Explicitly correlated CCSD theory

It is important to note that, at each level of coupled-cluster theory, we include through the exponential parameterization of Eq. (28) all possible determinants that can be generated within a given orbital basis, that is, all determinants that enter the FCI wave function in the same orbital basis. Thus, the improvement in the sequence CCSD, CCSDT, and so on does not occur because more determinants are included in the description but from an improved representation of their expansion coefficients. For example, in CCS theory, the doubly-excited determinants are represented by ]HF), whereas the same determinants are represented by (T2 + Tf) HF) in CCSD theory. Thus, in CCSD theory, the weight of each doubly-excited determinant is obtained as the sum of a connected doubles contribution from T2 and a disconnected singles contribution from Tf/2. This parameterization of the wave function is not only more compact than the linear parameterization of configuration-interaction (Cl) theory, but it also ensures size-extensivity of the calculated electronic energy. 

The hierarchy of coupled-cluster models provides a clear route towards the exact solution of the Schrodinger equation, but the slow basis-set convergence limits the accuracy sometimes even for small molecules. The way to overcome this problem is to combine the coupled-cluster model with the explicitly correlated approach. It can be done, in principle, for any model within the coupled-cluster hierarchy. The main task of this work is, however, the implementation of explicitly correlated CCSD model, hence the discussion will be focused on this particular model. 

The main idea of explicitly correlated CCSD theory is to extend the conventional space of excitations such that pairs of occupied orbitals are replaced by an explicitly correlated geminal function 

Similar to the MP2-E12 formalism, the strong orthogonality projector in the geminal basis leads to many-electron integrals in the amplitude equations. Explicit evaluation of these integrals severely restricts the range of application and the successful approaches are thc e that involve two-electron integration at most. The implementation of the CCSD(F12) in Turbomole fulfills this requirement. 

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It is worth to mention that in the UHF formalism, due to the orthogonality of the spin functions, it is not allowed to excite an electron occupying a spin-orbital to the / spin-orbital (and vice versa). As it was already mentioned in Subsection 3.2, in the case of explicitly correlated CCSD theory, the cluster operator is supplemented with the additional excitation operator T2/ [Eq. (40)]. This operator is responsible for the explicitly correlated treatment and involves additional excitations (F12 excitations) into the complementary basis. In the spin-free formalism its mathematical form was already shown and briefly discussed [Eq. (40)], in the spin-orbital formalism this operator can be introduced as... 

One of the special cases of coupled-cluster theory is the singles-and-doubles (CCSD) model [37]. The cluster operator Eq. (29) is restricted to contain only the singles and doubles excitation operators. The importance of this model can be seen from the fact that, for any coupled-cluster wave function, the singles and doubles amplitudes are the only ones that contribute directly to the coupled-cluster energy. In the explicitly correlated CCSD model the conventional cluster operator containing the T and T2 operators is supplemented with an additional term that takes care of the explicit correlation (written with red font)... 

Neiss, C., Hattig, C. Frequency-dependent nonlinear optical properties with explicitly correlated coupled-cluster response theory using the CCSD(R12) model. J. Chem. Phys. 2007, 126, 154101. 

Marshall, M. S., and Sherrill, C. D. (2011]. Dispersion-weighted explicitly correlated coupled-cluster theory [DW-CCSD(T ]-F12], /. Chem. Theory Comput. 7, pp. 3978-3982, doi 10.1021/ct200600p. 

The thesis begins with Section 2, where a brief history about the explicitly correlated approaches is presented. This is followed by Section 3 with general remarks about standard and explicitly correlated coupled-cluster theories. In Section 4, the details about the CCSD(F12) model relevant to the implementation in TuRBOMOLE are presented. The usefulness of the developed tool is illustrated with the application to the problems that are of interest to general chemistry. A very accurate determination of the reactions barrier heights of two CH3+CH4 reactions has been carried out (Section 5) and the atomization energies of 106 medium-size and small molecules were computed and compared with available experimental thermochemical data (Section 6). The ionization potentials and electron affinities of the atoms H, C, N, O and F were obtained and an agreement with the experimental values of the order of a fraction of a meV was reached (Section 7). Within all applications, the CCSD(F12) calculation was only a part of the whole computational procedure. The contributions from various levels of theory were taken into account to provide the final result, that could be successfully compared to the experiment. 

A second purpose of the present work is to assess the performance of the explicitly correlated coupled-cluster model CCSD(F12) that we have recently implemented in the TuR-BOMOLE program package [68, 69]. This model has the potential to yield electronic molecular energies at the level of coupled-cluster theory with single and double excitations (CCSD [37, 70]) at the limit of a complete one-particle basis set. In conjunction with corrections for higher excitations (connected triples and connected quadruples) it should be possible to compute the barrier height for the above reaction with an accuracy of about 1-2 kJ mol that is, with an error of about 0.5-1.0%. 

All explicitly correlated calculations were performed at the CCSD(F12) level of theory, as implemented in the TurbomOLE program [58, 69]. The Slater-type correlation factor was used with the exponent 7 = 1.0 aQ. It was approximated by a linear combination of six Gaussian functions with linear and nonlinear coefficients taken from Ref. [44]. The CCSD(F12) electronic energies were computed in an all-electron calculation with the d-aug-cc-pwCV5Z basis set [97]. For all cases we used full CCSD(F12) model (see Subsection 4.9 for the discussion about models implemented in Turbomole), the open-shell species were computed with a UHF reference wave function. The explicitly correlated contributions to the relative quantities are collected in Tables 10 and 11 under the label F12 . 

The electron affinities and ionization potential of the H, C, N, O and F atoms were computed by using conventional coupled-cluster methods supplemented with the explicitly correlated treatment at the CCSD(F12) level of theory. Agreement with experimental values of the order of magnitude of a fraction of meV was reached for hydrogen, carbon and nitrogen. 

A very accurate determination of the interaction-induced polarizability of He2 at the experimental internuclear separation of 5.6 ao was reported by Jaszunski et al The authors used a very large Ils8p6d5f4g3h basis set for He and high-precision explicitly correlated R12 methods. Their most accurate results for the mean and the anisotropy polarizability were calculated at the CCSD(T)-R12 level of theory and are are aint= 0.00104 and Aa = 0.06179 e ao Eh These values represent reference estimates of the interaction-induced dipole polarizability of two helium atoms. 

Figure 1 A family tree of quantum chemistry DFT, density functional theory QMC, quantum Monte Carlo RRV, Rayleigh-Ritz variational theory X-a, X-alpha method KS, Kohn-Sham approach LDA, BP, B3LYP, density functional approximations VQMC, variational QMC DQMC, diffusion QMC FNQMC, fixed-node QMC PIQMC, path integral QMC EQMC, exact QMC HF, Hartree-Fock EC, explicitly correlated functions P, perturbational MP2, MP4, Maller-Plesset perturbational Cl, configuration interaction MRCI, multireference Cl FCI, full Cl CC, CCSD(T), coupled-cluster approaches. Other acronyms are defined in the text.

Another series of composite computational methods, Weizmann-n (Wn), with n = 1-4, have been recently proposed by Martin and co-workers W1 and W2 in 1999 and W3 and W4 in 2004. These models are particularly accurate for thermochemical calculations and they aim at approximating the CBS limit at the CCSD(T) level of theory. In all Wn methods, the core-valence correlations, spin-orbit couplings, and relativistic effects are explicitly included. Note that in G2, for instance, the single-points are performed with the frozen core (FC) approximation, which was discussed in the previous section. In other words, there is no core-valence effect in the G2 theory. Meanwhile, in G3, the corevalence correlation is calculated at the MP2 level with a valence basis set. In the Wn methods, the core-valence correlation is done at the more advanced CCSD(T) level with a specially designed core-valence basis set. 

The Coupled-clusters (CC) method[7] based on the cluster expansion of the wavefunction has been established as a highly reliable method for calculations of ground state properties of small molecules with the spectroscopic accuracy. When this method is used together with a flexible basis set it recovers the dominant part of the electron correlation. Typically, CC variant explicitly considering single and double excitations (CCSD) is used. In order to save computer time the contributions from triple excitations are often calculated at the perturbation theory level (notation CCSD(T) is used in this case). CCSD(T) method can be routinely used only for systems with about 10 atoms at present. Therefore, it cannot be used directly in zeolite modeling, however, results obtained at CCSD(T) level for small model systems can serve as an important benchmark when discussing the reliability of more approximate methods. 

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