, along with a suite of correlated many-body methods, continues to be a hallmark of the Q-Chem software. The many-body methods include novel variants of both coupled-cluster and configuration-interaction approaches along with methods based on the algebraic diagrammatic construction and variational reduced density-matrix methods. Methods highlighted in Q-Chem 5 include a suite of tools for modeling core-level spectroscopy, methods for describing metastable resonances, methods for computing vibronic spectra, the nuclear-electronic orbital method, and several different energy decomposition analysis techniques. High-performance capabilities including multithreaded parallelism and support for calculations on graphics processing units are described. Q-Chem boasts a community of well over 100 active academic developers, and the continuing evolution of the software is supported by an "open teamware" model and an increasingly modular design.

Q-MP2-OS: Møller-Plesset Correlation Energy by Quadrature.

We present a quadrature-based algorithm for computing the opposite-spin component of the MP2 correlation energy which scales quadratically with basis set size and is well-suited to large-scale parallelization. The key ideas, which are rooted in the earlier work of Hirata and co-workers, are to abandon all two-electron integrals, recast the energy as a seven-dimensional integral, approximate that integral by quadrature, and employ a cutoff strategy to minimize the number of intermediate quantities. We discuss our implementation in detail and show that it parallelizes almost perfectly on 840 cores for cyclosporine (a molecule with roughly 200 atoms), exhibits [Formula: see text] scaling for a sequence of polyglycines, and is principally limited by the accuracy of its quadrature.

Simple Models for Difficult Electronic Excitations.

We present a single-determinant approach to three challenging topics in the chemistry of excited states: double excitations, charge-transfer states, and conical intersections. The results are obtained by using the Initial Maximum Overlap Method (IMOM) which is a modified version of the Maximum Overlap Method (MOM). The new algorithm converges better than the original, especially for these difficult problems. By considering several case studies, we show that a single-determinant framework provides a simple and accurate alternative for modeling excited states in cases where other low-cost methods, such as CIS and TD-DFT, either perform poorly or fail completely.

Efficient Method for Calculating Effective Core Potential Integrals.

Effective core potential (ECP) integrals are among the most difficult one-electron integrals to calculate due to the projection operators. The radial part of these operators may include r0, r-1, and r-2 terms. For the r0 terms, we exploit a simple analytic expression for the fundamental projected integral to derive new recurrence relations and upper bounds for ECP integrals. For the r-1 and r-2 terms, we present a reconstruction method that replaces these terms by a sum of r0 terms and show that the resulting errors are chemically insignificant for a range of molecular properties. The new algorithm is available in Q-Chem 5.0 and is significantly faster than the ECP implementations in Q-Chem 4.4, GAMESS (US) and Dalton 2016.

Excitation Number: Characterizing Multiply Excited States.

How many electrons are excited in an electronic transition? In this Letter, we introduce the excitation number η to answer this question when the initial and final states are each modeled by a single-determinant wave function. We show that calculated η values lie close to positive integers, leading to unambiguous assignments of the number of excited electrons. This contrasts with previous definitions of excitation quantities which can lead to mis-assignments. We consider several examples where η provides improved excited-state characterizations.

MP2[V]--A Simple Approximation to Second-Order Møller-Plesset Perturbation Theory.

We propose a simplified variant of the dual-basis MP2[K] scheme [ J. Chem. Phys. 2011, 134, 081103] that bootstraps a small-basis MP2 result to a large-basis one. This simplified method, which we call MP2[V], assumes the occupied orbitals are adequately described by the smaller basis, and, therefore, only the relaxation of the virtual orbitals is considered when shifting to the larger basis. Numerical tests on several organic reactions and noncovalent interactions show that MP2[V] yields absolute and relative energies that are in excellent agreement with the conventional large-basis MP2 calculations but in a small fraction of the time.

Accurate Electron Densities at Nuclei Using Small Ramp-Gaussian Basis Sets.

Electron densities at nuclei are difficult to calculate accurately with all-Gaussian basis sets because they lack an electron-nuclear cusp. The newly developed mixed ramp-Gaussian basis sets, such as R-31G, possess electron-nuclear cusps due to the presence of ramp functions in the basis. The R-31G basis set is a general-purpose mixed ramp-Gaussian basis set modeled on the 6-31G basis set. The prediction of electron densities at nuclei using R-31G basis sets for Li-F outperforms Dunning, Pople, and Jensen general purpose all-Gaussian basis sets of triple-ζ quality or lower and the cc-pVQZ basis set. It is of similar quality to the specialized pcJ-0 basis set which was developed with partial decontraction of core functions and extra high exponent s-Gaussians to predict electron density at the nucleus. These results show significant advantages in the properties of mixed ramp-Gaussian basis sets compared to all-Gaussian basis sets.

Communication: Hartree-Fock description of excited states of H2.

Hartree-Fock (HF) theory is most often applied to study the electronic ground states of molecular systems. However, with the advent of numerical techniques for locating higher solutions of the self-consistent field equations, it is now possible to examine the extent to which such mean-field solutions are useful approximations to electronic excited states. In this Communication, we use the maximum overlap method to locate 11 low-energy solutions of the HF equation for the H2 molecule and we find that, with only one exception, these yield surprisingly accurate models for the low-lying excited states of this molecule. This finding suggests that the HF solutions could be useful first-order approximations for correlated excited state wavefunctions.

Mixed Ramp-Gaussian Basis Sets.

We discuss molecular orbital basis sets that contain both Gaussian and polynomial (ramp) functions. We show that, by modeling ramp-Gaussian products as sums of ramps, all of the required one- and two-electron integrals can be computed quickly and accurately. To illustrate our approach, we construct R-31+G, a mixed ramp-Gaussian basis in which the core basis functions of the 6-31+G basis are replaced by ramps. By performing self-consistent Hartree-Fock calculations, we show that the thermochemical predictions of R-31+G and 6-31+G are similar but the former has the potential to be significantly faster.

Performance of Density Functional Theory Procedures for the Calculation of Proton-Exchange Barriers: Unusual Behavior of M06-Type Functionals.

We have examined the performance of a variety of density functional theory procedures for the calculation of complexation energies and proton-exchange barriers, with a focus on the Minnesota-class of functionals that are generally highly robust and generally show good accuracy. A curious observation is that M05-type and M06-type methods show an atypical decrease in calculated barriers with increasing proportion of Hartree-Fock exchange. To obtain a clearer picture of the performance of the underlying components of M05-type and M06-type functionals, we have investigated the combination of MPW-type and PBE-type exchange and B95-type and PBE-type correlation procedures. We find that, for the extensive E3 test set, the general performance of the various hybrid-DFT procedures improves in the following order: PBE1-B95 → PBE1-PBE → MPW1-PBE → PW6-B95. As M05-type and M06-type procedures are related to PBE1-B95, it would be of interest to formulate and examine the general performance of an alternative Minnesota DFT method related to PW6-B95.

Effective fragment potential method in Q-CHEM: a guide for users and developers.

A detailed description of the implementation of the effective fragment potential (EFP) method in the Q-CHEM electronic structure package is presented. The Q-CHEM implementation interfaces EFP with standard quantum mechanical (QM) methods such as Hartree-Fock, density functional theory, perturbation theory, and coupled-cluster methods, as well as with methods for electronically excited and open-shell species, for example, configuration interaction, time-dependent density functional theory, and equation-of-motion coupled-cluster models. In addition to the QM/EFP functionality, a "fragment-only" feature is also available (when the system is described by effective fragments only). To aid further developments of the EFP methodology, a detailed description of the C++ classes and EFP module's workflow is presented. The EFP input structure and EFP job options are described. To assist setting up and performing EFP calculations, a collection of Perl service scripts is provided. The precomputed EFP parameters for standard fragments such as common solvents are stored in Q-CHEM's auxiliary library; they can be easily invoked, similar to specifying standard basis sets. The instructions for generating user-defined EFP parameters are given. Fragments positions can be specified by their center of mass coordinates and Euler angles. The interface with the IQMOL and WEBMO software is also described.

Communication: efficient counterpoise corrections by a perturbative approach.

We investigate the use of Hartree-Fock and density functional perturbative corrections for estimating the counterpoise correction (CPC) for interaction energies at the self-consistent field level. We test our approach using several popular basis sets on the S22 set of weakly bound systems, which can exhibit large basis set superposition errors. Our results show that the perturbative approaches typically recover over 95% of the CPC and can be up to twelve times faster to compute than the conventional methods and therefore provide an attractive alternative to calculating CPCs in the conventional way.

Hartree-Fock perturbative corrections for total and reaction energies.

We have performed an assessment of the Hartree-Fock perturbative correction (HFPC) on a large and diverse set of molecules and reactions. Errors in both absolute and reaction energies with respect to converged secondary basis Hartree-Fock results are reported for a wide spectrum of primary/secondary basis set combinations. These results show that using an adequate primary basis, HFPC can accurately reproduce secondary basis energies at a substantially reduced cost. Comparisons of HFPC with the related dual basis Hartree-Fock (DBHF) scheme are also made for several molecules and target secondary basis sets. Our results indicate that HFPC is faster and more accurate than DBHF for approaching triple-zeta basis sets. For quadruple-zeta secondary basis sets, HFPC is capable of yielding more accurate energies at a marginally increased cost over DBHF.

Density functional triple jumping.

We propose a density functional perturbative scheme to approximate the energy of a high-level DFT calculation at a significantly reduced cost. Our approach involves performing a primary SCF calculation using a crude functional, basis set and quadrature grid, followed by a single step using a more sophisticated secondary functional, basis and grid. Unlike the earlier dual-level DFT approach of Nakajima and Hirao, we use Roothaan diagonalization instead of perturbation theory to incorporate the effects of the secondary basis set. We show that energies at the popular B3LYP/6-311+G(3df,2p)/(75,302) level can be accurately estimated from primary calculations at the relatively economical BLYP/6-31G(d)/SG-0 level.

Approaching the Hartree-Fock limit by perturbative methods.

We describe perturbative methods for improving finite-basis Hartree-Fock calculations toward the complete-basis limit. The best method appears to offer quadratic error reduction and preliminary numerical applications demonstrate that remarkably accurate Hartree-Fock energies can be obtained.

Self-consistent-field calculations of core excited states.

The accuracy of core excitation energies and core electron binding energies computed within a Delta self-consistent-field framework is assessed. The variational collapse of the core excited state is prevented by maintaining a singly occupied core orbital using an overlap criterion called the maximum overlap method. When applied to a wide range of small organic molecules, the resulting core excitation energies are not systematically underestimated as observed in time-dependent density functional theory and agree well with experiment. The accuracy of this approach for core excited states is illustrated by the calculation of the pre-edge features in x-ray absorption spectra of plastocyanin, which shows that accurate results can be achieved with Delta self-consistent-field calculations when used in conjunction with uncontracted basis functions.

The role of exchange in systematic DFT errors for some organic reactions.

Serious (up to 87 kJ mol(-1)) systematic DFT errors in a series of isodesmic reactions are found to be due to the DFT exchange component, and can be largely corrected by substitution of the DFT exchange energy with the Fock exchange energy.

Self-consistent field calculations of excited states using the maximum overlap method (MOM).

We present a simple algorithm, which we call the maximum overlap method (MOM), for finding excited-state solutions to self-consistent field (SCF) equations. Instead of using the aufbau principle, the algorithm maximizes the overlap between the occupied orbitals on successive SCF iterations. This prevents variational collapse to the ground state and guides the SCF process toward the nearest, rather than the lowest energy, solution. The resulting excited-state solutions can be treated in the same way as the ground-state solution and, in particular, derivatives of excited-state energies can be computed using ground-state code. We assess the performance of our method by applying it to a variety of excited-state problems including the calculation of excitation energies, charge-transfer states, and excited-state properties.