Implementation of the CCSD(T)-F12 method using cusp conditions

The explicitly-correlated coupled-cluster singles and doubles with perturbative triples method (CCSD(T)-F12) is implemented using the cusp conditions. Numerical tests for a set of 16 molecules have shown agreement of correlation energies within 1 mE(h) between the cusp-condition and fully-optimized CCSD(T)-F12 methods. Benchmark calculations on 13 chemical reactions with the cusp-condition CCSD(T)-F12 method reproduce experimental enthalpies within 2 kJ mol(-1). It is also shown that regular unitary-invariant ansatz cannot exactly satisfy singlet and triplet cusp conditions in open-shell situations. We present an extended ansatz which can handle both conditions exactly.     

A simple and efficient CCSD(T)-F12 approximation

A new explicitly correlated CCSD(T)-F12 approximation is presented and tested for 23 molecules and 15 chemical reactions. The F12 correction strongly improves the basis set convergence of correlation and reaction energies. Errors of the Hartree-Fock contributions are effectively removed by including MP2 single excitations into the auxiliary basis set. Using aug-cc-pVTZ basis sets the CCSD(T)-F12 calculations are more accurate and two orders of magnitude faster than standard CCSD(T)/aug-cc-pV5Z calculations.     

Toward more efficient CCSD(T) calculations of intermolecular interactions in model hydrogen-bonded and stacked dimers

Interaction energies of the model H-bonded complexes, the formamide and formamidine dimers, as well as the stacked formaldehyde and ethylene dimers are calculated by the coupled cluster CCSD(T) method. These systems serve as a model for H-bonded and stacking interactions, typical in molecules participating in biological systems. We use the optimized virtual orbital space (OVOS) technique, by which the dimension of the space of virtual orbitals in coupled cluster CCSD(T) calculations can be significantly reduced. We demonstrate that when the space of virtual orbitals is reduced to 50% of the full space, which means reducing computational demands by 1 order of magnitude, the interaction energies for both H-bonded and stacked dimers are affected by no more than 0.1 kcal/mol. This error is much smaller than the error when interaction energies are calculated using limited basis sets.     

Calibration study of the CCSD(T)-F12a/b methods for C2 and small hydrocarbons

Explicitly correlated CCSD(T)-F12a/b methods combined with basis sets specifically designed for this technique have been tested for their ability to reproduce standard CCSD(T) benchmark data covering 16 small molecules composed of hydrogen and carbon. The standard method calibration set was obtained with very large one-particle basis sets, including some aug-cc-pV7Z and aug-cc-pV8Z results. Whenever possible, the molecular properties (atomization energies, structures, and harmonic frequencies) were extrapolated to the complete basis set limit in order to facilitate a direct comparison of the standard and explicitly correlated approaches without ambiguities arising from the use of different basis sets. With basis sets of triple-ζ quality or better, the F12a variant was found to overshoot the presumed basis set limit, while the F12b method converged rapidly and uniformly. Extrapolation of F12b energies to the basis set limit was found to be very effective at reproducing the best standard method atomization energies. Even extrapolations based on the small cc-pVDZ-F12/cc-pVTZ-F12 combination proved capable of a mean absolute deviation of 0.20 kcal/mol. The accuracy and simultaneous cost savings of the F12b approach are such that it should enable high quality property calculations to be performed on chemical systems that are too large for standard CCSD(T).     

Probing phenylalanine/adenine pi-stacking interactions in protein complexes with explicitly correlated and CCSD(T) computations

To examine the effects of pi-stacking interactions between aromatic amino acid side chains and adenine bearing ligands in crystalline protein structures, 26 toluene/(N9-methyl)adenine model configurations have been constructed from protein/ligand crystal structures. Full geometry optimizations with the MP2 method cause the 26 crystal structures to collapse to six unique structures. The complete basis set (CBS) limit of the CCSD(T) interaction energies has been determined for all 32 structures by combining explicitly correlated MP2-R12 computations with a correction for higher-order correlation effects from CCSD(T) calculations. The CCSD(T) CBS limit interaction energies of the 26 crystal structures range from -3.19 to -6.77 kcal mol (-1) and average -5.01 kcal mol (-1). The CCSD(T) CBS limit interaction energies of the optimized complexes increase by roughly 1.5 kcal mol (-1) on average to -6.54 kcal mol (-1) (ranging from -5.93 to -7.05 kcal mol (-1)). Corrections for higher-order correlation effects are extremely important for both sets of structures and are responsible for the modest increase in the interaction energy after optimization. The MP2 method overbinds the crystal structures by 2.31 kcal mol (-1) on average compared to 4.50 kcal mol (-1) for the optimized structures.     

Magnitude and nature of carbohydrate-aromatic interactions in fucose-phenol and fucose-indole complexes: CCSD(T) level interaction energy calculations

The CH/π contact structures of the fucose-phenol and fucose-indole complexes and the stabilization energies by formation of the complexes (E(form)) were studied by ab initio molecular orbital calculations. The three types of interactions (CH/π and OH/π interactions and OH/O hydrogen bonds) were compared and evaluated in a single molecular system and at the same level of theory. The E(form) calculated for the most stable CH/π contact structure of the fucose-phenol complex at the CCSD(T) level (-4.9 kcal/mol) is close to that for the most stable CH/π contact structure of the fucose-benzene complex (-4.5 kcal/mol). On the other hand the most stable CH/π contact structure of the fucose-indole complex has substantially larger E(form) (-6.5 kcal/mol). The dispersion interaction is the major source of the attraction in the CH/π contact structures of the fucose-phenol and fucose-indole complexes as in the case of the fucose-benzene complex. The electrostatic interactions in the CH/π contact structures are small (less than 1.5 kcal/mol). The nature of the interactions between the nonpolar surface of the carbohydrate and aromatic rings is completely different from that of the conventional hydrogen bonds where the electrostatic interaction is the major source of the attraction. The distributed multipole analysis and DFT-SATP analysis show that the dispersion interactions in the CH/π contact structure of fucose-indole complex are substantially larger than those in the CH/π contact structures of fucose-benzene and fucose-phenol complexes. The large dispersion interactions are responsible for the large E(form) for the fucose-indole complex.     

CCSDT and CCSD(T) calculations on model H-bonded and stacked complexes

The CCSD(T) and CCSDT interaction energies were determined for model planar H-bonded complexes (formamide…formamide, formamidine…formamidine) and stacked complexes (ethylene…ethylene, formaldehyde…formaldehyde). Various basis sets from the 6-31G*(0.25) to aug-cc-pVDZ were used. Difference between CCSD(T) and CCSDT interaction energies were small and become negligible (bellow 0.1 kcal/mol) if the aug-cc-pVDZ (or aug-cc-pVDZ/cc-pVDZ) basis set was applied. This result strongly supports the use of the CCSD(T) method for determination of true stabilization energies of extended complexes.     

Studies on the intermolecular interactions of metal chelate complexesVIII: On the interactions of dithiocarbamato-copper(II) and -copper(I) complexes with haloalkanes

The interaction of Cu(II)(dtc)₂ and Cu(I)(dtc) complexes with haloalkanes were studied by the EPR method. It was found that the Cu(II)(dtc)₂ complex reacted with haloalkanes only in the presence of weak Lewis bases which formed adducts with it. The intermediate reaction product is the mixed-ligand complex Cu(II)(dtc)X_n (X = Cl, Br, n = 1 or 2); the final products being CuX₂B_n (B = Lewis bases, n = 1 or 2) and unstable resin-like residue. Cu(I)(dtc) reacted with haloalkanes without any promoters giving the mixed-ligand complex Cu(II)(dtc)X_n as product. Free radicals were detected in the reaction of Cu(I)(dtc) using the method of “radical scavenger” and were not found in the reaction of Cu(II)(dtc)₂. The reported results confirmed one of the two reaction mechanisms proposed in the previous studies. The role of the solvent on the EPR parameters of the mixed-ligand Cu(II)(dtc)X complex is also discussed.     

Coupled-cluster theory with simplified linear-r(12) corrections: the CCSD(R12) model

A simplified singles-and-doubles linear-r(12) corrected coupled-cluster model, denoted CCSD(R12), is proposed and compared with the complete singles-and-doubles linear-r(12) coupled-cluster method CCSD-R12. An orthonormal auxiliary basis set is used for the resolution-of-the-identity approximation to calculate three-electron integrals needed in the linear-r(12) Ansatz. Basis-set convergence is investigated for a selected set of atoms and small molecules. In a large basis, the CCSD(R12) model provides an excellent approximation to the full linear-r(12) energy contribution, whereas the magnitude of this contribution is significantly overestimated at the level of second-order perturbation theory.     

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.     

Electron affinities of small uracil-water complexes: a comparison of benchmark CCSD(T) calculations with DFT

We present benchmark CCSD(T) calculations of the adiabatic electron affinities (AEA) and the vertical detachment energies (VDE) of the uracil molecule interacting with one to three water molecules. Calculations with rather large aug-cc-pVTZ basis set were only tractable when the space of virtual orbitals was reduced to about 60% of the full space employing the OVOS (Optimized Virtual Orbital Space) technique. Because of the microhydration, the valence-bound uracil anion is stabilized leading to gradually more positive values of both AEA and VDE with increasing number of participating water molecules. This agrees with experimental findings. Upon hydration by three water molecules, the electron affinity of uracil increased in comparison with AEA of the isolated molecule by about 250 up to 570 meV, depending on the geometry of the complex. CCSD(T) results confirm trends determined by DFT calculations of the microhydrated uracil and its anion, even if electron affinities of the free and hydrated uracil molecule are overestimated by DFT by up to 300 meV.     

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.     

基于图形处理单元与密度拟合近似的单精度耦合簇CCSD和CCSD(T)程序

作者此前工作表明, 在耦合簇CCSD (Coupled-Cluster approaches within the singles and doubles approximation)与CCSD(T) (CCSD approaches augmented by a perturbative treatment of triple excitations)计算中结合单精度数与消费型图形处理单元(GPU), 可以显著提高计算速度. 然而由于CCSD(T)计算对内存的巨大需求以及消费型GPU的内存限制, 在利用消费型GPU进行加速时, 不考虑利用空间对称性的情况下, 此前开发的CCSD(T)程序仅能用于计算300~400个基函数的体系. 利用密度拟合(Density-Fitting, DF)处理双电子积分可以显著降低CCSD(T)计算过程中的内存需求, 本工作发展了基于密度拟合近似并结合单精度数进行运算的DF-CCSD(T)程序, 该程序可用于包含700个基函数的无对称性体系的单点能计算, 以及包含1700个基函数的有对称性体系. 本工作所使用的计算节点配置了型号为Intel I9-10900k的CPU和型号为RTX3090的GPU, 与用双精度数在CPU上的计算相比, 利用单精度数结合GPU进行运算可以将CCSD的计算速度提升16倍, (T)部分可提升40倍左右, 而使用单精度数引入的误差可忽略不计. 在程序开发过程中, 作者发展了一套可利用GPU或CPU结合单精度数或双精度数进行含空间对称性的矩阵操作代码库. 基于该套代码库, 可以显著降低开发含空间对称性的耦合簇代码的难度.     

Nanoparticulate metal complexes prepared with compressed carbon dioxide: correlation of particle morphology with precursor structure

Nanoparticles of metal complexes are rare because of the limited methods for their preparation. Described in this communication is the use of precipitation with compressed antisolvent (PCA) to process nanoparticles of neutral metal-salen complexes. PCA is a semicontinuous technique utilizing supercritical carbon dioxide (scCO2) as the precipitant. The scCO2 dissolves into a solution of the complex and simultaneously extracts the solvent, resulting in formation of nanoparticles of uniform morphology. Spectroscopic and analytical methods were used to support the assignment that the nanoparticles are composed of metal complexes. Scanning electron microscopy reveals that planar complexes, such as [NiIIsalen], afford rodlike particles with average diameter and length of 85 and 700 nm, respectively. Complexes with nonplanar molecular structures produce nanoparticles with varied structures; for instance, PCA of the six-coordinate [Ru(salen)(NO)(Cl)] complex yields spherical nanoparticles with average diameters of 50 nm. These findings suggest a correlation between the structures of molecular precursors and the morphology of the processed particles, which possibly could be used as a bottom-up approach to the preparation of functional nanoparticles.     

Crystal structures and intermolecular interactions of two copper(II) complexes with an asymmetric diazine ligand

Reactions of asymmetric ligand N-phenylacetyl picoloylhydrazide (HL) and copper(II) acetate/chloride give two complexes CuL₂ (1) and Cu₂Cl₂L₂ (2). The coordination geometries of Cu(II) in 1 and 2 are a severely distorted octahedron and a distorted square pyramid, respectively. The binuclear copper complex 2 contains a centrosymmetric Cu₂(μ-Cl)₂ core. Individual molecules of 1 and 2 further self-assemble through non-covalent intermolecular bonds in the solid state to form extended 2-D polymers. The magnetic properties, IR, EA, and solid-state photoluminescence properties of the title complexes are presented.     

Reaction of carbon dioxide with arylcopper(I) complexes containing tertiary phosphines

ArCu^I compounds (Ar = phenyl, o-, m-, or p-tolyl) do not react with carbon dioxide at appreciable rate, but in the presence of triphenylphosphine in appropriate solvents undergo insertion to give the corresponding carboxylato complexes ArCO₂Cu(PPh₃)₂. In the presence of diphos the carbon dioxide absorption was very slow and mainly gave (ArCO₂Cu)₂diphos₃ complexes. The (diphenylphosphino)methane (DPM) derivatives [ArCO₂CuDPM]₂ and (ArCO₂Cu)₂DPM were prepared by other methods for comparison. The X-ray structure of HCO₂Cu(PPh₃)₂ is reported.     

Intermolecular interaction between hexafluorobenzene and benzene: Ab initio calculations including CCSD(T) level electron correlation correction

The intermolecular interaction energy of hexafluorobenzene-benzene has been calculated with the ARS-E model (a model chemistry for the evaluation of the intermolecular interaction energy between aromatic systems using extrapolation), which was formerly called the AIMI model. The CCSD(T) interaction energy at the basis-set limit has been estimated from the MP2 interaction energy at the basis-set limit and the CCSD(T) correction term obtained using a medium-sized basis set. The slipped-parallel (Cs) complex has the largest (most negative) interaction energy (-5.38 kcal/mol). The sandwich (C6v) complex is slightly less stable (-5.07 kcal/mol). The interaction energies of two T-shaped (C2v) complexes are very small (-1.74 and -0.88 kcal/mol). The calculated interaction energy of the slipped-parallel complex is about twice as large as that of the benzene dimer. The dispersion interaction is found to be the major source of attraction in the complex, although electrostatic interaction also contributes to the attraction. The dispersion interaction increases the relative stability of the slipped-parallel benzene dimer and the hexafluorobenzene-benzene complex compared to T-shaped ones. The electrostatic interaction is repulsive in the slipped-parallel benzene dimer, whereas it stabilizes the slipped-parallel hexafluorobenzene-benzene complex. Both electrostatic and dispersion interactions stabilize the slipped-parallel hexafluorobenzene-benzene complex, which is the cause of the preference of the slipped-parallel orientation and the larger interaction energy of the complex compared to the benzene dimer.     

Communication: Second-order multireference perturbation theory with explicit correlation: CASPT2-F12

An explicitly correlated complete active space second-order perturbation (CASPT2-F12) method is presented which strongly accelerates the convergence of CASPT2 energies and properties with respect to the basis set size. A Slater-type geminal function is employed as a correlation factor to represent the electron-electron cusp of the wave function. The explicitly correlated terms in the wave function are internally contracted. The required density matrix elements and coupling coefficients are the same as in conventional CASPT2, and the additional computational effort for the F12 correction is small. The CASPT2-F12 method is applied to the singlet-triplet splitting of methylene, the dissociation energy of ozone, and low-lying excited states of pyrrole.