CCSD model

In Table 3 we have listed the results of a basis set and correlation study for the hyperpolarizability dispersion coefficients. In a previous investigation of the basis set effects on the dispersion coefficients for the first hyperpolarizability (3 of ammonia [22] we found quite different trends for the static hyperpolarizability and for the dispersion coefficients. While the static hyperpolarizability was very sensitive to the inclusion of diffuse functions, the dispersion coefficients remained almost unchanged on augmentation of the basis set with additional diffuse functions, but the results obtained with the CC2 and CCSD models, which include dynamic electron correlation, showed large changes with an increase of the... 

The CCSD model gives for static and frequency-dependent hyperpolarizabilities usually results close to the experimental values, provided that the effects of vibrational averaging and the pure vibrational contributions have been accounted for. Zero point vibrational corrections for the static and the electric field induced second harmonic generation (ESHG) hyperpolarizability of methane have recently been calculated by Bishop and Sauer using SCF and MCSCF wavefunctions [51]. 

Table 2 shows transition moments calculated by the different EOM-CCSD models. As has been discussed above, the right-hand transition moment 9 is size intensive but the left-hand transition moment 9 in model I and model II is not size intensive. Model II is much improved as far as size intensivity is concerned because of the elimination of the apparent unlinked terms. The apparent unlinked terms are a product of the size-intensive quantity ro and size-extensive quantities and therefore are size extensive. The difference between the values of model I and model II, as summarized in the fifth column, reveals strict size extensivity. Complete elimination of unlinked diagrams by using A amplitudes brings strict size intensivity for the transition moment and therefore the transition probabilities calculated by model III are strictly size intensive. 

All the calculations have been carried out using a local version of the Dalton program package [26]. The implementation of the Cauchy moments for the CCS, CC2, and CCSD models has been described in Ref. [4]. The CC3 Cauchy moments have been implemented by us following the outline presented in the previous section. 

In Table 1 we have collected the Cauchy moments, S(k), calculated for the Ne atom using the CCSD model and the n-aug-cc-pVAZ basis-set family. As can be seen from the table, single augmentation is not sufficient for the calculation of the Cauchy moments. On the other hand, going beyond the double augmentation... 

Table 1. The basis-set convergence of the Cauchy moments S(k) [a.u.] for Ne calculated with CCSD model and the n-aug-cc-VXZ basis-set family (all electrons correlated)...

The Cauchy moments of Kr (Table 4) have been calculated using the largest currently available correlation-consistent basis augmented with diffuse functions - d-aug-cc-pV5Z basis. As for the Ar atom, the CCS and CC2 results overestimate the DOSD values for smaller k and underestimate for larger k. The CCSD model behaves in the same manner. The CC3 model systematically... 

Figure 4. Hierarchy of the SF models. Similar to the non-SF SR methods, the SF models converge to the exact n-electron wavefunction when the spin-flipping operator 0 includes up to n-tuple excitations. For example, the SF-CCSD model...

CCSD Model. Coupled Cluster Singles and Doubles. A Correlated Model. 

G. E. Scuseria, A. C. Scheiner, T. J. Lee, J. E. Rice, and H. F. Schaefer, J. Chem. Phys., 86, 2881 (1987). The Closed-Shell Coupled-Cluster Single and Double Excitation (CCSD) Model for the Description of Electron Correlation. A Comparison with Configuration Interaction (CISD) Results. 

H. Koch, O. Christiansen, R. Kobayashi, P. Jorgensen, and T. Helgaker, Chem. Phys. Lett., 228, 233 (1994). A Direct Atomic Orbital Driven Implementation of the Coupled Cluster Singles and Doubles (CCSD) Model. 

The situation is somewhat different for the convergence with the wavefunction model, i.e. the treatment of electron correlation. As an anisotropic and nonlinear property the first dipole hyperpolarizability is considerably more sensitive to the correlation treatment than linear dipole polarizabilities. Uncorrelated methods like HF-SCF or CCS yield for /3 results which are for small molecules at most qualitatively correct. Also CC2 is for higher-order properties not accurate enough to allow for detailed quantitative studies. Thus the CCSD model is the lowest level which provides a consistent and accurate treatment of dynamic electron correlation effects for frequency-dependent properties. With the CC3 model which also includes the effects of connected triples the electronic structure problem for j8 seems to be solved with an accuracy that surpasses that of the latest experiments (vide infra). 

In Table 2, we have listed the values of (w) calculated using the CCSD model with different basis sets, including a stripped down aug-cc-pVTZ basis. At 1.17 eV, the CCSD model gives a value of 11.4 x 10 esu, somewhat lower than the B3LYP result of 12.9 x 10 esu and much lower than the experimental value of 15.4 X 10 esu. Since, at higher frequencies, the discrepancy between the B3LYP and CCSD models increases even more, it appears that the very large differences... 

Coupled-cluster theory provides the most important hierarchy of models in ab initio quantum chemistry. At each new excitation level of this hierarchy, a significant improvement is observed in the calculated energies and properties. Typically, the CCSD model reduces the error in the calculated properties by a factor of three or... 

The MP2 energy is always lower than the Hartree-Fock energy and usually represents a rather good approximation to the total electronic energy. The MP2 model usually works well whenever the Hartree-Fock wave function is a reasonable one, typically recovering about 90% of the total correlation energy, at a cost that scales formally as K5. Still, it is less robust and somewhat less generally applicable than the CCSD model. 

Polarizabilities and Hyperpolarizabilities of Larger Molecules. - Ab Initio Calculations. At the most highly correlated level Christiansen et al.157 have used the CCS, CC2 and CCSD models to calculate the static polarizability of furan. Dispersion effects are included to make an estimate of the frequency-dependent polarizability. 

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. 

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)... 

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