Basis Set Size Research Articles

CASPT2 molecular geometries of Fe(II) spin-crossover complexes.

Using fully internally contracted (FIC)-CASPT2 analytical gradients, geometry optimizations of spin-crossover complexes are reported. This approach is tested on a series of Fe(II) complexes with different sizes, ranging from 13 to 61 atoms. A combination of active space and basis set choices are employed to investigate their role in determining reliable molecular geometries. The reported strategy demonstrates that a wave function-based level of theory can be used to optimize the geometries of metal complexes in reasonable times and enables one to treat the molecular geometry and electronic structure of the complexes using the same level of theory. For a series of smaller Fe(II) SCO complexes, strong field ligands in the LS state result in geometries with the largest differences between DFT and CASPT2; however, good agreement overall is observed between DFT and CASPT2. For the larger complexes, moderate sized basis sets yield geometries that compare well with DFT and available experimental data. We recommend using the (10e,12o) active space since convergence to a minimum structure was more efficient than with truncated active spaces despite having similar Fe-ligand bond distances.

Origin-Independent Densities of Static and Dynamic Molecular Polarizabilities.

The notion of the electric dipole polarizability density function of atoms and molecules has been considered. The current density induced by the time derivative of the electric field of monochromatic light allows for a new definition of the electric dipole polarizability density, which is translationally invariant. This translational invariance provides the physical meaning that was lacking in previous defined polarizability densities. The new polarizability density has been implemented at the TD-DFT level of theory. The origin independence has been proven in silico to hold regardless of the basis set size. Two emblematic molecules, i.e., CO and N2, which in many respects display similar electric response, have been studied in detail. The substantial differences, which have been highlighted in the topology of the parallel and perpendicular polarizability density tensor components of CO and N2, are grossly hidden by compensation, when integration is carried out to get the molecular properties.

Energies, structures, and harmonic frequencies of small water clusters from the direct random phase approximation.

The binding energies, structures, and vibrational frequencies of water clusters up to 20 molecules are computed at the direct random phase approximation (RPA) level of theory and compared to theoretical benchmarks. Binding energies of the WATER27 set, which includes neutral and positively and negatively charged clusters, are predicted to be too low in the complete basis set limit by an average of 7 kcal/mol (9%) and are worse than the results from the best density functional theory methods or from the Møller-Plesset theory. The RPA shows significant basis set size dependence for binding energies. The order of the relative energies of the water hexamer and dodecamer isomers is predicted correctly by the RPA. The mean absolute deviation for angles and distances for neutral clusters up to the water hexamer are 0.2° and 0.6pm, respectively, using quintuple-ζ basis sets. The relative energetic order of the hexamer isomers is preserved upon optimization. Vibrational frequencies for these systems are underestimated by several tens of wavenumbers for large basis sets, and deviations increase with the basis set size. Overall, the direct RPA method yields accurate structural parameters but systematically underestimates binding energies and shows strong basis set size dependence.

Reliable TDDFT Protocol Based on a Local Hybrid Functional for the Prediction of Vibronic Phosphorescence Spectra Applied to Tris(2,2'-bipyridine)-Metal Complexes.

An efficient computational protocol for the prediction of vibrationally resolved phosphorescence spectra is developed and validated for five tris(2,2'-bipyridine)-metal complexes ([M(bpy)3]n+, where M = Zn, Ru, Rh, Os, Ir). The outstanding feature of this protocol is the use of full linear-response time-dependent density functional theory (TDDFT) for the excited-state triplet calculation, i.e., the commonly seen strategies employing the Tamm-Dancoff approximation (TDA) or unrestricted density functional theory (DFT) calculations for the T1 state are not needed. This is achieved by the use of a local hybrid functional (LH12ct-SsirPW92) that features a real-space dependent admixture of exact exchange governed by a local mixing function. The excellent performance of this LH for triplet excitation energies known from previous studies transfers to a remarkable mean absolute error of 0.06 eV for the phosphorescence 0-0 energies investigated herein, while the popular B3PW91 functional gives an error of 0.27 eV in TDDFT and 0.09 eV in unrestricted DFT calculations, respectively. The advantages of the local hybrid are particularly apparent for excited states with a mixed-valence character. The influence of spin-orbit coupling was found to be significant for [Os(bpy)3]2+ red-shifting the 0-0 energy for phosphorescence by 0.17 eV, while the effect is negligible for the other complexes (<0.03 eV). The influence of the basis-set and integration-grid sizes is evaluated, and a computationally lighter protocol is validated that leads to drastic savings in computation time with negligible loss in accuracy.

Distortion of yttrium bisphthalocyanine (YPc2) upon noncovalent interaction with carbon nanotubes: A DFT study

We performed a DFT analysis of the changes in geometry and some electronic properties of yttrium bisphthalocyanine YPc2 when forming noncovalent hybrids with two single-walled carbon nanotube models, having armchair (ANT) and zigzag (ZNT) chiralities. The Perdew-Burke-Ernzerhof functional in combination with a long-range dispersion correction by Grimme (PBE-D) and the double numerical basis sets of variable size (DN, DND and DNP) was used. We found that YPc2 molecule suffers binding distortion in order to increase the area of contact with nanotube sidewall, which enhances interaction between the two components. YPc2 bonding is stronger with ZNT nanotube, of −43.8 kcal/mol, vs. −37.1 kcal/mol for YPc2 + ANT complex (as calculated with DNP basis set). The height of bisphthalocyanine molecule in the dyads dramatically increases compared to that calculated for isolated YPc2, whereas the calculated N-Y-N angles in YN4 coordination change very insignificantly. In all cases, a minor HOMO-LUMO gap narrowing is observed. For YPc2 + ANT the gap tends to match the one for isolated YPc2, whereas for YPc2 + ZNT it is much closer to the band gap of ZNT. The calculated HOMO, LUMO and spin density plots are analyzed.

Origin invariant full optical rotation tensor in the length dipole gauge without London atomic orbitals.

We present an origin-invariant approach to compute the full optical rotation tensor (Buckingham/Dunn tensor) in the length dipole gauge without recourse to London atomic orbitals, called LG(OI). The LG(OI) approach is simpler and less computationally demanding than the more common length gauge (LG)-London and modified velocity gauge (MVG) approaches, and it can be used with any approximate wave function or density functional method. We report an implementation at the coupled cluster with single and double excitations level (CCSD), for which we present the first simulations of the origin-invariant Buckingham/Dunn tensor in the LG. We compare LG(OI) and MVG results on a series of 22 organic molecules, showing good linear correlation between the approaches, although for small tensor elements, they provide values of opposite sign. We also attempt to decouple the effects of electron correlation and basis set incompleteness on the choice of gauge for specific rotation calculations on simple test systems. The simulations show a smooth convergence of the LG(OI) and MVG results with the basis set size toward the complete basis set limit. However, these preliminary results indicate that CCSD may not be close to a complete description of the electron correlation effects on this property even for small molecules and that basis set incompleteness may be a less important cause of discrepancy between choices of gauge than electron correlation incompleteness.

(T)+EOM Quartic Force Fields for Theoretical Vibrational Spectroscopy of Electronically Excited States.

(T)+EOM quartic force fields (QFFs) are proposed for ab initio rovibrational properties of electronically excited states of small molecules. The (T)+EOM method is a simple treatment of the potential surface of the excited state using a composite energy from the CCSD(T) energy for the ground-state configuration and the EOM-CCSD excitation energy for the target state. The method is benchmarked with two open-shell species, HOO and HNF, and two closed-shell species, HNO and HCF. A (T)+EOM QFF with a complete basis set extrapolation (C) and corrections for core correlation (cC) and scalar relativity (R), dubbed (T)+EOM/CcCR, achieves a mean absolute error (MAE) as low as 1.6 cm-1 for the à 2A' state of HOO versus an established benchmark QFF with CCSD(T)-F12b/cc-pVTZ-F12 (F12-TZ) for this variationally accessible electronically excited state. The MAE for anharmonic frequencies for (T)+EOM/CcCR versus F12-TZ for HNF is 7.5 cm-1. The closed-shell species are compared directly with the experiment, where a simpler (T)+EOM QFF using the aug-cc-pVTZ basis set compares more favorably than the more costly (T)+EOM/CcCR, suggesting a possible influence of decreasing accuracy with basis set size. Scans along internal coordinates are also provided which show reasonable modeling of the potential surface by (T)+EOM compared to benchmark QFFs computed for variationally accessible electronic states. The agreement between (T)+EOM/CcCR with F12-TZ and CcCR benchmarks is also shown to be quite accurate for rotational constants and geometries, with an MAE of 0.008 MHz for the rotational constants of (T)+EOM/CcCR versus CcCR for à 2A' HOO and agreement within 0.003 Šfor bond lengths.

Focal-point approach with pair-specific cusp correction for coupled-cluster theory.

We present a basis set correction scheme for the coupled-cluster singles and doubles (CCSD) method. The scheme is based on employing frozen natural orbitals (FNOs) and diagrammatically decomposed contributions to the electronic correlation energy, which dominate the basis set incompleteness error (BSIE). As recently discussed in the work of Irmler et al. [Phys. Rev. Lett. 123, 156401 (2019)], the BSIE of the CCSD correlation energy is dominated by the second-order Møller-Plesset (MP2) perturbation energy and the particle-particle ladder term. Here, we derive a simple approximation to the BSIE of the particle-particle ladder term that effectively corresponds to a rescaled pair-specific MP2 BSIE, where the scaling factor depends on the spatially averaged correlation hole depth of the coupled-cluster and first-order pair wavefunctions. The evaluation of the derived expressions is simple to implement in any existing code. We demonstrate the effectiveness of the method for the uniform electron gas. Furthermore, we apply the method to coupled-cluster theory calculations of atoms and molecules using FNOs. Employing the proposed correction and an increasing number of FNOs per occupied orbital, we demonstrate for a test set that rapidly convergent closed and open-shell reaction energies, atomization energies, electron affinities, and ionization potentials can be obtained. Moreover, we show that a similarly excellent trade-off between required virtual orbital basis set size and remaining BSIEs can be achieved for the perturbative triples contribution to the CCSD(T) energy employing FNOs and the (T*) approximation.

Strategy for constructing compact numerical atomic orbital basis sets by incorporating the gradients of reference wavefunctions

We develop an algorithm to construct high-quality numerical atomic orbital (NAO) basis sets suitable for large-scale, efficient density-functional calculations. The key idea behind this algorithm is that, in addition to fitting the reference wavefunctions themselves generated by plane-wave based calculations of chosen target systems, the first derivatives of the reference wavefunctions are also taken into account as the fitting target. By doing so, the quality of the generated NAO basis sets is significantly improved in the sense that the same level of numerical precision can be achieved with smaller basis set sizes or with reduced cutoff radii of the NAOs.

Performance of a black-box-type rovibrational method in comparison with a tailor-made approach: Case study for the methane-water dimer.

The present work intends to join and respond to the excellent and thoroughly documented rovibrational study of X. G. Wang and T. Carrington, Jr. [J. Chem. Phys. 154, 124112 (2021)] that used an approach tailored for floppy dimers with an analytic dimer Hamiltonian and a non-product basis set including Wigner D functions. It is shown in the present work that the GENIUSH black-box-type rovibrational method can approach the performance of the tailor-made computation for the example of the floppy methane-water dimer. Rovibrational transition energies and intensities are obtained in the black-box-type computation with a twice as large basis set and in excellent numerical agreement in comparison with the more efficient tailor-made approach.

Dynamic transverse magnetic susceptibility in the projector augmented-wave method: Application to Fe, Ni, and Co

We present a first principles implementation of the dynamic transverse magnetic susceptibility in the framework of linear response time-dependent density functional theory. The dynamic susceptibility allows one to obtain the magnon dispersion as well as magnon lifetimes for a particular material, which strongly facilitates the interpretation of inelastic neutron scattering experiments as well as other spectroscopic techniques. We apply the method to Fe, Ni, and Co and perform a thorough convergence analysis with respect the basis set size, $k$-point sampling, spectral smearing and unoccupied bands. In particular, it is shown that while the gap error (acoustic magnon energy at $\mathbf{q}=\mathbf{0}$) is highly challenging to converge, the spin-wave stiffness and the dispersion relation itself are much less sensitive to convergence parameters. Our final results agrees well with experimentally extracted magnon dispersion relations except for Ni, where it is well-known that the exchange splitting energy is poorly represented in the local density approximation. We also find good agreement with previous first principles calculations and explain how differences in the calculated dispersion relations can arise from subtle differences in computational approaches.

Dynamical correlation energy of metals in large basis sets from downfolding and composite approaches.

Coupled-cluster theory with single and double excitations (CCSD) is a promising ab initio method for the electronic structure of three-dimensional metals, for which second-order perturbation theory (MP2) diverges in the thermodynamic limit. However, due to the high cost and poor convergence of CCSD with respect to basis size, applying CCSD to periodic systems often leads to large basis set errors. In a common "composite" method, MP2 is used to recover the missing dynamical correlation energy through a focal-point correction, but the inadequacy of finite-order perturbation theory for metals raises questions about this approach. Here, we describe how high-energy excitations treated by MP2 can be "downfolded" into a low-energy active space to be treated by CCSD. Comparing how the composite and downfolding approaches perform for the uniform electron gas, we find that the latter converges more quickly with respect to the basis set size. Nonetheless, the composite approach is surprisingly accurate because it removes the problematic MP2 treatment of double excitations near the Fermi surface. Using this method to estimate the CCSD correlation energy in the combined complete basis set and thermodynamic limits, we find that CCSD recovers 85%-90% of the exact correlation energy at rs = 4. We also test the composite approach with the direct random-phase approximation used in place of MP2, yielding a method that is typically (but not always) more cost effective due to the smaller number of orbitals that need to be included in the more expensive CCSD calculation.

A critical analysis of least-squares tensor hypercontraction applied to MP3.

The least-squares tensor hypercontraction (LS-THC) approach is a promising method of reducing the high polynomial scaling of wavefunction methods, for example, those based on many-body perturbation theory or coupled cluster. Here, we focus on LS-THC-MP3 and identify four variants with differing errors and efficiency characteristics. The performance of LS-THC-MP3 is analyzed for regular test systems with up to 40 first-row atoms. We also analyze the size-extensivity/size-consistency and grid- and basis set dependence of LS-THC-MP3. Overall, the errors observed are favorably small in comparison with standard density fitting, and a more streamlined method of generating grids via pruning is suggested. A practical crossover (the point at which LS-THC-MP3 is cheaper than the canonical method) is achieved around 240 correlated electrons. Despite several drawbacks of LS-THC that have been identified: an initial non-linearity of error when increasing system size, poor description of angular correlation, and a potentially large increase in error with the basis set size, the results show that LS-THC has significant potential for practical application to MP3 and other wavefunction methods.

Comparison of many-particle representations for selected-CI I: A tree based approach.

The full configuration interaction (FCI) method is only applicable to small molecules with few electrons in moderate size basis sets. One of the main alternatives to obtain approximate FCI energies for bigger molecules and larger basis sets is selected CI. However, due to: (a) the lack of a well-defined structure in a selected CI Hamiltonian, (b) the potentially large number of electrons together with c) potentially large orbital spaces, a computationally and memory efficient algorithm is difficult to construct. In the present series of papers, we describe our attempts to address these issues by exploring tree-based approaches. At the same time, we devote special attention to the issue of obtaining eigenfunctions of the total spin squared operator since this is of particular importance in tackling magnetic properties of complex open shell systems. Dedicated algorithms are designed to tackle the CI problem in terms of determinant, configuration (CFG) and configuration state function many-particle bases by effective use of the tree representation. In this paper we describe the underlying logic of our algorithm design and discuss the advantages and disadvantages of the different many particle bases. We demonstrate by the use of small examples how the use of the tree simplifies many key algorithms required for the design of an efficient selected CI program. Our selected CI algorithm, called the iterative configuration expansion, is presented in the penultimate part. Finally, we discuss the limitations and scaling characteristics of the present approach.

Assessment of the Ab Initio Bethe-Salpeter Equation Approach for the Low-Lying Excitation Energies of Bacteriochlorophylls and Chlorophylls.

Bacteriochlorophyll and chlorophyll molecules are crucial building blocks of the photosynthetic apparatus in bacteria, algae, and plants. Embedded in transmembrane protein complexes, they are responsible for the primary processes of photosynthesis: excitation energy and charge transfer. Here, we use ab initio many-body perturbation theory within the GW approximation and Bethe–Salpeter equation (BSE) approach to calculate the electronic structure and optical excitations of bacteriochlorophylls a, b, c, d, and e and chlorophylls a and b. We systematically study the effects of the structure, basis set size, partial self-consistency in GW, and the underlying exchange–correlation approximation and compare our calculations with results from time-dependent density functional theory, multireference RASPT2, and experimental literature results. We find that optical excitations calculated with GW+BSE are in excellent agreement with experimental data, with an average deviation of less than 100 meV for the first three bright excitations of the entire family of (bacterio)chlorophylls. Contrary to state-of-the-art time-dependent density functional theory (TDDFT) with an optimally tuned range-separated hybrid functional, this accuracy is achieved in a parameter-free approach. Moreover, GW+BSE predicts the energy differences between the low-energy excitations correctly and eliminates spurious charge transfer states that TDDFT with (semi)local approximations is known to produce. Our study provides accurate reference results and highlights the potential of the GW+BSE approach for the simulation of larger pigment complexes.

Highly Efficient Resolution-of-Identity Density Functional Theory Calculations on Central and Graphics Processing Units.

We present an efficient method to evaluate Coulomb potential matrices using the resolution of identity approximation and semilocal exchange-correlation potentials on central (CPU) and graphics processing units (GPU). The new GPU-based RI-algorithm shows a high performance and ensures the favorable scaling with increasing basis set size as the conventional CPU-based method. Furthermore, our method is based on the J-engine algorithm [White; , Head-Gordon, J. Chem. Phys. 1996, 7, 2620], which allows for further optimizations that also provide a significant improvement of the corresponding CPU-based algorithm. Due to the increased performance for the Coulomb evaluation, the calculation of the exchange-correlation potential of density functional theory on CPUs quickly becomes a bottleneck to the overall computational time. Hence, we also present a GPU-based algorithm to evaluate the exchange-correlation terms, which results in an overall high-performance method for density functional calculations. The algorithms to evaluate the potential and nuclear derivative terms are discussed, and their performance on CPUs and GPUs is demonstrated for illustrative calculations.

A periodic equation-of-motion coupled-cluster implementation applied to F-centers in alkaline earth oxides.

We present an implementation of the equation of motion coupled-cluster singles and doubles (EOM-CCSD) theory using periodic boundary conditions and a plane wave basis set. Our implementation of EOM-CCSD theory is applied to study F-centers in alkaline earth oxides employing a periodic supercell approach. The convergence of the calculated electronic excitation energies for neutral color centers in MgO, CaO, and SrO crystals with respect to the orbital basis set and system size is explored. We discuss extrapolation techniques that approximate excitation energies in the complete basis set limit and reduce finite size errors. Our findings demonstrate that EOM-CCSD theory can predict optical absorption energies of F-centers in good agreement with experiment. Furthermore, we discuss calculated emission energies corresponding to the decay from triplet to singlet states responsible for the photoluminescence properties. Our findings are compared to experimental and theoretical results available in the literature.

Accuracy of DLPNO-CCSD(T): Effect of Basis Set and System Size.

The DLPNO-CCSD(T) method is designed to study large molecular systems at significantly reduced cost relative to its canonical counterpart. However, the error in this approach is also size-extensive and relies on cancellation of errors for the calculation of relative energies. This work provides a direct comparison of canonical CCSD(T) and TightPNO DLPNO-CCSD(T) calculations of reaction energies and barriers of a broad range of chemical reactions. The dataset includes acidities, anion binding affinities, enolization, Diels-Alder, nucleophilic substitution, and atom transfer reactions and complements existing theoretical datasets in terms of system size as well as new reaction types (e.g., anion binding affinities and chlorine atom transfer reactions). The performance of DLPNO-CCSD(T) was further examined with respect to systematic variation of basis set and system size and amounts of nonbonded interaction present in the system. The errors in the DLPNO-CCSD(T) were found to be relatively insensitive to the choice of basis set for small systems but increase monotonically with system size. Additionally, calculations of barriers appear to be more challenging than reaction energies with errors exceeding 5 kJ mol-1 for many Diels-Alder reactions. Further tests on three realistic organic reactions reveal the impact of the DLPNO approximation in calculating absolute and relative barriers that are important for predictions such as stereoselectivity.

CC-stretched formic acid: isomerisation, dimerisation, and carboxylic acid complexation.

The cis-trans-isomerism of the propiolic acid monomer (HC[triple bond, length as m-dash]C-COOH) is examined with linear Raman jet spectroscopy, yielding the first environment-free vibrational band centres of a higher-energy cis-rotamer beyond formic acid (HCOOH) in addition to all fundamentals and a large number of hot and combination/overtone bands of the trans-conformer. Two near-isoenergetic trans-fundamentals of different symmetry (CC[double bond, length as m-dash]O bend and OH torsion) prove to be a sensitive benchmarking target, as their energetic order is susceptible to the choice of electronic structure method, basis set size, and inclusion of vibrational anharmonicity. For the infrared- and Raman-active C[double bond, length as m-dash]O stretching fundamentals of the cyclic (C2h) trans-propiolic acid dimer, resonance couplings are found that in part extend to the Cs-symmetric heterodimer of trans-propiolic and trans-formic acid. Exploratory vibrational perturbation theory (VPT2) calculations show that all perturbing states involve displacements of the OH moieties located on the doubly hydrogen bonded ring. The comparison of the infrared spectra of the propiolic acid dimer and its heterodimer with formic acid to that of several other carboxylic acid dimers from the literature reveals a notable similarity regarding a non-fundamental dimer band around 1800 cm-1, which in most cases is so far unassigned. VPT2 calculations and a simple harmonic model suggest an assignment to a combination vibration of the symmetric and antisymmetric OH torsion.

Prediction of the standard potentials for one-electron oxidation of N,N,N',N' tetrasubstituted p-phenylenediamines by calculation.

The formal potentials for the reversible one-electron oxidation of N,N,N',N' tetrasubstituted p-phenylenediamines in acetonitrile have been applied as a test set for benchmarking computational methods for a series of compounds with only small structural differences. The aim of the study is to propose a simple method for calculating the standard oxidation potentials, and therefore, the protocol is progressively developed by adding more terms in the energy expression. In addition, the effect of including implicit solvation models (IEFPCM, CPCM, and SMD), larger basis sets, and correlation methods are investigated. The oxidation potentials calculated using the G3MP2B3 approach with IEFPCM resulted in the best fit (R2 = 0.9624), but the slope of the correlation line, 0.74, is far from the optimal value, 1.00. B3LYP/6-311++G(d,p) and TPSSh/6-311++G(2d,p) yielded only slightly less consistent data (R2 = 0.9388 and R2 = 0.9425), but with much better slopes, 1.00 and 0.94, respectively. We conclude that it is important to investigate the basis set size and treatment of electron correlation when calculating oxidation potentials for N,N,N',N' tetrasubstituted p-phenylenediamines.