Diffuse Basis Functions Research Articles

Importance of Orbital Invariance in Quantifying Electron-Hole Separation and Exciton Size.

A fundamental tenet of quantum mechanics is that properties should be independent of representation. In self-consistent field methods such as density functional theory, this manifests as a requirement that properties be invariant with respect to unitary transformations of the occupied molecular orbitals and (separately) the unoccupied molecular orbitals. Various ad hoc measures of excited-state charge separation that are commonly used to analyze time-dependent density-functional calculations violate this requirement, as they are based on incoherent averages of excitation amplitudes rather than expectation values involving coherent superpositions. As a result, these metrics afford markedly different values in various common representations, including canonical molecular orbitals, Boys-localized orbitals, and natural orbitals. Numerical values can be unstable with respect to basis-set expansion and may afford nonsensical results in the presence of extremely diffuse basis functions. In contrast, metrics based on well-defined expectation values are stable, representation-invariant, and physically interpretable. Use of natural transition orbitals improves the stability of the incoherent averages, but numerical values can only be interpreted as expectation value in the absence of superposition. To satisfy this condition, the particle and hole density matrices must each be dominated by a single eigenvector so that the transition density is well described by a single pair of natural transition orbitals. Counterexamples are readily found where this is not the case. Our results suggest that ad hoc charge-transfer diagnostics should be replaced by rigorous expectation values, which are no more expensive to compute.

Semiparametric maximum likelihood reconstruction of stochastic differential equations driven by white and correlated noise.

A semiparametric methodology for reconstructing Markovian and non-Markovian Langevin equations from time series data using unscented Kalman filtering is introduced and explored. The drift function and the logarithm of the diffusion function are expanded into sets of polynomial basis functions. In contrast to the more common state augmentation approach, the Kalman filter is here used only for state estimation and propagation; the model parameters are determined by maximum likelihood based on the predictive distribution generated by the Kalman filter. Model selection regarding the number of included drift and diffusion basis functions is performed with the Bayesian information criterion. The method is successfully validated on various simulated datasets with known system dynamics; it achieves accurate identification of drift and diffusion functions, also outside the prescribed model class, from datasets of moderate length with medium computational cost.

Accuracy of projected atomic virtual orbital space in embedding applications

ABSTRACT The performance of the recently suggested projected atomic orbitals (PAO) approach as the virtual space in excited state projection-based embedding applications is analysed. The role of the parameters in the PAO generation is discussed and the impact of different choices is evaluated on the ground state of the stacked formaldehyde and pyrrole dimers. A comparison of excitation energies obtained with PAOs and localised virtual orbitals is given and the role of diffuse basis functions is discussed using a benchmark set from previous studies.

Diffuse basis functions for explicitly correlated calculations on the heavy p-block: aug-cc-pVnZ-PP-F12 sets for Ga-Kr, In-Xe, and Tl-Rn.

New aug-cc-pVnZ-PP-F12 basis sets (n = D, T, Q) for the heavy p-block elements, Ga-Kr, In-Xe, and Tl-Rn, have been developed by augmenting the cc-pVnZ-PP-F12 sets with additional higher angular momentum diffuse functions. These basis sets have been optimized for use in explicitly correlated F12 calculations, and matching auxiliary basis sets for density fitting of conventional and F12 integrals have also been developed. The new sets have been validated with benchmark CCSD(T)-F12b calculations of electron affinities, where an accelerated convergence to the complete basis set limit is evident. The effect of the additional diffuse functions on electron affinities is shown to be comparable to the effect of correlating the outer-core d electrons.

Development of a universal method for vibrational analysis of the terminal alkyne C≡C stretch.

The terminal alkyne C≡C stretch has a large Raman scattering cross section in the "silent" region for biomolecules. This has led to many Raman tag and probe studies using this moiety to study biomolecular systems. A computational investigation of these systems is vital to aid in the interpretation of these results. In this work, we develop a method for computing terminal alkyne vibrational frequencies and isotropic transition polarizabilities that can easily and accurately be applied to any terminal alkyne molecule. We apply the discrete variable representation method to a localized version of the C≡C stretch normal mode. The errors of (1) vibrational localization to the terminal alkyne moiety, (2) anharmonic normal mode isolation, and (3) discretization of the Born-Oppenheimer potential energy surface are quantified and found to be generally small and cancel each other. This results in a method with low error compared to other anharmonic vibrational methods like second-order vibrational perturbation theory and to experiments. Several density functionals are tested using the method, and TPSS-D3, an inexpensive nonempirical density functional with dispersion corrections, is found to perform surprisingly well. Diffuse basis functions are found to be important for the accuracy of computed frequencies. Finally, the computation of vibrational properties like isotropic transition polarizabilities and the universality of the localized normal mode for terminal alkynes are demonstrated.

Diffusion Monte Carlo method for barrier heights of multiple proton exchanges and complexation energies in small water, ammonia, and hydrogen fluoride clusters.

Proton exchange reactions are of key importance in many processes in water. However, it is nontrivial to achieve reliable barrier heights for multiple proton exchanges and complexation energies in hydrogen-bonded systems theoretically. Performance of the fixed-node diffusion quantum Monte Carlo (FN-DMC) with the single-Slater-Jastrow trial wavefunction on total energies, barrier heights of multiple proton exchanges, and complexation energies of small water, ammonia, and hydrogen fluoride clusters is investigated in this study. Effects of basis sets and those of locality approximation (LA), T-move approximation (T-move), and determinant localization approximation (DLA) schemes in dealing with the nonlocal part of pseudopotentials on FN-DMC results are evaluated. According to our results, diffuse basis functions are important in achieving reliable barrier heights and complexation energies with FN-DMC, although the cardinal number of the basis set is more important than diffuse basis functions on total energies of these systems. Our results also show that the time step bias with DLA and LA is smaller than T-move; however, the time step bias of DMC energies with respect to time steps using the T-move is roughly linear up to 0.06a.u., while this is not the case with LA and DLA. Barrier heights and complexation energies with FN-DMC using these three schemes are always within chemical accuracy. Taking into account the fact that T-move and DLA are typically more stable than LA, FN-DMC calculations with the T-move or DLA scheme and basis sets containing diffuse basis functions are suggested for barrier heights of multiple proton exchanges and complexation energies of hydrogen-bonded clusters.

Computational study of physicochemical, optical, and thermodynamic properties of 2,2-dimethylchromene derivatives.

A large number of heterocyclic compounds are used as drugs, mainly due to the duality of lipophilicity playing in hydrophobic interactions and solubility with at least one hydrogen bond acceptor. The study of electronic properties is then important to better understand not only these charge distribution effects but also some other physicochemical properties involved in bioactivity to directly assess the bioavailability of these compounds and a possible classification in related applications. Phytomolecules such as chromenes are very accessible molecules exhibiting a bioactivity. Our study is focused on the impact of a number of functional groups acting on some 2,2-dimethylchromene derivatives, namely their global reactivity from the frontier molecular orbitals and local reactivity from the Fukui functions, where the carbonyl group acting as an electron withdrawal group has the most relevant effect, the solubility from the partition coefficient Log P strongly depending on the charge distribution and electronegative sites, the optical effects from the delocalization in the vinyl group, as well as the evaluation of the entropy associated with the molecular flexibility also acting on the bioactivity. Despite the effects of the wave function or density methods on the order of magnitude of these properties, these compounds are consistent with the rules for a potential oral drug candidate. The calculations of the electronic properties were performed through two levels of theory: Hartree-Fock level as a wave function-based method as an ab initio reference including some physically consistent eigenvalues and density functional theory DFT as a correlation consistent method using different functionals: hybrid or with a long-range correction. The basis set used is a 6-311++G(d,p) Pople basis set including diffuse and polarization basis functions. The basis set is adapted to the size of the molecules and consequently to the degree of electronic localization. Gaussian 09 software was used for the computation.

Using Koopmans' theorem for constructing basis sets: approaching high Rydberg excited states of lithium with a compact Gaussian basis.

For accurate ab initio description of Rydberg excited states, this study suggests generating appropriate diffuse basis functions by cheap variational optimization of virtual orbitals of the corresponding ion core. By following this approach, dozens of converged correlated lithium Rydberg states, namely, all the states up to 24 2S, 25 2P, 14 2D, 16 2F and 16 2G, not yet achieved via other ab initio approaches, could be obtained at the EOM-CCSD level of theory with compact and mostly state-selective contracted Gaussian basis sets. Despite its small size and Gaussian character, the optimized basis leads to highly accurate excitation energies that differ merely in the order of meV from the reference state-of-the-art explicitly correlated Gaussian method and even surpass Full-CI results on the Slater basis by an order of magnitude.

Employing Pseudopotentials to Tackle Excited-State Electron Spill-Out in Frozen Density Embedding Calculations.

In frozen density embedding (FDE), the properties of a target molecule are computed in the presence of an effective embedding potential, which accounts for the attractive and repulsive contributions of the environment. The formally exact embedding potential, however, is in practice calculated using explicit kinetic-energy functionals for which the resulting potentials are in many cases not repulsive enough to account fully for Pauli repulsion by the electrons of the environment and to compensate thereby the strong electron-nuclear attraction. For the excited states on the target molecule, this leads to charge spill-out when diffuse basis functions are included, which allow that valence electrons are excited to those regions of the environment where the strong nuclear attraction is not sufficiently compensated by repulsive contributions. To reduce this insufficiency, we propose in the present work the inclusion of atomic all-electron pseudopotentials for all environment atoms on top of the conventional embedding potential. In the current work, the pseudopotentials are applied for computing vertical excitation energies of local excited states in complex systems employing the second-order algebraic diagrammatic construction (ADC(2)) scheme. The proposed approach leads to significantly reduced charge spill-out and an improved agreement of FDE and supermolecular calculations in the frozen solvent approximation. In particular, when diffuse functions are employed, the mean absolute deviation (MAD) is reduced from 0.27 to 0.05 eV for the investigated cases.

Application of diffuse source basis functions for improved near well upscaling

Application of diffuse source basis functions for improved near well upscaling

Property-optimized Gaussian basis sets for lanthanides

Property-optimized Gaussian basis sets of split-valence, triple-zeta valence, and quadruple-zeta valence quality are developed for the lanthanides Ce-Lu for use with small-core relativistic effective core potentials. They are constructed in a systematic fashion by augmenting def2 orbital basis sets with diffuse basis functions and minimizing negative static isotropic polarizabilities of lanthanide atoms with respect to basis set exponents within the unrestricted Hartree-Fock method. The basis set quality is assessed using a test set of 70 molecules containing the lanthanides in their common oxidation states and f electron occupations. 5d orbital occupation turns out to be the determining factor for the basis set convergence of polarizabilities in lanthanide atoms and the molecular test set. Therefore, two series of property-optimized basis sets are defined. The augmented def2-SVPD, def2-TZVPPD, and def2-QZVPPD basis sets balance the accuracy of polarizabilities across lanthanide oxidation states. The relative errors in atomic and molecular polarizability calculations are ≤8% for augmented split-valence basis sets, ≤ 2.5% for augmented triple-zeta valence basis sets, and ≤1% for augmented quadruple-zeta valence basis sets. In addition, extended def2-TZVPPDD and def2-QZVPPDD are provided for accurate calculations of lanthanide atoms and neutral clusters. The property-optimized basis sets developed in this work are shown to accurately reproduce electronic absorption spectra of a series of LnCp3 '- complexes (Cp' = C5H4SiMe3, Ln = Ce-Nd, Sm) with time-dependent density functional theory.

Automated fragmentation quantum mechanical calculation of 13C and 1H chemical shifts in molecular crystals.

In this work, the automated fragmentation quantum mechanics/molecular mechanics (AF-QM/MM) approach was applied to calculate the 13C and 1H nuclear magnetic resonance (NMR) chemical shifts in molecular crystals. Two benchmark sets of molecular crystals were selected to calculate the NMR chemical shifts. Systematic investigation was conducted to examine the convergence of AF-QM/MM calculations and the impact of various density functionals with different basis sets on the NMR chemical shift prediction. The result demonstrates that the calculated NMR chemical shifts are close to convergence when the distance threshold for the QM region is larger than 3.5 Å. For 13C chemical shift calculations, the mPW1PW91 functional is the best density functional among the functionals chosen in this study (namely, B3LYP, B3PW91, M06-2X, M06-L, mPW1PW91, OB98, and OPBE), while the OB98 functional is more suitable for the 1H NMR chemical shift prediction of molecular crystals. Moreover, with the B3LYP functional, at least a triple-ζ basis set should be utilized to accurately reproduce the experimental 13C and 1H chemical shifts. The employment of diffuse basis functions will further improve the accuracy for 13C chemical shift calculations, but not for the 1H chemical shift prediction. We further proposed a fragmentation scheme of dividing the central molecule into smaller fragments. By comparing with the results of the fragmentation scheme using the entire central molecule as the core region, the AF-QM/MM calculations with the fragmented central molecule can not only achieve accurate results but also reduce the computational cost. Therefore, the AF-QM/MM approach is capable of predicting the 13C and 1H NMR chemical shifts for molecular crystals accurately and effectively, and could be utilized for dealing with more complex periodic systems such as macromolecular polymers and biomacromolecules. The AF-QM/MM program for molecular crystals is available at https://github.com/shiman1995/NMR.

Curing basis set overcompleteness with pivoted Cholesky decompositions.

The description of weakly bound electronic states is especially difficult with atomic orbital basis sets. The diffuse atomic basis functions that are necessary to describe the extended electronic state generate significant linear dependencies in the molecular basis set, which may make the electronic structure calculations ill-convergent. We propose a method where the overcomplete molecular basis set is pruned by a pivoted Cholesky decomposition of the overlap matrix, yielding an optimal low-rank approximation that is numerically stable, the pivot indices determining a reduced basis set that is complete enough to describe all the basis functions in the original overcomplete basis. The method can be implemented either by a simple modification to the usual canonical orthogonalization procedure, which hides the excess functions and yields fewer efficiency benefits, or by generating custom basis sets for all the atoms in the system, yielding significant cost reductions in electronic structure calculations. The pruned basis sets from the latter choice allow accurate calculations to be performed at a lower cost even at the self-consistent field level, as illustrated on a solvated (H2O)24 - anion. Our results indicate that the Cholesky procedure allows one to perform calculations with accuracies close to standard augmented basis sets with cost savings which increase with the size of the basis set, ranging from 9% fewer functions in single-ζ basis sets to 28% fewer functions in triple-ζ basis sets.

VeloxChem: A Python‐driven density‐functional theory program for spectroscopy simulations in high‐performance computing environments

AbstractAn open‐source program named VeloxChem has been developed for the calculation of electronic real and complex linear response functions at the levels of Hartree–Fock and Kohn–Sham density functional theories. With an object‐oriented program structure written in a Python/C++ layered fashion, VeloxChem enables time‐efficient prototyping of novel scientific approaches without sacrificing computational efficiency, so that molecular systems involving up to and beyond 500 second‐row atoms (or some 10,000 contracted and in part diffuse Gaussian basis functions) can be routinely addressed. In addition, VeloxChem is equipped with a polarizable embedding scheme for the treatment of the classical electrostatic interactions with an environment that in turn is modeled by atomic site charges and polarizabilities. The underlying hybrid message passing interface (MPI)/open multiprocessing (OpenMP) parallelization scheme makes VeloxChem suitable for execution in high‐performance computing cluster environments, showing even slightly beyond linear scaling for the Fock matrix construction with use of up to 16,384 central processing unit (CPU) cores. An efficient—with respect to convergence rate and overall computational cost—multifrequency/gradient complex linear response equation solver enables calculations not only of conventional spectra, such as visible/ultraviolet/X‐ray electronic absorption and circular dichroism spectra, but also time‐resolved linear response signals as due to ultra‐short weak laser pulses. VeloxChem distributed under the GNU Lesser General Public License version 2.1 (LGPLv2.1) license and made available for download from the homepage https://veloxchem.org.This article is categorized under: Software > Quantum Chemistry Electronic Structure Theory > Density Functional Theory Theoretical and Physical Chemistry > Spectroscopy

Theoretical study on divergence problems of single reference perturbation theories

Divergences of the single reference perturbation theories due to the addition of diffusion basis functions have been investigated for both closed-shell and open-shell molecular systems. It is found that the oscillatory range of perturbation energies of open-shell systems increases as the spin multiplicity of systems changes from 2 to 4. Feenberg transformation is exploited to treat the divergence problems. It is found numerically that within the interval of Feenberg parameter there exists a minimum perturbation order at which the perturbation series become convergent. It is also found for the open-shell systems that the magnitude of the corresponding Feenberg parameter becomes larger as the spin multiplicity of the system of interest changes from 2 to 4.

Nature of the 11Bu and 21Ag Excited States of Butadiene and the Goldilocks Principle of Basis Set Diffuseness.

Butadiene, being the simplest conjugated organic molecule, has been studied extensively by experiments and various high-level theoretical methods. Previous studies conclude that the complete active space (CAS) self-consistent field (SCF) method was unable to obtain the correct 11Bu and 21Ag state ordering and that it introduces artificial valence-Rydberg mixing into the 11Bu state because of the lack of external correlation. Basis sets and initial guesses specifically constructed for this problem were able to improve the vertical excitation energy of the 11Bu state but did not resolve the controversy of the nature of the 11Bu state. In the present work, we demonstrate that, using standard intermediately diffuse basis sets such as jul-cc-pVTZ and ma-TZVP, CASSCF is able to obtain the 11Bu and 21Ag states of trans-butadiene with much improved vertical excitation energy and reasonable wave function characteristics, and it provides a good reference wave function (capable of yielding quantitative excitation energies for excited states) for three methods that treat electron correlation in different ways, namely, multiconfiguration pair-density functional theory (MC-PDFT), CAS second-order perturbation theory (PT2), and multistate (MS) CASPT2. We demonstrate that a combined analysis of the orbital second moments, state second moments, and MC-PDFT energy components is a very useful approach in determining excited-state characteristics and assigning states, and we show that basis sets without diffuse functions or with very diffuse basis functions are not stable or accurate in predicting the excited states of butadiene. We show that the 21Ag state is valence-like and has an atypical double/single excitation character and the 11Bu state has a certain degree of Rydberg character that is not artificial, settling the decades of controversy of the characters of these states.

Laplacian spectral basis functions

Representing a signal as a linear combination of a set of basis functions is central in a wide range of applications, such as approximation, de-noising, compression, shape correspondence and comparison. In this context, our paper addresses the main aspects of signal approximation, such as the definition, computation, and comparison of basis functions on arbitrary 3D shapes. Focusing on the class of basis functions induced by the Laplace–Beltrami operator and its spectrum, we introduce the diffusion and Laplacian spectral basis functions, which are then compared with the harmonic and Laplacian eigenfunctions. As main properties of these basis functions, which are commonly used for numerical geometry processing and shape analysis, we discuss the partition of the unity and non-negativity; the intrinsic definition and invariance with respect to shape transformations (e.g., translation, rotation, uniform scaling); the locality, smoothness, and orthogonality; the numerical stability with respect to the domain discretisation; the computational cost and storage overhead. Finally, we consider geometric metrics, such as the area, conformal, and kernel-based norms, for the comparison and characterisation of the main properties of the Laplacian basis functions.

Off‐center Gaussian functions: Applications toward larger basis sets, post‐second‐order correlation treatment, and truncated virtual orbital space in investigations of noncovalent interactions

AbstractThis work is a continuation of our pilot study on the use of off‐center s‐type Gaussian functions in noncovalent interaction calculations. A grid of s‐type Gaussians surrounding the molecule is intended to substitute for the presence of diffuse basis functions, which are important for accurate description of noncovalent interactions. The advantage over the use of diffuse functions is reduction or elimination of linear dependency issues in the atomic orbital basis set, which often causes convergence problems. Placement of a grid of s‐functions on the surface of the molecule allows more favorable scaling of the total number of basis functions with the molecular size to be achieved. In this article, we present several innovations on the concept, namely the parametrization and assessment of grids with triple‐ζ quality basis set; use in post‐second‐order correlation treatment (methods such as Møller–Plesset third‐order or coupled‐cluster); and combination with modified virtual orbital approaches, namely the frozen natural orbitals and optimized virtual orbital space methods.

Detailed Wave Function Analysis for Multireference Methods: Implementation in the Molcas Program Package and Applications to Tetracene.

High-level multireference computations on electronically excited and charged states of tetracene are performed, and the results are analyzed using an extensive wave function analysis toolbox that has been newly implemented in the Molcas program package. Aside from verifying the strong effect of dynamic correlation, this study reveals an unexpected critical influence of the atomic orbital basis set. It is shown that different polarized double-ζ basis sets produce significantly different results for energies, densities, and overall wave functions, with the best performance obtained for the atomic natural orbital (ANO) basis set by Pierloot et al. Strikingly, the ANO basis set not only reproduces the energies but also performs exceptionally well in terms of describing the diffuseness of the different states and of their attachment/detachment densities. This study, thus, not only underlines the fact that diffuse basis functions are needed for an accurate description of the electronic wave functions but also shows that, at least for the present example, it is enough to include them implicitly in the contraction scheme.

Artificial nodes in the H2+ wave functions expanded using Gaussian-type orbitals or Laguerre-type orbitals

In Hartree-Fock calculations for atoms and molecules, multicenter-expansion methods are commonly used, especially those based on linear combinations of atomic orbitals (LCAO) using Gaussian-type orbitals (GTOs) or Slater-type orbitals (STOs). Artificial nodes, which are not expected to exist in real wave functions, arise in many calculations. For the hydrogen molecular ion (H2+) there are no nodes in the exact solution of its ground state ((1σg) 2Σg). The occurrence of artificial nodes and the cause of the problem are investigated for this state. Calculations of LCAO with GTOs and of single-center expansion (SCE) with Laguerre-type orbitals (LTOs) have been performed as part of this study. It is found that an excess of diffuse basis functions is responsible for artificial nodes.