A diagonal orbital-invariant explicitly-correlated coupled-cluster method
Graphical abstract
A novel, efficient explicitly-correlated coupled-cluster method that returns quintuple-zeta quality energy differences using triple-zeta orbital basis sets.
Introduction
The past five years have seen a rapid development in R12 explicitly-correlated methods [1], [2], [3], [4], [5], which are designed to overcome the painfully slow basis set convergence of standard orbital based computational approaches [6]. At the level of second order Møller–Plesset perturbation theory (MP2), near basis set limit correlation energies may now be obtained using orbital basis sets of only triple- quality [7]. Similarly, the recently developed explicitly-correlated coupled-cluster method CCSD(F12) yields correlation energies equivalent to quintuple- quality conventional CCSD calculation, with a triple- quality orbital basis [8]. Since each CCSD(F12) iteration requires approximately three times as many floating point operations as that of an orbital-only calculation, the cost to accuracy ratio is favourable and the F12 methods are now an attractive alternative to conventional approaches for many thermochemical applications. However, a number of issues remain to be addressed. One such issue is that of geminal basis set superposition error (BSSE), which has been observed to be critical for MP2 calculations of weak interactions [9]. Geminal BSSE can be removed by excluding inter-fragment F12 excitations, which may be achieved through using the original diagonal (non-invariant) ansatz [1]. Combining the diagonal ansatz with localised orbitals gives much improved predictions of Van der Waals (VdW) binding energies [9] and also for reaction enthalpies [10]. For methods using canonical Hartree–Fock orbitals, however, the orbital variance of the diagonal approach leads to a violation of size consistency [11]. Recently, Ten-no has recommended a further simplification that reintroduces orbital invariance by constraining the coefficients of the F12 geminal basis functions to have the same value, which is chosen such that they obey the coalescence conditions [12], [13]. For MP2-F12, Ten-no shows that, although there is a stronger dependence on the exponent of the geminal basis functions, the loss in accuracy due to the reduced variational flexibility is small. Moreover, the diagonal, fixed-amplitude MP2-F12 method scales as with the size of the one-particle basis, compared to for the original orbital-invariant method where all F12 excitation coefficients are optimised. In this Letter we extend this approach to the coupled-cluster level. We assess the diagonal, fixed-amplitude method using the same set of 23 molecules and 15 reaction enthalpies as we used in Ref. [8] (Sections 3 Molecular correlation energies, 4 Reaction enthalpies) and also investigate its performance for weak VdW interaction energies (Section 5).
Section snippets
Theory
The CCSD-F12 wave function, , is defined by the singles, doubles and F12 doubles excitation operators, .The indices indicate occupied orbitals and refer to virtual orbitals and a formally complete one-electron basis has been introduced to express the F12 double excitations in the occupation vector formalism: . The spin-free excitation operators have the
Molecular correlation energies
In Fig. 1 we plot the basis set convergence of the frozen-core (fc) CCSD(F12)-, -, -c and F12-0 correlation energies of H2O as a function of the geminal exponent . For all our computations we use augmented correlation-consistent basis sets [23], [24], since previous investigations have revealed that diffuse polarisation functions are important in F12 calculations [16]. As the complementary auxiliary basis set (CABS) for the approximate RI, we used a 21s14p8d7f5g4h3i2k basis
Reaction enthalpies
To assess the performance of the diagonal, orbital-invariant fc-CCSD(T)(F12)-c method for thermochemistry we use Pflüger and Werner’s [27] test set of 15 reactions involving the 23 molecules of Section 3. In Table 2, Table 3 we present basis set errors for the CCSD and (T) correlation contributions to the reaction enthalpies. We report the results of fc-CCSD(T)(F12)- and -c calculations using , and . For reaction 12, the CCSD reference is computed from 56 extrapolation
Van der Waals interactions of He2 and Ne2
The basis set limit fc-CCSD(T) spectroscopic parameters for He2 and Ne2 have been thoroughly investigated by Cybulski and Toczyłowski using special bond functions [30] and the basis set limit interaction energy of He2 is known to high accuracy from extrapolations by Van Mourik and Dunning using doubly augmented basis sets [31] and from CCSD(T)-R12 calculations with very large basis sets, performed by one of us [32]. The fc-CCSD(T) equilibrium bond lengths and interaction energies are 5.62 and
Conclusion
By building on Ten-no’s fixed-amplitude MP2-F12 approach, we have introduced a diagonal orbital-invariant explicitly-correlated coupled-cluster method where the F12 amplitudes are fixed according to the coalescence conditions. This method is free from geminal BSSE and provides stable predictions for weak VdW interactions and yields aug-cc-pV5Z quality reaction enthalpies using aug-cc-pVTZ basis sets. The slow convergence of the (T) contribution with orbital basis appears to be a limiting factor
Acknowledgements
D.P.T. acknowledges financial support from the Deutsche Forschungsgemeinschaft (DFG). The research of W.K. has been supported by the DFG through the Center for Functional Nanostructures (CFN, Project No. C3.3). It has been further supported by a grant from the Ministry of Science, Research and the Arts of Baden-Württemberg (Az: 7713.14-300). C.H. also acknowledges support by the DFG through the Priority Programme 1145 (Project No. HA 2588/3-2).
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