Pair Natural Orbitals Research Articles

Scalable Electron Correlation Methods. 6. Local Spin-Restricted Open-Shell Second-Order Møller-Plesset Perturbation Theory Using Pair Natural Orbitals: PNO-RMP2.

We present a (near) linear scaling implementation of high-spin open-shell Møller-Plesset perturbation theory using pair natural orbitals (PNO-RMP2). The theory is based on a new variant of open-shell MP2 which is fully spin-adapted and uses a single set of spin-free amplitudes, as in closed-shell MP2. This method, denoted SROMP2, is invariant to unitary orbital transformations within the closed, open, and virtual orbital subspaces. Accordingly, only a single set of PNOs per spatial orbital pair is needed, and the efficiency is similar to closed-shell calculations. The PNOs are obtained using a semicanonical approximation with large domains of projected atomic orbitals (PAOs). Linear scaling is achieved provided that the open-shell orbitals are local, and distant pairs are treated by multipole approximations. The method is efficiently parallelized. The convergence of ionization and reaction energies as a function of the PAO and PNO domain sizes is demonstrated and found to be very similar as for closed-shell calculations. The suitability of the PNOs for explicitly correlated PNO-RCCSD-F12 calculations is also tested. So far, this method is only simulated using a conventional program with appropriate projections to the PAO and PNO subspaces. It is demonstrated for radical stabilization energies as well as ionization potentials that the errors caused by the local domain approximations with our default thresholds are negligible.

Effect of Electron Correlation on Intermolecular Interactions: A Pair Natural Orbitals Coupled Cluster Based Local Energy Decomposition Study.

The development of post-Hartree-Fock (post-HF) energy decomposition schemes that are able to decompose the HF and correlation components of the interaction energy into chemically meaningful contributions is a very active field of research. One of the challenges is to provide a clear-cut quantification to the elusive London dispersion component of the intermolecular interaction. London dispersion is well-known to be a pure correlation effect, and as such it is not properly described by mean field theories. In this context, we have recently developed the local energy decomposition (LED) analysis, which provides a chemically meaningful decomposition of the interaction energy between two or more fragments computed at the domain-based local pair natural orbitals coupled cluster (DLPNO-CCSD(T)) level of theory. In this work, this scheme is used in conjunction with other interpretation tools to study a series of molecular adducts held together by intermolecular interactions of different natures. The HF and correlation components of the interaction energy are thus decomposed into a series of chemically meaningful contributions. Emphasis is placed on discussing the physical effects associated with the inclusion of electron correlation. It is found that four distinct physical effects can contribute to the magnitude of the correlation part of intermolecular binding energies (Δ EintC): (i) London dispersion, (ii) the correlation correction to the reference induction energy, (iii) the correlation correction to the electron sharing process, and (iv) the correlation correction to the permanent electrostatics. As expected, the largest contribution to the correlation binding energy of neutral, apolar molecules is London dispersion, as in the argon dimer case. In contrast, the correction for the HF induction energy dominates Δ EintC in systems in which an apolar molecule interacts with charged or strongly polar species, as in Ar-Li+. This effect has its origin in the systematic underestimation of polarizabilities at the HF level of theory. For similar reasons, electron sharing largely contributes to the correlation binding energy of covalently bound molecules, as in the beryllium dimer case. Finally, the correction for HF permanent electrostatics significantly contributes to Δ EintC in molecules with strong dipoles, such as water and hydrogen fluoride dimers. This effect originates from the characteristic overestimation of dipole moments at the HF level of theory, leading in some cases to positive Δ EintC values. Our results are apparently in contrast to the widely accepted view that Δ EintC is typically dominated by London dispersion, at least, in the strongly interacting region. Clearly, post-HF energy decomposition schemes are very powerful tools to analyze, categorize, and understand the various contributions to the intermolecular interaction energy. Hopefully, this will eventually lead to insights that are helpful in designing systems with tailored properties. All analysis tools presented in this work will be available free of charge in the next release of the ORCA program package.

Correction to Scalable Electron Correlation Methods. 4. Parallel Explicitly Correlated Local Coupled Cluster with Pair Natural Orbitals (PNO-LCCSD-F12).

Correction to Scalable Electron Correlation Methods. 4. Parallel Explicitly Correlated Local Coupled Cluster with Pair Natural Orbitals (PNO-LCCSD-F12).

State-Averaged Pair Natural Orbitals for Excited States: A Route toward Efficient Equation of Motion Coupled-Cluster.

A reduced-complexity variant of equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) method is formulated in terms of state-averaged excited-state pair natural orbitals (PNO) designed to describe manifolds of excited states. State-averaged excited-state PNOs for the target manifold are determined by averaging CIS(D) pair densities over the model manifold. The performance of the PNO-EOM-CCSD approach has been tested with the help of a distributed-memory parallel canonical EOM-CCSD implementation within the Massively Parallel Quantum Chemistry program that allows treatment of systems with 50+ atoms using realistic basis sets with 1000+ functions. The use of state-averaged PNOs offers several potential advantages relative to the recently proposed state-specific PNOs: our approach is robust with respect to root flipping and state degeneracies, it is more economical when computing large manifolds of states, and it simplifies evaluation of transition-specific observables such as dipole moments. With the PNO truncation threshold of 10-7, the errors in excitation energies are on average below 0.02 eV for the first six singlet states of 28 organic molecules included in the standard test set of Thiel and co-workers ( J. Chem. Phys. 2008, 128, 134110) with 50-70 state-averaged PNOs per pair.

Optimized Pair Natural Orbitals for the Coupled Cluster Methods.

We present the coupled-cluster singles and doubles method formulated in terms of truncated pair natural orbitals (PNO) that are optimized to minimize the effect of truncation. Compared to the standard ground-state PNO coupled-cluster approaches, in which truncated PNOs derived from first-order Møller-Plesset (MP1) amplitudes are used to compress the CC wave operator, the iteratively optimized PNOs ("iPNOs") offer moderate improvement for small PNO ranks but rapidly increase their effectiveness for large PNO ranks. The error introduced by PNO truncation in the CCSD energy is reduced by orders of magnitude in the asymptotic regime, with an insignificant increase in PNO ranks. The effect of PNO optimization is particularly effective when combined with Neese's perturbative correction for the PNO incompleteness of the CCSD energy. The use of the perturbative correction in combination with the PNO optimization procedure seems to produce the most precise approximation to the canonical CCSD energies for small and large PNO ranks. For the standard benchmark set of noncovalent binding energies, remarkable improvements with respect to the standard PNO approach range from a factor of 3 with PNO truncation threshold τPNO = 10-6 (with the maximum PNO truncation error in the binding energy of only 0.1 kcal/mol) to more than 2 orders of magnitude with τPNO = 10-9.

A near-linear scaling equation of motion coupled cluster method for ionized states.

In this work, a domain-based local pair natural orbital (DLPNO) version of the equation of motion coupled cluster theory with single and double excitations for ionization potentials (IP-EOM-CCSD) equations has been formulated and implemented. The method uses ground state localized occupied and pair natural virtual orbitals and applies the DLPNO machinery to arrive at a linear scaling implementation of the IP-EOM-CCSD method. The accuracy of the method is controllable using ground state truncation parameters. Using default thresholds, the method predicts ionization potential (IP) values with good accuracy (mean absolute error of 0.08 eV). We demonstrate that our code can be used to compute IP values for systems with more than 1000 atoms and 10 000 basis functions.

A pair natural orbital based implementation of CCSD excitation energies within the framework of linear response theory.

We present a pair natural orbital (PNO)-based implementation of coupled cluster singles and doubles (CCSD) excitation energies that builds upon the previously proposed state-specific PNO approach to the excited state eigenvalue problem. We construct the excited state PNOs for each state separately in a truncated orbital specific virtual basis and use a local density-fitting approximation to achieve an at most quadratic scaling of the computational costs for the PNO construction. The earlier reported excited state PNO construction is generalized such that a smooth convergence of the results for charge transfer states is ensured for general coupled cluster methods. We investigate the accuracy of our implementation by applying it to a large and diverse test set comprising 153 singlet excitations in organic molecules. Already moderate PNO thresholds yield mean absolute errors below 0.01 eV. The performance of the implementation is investigated through the calculations on alkene chains and reveals an at most cubic cost-scaling for the CCSD iterations with the system size.

Pair Natural Orbital Restricted Open-Shell Configuration Interaction (PNO-ROCIS) Approach for Calculating X-ray Absorption Spectra of Large Chemical Systems.

In this work, the efficiency of first-principles calculations of X-ray absorption spectra of large chemical systems is drastically improved. The approach is based on the previously developed restricted open-shell configuration interaction singles (ROCIS) method and its parametrized version, based on a density functional theory (DFT) ground-state determinant ROCIS/DFT. The combination of the ROCIS or DFT/ROCIS methods with the well-known machinery of the pair natural orbitals (PNOs) leads to the new PNO-ROCIS and PNO-ROCIS/DFT variants. The PNO-ROCIS method can deliver calculated metal K-, L-, and M-edge XAS spectra orders of magnitude faster than ROCIS while maintaining an accuracy with calculated spectral parameters better than 1% relative to the original ROCIS method (referred to as canonical ROCIS). The method is of a black box character, as it does not require any user adjustments, while it scales quadratically with the system size. It is shown that for large systems, the size of the virtual molecular orbital (MO) space is reduced by more than 90% with respect to the canonical ROCIS method. This allows one to compute the X-ray absorption spectra of a variety of large "real-life" chemical systems featuring hundreds of atoms using a first-principles wave-function-based approach. Examples chosen from the fields of bioinorganic and solid-state chemistry include the Co K-edge XAS spectrum of aquacobalamin [H2OCbl]+, the Fe L-edge XAS spectrum of deoxymyoglobin (DMb), the Ti L-edge XAS spectrum of rutile TiO2, and the Fe M-edge spectrum of α-Fe2O3 hematite. In the largest calculations presented here, molecules with more than 700 atoms and cluster models with more than 50 metal centers were employed. In all the studied cases, very good to excellent agreement with experiment is obtained. It will be shown that the PNO-ROCIS method provides an unprecedented performance of wave-function-based methods in the field of computational X-ray spectroscopy.

Communication: Exact analytical derivatives for the domain-based local pair natural orbital MP2 method (DLPNO-MP2).

Electron correlation methods based on pair natural orbitals (PNOs) have gained an increasing degree of interest in recent years, as they permit energy calculations to be performed on systems containing up to many hundred atoms, while maintaining chemical accuracy for reaction energies. We present an approach for taking exact analytical first derivatives of the energy contributions in the simplest method of the family of Domain-based Local Pair Natural Orbital (DLPNO) methods, closed-shell DLPNO-MP2. The Lagrangian function contains constraints to account for the relaxation of PNOs. RI-MP2 reference geometries are reproduced accurately, as exemplified for four systems with a substantial degree of nonbonding interactions. By the example of electric field gradients, we demonstrate that omitting PNO-specific constraints can lead to dramatic errors for orbital-relaxed properties.

Assessment of DFT for endohedral complexes' dipole moment: PNO-LCCSD-F12 as a reference method.

We present a systematic evaluation of the performance of a wide range of exchange-correlation functionals and related dispersion correction schemes for the computation of dipole moments of endohedral complexes, formed through the encapsulation of an AB molecule (AB = LiF, HCl) inside carbon nanotubes (CNTs) of different diameter. The consistency and accuracy of (i) generalized gradient approximation, (ii) meta GGA, (iii) global hybrid, and (iv) range-separated hybrid density functionals are assessed. In total, 37 density functionals are tested. The results obtained using the highly accurate pair natural orbitals based explicitly correlated local coupled cluster singles doubles (PNO-LCCSD-F12) method of Werner and co-workers [Schwilk et al., J. Chem. Theory Comput., 2017, 13, 3650; Ma et al., J. Chem. Theory Comput., 2017, 13, 4871] with the aug-cc-pVTZ basis set serve as a reference. The static electric dipole moment is computed via the finite field response or, when possible, as the expectation value of the dipole operator. Among others, it is shown that functionals belonging to the class of range-separated hybrids, provide results closest to the coupled cluster reference data. In particular, the ωB97X as well as the M11 functional may be considered as a promising choice for computing electric properties of noncovalent endohedral complexes. On the other hand, the worst performance was found for the functionals which do not include the Hartree-Fock exchange. The analysis of both the coupled cluster and the DFT results indicates a strong coupling of dispersion and polarization that may also explain why lower level DFT methods, as well as Hartree-Fock and MP2, cannot yield dipole moments beyond a qualitative agreement with the higher order reference data. Interestingly, the much smaller and less systematically constructed basis sets of Pople of moderate size provide results of accuracy at least comparable with the extended Dunning's aug-cc-pVTZ basis set.

Scalable Electron Correlation Methods. 5. Parallel Perturbative Triples Correction for Explicitly Correlated Local Coupled Cluster with Pair Natural Orbitals.

A well-parallelized perturbative triples correction implementation for the pair natural orbital based coupled cluster method PNO-LCCSD(T)-F12 is presented. A composite approach is adopted in addressing the coupling due to off-diagonal Fock matrix elements, in which the local triples amplitudes are iteratively solved using small domains of triples natural orbitals, and a semicanonical (T0) domain correction with larger domains is applied to reduce the domain errors. This treatment adds only about 20% to the computational cost of (T0) calculations with large domains, and the errors due to the use of small domains in the iterations are very small. In addition, a two-step triple list selection method is applied: First, an initial triple list is generated using LCCSD-F12 pair energy criteria, and the (T0) triples energies are computed using small domains. Second, this list is reduced by neglecting triples with small energy contributions, and the final calculation with large domains is only carried out for the reduced list. The cost of the (T) calculation scales asymptotically linear with the molecular size and shows excellent parallelization efficiency up to hundreds of CPU cores. The convergence of the (T) contribution to the relative energies of large molecular systems is carefully tested, and for most of the cases the results obtained with our default thresholds agree within ∼0.5 kcal mol-1 with those computed with very tight thresholds. For all tested molecular systems where canonical calculations are still feasible, the PNO-LCCSD(T)-F12 relative energies also agree within 0.5 kcal mol-1 with the canonical CCSD(T)-F12 results using the same F12 treatment. The (T) calculation generally takes 3-5 times the cost of the preceding PNO-LCCSD-F12 calculation, primarily due to the large number of triples required in obtaining accurate relative energies. We find that for large molecular systems the triple selection criteria used in previous local triples methods are insufficient, and a much larger number of triples are required than it was assumed so far. Still, using a commodity computer cluster, the PNO-LCCSD(T)-F12 calculation of molecules with ∼100 atoms using augmented triple-ζ basis sets can be carried out in a few hours of elapsed time.

Combining Accuracy and Efficiency: An Incremental Focal-Point Method Based on Pair Natural Orbitals.

In this work, we present a new pair natural orbitals (PNO)-based incremental scheme to calculate CCSD(T) and CCSD(T0) reaction, interaction, and binding energies. We perform an extensive analysis, which shows small incremental errors similar to previous non-PNO calculations. Furthermore, slight PNO errors are obtained by using TPNO = TTNO with appropriate values of 10-7 to 10-8 for reactions and 10-8 for interaction or binding energies. The combination with the efficient MP2 focal-point approach yields chemical accuracy relative to the complete basis-set (CBS) limit. In this method, small basis sets (cc-pVDZ, def2-TZVP) for the CCSD(T) part are sufficient in case of reactions or interactions, while some larger ones (e.g., (aug)-cc-pVTZ) are necessary for molecular clusters. For these larger basis sets, we show the very high efficiency of our scheme. We obtain not only tremendous decreases of the wall times (i.e., factors >102) due to the parallelization of the increment calculations as well as of the total times due to the application of PNOs (i.e., compared to the normal incremental scheme) but also smaller total times with respect to the standard PNO method. That way, our new method features a perfect applicability by combining an excellent accuracy with a very high efficiency as well as the accessibility to larger systems due to the separation of the full computation into several small increments.

Scalable Electron Correlation Methods. 4. Parallel Explicitly Correlated Local Coupled Cluster with Pair Natural Orbitals (PNO-LCCSD-F12).

We present an efficient explicitly correlated pair natural orbital local coupled cluster (PNO-LCCSD-F12) method. The method is an extension of our previously reported PNO-LCCSD approach ( Schwilk et al., J. Chem. Theory Comput. 2017 , 13 , 3650 - 3675 ). Near linear scaling with the molecular size is achieved by using pair, domain, and projection approximations, local density fitting and local resolution of the identity, and by exploiting the sparsity of the local molecular orbitals as well as of the projected atomic orbitals. The effect of the various domain approximations is tested for a wide range of chemical reactions and intermolecular interactions. In accordance with previous findings, it is demonstrated that the F12 terms significantly reduce the domain errors. The convergence of the reaction and interaction energies with respect to the parameters that determine the domain sizes and pair approximations is extensively tested. The results obtained with our default thresholds agree within a few tenths of a kcal mol-1 with the ones computed with very tight options. For cases where canonical calculations are still feasible, the relative energies of local and canonical calculations agree within similar error bounds. The PNO-LCCSD-F12 method needs only 25-40% more computer time than a corresponding PNO-LCCSD calculation while greatly improving the accuracy. Our program is well parallelized and capable of computing accurate correlation energies for molecules with more than 150 atoms using augmented triple-ζ basis sets and more than 5000 basis functions. Using several nodes of a small computer cluster, such calculations can be carried out within a few hours.

Scalable Electron Correlation Methods. 3. Efficient and Accurate Parallel Local Coupled Cluster with Pair Natural Orbitals (PNO-LCCSD).

A well-parallelized local singles and doubles coupled-cluster (LCCSD) method using pair natural virtual orbitals (PNOs) is presented. The PNOs are constructed using large domains of projected atomic orbitals (PAOs) and orbital specific virtual orbitals (OSVs). We introduce a hierarchy of close, weak, and distant pairs, based on pair energies evaluated by local Møller-Plesset perturbation theory (LMP2). In contrast to most previous implementations of LCCSD methods, the close and weak pairs are not approximated by LMP2 but treated by higher-order methods. This leads to greatly improved accuracy for large systems, in particular when long-range dispersion interactions are important. Close pair amplitudes are optimized using approximate LCCSD equations, in which slowly decaying terms that mutually cancel at long-range are neglected. For weak pairs, the same approximations are used, but in addition, the nonlinear terms are neglected (coupled electron pair approximation). Distant pairs are treated by spin-component scaled (SCS)-LMP2 using multipole approximations. For efficiency reasons, various projection approximations are also introduced. The impact of these approximations on absolute and relative energies depends on the PNO domain sizes. The errors are found to be negligible, provided that sufficiently large PNO domains are used for close and weak pairs. For the selection of these domains the usual natural orbital occupation number criterion is found to be insufficient, and an additional energy criterion is used. For extended one-dimensional systems, the computational effort of the method scales nearly linearly with the number of correlated electrons, but the linear scaling regime is usually not reached in real-life applications for three-dimensional systems. Nevertheless, due to the parallelization that is efficient up to about 100-200 CPU cores (dependent on the molecular size), accurate calculations for three-dimensional molecules with about 100 atoms and augmented triple-ζ basis sets (e.g., cc-pVTZ-F12) can be carried out in 1-3 h of elapsed time (depending on the molecular structure and the number of CPU cores, excluding the time for Hartree-Fock). The convergence of the results with respect to the thresholds and options that control the domain, pair, and projection approximations is extensively tested. Benchmark examples for several difficult and large cases are presented, which demonstrate that the errors of relative energies (e.g., reaction energies, barrier heights) caused by the pair and projection approximations can be reduced to below 1 kJ mol-1. The remaining errors are mainly caused by the finite PAO domains. The larger these are made, the more intramolecular or intermolecular basis set superposition errors are introduced, and the canonical results are approached only very slowly. This problem is eliminated in the explicitly correlated variant of our method (PNO-LCCSD-F12), which will be described in a separate paper, along with a larger set of benchmark calculations.

SparseMaps-A systematic infrastructure for reduced scaling electronic structure methods. V. Linear scaling explicitly correlated coupled-cluster method with pair natural orbitals.

In this work, we present a linear scaling formulation of the coupled-cluster singles and doubles with perturbative inclusion of triples (CCSD(T)) and explicitly correlated geminals. The linear scaling implementation of all post-mean-field steps utilizes the SparseMaps formalism [P. Pinski et al., J. Chem. Phys. 143, 034108 (2015)]. Even for conservative truncation levels, the method rapidly reaches near-linear complexity in realistic basis sets, e.g., an effective scaling exponent of 1.49 was obtained for n-alkanes with up to 200 carbon atoms in a def2-TZVP basis set. The robustness of the method is benchmarked against the massively parallel implementation of the conventional explicitly correlated coupled-cluster for a 20-water cluster; the total dissociation energy of the cluster (∼186 kcal/mol) is affected by the reduced scaling approximations by only ∼0.4 kcal/mol. The reduced scaling explicitly correlated CCSD(T) method is used to examine the binding energies of several systems in the L7 benchmark data set of noncovalent interactions.

A new near-linear scaling, efficient and accurate, open-shell domain-based local pair natural orbital coupled cluster singles and doubles theory.

The Coupled-Cluster expansion, truncated after single and double excitations (CCSD), provides accurate and reliable molecular electronic wave functions and energies for many molecular systems around their equilibrium geometries. However, the high computational cost, which is well-known to scale as O(N6) with system size N, has limited its practical application to small systems consisting of not more than approximately 20-30 atoms. To overcome these limitations, low-order scaling approximations to CCSD have been intensively investigated over the past few years. In our previous work, we have shown that by combining the pair natural orbital (PNO) approach and the concept of orbital domains it is possible to achieve fully linear scaling CC implementations (DLPNO-CCSD and DLPNO-CCSD(T)) that recover around 99.9% of the total correlation energy [C. Riplinger et al., J. Chem. Phys. 144, 024109 (2016)]. The production level implementations of the DLPNO-CCSD and DLPNO-CCSD(T) methods were shown to be applicable to realistic systems composed of a few hundred atoms in a routine, black-box fashion on relatively modest hardware. In 2011, a reduced-scaling CCSD approach for high-spin open-shell unrestricted Hartree-Fock reference wave functions was proposed (UHF-LPNO-CCSD) [A. Hansen et al., J. Chem. Phys. 135, 214102 (2011)]. After a few years of experience with this method, a few shortcomings of UHF-LPNO-CCSD were noticed that required a redesign of the method, which is the subject of this paper. To this end, we employ the high-spin open-shell variant of the N-electron valence perturbation theory formalism to define the initial guess wave function, and consequently also the open-shell PNOs. The new PNO ansatz properly converges to the closed-shell limit since all truncations and approximations have been made in strict analogy to the closed-shell case. Furthermore, given the fact that the formalism uses a single set of orbitals, only a single PNO integral transformation is necessary, which offers large computational savings. We show that, with the default PNO truncation parameters, approximately 99.9% of the total CCSD correlation energy is recovered for open-shell species, which is comparable to the performance of the method for closed-shells. UHF-DLPNO-CCSD shows a linear scaling behavior for closed-shell systems, while linear to quadratic scaling is obtained for open-shell systems. The largest systems we have considered contain more than 500 atoms and feature more than 10 000 basis functions with a triple-ζ quality basis set.

Communication: Multipole approximations of distant pair energies in local correlation methods with pair natural orbitals.

The accuracy of multipole approximations for distant pair energies in local second-order Møller-Plesset perturbation theory (LMP2) as introduced by Hetzer et al. [Chem. Phys. Lett. 290, 143 (1998)] is investigated for three chemical reactions involving molecules with up to 92 atoms. Various iterative and non-iterative approaches are compared, using different energy thresholds for distant pair selection. It is demonstrated that the simple non-iterative dipole-dipole approximation, which has been used in several recent pair natural orbitals (PNO)-LMP2 and PNO-LCCSD (local coupled-cluster with singles and doubles) methods, may underestimate the distant pair energies by up to 50% and can lead to significant errors in relative energies, unless very tight thresholds are used. The accuracy can be much improved by including higher multipole orders and by optimizing the distant pair amplitudes iteratively along with all other amplitudes. A new approach is presented in which very small special PNO domains for distant pairs are used in the iterative approach. This reduces the number of distant pair amplitudes by 3 orders of magnitude and keeps the additional computational effort for the iterative optimization of distant pair amplitudes minimal.

Local complete active space second-order perturbation theory using pair natural orbitals (PNO-CASPT2).

We present a CASPT2 method which exploits local approximations to achieve linear scaling of the computational effort with the molecular size, provided the active space is small and local. The inactive orbitals are localized, and the virtual space for each electron pair is spanned by a domain of pair-natural orbitals (PNOs). The configuration space is internally contracted, and the PNOs are defined for uniquely defined orthogonal pairs. Distant pair energies are obtained by multipole approximations, so that the number of configurations that are explicitly treated in the CASPT2 scales linearly with molecular size (assuming a constant active space). The PNOs are generated using approximate amplitudes obtained in a pair-specific semi-canonical basis of projected atomic orbitals (PAOs). The evaluation and transformation of the two-electron integrals use the same parallel local density fitting techniques as recently described for linear-scaling PNO-LMP2 (local second-order Møller-Plesset perturbation theory). The implementation of the amplitude equations, which are solved iteratively, employs the local integrated tensor framework. The efficiency and accuracy of the method are tested for excitation energies and correlation energies. It is demonstrated that the errors introduced by the local approximations are very small. They can be well controlled by few parameters for the distant pair approximation, initial PAO domains, and the PNO domains.

Analytic energy derivatives for the calculation of the first-order molecular properties using the domain-based local pair-natural orbital coupled-cluster theory

The domain-based local pair-natural orbital coupled-cluster (DLPNO-CC) theory has recently emerged as an efficient and powerful quantum-chemical method for the calculation of energies of molecules comprised of several hundred atoms. It has been demonstrated that the DLPNO-CC approach attains the accuracy of a standard canonical coupled-cluster calculation to about 99.9% of the basis set correlation energy while realizing linear scaling of the computational cost with respect to system size. This is achieved by combining (a) localized occupied orbitals, (b) large virtual orbital correlation domains spanned by the projected atomic orbitals (PAOs), and (c) compaction of the virtual space through a truncated pair natural orbital (PNO) basis. In this paper, we report on the implementation of an analytic scheme for the calculation of the first derivatives of the DLPNO-CC energy for basis set independent perturbations within the singles and doubles approximation (DLPNO-CCSD) for closed-shell molecules. Perturbation-independent one-particle density matrices have been implemented in order to account for the response of the CC wave function to the external perturbation. Orbital-relaxation effects due to external perturbation are not taken into account in the current implementation. We investigate in detail the dependence of the computed first-order electrical properties (e.g., dipole moment) on the three major truncation parameters used in a DLPNO-CC calculation, namely, the natural orbital occupation number cutoff used for the construction of the PNOs, the weak electron-pair cutoff, and the domain size cutoff. No additional truncation parameter has been introduced for property calculation. We present benchmark calculations on dipole moments for a set of 10 molecules consisting of 20-40 atoms. We demonstrate that 98%-99% accuracy relative to the canonical CCSD results can be consistently achieved in these calculations. However, this comes with the price of tightening the threshold for the natural orbital occupation number cutoff by an order of magnitude compared to the DLPNO-CCSD energy calculations.

Towards a pair natural orbital coupled cluster method for excited states.

The use of back-transformed pair natural orbitals in the calculation of excited state energies, ionization potentials, and electron affinities is investigated within the framework of equation of motion coupled cluster theory and its similarity transformed variant. Possible approaches to a more optimal use of pair natural orbitals in these methods are indicated.