Resolution-of-the-identity Approximation Research Articles

Paramagnetic NMR Shielding Tensors and Ring Currents: Efficient Implementation and Application to Heavy Element Compounds.

We present an efficient implementation of paramagnetic NMR shielding tensors and shifts in a nonrelativistic and scalar-relativistic density functional theory framework. For the latter, we make use of the scalar exact two-component Hamiltonian in its local approximation, and generally we apply the well established (multipole-accelerated) resolution of the identity approximation and the seminumerical exchange approximation. The perturbed density matrix of a paramagnetic NMR shielding calculation is further used to study the magnetically induced current density and ring currents of open-shell systems as illustrated for [U@Bi12]3-. [U@Bi12]3- features delocalized highest occupied molecular orbitals and sustains a net diatropic ring current of ca. 18 nA/T through the Bi12 torus similar to the all-metal aromatic heavy-element cluster [Th@Bi12]4-.

PDG: A Composite Method Based on the Resolution of the Identity.

The Gaussian-3 (G3) composite approach for thermochemical properties is revisited in light of the enhanced computational efficiency and reduced memory costs by applying the resolution-of-the-identity (RI) approximation for two-electron repulsion integrals (ERIs) to the computationally demanding component methods in the G3 model: the energy and gradient computations via the second-order Møller-Plesset perturbation theory (MP2) and the energy computations using the coupled-cluster singles-doubles method augmented with noniterative triples corrections [CCSD(T)]. Efficient implementation of the RI-based methods is achieved by employing a hybrid distributed/shared memory model based on MPI and OpenMP. The new variant of the G3 composite approach based on the RI approximation is termed the RI-G3 scheme, or alternatively the PDG method. The accuracy of the new RI-G3/PDG scheme is compared to the "standard" G3 composite approach that employs the memory-expensive four-center ERIs in the MP2 and CCSD(T) calculations. Taking the computation of the heats of formation of the closed-shell molecules in the G3/99 test set as a test case, it is demonstrated that the RI approximation introduces negligible changes to the mean absolute errors relative to the standard G3 model (less than 0.1 kcal/mol), while the standard deviations remain unaltered. The efficiency and memory requirements for the RI-MP2 and RI-CCSD(T) methods are compared to the standard MP2 and CCSD(T) approaches, respectively. The hybrid MPI/OpenMP-based RI-MP2 energy plus gradient computation is found to attain a 7.5× speedup over the standard MP2 calculations. For the most demanding CCSD(T) calculations, the application of the RI approximation is found to nearly halve the memory demand, confer about a 4-5× speedup for the CCSD iterations, and reduce the computational time for the compute-intensive triples correction step by several hours.

All-electron periodic G0W0 implementation with numerical atomic orbital basis functions: Algorithm and benchmarks

We present an all-electron, periodic ${G}_{\text{0}}{W}_{\text{0}}$ implementation within the numerical atomic orbital (NAO) basis framework. A localized variant of the resolution-of-the-identity (RI) approximation is employed to significantly reduce the computational cost of evaluating and storing the two-electron Coulomb repulsion integrals. We demonstrate that the error arising from localized RI approximation can be reduced to an insignificant level by enhancing the set of auxiliary basis functions, used to expand the products of two single-particle NAOs. An efficient algorithm is introduced to deal with the Coulomb singularity in the Brillouin zone sampling that is suitable for the NAO framework. We perform systematic convergence tests and identify a set of computational parameters, which can serve as the default choice for most practical purposes. Benchmark calculations are carried out for a set of prototypical semiconductors and insulators, and compared to independent reference values obtained from an independent ${G}_{\text{0}}{W}_{\text{0}}$ implementation based on linearized augmented plane waves (LAPWs) plus high-energy localized orbitals (HLOs) basis set, as well as experimental results. With a moderate (FHI-aims tier 2) NAO basis set, our ${G}_{\text{0}}{W}_{\text{0}}$ calculations produce band gaps that typically lie in between the standard LAPW and the $\mathrm{LAPW}+\mathrm{HLO}$ results. Complementing tier 2 with highly localized Slater-type orbitals (STOs), we find that the obtained band gaps show an overall convergence towards the $\mathrm{LAPW}+\mathrm{HLO}$ results. The algorithms and techniques developed in this work pave the way for efficient implementations of correlated methods within the NAO framework.

Efficient evaluation of three-centre two-electron integrals over London orbitals

The nonperturbative calculation of molecular properties in magnetic fields requires the evaluation of integrals over complex-valued Gaussian-type London atomic orbitals (LAOs). With these orbitals, the calculation of four-centre electron-repulsion integrals (ERIs) is particularly demanding, because their permutational symmetry is lowered, and because complex algebra is required. We have implemented the resolution-of-the-identity (RI) approximation for LAOs in the TURBOMOLE program package. With respect to LAOs, employing the RI approximation is particularly beneficial, because the auxiliary basis set may always be chosen to be real-valued. As a consequence, the two-centre integrals in the RI approximation remain real-valued, and the three-centre integrals possess the same permutational symmetry as their real-valued counterparts. Compared to a direct calculation of four-centre ERIs over LAOs, using the RI approximation thus not only reduces the scaling of the integral evaluation, but also increases the efficiency by an additional factor of at least two. By using other well-established methods such as Cauchy–Schwarz screening, the difference-density approach, and Pulay's direct inversion in the iterative subspace (DIIS), the efficiency of nonperturbative calculations in magnetic fields can be increased even further.

Development of the FMO/RI-MP2 Fully Analytic Gradient Using a Hybrid-Distributed/Shared Memory Programming Model.

The fully analytic gradient of the second-order Møller-Plesset perturbation theory (MP2) with the resolution-of-the-identity (RI) approximation in the fragment molecular orbital (FMO) framework is derived and implemented using a hybrid multilevel parallel programming model, a combination of the general distributed data interface (GDDI) and the OpenMP API. The FMO/MP2 analytic gradient contains three parts, i.e., the internal fragment component, the electrostatic potential (ESP) component, and the response terms. The RI approximation is applied to the internal fragment MP2 gradient term, whose MP2 densities and monomer MP2 Lagrangians are shared with the ESP and the response terms. The FMO/RI-MP2 analytic gradient implementation is validated against the numerical gradient (with errors ∼10-6-10-5 Hartree/Bohr) and the energy conservation in molecular dynamics (MD) simulations using NVE ensembles. The RI approximation introduces an error of ∼10-5 Hartree/Bohr with a speedup of 4.0-8.0× compared with the currently available GDDI FMO/MP2 gradient. The node linear scaling of the fragmentation framework due to multilevel parallelism is well-preserved and is demonstrated in single-point gradient calculations of large water clusters (e.g., 1120 and 2165 molecules) using 300-800 KNL compute nodes with a parallel efficiency of more than 90%.

Resolution-of-the-identity approximation for complex-scaled basis functions.

A resolution-of-the-identity (RI) approximation for two-electron integrals over Gaussian basis functions with a complex-scaled exponent is presented. Such functions are used in non-Hermitian quantum mechanics to represent electronic resonances by L2 integrable wave functions with complex energies. We have implemented this new RI approximation for second-order Møller-Plesset perturbation (MP2) theory as well as for the Coulomb and exchange contributions in Hartree-Fock (HF) theory. We discuss the differences to the standard RI approximation of Hermitian quantum mechanics and demonstrate the utility of the non-Hermitian RI-MP2 and RI-HF methods by computations of the orientation-dependent ionization rates of CO, C6H6, and C10H8 in static electric fields. Our results illustrate that RI-MP2 correctly describes correlation effects in molecular electronic resonances while the computational cost is low enough to allow for investigations of medium-sized molecules.

Cluster perturbation theory. III. Perturbation series for coupled cluster singles and doubles excitation energies.

The cluster perturbation series, CPS(D), for coupled cluster singles and doubles excitation energies is considered. It is demonstrated that the second-order model CPS(D-2) is identical to the configuration interaction singles with perturbative doubles, CIS(D) model. The third-order model, CPS(D-3), provides excitation energies of coupled cluster singles and doubles (CCSD) quality in the sense that the difference between CPS(D-3) and CCSD excitation energies is of the same size or smaller than the effect of adding triples corrections to CCSD excitation energies. We further show that the third-order corrections can be efficiently implemented, in particular, when the resolution of the identity approximation is used for integrals. We also show that the CPS(D-3) excitation energies can be determined for system sizes that are far beyond what can be considered in conventional CCSD excitation energy calculations.

Compressing the Four-Index Two-Electron Repulsion Integral Matrix using the Resolution-of-the-Identity Approximation Combined with the Rank Factorization Approximation.

The four-index two-electron repulsion integral (4-2ERI) matrix is compressed using the resolution-of-the-identity (RI) approximation combined with the rank factorization approximation (RFA). The 4-2ERI is first approximated by the RI product. Then, the singular value decomposition (SVD) approximation is used to eliminate low-weighted singular vectors. The SVD RI approximation maintains the canonical form of the RI approximation and introduces a tunable compression factor. The characteristics of the SVD RI approximation along with the stochastic RI and natural auxiliary function approximation were numerically examined by applying these methods to the closed-shell second-order Møller-Plesset perturbation theory (MP2). The results show that, while the SVD RI approximation yields large errors for absolute properties (e.g., the correlation energy), it provides accurate relative properties (potential energy surface, binding energy) of the applied ab initio method (e.g., RHF, MP2).

Relativistic Cholesky-decomposed density matrix MP2

In the present article, we introduce the relativistic Cholesky-decomposed density (CDD) matrix second-order Møller-Plesset perturbation theory (MP2) energies. The working equations are formulated in terms of the usual intermediates of MP2 when employing the resolution-of-the-identity approximation (RI) for two-electron integrals. Those intermediates are obtained by substituting the occupied and virtual quaternion pseudo-density matrices of our previously proposed two-component (2C) atomic orbital-based MP2 (Helmich-Paris et al., 2016) by the corresponding pivoted quaternion Cholesky factors. While working within the Kramers-restricted formalism, we obtain a formal spin-orbit overhead of 16 and 28 for the Coulomb and exchange contribution to the 2C MP2 correlation energy, respectively, compared to a non-relativistic (NR) spin-free CDD-MP2 implementation. This compact quaternion formulation could also be easily explored in any other algorithm to compute the 2C MP2 energy. The quaternion Cholesky factors become sparse for large molecules and, with a block-wise screening, block sparse-matrix multiplication algorithm, we observed an effective quadratic scaling of the total wall time for heavy-element containing linear molecules with increasing system size. The total run time for both NR and 2C calculations was dominated by the contraction to the exchange energy. We have also investigated a bulky Te-containing supramolecular complex. For such bulky, three-dimensionally extended molecules the present screening scheme has a much larger prefactor and is less effective.

Efficient Implementation of NOCI-MP2 Using the Resolution of the Identity Approximation with Application to Charged Dimers and Long C-C Bonds in Ethane Derivatives.

An efficient implementation of the perturb-then-diagonalize nonorthogonal configuration interaction method with second-order Møller-Plesset perturbation theory (NOCI-MP2) is presented. Relative to other low scaling multireference perturbation theories, NOCI-MP2 often requires a much smaller active space because of the use of nonorthogonal reference configurations. Reworking the NOCI-MP2 equations with the resolution of the identity (RI) approximation enables the method to have the same memory requirements and computational scaling as single reference RI-MP2. The working equations are extended to include single substitutions as required when the reference determinants do not satisfy the Hartree-Fock equations. A detailed computational algorithm is presented along with timings to establish the performance of the implementation. NOCI-MP2 is applied to the binding energy and charge resonance energy in dication and monocation π dimers, as well as didiamantane ethane, and hexaphenylethane. A well-defined set of nonorthogonal determinants are obtained using absolutely localized molecular orbitals (ALMOs), as solutions to the self-consistent field for molecular interactions (SCF-MI) equations corresponding to covalent and ionic determinants. Agreement with experimental information where available, and other multireference methods, is satisfactory, with the use of an 0.3 au level shift to guard against large MP2 amplitudes. For didiamantane ethane and hexaphenylethane, large dispersion forces help stabilize the molecules despite the steric repulsion. By contrast, in the case of hexaphenylethane, the energy penalty from the geometric distortion of the fragments significantly weakens the bond.

Assessment of the overlap metric in the context of RI-MP2 and atomic batched tensor decomposed MP2

The resolution-of-the-identity approximation (RI) is a standard tool to accelerate the evaluation of two electron repulsion integrals. However introducing further approximations on top of for example RI-MP2, makes it important to check again the made choices. Usually the so called Coulomb metric is used. An alternative is the less accurate overlap metric, which has the benefit of a faster decay with distance. We encountered both choices in our atomic batched tensor decomposed MP2 (AB-MP2) and went with the Coulomb metric for the safe choice. In this work we re-investigate the choice of Coulomb and overlap metric for RI-MP2 and AB-MP2.

On Resolution-of-the-Identity Electron Repulsion Integral Approximations and Variational Stability.

The definiteness of the Mulliken and Dirac electron repulsion integral (ERI) matrices is examined for different classes of resolution-of-the-identity (RI) ERI approximations with particular focus on local fitting techniques. For global RI, robust local RI, and nonrobust local RI we discuss the definiteness of the approximated ERI matrices as well as the resulting bounds of Hartree, exchange, and total energies. Lower bounds of Hartree and exchange energy contributions are crucial as their absence may lead to variational instabilities, causing severe convergence problems or even convergence to a spurious state in self-consistent-field optimizations. While the global RI approximation guarantees lower bounds of Hartree and exchange energies, local RI approximations are generally unbounded. The robust local RI approximation guarantees a lower bound of the exchange energy but not of the Hartree energy. The nonrobust local RI approximation guarantees a lower bound of the Hartree energy but not of the exchange energy. These issues are demonstrated by sample calculations on carbon dioxide and benzene using the pair atomic RI approximation.

Correlation Energies from the Two-Component Random Phase Approximation.

The correlation energy within the two-component random phase approximation accounting for spin-orbit effects is derived. The resulting plasmon equation is rewritten-analogously to the scalar relativistic case-in terms of the trace of two Hermitian matrices for (Kramers-restricted) closed-shell systems and then represented as an integral over imaginary frequency using the resolution of the identity approximation. The final expression is implemented in the TURBOMOLE program suite. The code is applied to the computation of equilibrium distances and vibrational frequencies of heavy diatomic molecules. The efficiency is demonstrated by calculation of the relative energies of the Oh-, D4h-, and C5v-symmetric isomers of Pb6. Results within the random phase approximation are obtained based on two-component Kohn-Sham reference-state calculations, using effective-core potentials. These values are finally compared to other two-component and scalar relativistic methods, as well as experimental data.

Efficient evaluation of Coulomb integrals in a mixed Gaussian and plane-wave basis using the density fitting and Cholesky decomposition

The objective of this paper is to show that the density fitting (resolution of the identity approximation) can also be applied to Coulomb integrals of the type (k(1)(1)k(2)(1)|g(1)(2)g(2)(2)), where k and g symbols refer to plane-wave functions and gaussians, respectively. We have shown how to achieve the accuracy of these integrals that is needed in wave-function MO and density functional theory-type calculations using mixed Gaussian and plane-wave basis sets. The crucial issues for achieving such a high accuracy are application of constraints for conservation of the number electrons and components of the dipole moment, optimization of the auxiliary basis set, and elimination of round-off errors in the matrix inversion.

RI-MP2 Gradient Calculation of Large Molecules Using the Fragment Molecular Orbital Method.

The second-order Møller-Plesset perturbation theory (MP2) gradient using resolution of the identity approximation (RI-MP2 gradient) was combined with the fragment molecular orbital (FMO) method to evaluate the gradient including electron correlation for large molecules. In this study, we adopted a direct implementation of the RI-MP2 gradient, in which a characteristic feature of the FMO scheme was utilized. Test calculations with a small peptide presented a computational advantage of the RI-MP2 gradient over the canonical MP2 gradient. In addition, it was shown that the error of the RI-MP2 gradient, caused by RI approximation, was negligible. As an illustrative example, we performed gradient calculations for two biomolecules-a prion protein with GN8 and a human immunodeficiency virus type 1 (HIV1) protease with lopinavir (LPV). These calculations demonstrated that the gradient including the correlation effect could be evaluated with only about twice the computational effort of the Hartree-Fock (HF) gradient.

Correlation consistent basis sets for molecular core-valence effects with explicitly correlated wave functions: The atoms B–Ne and Al–Ar

Correlation consistent basis sets have been optimized for accurately describing core-core and core-valence correlation effects with explicitly correlated F12 methods. The new sets, denoted cc-pCVnZ-F12 (n=D, T, Q) and aug-cc-pC(F12)VnZ (n=D, T, Q, 5), were developed by augmenting the cc-pVnZ-F12 and aug-cc-pVnZ families of basis sets with additional functions whose exponents were optimized based on the difference between all-electron and valence-electron correlation energies. The number of augmented functions added is fewer, in general, than in the standard cc-pCVnZ and cc-pwCVnZ families of basis sets. Optimal values of the geminal Slater exponent for use with these basis sets in MP2-F12 calculations are presented and are also recommended for CCSD-F12b calculations. Auxiliary basis sets for use in the resolution of the identity approximation in explicitly correlated calculations have also been optimized and matched to the new cc-pCVnZ-F12 series of orbital basis sets. The cc-pCVnZ-F12 basis sets, along with the new auxiliary sets, were benchmarked in CCSD(T)-F12b calculations of spectroscopic properties on a series of homo- and heteronuclear first and second row diatomic molecules. Comparing the effects of correlating the outer core electrons in these molecules with those from conventional CCSD(T) at the complete basis set limit, which involved calculations with new cc-pCV6Z basis sets for the second row elements that were also developed in the course of this work, it is observed that the F12 values are reasonably well converged already at just the triple-zeta level.

Correlation consistent basis sets for explicitly correlated wavefunctions: valence and core–valence basis sets for Li, Be, Na, and Mg

Correlation consistent basis sets for the alkali and alkaline earth metals Li, Be, Na, and Mg have been optimized for use with explicitly correlated F12 methods. These include orbital basis sets for valence-only (denoted cc-pVnZ-F12) and core-valence (cc-pCVnZ-F12) correlation, along with compact complementary auxiliary basis sets for use in the resolution of the identity approximation. Additional auxiliary basis sets that can be employed in the density fitting of two-electron integrals in both explicitly correlated methods and in more conventional correlated methods, such as density fitted second-order Møller-Plesset perturbation theory (DF-MP2), have also been developed by adding additional diffuse or core-valence functions to the cc-pVnZ/MP2FIT sets of Hättig. Explicitly correlated calculations with the approximate CCSD(T)-F12b method have been carried out with these basis sets on a series of sixteen test molecules to demonstrate their accuracy for optimized geometries, harmonic vibrational frequencies, and atomization energies. Results comparable to conventional CCSD(T) quintuple-zeta, which are near the complete basis set limits for these molecules, are obtained using CCSD(T)-F12b with just triple-zeta quality basis sets. The effects on the spectroscopic constants from correlating the outer core electrons are accurately recovered with just the cc-pCVDZ-F12 basis sets.

Interaction Analysis of the Native Structure of Prion Protein with Quantum Chemical Calculations.

We examined the solvent interaction and intramolecular interaction of the native structure of prion protein (PrP) using quantum chemical calculations based on the fragment molecular orbital (FMO) method. The influence due to the geometrical fluctuation was taken into account by performing calculations on forty different conformations. Each FMO calculation was carried out at the MP2 level of theory with the cc-pVDZ in which the resolution of the identity approximation was employed to reduce the computational cost. The solvent interaction energies obtained from the calculations provided information about the hydrophilicity of the three α-helices. We examined the roles of the charged residues in retaining the native structure of PrP with the calculated intramolecular interaction energies. The analysis, focused on van der Waals interaction, showed that the hydrophobic residues were important for the stability of the native structure. Our results were also discussed in relation to the identified pathogenetic mutations of prion diseases. Additionally, we examined the distribution of the calculated values with 40 structures, in which we demonstrated the influence of geometrical fluctuations on quantum chemical calculations.

Specificity of 13C NMR Shielding Calculations in Thiocarbonyl Compounds

Isotropic 13 C nuclear magnetic shielding values (σ iso ) and corresponding chemical shift magnitudes (δ C ) have been calculated for the large series of the thiocarbonyl compounds using DFT (RI-BP86 and B3LYP) and ab initio (RHF and MP2, with or without resolution of the identity approximation) levels of theory and different basis sets. As expected, the Hartree–Fock method and hybrid B3LYP approach essentially overestimate deshielding of the C═S carbon nuclei. The RI-BP86 level of approximation provides a noticeably better agreement with the experiment and can be used for calculations of the theoretical 13 C NMR spectra for the most classes of the thiocarbonyl compounds. Geometry optimizations and the nuclear magnetic shielding calculations using the MP2 level of theory provide the most reliable results. The resolution of the identity algorithm implemented into the TURBOMOLE program set does not affect the calculated σ iso and δ C values compared to the strict MP2 nuclear magnetic shielding calculations (using the GAUSSIAN-03 program packet). A poor agreement with the experiment found in the case of the thiourea and thiosemicarbazide derivatives is referred to the different conformations of these species in the gas phase, solid state, and solution. Supplementary materials are available for this article. Please go to the publisher's online edition of Phosphorus, Sulfur, and Silicon to view the free supplemental resource.

Slater-type geminals in explicitly-correlated perturbation theory: application to n-alkanols and analysis of errors and basis-set requirements

In the recent years, Slater-type geminals (STGs) have been used with great success to expand the first-order wave function in an explicitly-correlated perturbation theory. The present work reports on this theory's implementation in the framework of the Turbomole suite of programs. A formalism is presented for evaluating all of the necessary molecular two-electron integrals by means of the Obara-Saika recurrence relations, which can be applied when the STG is expressed as a linear combination of a small number (n) of Gaussians (STG-nG geminal basis). In the Turbomole implementation of the theory, density fitting is employed and a complementary auxiliary basis set (CABS) is used for the resolution-of-the-identity (RI) approximation of explicitly-correlated theory. By virtue of this RI approximation, the calculation of molecular three- and four-electron integrals is avoided. An approximation is invoked to avoid the two-electron integrals over the commutator between the operators of kinetic energy and the STG. This approximation consists of computing commutators between matrices in place of operators. Integrals over commutators between operators would have occurred if the theory had been formulated and implemented as proposed originally. The new implementation in Turbomole was tested by performing a series of calculations on rotational conformers of the alkanols n-propanol through n-pentanol. Basis-set requirements concerning the orbital basis, the auxiliary basis set for density fitting and the CABS were investigated. Furthermore, various (constrained) optimizations of the amplitudes of the explicitly-correlated double excitations were studied. These amplitudes can be optimized in orbital-variant and orbital-invariant manners, or they can be kept fixed at the values governed by the rational generator approach, that is, by the electron cusp conditions. Electron-correlation effects beyond the level of second-order perturbation theory were accounted for by conventional coupled-cluster calculations with single, double and perturbative triple excitations [CCSD(T)]. The explicitly-correlated perturbation theory results were combined with CCSD(T) results and compared with literature data obtained by basis-set extrapolation.