Toward spectroscopic accuracy of ab initio calculations of vibrational frequencies and related quantities: a case study of the HF molecule

Calculations at various coupled-cluster (CC) levels with and without the inclusion of linear r _{i j} -dependent terms are performed for the HF molecule in its ground state with a systematic variation of basis sets. The main emphasis is on spectroscopic properties such as the equilibrium distance r _e and the harmonic vibration frequency ω_e. Especially with the R12 methods (including linear r _{i j} -dependent terms), convergence to the basis set limit is reached. However, the results (at the basis set limit) are rather sensitive to the level of the treatment of electron correlation. The best results are found for the CCSDT1-R12 and CCSD[T]-R12 methods (CCSD[T] was previously called CCSD+T(CCSD)), while CCSD(T) overestimates ω_e by ≈6 cm⁻¹. The good agreement of conventional CCSD(T) with experiment for basis sets far from saturation (e.g. truncated at g-functions) is probably the result of a compensation of errors. The contribution of core-correlation is non-negligible and must be included (effect on ω_e≈5 cm⁻¹). Relativistic effects are also important (23 cm⁻¹), while adiabatic effects are much smaller (<1cm⁻¹) and non-adiabatic effects on ω_e can be simulated in replacing nuclear by atomic masses; for rotation nuclear masses appear to be the better choice, at least for hydrides. From a potential curve based on calculations with the CCSDT1-R12 method with relativistic corrections, the IR spectrum is computed quantum-mechanically. Both the band heads and the rotational structures of the observed spectra are reproduced with a relative error of ≈10⁻⁴ for the three isotopomers HF, DF, and TF. Received: 3 July 1998 / Accepted: 4 August 1998 / Published online: 28 October 1998     

Analysis of coupled cluster methods. II. What is the best way to account for triple excitations in coupled cluster theory?

Summary Various coupled cluster (CC) and quadratic CI (QCI) methods are compared in terms of sixth, seventh, eighth, and infinite order Møller-Plesset (MPn, n=6, 7, 8, ) perturbation theory. By partitioning the MPn correlation energy into contributions resulting from combinations of single (S), double (D), triple (T), quadruple (Q), pentuple (P), hextuple (H), etc. excitations, it has been determined how many and which of these contributions are covered by CCSD, QCISD, CCSD(T), QCISD(T), CCSD(TQ), QCISD(TQ), and CCSDT. The analysis shows that QCISD is inferior to CCSD because of three reasons: a) With regard to the total number of energy contributions QCI rapidly falls behind CC for largen. b) Part of the contributions resulting from T, P, and higher odd excitations are delayed by one order of perturbation theory. c) Another part of the T, P, etc. contributions is missing altogether. The consequence of reason a) is that QCISD(T) covers less infinite order effects than CCSD does, and QCISD(TQ) less than CCSD(T), which means that the higher investment on the QCI side (QCISD(T) :O(M ⁷), CCSD :O(M ⁶), QCISD(TQ) :O(M ⁸), CCSD(T) :O(M ⁷),M: number of basis functions) does not compensate for its basic deficiencies. Another deficiency of QCISD(T) is that it does not include a sufficiently large number of TT coupling terms to prevent an exaggeration of T effects in those cases where T correlation effects are important. The best T method in terms of costs and efficiency should be CCSD(T).     

New coupled cluster approaches based on the unrestricted Hartree-Fock reference for treating molecules with multireference character

Here we review the basic formalism, implementation details, and performance of two newly developed coupled cluster (CC) methods based on the unrestricted Hartree-Fock (UHF) reference for treating molecules with multireference character. These two approaches can be considered to be approximations to the CC singles, doubles, and triples (CCSDT) method. The key concept of these two approaches is the corresponding orbitals, which are unitary transformations of canonical UHF molecular orbitals so that all spin orbitals are grouped into unique orbital pairs. In one approach called CCSDT(5P), a subset of triple excitations involving up to five-pair indices is included. In another approach called CCSD(T)-h, the contribution of connected triple excitations is treated in a hybrid way. With the concept of active corresponding orbitals, triple excitations can be automatically partitioned into two subsets, and the amplitudes of these two subsets are determined via solving different equations. Both CCSD(T)-h and CCSDT(5P) computationally scale as the seventh power of the system size. A survey of a number of applications demonstrates that CCSD(T)-h is an excellent approximation to the full CCSDT method, and CCSDT(5P) provides a good approximation to CCSDT for single-bond breaking processes. The overall performance of CCSDT(5P) is less accurate than that of CCSD(T)-h, but significantly better than that of the widely used CCSD(T).     

The Beryllium Pentamer: Trailing an Uneven Sequence of Dissociation Energies

Recent high‐resolution spectroscopic studies by Merritt, Bondybey, and Heaven (Science 2009 , 324, 1548) have heightened the anticipation that small beryllium clusters will soon be observed in the laboratory. Beryllium clusters are important discrete models for the theoretical study of metals. The trigonal bipyramidal Be₅ molecule is studied using high‐level coupled cluster methods. We obtain the optimized geometry, atomization and dissociation energies, and vibrational frequencies. The c～CCSDT(Q) method is employed to compute the atomization and dissociation energies. In this approach, complete basis set (CBS) extrapolations at the CCSD(T) level of theory are combined with an additive correction for the effect of iterative triple and perturbative quadruple excitations. Harmonic vibrational frequencies are obtained using analytic gradients computed at the CCSD(T) level of theory. We report an atomization energy of 129.6 kcal mol^?1 at the trigonal bipyramid global minimum geometry. The Be₅→Be₄+Be dissociation energy is predicted to be 39.5 kcal mol^?1. The analogous dissociation energies for the smaller beryllium clusters are 64.0 kcal mol^?1 (Be₄→Be₃+Be), 24.2 kcal mol^?1 (Be₃→Be₂+Be), and 2.7 kcal mol^?1 (Be₂→Be+Be). The trigonal bipyramidal Be₅ structure has an equatorial–equatorial bond length of 2.000 Å and an axial–equatorial distance of 2.060 Å. Harmonic frequencies of 730, 611, 456, 583, 488, and 338 cm^?1 are obtained at the CCSD(T)/cc‐pCVQZ level of theory. Quadruple excitations are found to make noticeable contributions to the energetics of the pentamer, which exhibits a significant level of static correlation.     

Performance of single-reference coupled-cluster methods for quasidegenerate problems: The H4 model

Summary Several coupled-cluster methods based on a single-determinantal reference function have been applied to the model system composed of four hydrogen atoms in a trapezoidal arrangement. For nondegenerate regions all methods with the exception of CCD provide results within 1 mhartree of the exact (FCI) value. For degenerate regions such an accuracy can be achieved with the inclusion of theT ₃ andT ₄ clusters, in an iterative and in a noniterative manner. We report results for CCSDT, CCSDTQ-1, CCSD+TQ*(CCSD), CCSD+Q(CCSDT) plus other methods. In particular, the ACP method which has been proposed to indirectly account forT ₄ terms is critically analyzed by including allT ₃ contributions.     

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.     

Nearly Degenerate Isomers of C(BH)2: Cumulene,Carbene, or Carbone?

The ground electronic state of C(BH)₂ exhibits both a linear minimum and a peculiar angle‐deformation isomer with a central B‐C‐B angle near 90°. Definitive computations on these species and the intervening transition state have been executed by means of coupled‐cluster theory including single and double excitations (CCSD), perturbative triples (CCSD(T)), and full triples with perturbative quadruples (CCSDT(Q)), in concert with series of correlation‐consistent basis sets (cc‐pVXZ, X=D, T, Q, 5, 6; cc‐pCVXZ, X=T, Q). Final energies were pinpointed by focal‐point analyses (FPA) targeting the complete basis‐set limit of CCSDT(Q) theory with auxiliary core correlation, relativistic, and non‐Born–Oppenheimer corrections. Isomerization of the linear species to the bent form has a minuscule FPA reaction energy of 0.02 kcal mol^?1 and a corresponding barrier of only 1.89 kcal mol^?1. Quantum tunneling computations reveal interconversion of the two isomers on a timescale much less than 1 s even at 0 K. Highly accurate CCSD(T)/cc‐pVTZ and composite c～CCSDT(Q)/cc‐pCVQZ anharmonic vibrational frequencies confirm matrix‐isolation infrared bands previously assigned to linear C(BH)₂ and provide excellent predictions for the heretofore unobserved bent isomer. Chemical bonding in the C(BH)₂ species was exhaustively investigated by the atoms‐in‐molecules (AIM) approach, molecular orbital plots, various population analyses, local mode vibrations and force constants, unified reaction valley analysis (URVA), and other methods. Linear C(BH)₂ is a cumulene, whereas bent C(BH)₂ is best characterized as a carbene with little carbone character. Weak B–B attraction is clearly present in the unusual bent isomer, but its strength is insufficient to form a CB₂ ring with a genuine boron–boron bond and attendant AIM bond path.     

Ab initio study on the reaction mechanism of ozone with bromine atom

Ab initio calculations of the potential energy surface (PES) for the Br+O₃ reaction have been performed using the MP2, CCSD(T), and QCISD(T) methods with 6‐31G(d), 6‐311G(d), and 6‐311+G(3df). The reaction begins with a transition state (TS) when the Br atom attacks a terminal oxygen of ozone, producing an intermediate, the bromine trioxide (M), which immediately dissociates to BrO+O₂. The geometry optimizations of the reactants, products, and intermediate and transition states are carried out at the MP2/6‐31G(d) level. The reaction potential barrier is 3.09 kcal/mol at the CCSD(T)/6‐311+G(3df)//MP2 level, which shows that the bromine atom trends intensively to react with the ozone. The comparison of the Br+O₃ reaction with the F+O₃ and Cl+O₃ reactions indicates that the reactions of ozone with the halogen atoms have the similar reaction mechanism. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

A comparative basis-set study of NeH+ using coupled-cluster techniques

Unrestricted Hartree-Fock, coupled-cluster calculations are reported for the ground state of NeH⁺ using atomic basis sets of increasing size and accuracy for both Ne and H. The goal is to determine the basis set and coupled-cluster level of calculation needed to obtain a NeH⁺ potential energy curve of known accuracy. Here, it is shown that calculations using a quintuple zeta basis at the coupled-cluster singles and doubles level with noniterative triples, CCSD(T) , predict a Ne—H bond dissociation energy that is within about 0.01 eV of the exact Born–Oppenheimer molecular electronic structure result. Spectroscopic constants determined using the Simons–Parr–Finlan procedure are found to be in very good agreement with the experimental results. Calculations at the augmented quadruple zeta level for the two lowest triplet excited states of the NeH⁺ species are presented. Both of these states separate into ground-state Ne⁺ and H(1s). The resulting potential curves predict stable minima at the SCF, CCSD, and CCSD(T) levels with dissociation energies of about 0.07 eV. Spectroscopic constants from the potential curves and dissociation constants are reported. © 1994 John Wiley & Sons, Inc.     

Formation of Short Zn−Zn Bonds Stabilized by Simple Cyanide and Isocyanide Ligands

Cyanogen diluted in argon was reacted with laser ablated Zn atoms to produce the NCZnCN and NCZnZnCN cyanides and higher energy isocyanides ZnNC, CNZnNC, and CNZnZnNC, which were isolated in excess argon at 4 K. These reaction products, identified from the matrix infrared spectra of their ‐CN and ‐NC chromophore ligand stretching modes, were confirmed by ¹³C and ¹⁵N isotopic substitution and comparison with frequencies calculated by the B3LYP and CCSD(T) methods using the all electron aug‐cc‐pVTZ basis sets. The cyanide and isocyanide products were increased markedly by mercury arc UV photolysis, which covers the zinc atomic absorption. The above electronic structure calculations that produce appropriate ligand frequencies for these dizinc products also provide their Zn?Zn bond lengths: CCSD(T) calculations find a short 2.367 Å Zn?Zn bond in the NCZnZnCN cyanide, a shorter 2.347 Å Zn?Zn bond in the 37.4 kJ mol^?1 higher energy isocyanide CNZnZnNC, and a longer 4.024 Å bond in the dizinc van der Waals dimer. Thus, the diatomic cyanide (‐CN) and isocyanide (‐NC) ligands are as capable of stabilizing the Zn?Zn bond as many much larger ligands based on their measured and our calculated Zn?Zn bond lengths. This is the first example of dizinc complexes stabilized by different ligand isomers. Additional weaker bands in this region can be assigned to the analogous trizinc molecules NCZnZnZnCN and CNZnZnZnNC.     

A coupled cluster approach with a hybrid treatment of connected triple excitations: implementation and applications for open-shell systems

An implementation of the coupled cluster (CC) singles, doubles, and a hybrid treatment of connected triples [denoted as CCSD(T)-h], based on the unrestricted Hartree-Fock (UHF) reference, is presented. Based on the spin-integrated formulation, we have developed a computer program to achieve the automatic derivation and implementation of the CCSD(T)-h approach. The CCSD(T)-h approach computationally scales as the seventh power of the system size, and is affordable for many medium-sized systems. The present approach has been applied to study the equilibrium geometries and harmonic vibrational frequencies in a number of open-shell diatomic molecules and bond breaking potential energy profiles in several open-shell molecules, including CH(3), NH(2), and SiH(2). For all systems under study, the overall performance of the UHF-based CCSD(T)-h approach is very close to that of the corresponding CCSDT (CC singles, doubles, and triples), and much better than that of the UHF-based CCSD(T) (CC singles, doubles, and perturbative triples).     

Regioselectivity in cycloaddition reaction between phosphaacetylene and diazomethane: An ab initio study

The [3 + 2] cycloaddition reaction of phosphaacetylene with diazomethane was investigated by means of high level ab initio calculations. It was deduced that the aromatic diazaphosphole is formed via a nonaromatic intermediate. The regiospecificity of the reaction is thus determined by the energy difference between the two transition states that lead to the two possible regioisomeric intermediates. Of the transition states in the concerted pathways, the one leading to the regioisomer with two PC bonds ( 3 ) was found to be more stable at all the levels of theory investigated, including coupled-cluster singles doubles (CCSD)(T)/6-311 + G^*//Møller-Plessett(MP)2/6-311 + G^* (+ basis set superposition, BSSE, correction). The energy difference between the two transition states, however, is always less than 2 kcal/mol. When the free energies in the two reactions are calculated by use of the harmonic frequencies, the energy separation between the two transition structures remains practically unchanged. The free energy of activation ΔG^† was 21 kcal/mol at the CCSD(T)/6-311 + G^* level of theory and use of the MP2/6-31 + G^* frequencies. At the MP2 level, a rather stable complex is obtained in the initial phase of the reaction. However, the stability of the complexes decreases at the CCSD(T) level, and application of the BSSE correction results in unstable complexes. © 1997 by John Wiley & Sons, Inc.     

Theoretical reference values for the AE6 and BH6 test sets from explicitly correlated coupled-cluster theory

We propose a new computational protocol to obtain highly accurate theoretical reference data. This protocol employs the explicitly correlated coupled-cluster method with iterative single and double excitations as well as perturbative triple excitations, CCSD(T)(F12), using quadruple-z\zeta basis sets. Higher excitations are accounted for by conventional CCSDT(Q) calculations using double-z\zeta basis sets, while core/core-valence correlation effects are estimated by conventional CCSD(T) calculations using quadruple-z\zeta basis sets. Finally, scalar-relativistic effects are accounted for by conventional CCSD(T) calculations using triple-z\zeta basis sets. In the present article, this protocol is applied to the popular test sets AE6 and BH6. An error analysis shows that the new reference values obtained by our computational protocol have an uncertainty of less than 1 kcal/mol (chemical accuracy). Furthermore, concerning the atomization energies, a cancellation of the basis set incompleteness error in the CCSD(T)(F12) perturbative triples contribution with the corresponding error in the contribution from higher excitations is observed. This error cancellation is diminished by the CCSD(T*)(F12) method. Thus, we recommend the use of the CCSD(T*)(F12) method only for small- and medium-sized basis sets, while the CCSD(T)(F12) approach is preferred for high-accuracy calculations in large basis sets.     

Symmetry breaking and electron correlation in O2−, O2, and O2+: A comparison of coupled cluster and quadratic configuration interaction approaches

Potential energy curves are calculated for O₂⁻, O₂, and O₂⁺ at the CCSD, QCISD, CCSD(T), and QCISD(T) levels of theory using aug-cc-pVDZ and aug-cc-pVTZ basis sets with electron correlation built onto inversion symmetry constrained and relaxed UHF wave functions. The spectroscopic constant r_e, w_e, w_e, x_e, D_j, and α_e, are determined from the potential curves using standard second-order perturbation theory expressions and are compared with experimental values to assess the relative accuracy of the theoretical approaches. Comparison of corresponding symmetry-constrained and symmetry-relaxed calculations indicates that the CCSD method is generally superior to CCSD(T), QCISD, and QCISD(T) in recovering from a symmetry-broken reference function. © 1996 John Wiley & Sons, Inc.     

Critical comparison of various connected quadruple excitation approximations in the coupled-cluster treatment of bond breaking

To assess the limits of single-reference coupled-cluster (CC) methods for potential-energy surfaces, several methods have been considered for the inclusion of connected quadruple excitations. Most are based upon the factorized inclusion of the connected quadruple contribution (Qf) [J. Chem. Phys. 108, 9221 (1998)]. We compare the methods for the treatment of potential-energy curves for small molecules. These include CCSD(TQf), where the initial contributions of triple (T) and factorized quadruple excitations are added to coupled-cluster singles (S) and doubles (D), its generalization to CCSD(TQf), where instead of measuring their first contribution from orders in H, it is measured from orders in H=e(-(T1+T2))He(T1+T2); renormalized approximations of both, and CCSD2 defined in [J. Chem. Phys. 115, 2014 (2001)]. We also consider CCSDT, CCSDT(Qf), CCSDTQ, and CCSDTQP for comparison, where T, Q, and P indicate full triple, quadruple, and pentuple excitations, respectively. Illustrations for F2, the double bond breaking in water, and N2 are shown, including effects of quadruples on equilibrium geometries and vibrational frequencies. Despite the fact that no perturbative approximation, as opposed to an iterative approximation, should be able to separate a molecule correctly for a restricted-Hartree-Fock reference function, some of these higher-order approximations have a role to play in developing new, more robust procedures.     

Theoretical study of XCN− (XF,Cl) anions

Geometrical parameters, vibrational frequencies, zero‐point energy, dipole moment, NBO atomic charges of FCN^?, ClCN^? and their isomers, and the isomerization energy of FCN^? and ClCN^? are studied in detail using the ab initio MP2, CCSD, CCSD(T), and density functional B3LYP methods. The results are compared with the existing data reported using various theoretical methods in conjunction with different basis sets. The effect of electron correlation on the structural properties of these molecular anions is discussed. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Theoretical prediction of the noble gas complexes HeAuF and NeAuF

Ab initio calculations were carried out to investigate the structures and the stability of the noble gas complexes HeAuF and NeAuF through MP2 and CCSD(T) methods. The HeAuF was predicted to have a linear structure with weak He—Au covalent bonding, the distance of which is closer to the covalent limit in comparison with the corresponding van der Waals limit. The dissociation energy with respect to He + AuF was found to be 24 and 26 kJ·mol⁻¹ at the CCSD(T)/basis set B and B′ levels, respectively. However, similar calculations for NeAuF indicate that NeAuF is not a stable species.