Renormalized coupled-cluster methods exploiting left eigenstates of the similarity-transformed Hamiltonian

Completely renormalized (CR) coupled-cluster (CC) approaches, such as CR-CCSD(T), in which one corrects the standard CC singles and doubles (CCSD) energy for the effects of triply (T) and other higher-than-doubly excited clusters [K. Kowalski and P. Piecuch, J. Chem. Phys. 113, 18 (2000)], are reformulated in terms of the left eigenstates Phimid R:L of the similarity-transformed Hamiltonian of CC theory. The resulting CR-CCSD(T)(L) or CR-CC(2,3) and other CR-CC(L) methods are derived from the new biorthogonal form of the method of moments of CC equations (MMCC) in which, in analogy to the original MMCC theory, one focuses on the noniterative corrections to standard CC energies that recover the exact, full configuration-interaction energies. One of the advantages of the biorthogonal MMCC theory, which will be further analyzed and extended to excited states in a separate paper, is a rigorous size extensivity of the basic ground-state CR-CC(L) approximations that result from it, which was slightly violated by the original CR-CCSD(T) and CR-CCSD(TQ) approaches. This includes the CR-CCSD(T)(L) or CR-CC(2,3) method discussed in this paper, in which one corrects the CCSD energy by the relatively inexpensive noniterative correction due to triples. Test calculations for bond breaking in HF, F(2), and H(2)O indicate that the noniterative CR-CCSD(T)(L) or CR-CC(2,3) approximation is very competitive with the standard CCSD(T) theory for nondegenerate closed-shell states, while being practically as accurate as the full CC approach with singles, doubles, and triples in the bond-breaking region. Calculations of the activation enthalpy for the thermal isomerizations of cyclopropane involving the trimethylene biradical as a transition state show that the noniterative CR-CCSD(T)(L) approximation is capable of providing activation enthalpies which perfectly agree with experiment.     

Extensive generalization of renormalized coupled-cluster methods

The recently developed completely renormalized (CR) coupled-cluster (CC) methods with singles, doubles, and noniterative triples or triples and quadruples [CR-CCSD(T) or CR-CCSD(TQ), respectively], which are based on the method of moments of CC equations (MMCC) [K. Kowalski and P. Piecuch, J. Chem. Phys. 113, 18 (2000)], eliminate the failures of the standard CCSD(T) and CCSD(TQ) methods at larger internuclear separations, but they are not rigorously size extensive. Although the departure from strict size extensivity of the CR-CCSD(T) and CR-CCSD(TQ) methods is small, it is important to examine the possibility of formulating the improved CR-CC methods, which are as effective in breaking chemical bonds as the existing CR-CCSD(T) and CR-CCSD(TQ) approaches, which are as easy to use as the CR-CCSD(T) and CR-CCSD(TQ) methods, and which can be made rigorously size extensive. This may be particularly useful for the applications of CR-CC methods and other MMCC approaches in calculations of potential energy surfaces of large many-electron systems and van der Waals molecules, where the additive separability of energies in the noninteracting limit is very important. In this paper, we propose different types of CR-CC approximations, termed the locally renormalized (LR) CCSD(T) and CCSD(TQ) methods, which become rigorously size extensive if the orbitals are localized on nointeracting fragments. The LR-CCSD(T) and LR-CCSD(TQ) methods rely on the form of the energy expression in terms of the generalized moments of CC equations, derived in this work, termed the numerator-denominator-connected MMCC expansion. The size extensivity and excellent performance of the LR-CCSD(T) and LR-CCSD(TQ) methods are illustrated numerically by showing the results for the dimers of stretched HF and LiH molecules and bond breaking in HF and H2O.     

Application of renormalized coupled-cluster methods to potential function of water

The goal of this paper is to examine the performance of the conventional and renormalized single-reference coupled-cluster (CC) methods in calculations of the potential energy surface of the water molecule. A comparison with the results of the internally contracted multi-reference configuration interaction calculations including the quasi-degenerate Davidson correction (MRCI(Q)) and the spectroscopically accurate potential energy surface of water resulting from the use of the energy switching (ES) approach indicates that the relatively inexpensive completely renormalized (CR) CC methods with singles (S), doubles (D), and a non-iterative treatment of triples (T) or triples and quadruples (TQ), such as CR-CCSD(T), CR-CCSD(TQ), and the recently developed rigorously size extensive extension of CR-CCSD(T), termed CR-CC(2,3), provide substantial improvements in the results of conventional CCSD(T) and CCSD(TQ) calculations at larger internuclear separations. It is shown that the CR-CC(2,3) results corrected for the effect of quadruply excited clusters through the CR-CC(2,3)+Q approach can compete with the highly accurate MRCI(Q) data. The excellent agreement between the CR-CC(2,3)+Q and MRCI(Q) results suggests ways of improving the global potential energy surface of water resulting from the use of the ES approach in the regions of intermediate bond stretches and intermediate energies connecting the region of the global minimum with the asymptotic regions. Contribution to the Mark S. Gordon 65th Birthday Festschrift Issue.     

Computational studies on 3-aza-Cope rearrangements: protonation-induced switch of mechanism in the reaction of vinylpropargylamine

The 3-aza-Cope rearrangements of 3-azoniahexa-1,5-diene (1), 3-azoniahex-1-ene-5-yne (3), and 3-azahex-1-ene-5-yne (5) were investigated up to the coupled-cluster level, CCSD(T), by using a valence triple-zeta basis set. Activation barriers and geometrical parameters of the transition states are provided. Conformational studies were performed for all reactants and products of the reactions. Solvent effects were estimated from self-consistent reaction field calculations. In contrast to the other two species, the Cope rearrangement of 5 was found to proceed by a stepwise mechanism.     

NBO and NICS analysis of the allylic rearrangements (the Cope and 3-aza-Cope rearrangements) of hexa-1,5-diene and N-vinylprop-2-en-1-amine: A DFT study

In this work, ab initio density functional theory (DFT) calculations have been performed on the 3,3-sigmatropic rearrangements of hexa-1,5-diene (Cope) and N-vinylprop-2-en-1-amine (3-aza-Cope) in the gas phase. The barrier heights and heats of reactions calculated at the B3LYP/6-311G** level of theory were in good agreement with experimental data. Transition states optimized with B3LYP/6-311G** theory were used for calculating the nucleus independent chemical shift (NICS) and, a natural bond orbital (NBO) analysis was also performed at the same level of theory. Our results indicate that the aromaticities of the transition states are controlled by the out-of-plane component and that the chair-like transition state of the Cope rearrangement exhibits the strongest aromatic character. Analysis of donor-acceptor (bonding and anti-bonding) interactions of σ_3–4 → π*_1–2 suggests that the TS structure in the hexa-1,5-diene reaction (the Cope rearrangement) has more aromatic character than the N-vinylprop-2-en-1-amine reaction (the 3-aza-Cope rearrangement). The NBO results show that in the hexa-1,5-diene and N-vinylprop-2-en-1-amine rearrangements, activation energies are controlled by σ_3–4 → π*_1–2 and σ_3–4 → π*_1–2 resonance energies.      

Application of Cope rearrangement in synthesis

Cope rearrangement is one of the pericyclic reactions in many organic rearrangements. It has high stereoselectivity and has been widely used in organic synthesis chemistry. Herein, the discovery, mechanism, and application of Cope rearrangement were analyzed and discussed.     

On the accuracy of explicitly correlated coupled-cluster interaction energies--have orbital results been beaten yet?

The basis set convergence of weak interaction energies for dimers of noble gases helium through krypton is studied for six variants of the explicitly correlated, frozen geminal coupled-cluster singles, doubles, and noniterative triples [CCSD(T)-F12] approach: the CCSD(T)-F12a, CCSD(T)-F12b, and CCSD(T)(F12*) methods with scaled and unscaled triples. These dimers were chosen because CCSD(T) complete-basis-set (CBS) limit benchmarks are available for them to a particularly high precision. The dependence of interaction energies on the auxiliary basis sets has been investigated and it was found that the default resolution-of-identity sets cc-pVXZ/JKFIT are far from adequate in this case. Overall, employing the explicitly correlated approach clearly speeds up the basis set convergence of CCSD(T) interaction energies, however, quite surprisingly, the improvement is not as large as the one achieved by a simple addition of bond functions to the orbital basis set. Bond functions substantially improve the CCSD(T)-F12 interaction energies as well. For small and moderate bases with bond functions, the accuracy delivered by the CCSD(T)-F12 approach cannot be matched by conventional CCSD(T). However, the latter method in the largest available bases still delivers the CBS limit to a better precision than CCSD(T)-F12 in the largest bases available for that approach. Our calculations suggest that the primary reason for the limited accuracy of the large-basis CCSD(T)-F12 treatment are the approximations made at the CCSD-F12 level and the non-explicitly correlated treatment of triples. In contrast, the explicitly correlated second-order Mo?ller-Plesset perturbation theory (MP2-F12) approach is able to pinpoint the complete-basis-set limit MP2 interaction energies of rare gas dimers to a better precision than conventional MP2. Finally, we report and analyze an unexpected failure of the CCSD(T)-F12 method to deliver the core-core and core-valence correlation corrections to interaction energies consistently and accurately.     

Breaking bonds with the left eigenstate completely renormalized coupled-cluster method

The recently developed [P. Piecuch and M. Wloch, J. Chem. Phys. 123, 224105 (2005)] size-extensive left eigenstate completely renormalized (CR) coupled-cluster (CC) singles (S), doubles (D), and noniterative triples (T) approach, termed CR-CC(2,3) and abbreviated in this paper as CCL, is compared with the full configuration interaction (FCI) method for all possible types of single bond-breaking reactions between C, H, Si, and Cl (except H2) and the H2Si[Double Bond]SiH2 double bond-breaking reaction. The CCL method is in excellent agreement with FCI in the entire region R=1-3Re for all of the studied single bond-breaking reactions, where R and Re are the bond distance and the equilibrium bond length, respectively. The CCL method recovers the FCI results to within approximately 1 mhartree in the region R=1-3Re of the H-SiH3, H-Cl, H3Si-SiH3, Cl-CH3, H-CH3, and H3C-SiH3 bonds. The maximum errors are -2.1, 1.6, and 1.6 mhartree in the R=1-3Re region of the H3C-CH3, Cl-Cl, and H3Si-Cl bonds, respectively, while the discrepancy for the H2Si[Double Bond]SiH2 double bond-breaking reaction is 6.6 (8.5) mhartree at R=2(3)Re. CCL also predicts more accurate relative energies than the conventional CCSD and CCSD(T) approaches, and the predecessor of CR-CC(2,3) termed CR-CCSD(T).     

Asymmetric Catalytic Formal 1,4‐Allylation of β,γ‐Unsaturated α‐Ketoesters: Allylboration/Oxy‐Cope Rearrangement

A highly enantioselective formal conjugate allyl addition of allylboronic acids to β,γ‐unsaturated α‐ketoesters has been realized by employing a chiral Ni^II/N,N′‐dioxide complex as the catalyst. This transformation proceeds by an allylboration/oxy‐Cope rearrangement sequence, providing a facile and rapid route to γ‐allyl‐α‐ketoesters with moderate to good yields (65–92 %) and excellent ee values (90–99 % ee). The isolation of 1,2‐allylboration products provided insight into the mechanism of the subsequent oxy‐Cope rearrangement reaction: substrate‐induced chiral transfer and a chiral Lewis acid accelerated process. Based on the experimental investigations and DFT calculations, a rare boatlike transition‐state model is proposed as the origin of high chirality transfer during the oxy‐Cope rearrangement.     

Explicit correlation and basis set superposition error: the structure and energy of carbon dioxide dimer

We have investigated the slipped parallel and t-shaped structures of carbon dioxide dimer [(CO(2))(2)] using both conventional and explicitly correlated coupled cluster methods, inclusive and exclusive of counterpoise (CP) correction. We have determined the geometry of both structures with conventional coupled cluster singles doubles and perturbative triples theory [CCSD(T)] and explicitly correlated cluster singles doubles and perturbative triples theory [CCSD(T)-F12b] at the complete basis set (CBS) limits using custom optimization routines. Consistent with previous investigations, we find that the slipped parallel structure corresponds to the global minimum and is 1.09 kJ mol(-1) lower in energy. For a given cardinal number, the optimized geometries and interaction energies of (CO(2))(2) obtained with the explicitly correlated CCSD(T)-F12b method are closer to the CBS limit than the corresponding conventional CCSD(T) results. Furthermore, the magnitude of basis set superposition error (BSSE) in the CCSD(T)-F12b optimized geometries and interaction energies is appreciably smaller than the magnitude of BSSE in the conventional CCSD(T) results. We decompose the CCSD(T) and CCSD(T)-F12b interaction energies into the constituent HF or HF CABS, CCSD or CCSD-F12b, and (T) contributions. We find that the complementary auxiliary basis set (CABS) singles correction and the F12b approximation significantly reduce the magnitude of BSSE at the HF and CCSD levels of theory, respectively. For a given cardinal number, we find that non-CP corrected, unscaled triples CCSD(T)-F12b/VXZ-F12 interaction energies are in overall best agreement with the CBS limit.     

Coupled cluster and DFT calculations of 14N nuclear quadrupole coupling constants

The dependence of ¹⁴N quadrupole coupling constants calculated using coupled cluster theory on the level of approximation is examined for a series of small molecules. For HCN, HNC, CH₃CN, and CH₃NC, we use the coupled cluster singles‐and‐doubles with a noniterative perturbative triples correction—CCSD(T)—approach, and we analyze the basis set dependence of the results. For aziridine, diazirine, and cyclopropyl cyanide, we use the CCSD(T) approach, but smaller basis sets, and for the largest studied molecules—quinuclidine and hexamine—we present CCSD results. The differences between computed and experimental values for the best basis sets used are ≈ 5% at the CCSD level and decrease noticeably at the CCSD(T) level. The ‐ N≡C bonds are an exception—in this case the quadrupole coupling constants are very small, hence the differences between theory and experiment become larger (up to 9%). We also consider the performance of density functional theory, comparing the results for different density functionals with the coupled cluster values of the same constants. Most of the functionals provide results systematically improved with respect to the Hartree–Fock values, with ¹⁴N coupling constants in ‐ N≡C bonds being again an exception. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Understanding reaction mechanisms in organic chemistry from catastrophe theory applied to the electron localization function topology

Thom's catastrophe theory applied to the evolution of the topology of the electron localization function (ELF) gradient field constitutes a way to rationalize the reorganization of electron pairing and a powerful tool for the unambiguous determination of the molecular mechanisms of a given chemical reaction. The identification of the turning points connecting the ELF structural stability domains along the reaction pathway allows a rigorous characterization of the sequence of electron pair rearrangements taking place during a chemical transformation, such as multiple bond forming/breaking processes, ring closure processes, creation/annihilation of lone pairs, transformations of C-C multiple bonds into single ones. The reaction mechanism of some relevant organic reactions: Diels-Alder, 1,3-dipolar cycloaddition and Cope rearrangement are reviewed to illustrate the potential of the present approach.     

Simple coupled-cluster singles and doubles method with perturbative inclusion of triples and explicitly correlated geminals: The CCSD(T)R12 model

To approach the complete basis set limit of the "gold-standard" coupled-cluster singles and doubles plus perturbative triples [CCSD(T)] method, we extend the recently proposed perturbative explicitly correlated coupled-cluster singles and doubles method, CCSD(2)(R12) [E. F. Valeev, Phys. Chem. Chem. Phys. 8, 106 (2008)], to account for the effect of connected three-electron correlations. The natural choice of the zeroth-order Hamiltonian produces a perturbation expansion with rigorously separable second-order energy corrections due to the explicitly correlated geminals and conventional triple and higher excitations. The resulting CCSD(T)(R12) energy is defined as a sum of the standard CCSD(T) energy and an amplitude-dependent geminal correction. The method is technically very simple: Its implementation requires no modification of the standard CCSD(T) program and the formal cost of the geminal correction is small. We investigate the performance of the open-shell version of the CCSD(T)(R12) method as a possible replacement of the standard complete-basis-set CCSD(T) energies in the high accuracy extrapolated ab initio thermochemistry model of Stanton et al. [J. Chem. Phys. 121, 11599 (2004)]. Correlation contributions to the heat of formation computed with the new method in an aug-cc-pCVXZ basis set have mean absolute basis set errors of 2.8 and 1.0 kJmol when X is T and Q, respectively. The corresponding errors of the standard CCSD(T) method are 9.1, 4.0, and 2.1 kJmol when X=T, Q, and 5. Simple two-point basis set extrapolations of standard CCSD(T) energies perform better than the explicitly correlated method for absolute correlation energies and atomization energies, but no such advantage found when computing heats of formation. A simple Schwenke-type two-point extrapolation of the CCSD(T)(R12)aug-cc-pCVXZ energies with X=T,Q yields the most accurate heats of formation found in this work, in error on average by 0.5 kJmol and at most by 1.7 kJmol.     

Stepwise cope rearrangement of cyclo-biphenalenyl via an unusual multicenter covalent pi-bonded intermediate

Multicenter covalent pi-bonding between pi-conjugated radicals has been recently recognized as a novel and important bonding interaction. The Cope rearrangement of cyclo-biphenalenyl 9 is studied by exploring its potential energy surface with density functional theory (DFT), and it is found that pi-bonding plays a critical role in the rearrangement process. Affected by this, the rearrangement of 9 takes place by a stepwise mechanism through an unusual pi-intermediate 10, of C2h symmetry, which can be characterized as a 2 x 13pi + 2 x 2pi system. The pi-intermediate has a long inter-phenalenyl distance of R approximately 2.8 angstroms, which is shorter than the sum of the van der Waals radii displaying multicenter covalent pi-bonding between the two phenalenyl units. The energy of the pi-intermediate 10 is higher than that of the sigma-bonded reactant 9 by approximately 2 kcal/mol according to the employed spin-restricted DFT. NMR chemical shift calculations support the sigma-bonded 9 as the global minimum. The calculated activation barrier of approximately 6 kcal/mol for the Cope rearrangement is consistent with the stepwise mechanism. A covalent pi-bonding effect in the pi-intermediate 10 is demonstrated indirectly by the shortening of inter-naphthalene distance of the dianion and dication of the cyclophane 14 compared to that of its neutral counterpart. The unusual pi-bonded structure with a long inter-phenalenyl distance becomes the most stable structure for the ethano-bridged derivative 13, which should have observable paramagnetism according to the calculated paramagnetic susceptibility.