Local pair natural orbitals for excited states

We explore how in response calculations for excitation energies with wavefunction based (e.g., coupled cluster) methods the number of double excitation amplitudes can be reduced by means of truncated pair natural orbital (PNO) expansions and localized occupied orbitals. Using the CIS(D) approximation as a test model, we find that the number of double excitation amplitudes can be reduced dramatically with minor impact on the accuracy if the excited state wavefunction is expanded in state-specific PNOs generated from an approximate first-order guess wavefunction. As for ground states, the PNO truncation error can also for excitation energies be controlled by a single threshold related to generalized natural occupation numbers. The best performance is found with occupied orbitals which are localized by the Pipek-Mezey localization. For a large test set of excited states we find with this localization that already a PNO threshold of 10(-8)-10(-7), corresponding to an average of only 40-80 PNOs per pair, is sufficient to keep the PNO truncation error for vertical excitation energies below 0.01 eV. This is a significantly more rapid convergence with the number doubles amplitudes than in domain-based local response approaches. We demonstrate that the number of significant excited state PNOs scales asymptotically linearly with the system size in the worst case of completely delocalized excitations and sub-linearly whenever the chromophore does not increase with the system size. Moreover, we observe that the flexibility of state-specific PNOs to adapt to the character of an excitation allows for an almost unbiased treatment of local, delocalized and charge transfer excited states.     

Local explicitly correlated second- and third-order Møller-Plesset perturbation theory with pair natural orbitals

We present an algorithm for computing explicitly correlated second- and third-order M?ller-Plesset energies near the basis set limit for large molecules with a cost that scales formally as N(4) with system size N. This is achieved through a hybrid approach where locality is exploited first through orbital specific virtuals (OSVs) and subsequently through pair natural orbitals (PNOs) and integrals are approximated using density fitting. Our method combines the low orbital transformation costs of the OSVs with the compactness of the PNO representation of the doubles amplitude vector. The N(4) scaling does not rely upon the a priori definition of domains, enforced truncation of pair lists, or even screening and the energies converge smoothly to the canonical values with decreasing occupation number thresholds, used in the selection of the PNO basis. For MP2.5 intermolecular interaction energies, we find that 99% of benchmark basis set limit correlation energy contributions are recovered using an aug-cc-pVTZ basis and that on average only 50 PNOs are required to correlate the significant orbital pairs.     

Comparison of explicitly correlated local coupled-cluster methods with various choices of virtual orbitals

Explicitly correlated local coupled-cluster (LCCSD-F12) methods with pair natural orbitals (PNOs), orbital specific virtual orbitals (OSVs), and projected atomic orbitals (PAOs) are compared. In all cases pair-specific virtual subspaces (domains) are used, and the convergence of the correlation energy as a function of the domain sizes is studied. Furthermore, the performance of the methods for reaction energies of 52 reactions involving 58 small and medium sized molecules is investigated. It is demonstrated that for all choices of virtual orbitals much smaller domains are needed in the explicitly correlated methods than without the explicitly correlated terms, since the latter correct a large part of the domain error, as found previously. For PNO-LCCSD-F12 with VTZ-F12 basis sets on the average only 20 PNOs per pair are needed to obtain reaction energies with a root mean square deviation of less than 1 kJ mol(-1) from complete basis set estimates. With OSVs or PAOs at least 4 times larger domains are needed for the same accuracy. A new hybrid method that combines the advantages of the OSV and PNO methods is proposed and tested. While in the current work the different local methods are only simulated using a conventional CCSD program, the implications for low-order scaling local implementations of the various methods are discussed.     

Zero-field splittings from density functional calculations: analysis and improvement of known methods

Several different approaches have been proposed to calculate the zero-field splitting tensor with density functional methods. In this work, our own derivation is presented in some detail, to allow a theoretical analysis and a comparison with other methods [M. R. Pederson and S. N. Khanna, Phys. Rev. B 60, 9566 (1999); F. Neese, J. Am. Chem. Soc. 128, 10213 (2006); J. Chem. Phys. 127, 164112 (2007)]. Pederson's method can be improved by properly taking into account the quantum nature of spin when extracting the zero field splitting tensor from the magnetic anisotropy. A closed-shell molecule at large distance from an open shell complex will have a spurious contribution to the zero-field splitting tensor calculated with Neese's methods. We thus have analyzed his approach in some detail and found that it can be corrected if one properly transforms the equations used in wave function based theory to a sum-over-states type expression before one interprets it as an energy derivative. If improved along these lines, Neese's and Pederson's methods become identical down to the working equations. The theoretical analysis is illustrated by sample calculations on the well-studied Mn(III)-tris-acetylacetonato complex Mn(acac)(3), both as an isolated molecule and with a Pd(II) dichloro diammine complex at large distance as an innocent spectator.     

Ab initio and coupled-perturbed density functional theory estimation of zero-field splittings in MnII transition metal complexes

The paper presents a method comparison for the prediction of zero-field splitting (ZFS) parameters in a series of Mn (II) coordination complexes. The test set consists of Mn (II) complexes that are experimentally well-characterized by X-ray diffraction and high-field electron paramagnetic resonance. Their ZFS parameters have been calculated using density functional theory (DFT) as well as complete active space self-consistent field (CASSCF) methods. It is shown that the recently introduced coupled-perturbed spin-orbit coupling (CP-SOC) approach [ Neese, F. J. Chem. Phys. 2007, 127, 164112 ] together with hybrid-DFT functionals leads to a slope of the correlation line (plot of experimental vs calculated D values) that is essentially unity provided that the direct spin-spin interaction is properly included in the treatment. This is different from our previous DFT study on the same series of complexes where a severe overestimation of the D parameter has been found [ Zein, S. ; Duboc, C. ; Lubitz, W. ; Neese, F. Inorg. Chem. 2008, 47, 134 ]. CASSCF methods have been used to evaluate the ZFS in an "ab initio ligand-field" type treatment. The study demonstrates that a substantial part of the relevant physics is lost in such a treatment since only excitations within the manganese d-manifold are accounted for. Thus, a severe underestimation of the D parameter has been found. Because the CASSCF calculations in combination with quasidegenerate perturbation theory treats the SOC to all orders, we have nevertheless verified that second-order perturbation theory is an adequate approximation in the case of the high-spin d (5) configuration.     

Extension of the renormalized coupled-cluster methods exploiting left eigenstates of the similarity-transformed hamiltonian to open-shell systems: a benchmark study

The recently formulated completely renormalized coupled-cluster method with singles, doubles, and noniterative triples, exploiting the biorthogonal form of the method of moments of coupled-cluster equations (Piecuch, P.; W?och, M. J. Chem. Phys. 2005, 123, 224105; Piecuch, P.; W?och, M.; Gour, J. R.; Kinal, A. Chem. Phys. Lett. 2006, 418, 467), termed CR-CC(2,3), is extended to open-shell systems. Test calculations for bond breaking in the OH radical and the F2+ ion and singlet-triplet gaps in the CH2, HHeH, and (HFH)- biradical systems indicate that the CR-CC(2,3) approach employing the restricted open-shell Hartree--Fock (ROHF) reference is significantly more accurate than the widely used CCSD(T) method and other noniterative triples coupled-cluster approximations without making the calculations substantially more expensive. A few molecular examples, including the activation energies of the C2H4 + H --> C2H5 forward and reverse reactions and the triplet states of the CH2 and H2Si2O2 biradicals, are used to show that the dependence of the ROHF-based CR-CC(2,3) energies on the method of canonicalization of the ROHF orbitals is, for all practical purposes, negligible.     

Open-shell singlet character of stable derivatives of nonacene, hexacene and teranthene

The electronic ground states of the recently synthesized stable nonacene derivatives (J. Am. Chem. Soc. 2010, 132, 1261) are open-shell singlets with a polyradical nature instead of closed-shell singlets as originally assumed, according to the unrestricted broken spin-symmetry density functional theory (UBS-DFT) computations (at B3LYP/6-31G*). It is the bulky protecting groups, not the transfer from the open-shell singlet to closed-shell singlet ground state, that stabilizes these longest characterized acenes. Similar analyses also confirmed the open-shell singlet character of the hexacene and teranthene derivatives.     

A mathematical discussion of Pons Viver's implementation of Löwdin's spin projection operator

Recently, the molecular electronic structure theories for efficiently treating static (or strong) correlation in a black-box manner have attracted much attention. In these theories, a spin projection operator is used to recover the spin symmetry of a broken-symmetry Slater determinant. Very recently, Pons Viver proposed the practical and exact implementation of Löwdin's spin projection operator (Int. J. Quantum Chem. 2019, 119, e25770). In the present study, we attempt to supply mathematical proofs to Pons Viver's proposals and show a condition for establishing Pons Viver's implementation. Moreover, we explicitly derive the (spin projected) extended Hartree-Fock (EHF) equations on the basis of the model of common orbitals (ie, closed-shell orbitals used in the restricted open-shell Hartree-Fock (ROHF) method), which was combined by Pons Viver with the EHF method.     

The implementation of a self-consistent constricted variational density functional theory for the description of excited states

We present here the implementation of a self-consistent approach to the calculation of excitation energies within regular Kohn-Sham density functional theory. The method is based on the n-order constricted variational density functional theory (CV(n)-DFT) [T. Ziegler, M. Seth, M. Krykunov, J. Autschbach, and F. Wang, J. Chem. Phys. 130, 154102 (2009)] and its self-consistent formulation (SCF-CV(∞)-DFT) [J. Cullen, M. Krykunov, and T. Ziegler, Chem. Phys. 391, 11 (2011)]. A full account is given of the way in which SCF-CV(∞)-DFT is implemented. The SCF-CV(∞)-DFT scheme is further applied to transitions from occupied π orbitals to virtual π(?) orbitals. The same series of transitions has been studied previously by high-level ab initio methods. We compare here the performance of SCF-CV(∞)-DFT to that of time dependent density functional theory (TD-DFT), CV(n)-DFT and ΔSCF-DFT, with the ab initio results as a benchmark standard. It is finally demonstrated how adiabatic TD-DFT and ΔSCF-DFT are related through different approximations to SCF-CV(∞)-DFT.     

Coupled-cluster dynamic polarizabilities including triple excitations

Dynamic polarizabilities for open- and closed-shell molecules were obtained by using coupled-cluster (CC) linear response theory with full treatment of singles, doubles, and triples (CCSDT-LR) with large basis sets utilizing the NWChem software suite. By using four approximate CC methods in conjunction with augmented cc-pVNZ basis sets, we are able to evaluate the convergence in both many-electron and one-electron spaces. For systems with primarily dynamic correlation, the results for CC3 and CCSDT are almost indistinguishable. For systems with significant static correlation, the CC3 tends to overestimate the triples contribution, while the PS(T) approximation [J. Chem. Phys. 127, 164105 (2007)] produces mixed results that are heavily dependent on the accuracies provided by noniterative approaches used to correct the equation-of-motion CCSD excitation energies. Our results for open-shell systems show that the choice of reference (restricted open-shell Hartree-Fock versus unrestricted Hartree-Fock) can have a significant impact on the accuracy of polarizabilities. A simple extrapolation based on pentuple-zeta CCSD calculations and triple-zeta CCSDT calculations reproduces experimental results with good precision in most cases.     

Combining two-body density correlation functionals with multiconfigurational wave functions using natural orbitals and occupation numbers

We propose a procedure that combines multiconfigurational (MC) wave functions with two-body density correlation functionals by transforming the latter into functionals of the MC natural orbitals and occupation numbers. The method is tested with the spectroscopic constants of a set of 11 diatomics, the diradical-involved automerization barrier of cyclobutadiene, the energy difference between triplet and open-shell singlet states in He and the methylene molecule, and the magnetic coupling constants of several systems, such as NiO, KNiF(3), K(2)NiF(4), La(2)CuO(4), alpha-4-dehydrotoluene, 1,1('),5,5(')-tetramethyl-6,6(')-dioxo-3,3(')-biverdazyl, [Cu(2)Cl(6)](-2), copper(II) acetate monohidrate and H-He-H. The procedure is applied to the Colle-Salvetti [Theor. Chim. Acta 37, 329 (1975); 53, 55 (1979)], functional and to a size-consistent functional depending on the on-top pair density (F1-5-N(eff)). On average, the best results are provided by the transformed F1-5-N(eff) [J. Chem. Phys. 114, 2022 (2001)] functional.     

Quantum Monte Carlo calculations of the dissociation energy of the water dimer

We report diffusion quantum Monte Carlo (DMC) calculations of the equilibrium dissociation energy D(e) of the water dimer. The dissociation energy measured experimentally, D(0), can be estimated from D(e) by adding a correction for vibrational effects. Using the measured dissociation energy and the modern value of the vibrational energy Mas et al., [J. Chem. Phys. 113, 6687 (2000)] leads to D(e)=5.00+/-0.7 kcal mol(-1), although the result Curtiss et al., [J. Chem. Phys. 71, 2703 (1979)] D(e)=5.44+/-0.7 kcal mol(-1), which uses an earlier estimate of the vibrational energy, has been widely quoted. High-level coupled cluster calculations Klopper et al., [Phys. Chem. Chem. Phys. 2, 2227 (2000)] have yielded D(e)=5.02+/-0.05 kcal mol(-1). In an attempt to shed new light on this old problem, we have performed all-electron DMC calculations on the water monomer and dimer using Slater-Jastrow wave functions with both Hartree-Fock approximation (HF) and B3LYP density functional theory single-particle orbitals. We obtain equilibrium dissociation energies for the dimer of 5.02+/-0.18 kcal mol(-1) (HF orbitals) and 5.21+/-0.18 kcal mol(-1) (B3LYP orbitals), in good agreement with the coupled cluster results.     

Exponential transformation of molecular orbitals. II. General formulation for UHF calculations and application to diatomics and molecular fragments

The exponential transformation of the molecular orbitals, that has been previously used to achieve a process with a convergence of quadratic quality in SCF closed-shell calculations [J. Chem. Phys. 72 , 1452 (1980)] has been extended to UHF determinantal wave functions built from different orbitals for different spins. Explicit formulas are given for the first and second derivatives of the energy to be varied. The method is illustrated by UHF calculations for systems described as standard singlets (Li₂ and F₂) or triplets (NH) at the RHF approximation level, as well as for CH, CH₂, CH₃ molecular fragments in their valence states.     

Spin-adapted open-shell time-dependent density functional theory. III. An even better and simpler formulation

The recently proposed spin-adapted time-dependent density functional theory (S-TD-DFT) [Z. Li and W. Liu, J. Chem. Phys. 133, 064106 (2010)] resolves the spin-contamination problem in describing singly excited states of high spin open-shell systems. It is an extension of the standard restricted open-shell Kohn-Sham-based TD-DFT which can only access those excited states due to singlet-coupled single excitations. It is also far superior over the unrestricted Kohn-Sham-based TD-DFT (U-TD-DFT) which suffers from severe spin contamination for those excited states due to triplet-coupled single excitations. Nonetheless, the accuracy of S-TD-DFT for high spin open-shell systems is still inferior to TD-DFT for well-behaved closed-shell systems. The reason can be traced back to the violation of the spin degeneracy conditions (SDC) by approximate exchange-correlation (XC) functionals. Noticing that spin-adapted random phase approximation (S-RPA) can indeed maintain the SDC by virtue of the Wigner-Eckart theorem, a hybrid ansatz combining the good of S-TD-DFT and S-RPA can immediately be envisaged. The resulting formalism, dubbed as X-TD-DFT, is free of spin contamination and can also be viewed as a S-RPA correction to the XC kernel of U-TD-DFT. Compared with S-TD-DFT, X-TD-DFT leads to much improved results for the low-lying excited states of, e.g., N(2)(+), yet with much reduced computational cost. Therefore, X-TD-DFT can be recommended for routine calculations of excited states of high spin open-shell systems.     

Accurate thermochemistry from a parameterized coupled-cluster singles and doubles model and a local pair natural orbital based implementation for applications to larger systems

We have recently introduced a parameterized coupled-cluster singles and doubles model (pCCSD(α, β)) that consists of a bivariate parameterization of the CCSD equations and is inspired by the coupled electron pair approximations. In our previous work, it was demonstrated that the pCCSD(-1, 1) method is an improvement over CCSD for the calculation of geometries, harmonic frequencies, and potential energy surfaces for single bond-breaking. In this paper, we find suitable pCCSD parameters for applications in reaction thermochemistry and thermochemical kinetics. The motivation is to develop an accurate and economical methodology that, when coupled with a robust local correlation framework based on localized pair natural orbitals, is suitable for large-scale thermochemical applications for sizeable molecular systems. It is demonstrated that the original pCCSD(-1, 1) method and several other pCCSD methods are a significant improvement upon the standard CCSD approach and that these methods often approach the accuracy of CCSD(T) for the calculation of reaction energies and barrier heights. We also show that a local version of the pCCSD methodology, implemented within the local pair natural orbital (LPNO) based CCSD code in ORCA, is sufficiently accurate for wide-scale chemical applications. The LPNO based methodology allows us for routine applications to intermediate sized (20-100 atoms) molecular systems and is a significantly more accurate alternative to MP2 and density functional theory for the prediction of reaction energies and barrier heights.     

Closed-shell ring coupled cluster doubles theory with range separation applied on weak intermolecular interactions

We explore different variants of the random phase approximation to the correlation energy derived from closed-shell ring-diagram approximations to coupled cluster doubles theory. We implement these variants in range-separated density-functional theory, i.e., by combining the long-range random phase approximations with short-range density-functional approximations. We perform tests on the rare-gas dimers He(2), Ne(2), and Ar(2), and on the weakly interacting molecular complexes of the S22 set of Jurec?ka et al. [P. Jurec?ka, J. S?poner, J. C?erny?, and P. Hobza, Phys. Chem. Chem. Phys. 8, 1985 (2006)]. The two best variants correspond to the ones originally proposed by Szabo and Ostlund [A. Szabo and N. S. Ostlund, J. Chem. Phys. 67, 4351 (1977)]. With range separation, they reach mean absolute errors on the equilibrium interaction energies of the S22 set of about 0.4 kcal/mol, corresponding to mean absolute percentage errors of about 4%, with the aug-cc-pVDZ basis set.     

Spin-Projected EHF equations

Extended Hartree-Fock (EHF ) equations are developed for the general open-shell case using a modified pair-orthogonality-constrained variation (POCV ) method. The EHF energy is expressed in terms of corresponding orbitals that are required to remain orthogonal and paired for all arbitrary infinitesimal variations. The Euler equations for each set of orbitals are reduced to unique pseudosecular equations, the LCAO form of which may easily be derived. The Euler equations and the expressions obtained for the off-diagonal elements of the ?^γδ (γ, δ = a or b) matrices for the closed-shell case are identical to those obtained by Mayer, who used the generalized Brillouin theorem method. However, the present method yields equations for both closed- and open-shell cases and for any spin state.     

Coupled cluster approaches with an approximate account of triexcitations and the optimized inner projection technique

Summary A general orthogonally spin-adapted formalism for coupled cluster (CC) approaches, with an approximate account of triexcited configurations, and for optimized inner projection (OIP) technique is described. Modifying the linear part of the CC equations for pair clusters (CCD) we obtain the orthogonally spin-adapted, non-iterative version of the CCDT-1 method of Bartlett et al. [J. Chem. Phys. 80, 4371 (1984), 81, 5906 (1984), 82, 5761 (1985)]. Similar modification of an approximate coupled pair theory corrected for connected quadruply excited clusters (ACPQ) yields a new approach called ACPTQ. Both the CCDT-1 and ACPTQ methods can be formulated in terms of effective interaction matrix elements between the orthogonally spin-adapted biexcited singlet configurations. The same matrix elements also appear in the orthogonally spin-adapted form of the CCD + T(CCD) perturbative estimate of triply excited contributions due to Raghavachari [J. Chem. Phys. 82, 4607 (1985)] and Urban et al. [J. Chem. Phys. 83, 4041 (1985)], and in the OIP method when applied to the Pariser-Parr-Pople (PPP) model Hamiltonians. We use the diagrammatic approach based on the graphical methods of spin algebras to derive the explicit form of these interaction matrix elements. Finally, the relationship between different diagrammatic spin-adaptation procedures and their relative advantages are discussed in detail.Also affiliated with the Department of Chemistry, and Guelph-Waterloo Center for Graduate Work in Chemistry, Waterloo Campus, University of Waterloo, Waterloo, Ontario, Canada. Killam Research Fellow 1987–89