CCSD theory

In particular, if method A represents the CCSD theory, the corresponding QVMMCC energy is calculated by adding the correction (cf. eq (22))... 

Figures 7 and 8 plot deviations of total energies from FCI results for the various methods. It is clear that the CASSCF/L-CTD theory performs best out of all the methods smdied. (We recall that although the canonical transformation operator exp A does not explicitly include single excitations, the main effects are already included via the orbital relaxation in the CASSCF reference.) The absolute error of the CASSCF/L-CTD theory at equilibrium—1.57 mS (6-31G), 2.26 m j (cc-pVDZ)—is slightly better than that of CCSD theory—1.66m j (6-31G), 3.84 m j (cc-pVDZ) but unlike for the CCSD and CCSDT theories, the CASSCF/L-CTD error stays quite constant as the molecule is pulled apart while the CC theories exhibit a nonphysical turnover and a qualitatively incorrect dissociation curve. The largest error for the CASSCF/L-CTD method occurs at the intermediate bond distance of 1.8/ with an error of —2.34m (6-3IG), —2.42 mE j (cc-pVDZ). Although the MRMP curve is qualitatively correct, it is not quantitatively correct especially in the equilibrium region, with an error of 6.79 mEfi (6-3IG), 14.78 mEk (cc-pVDZ). One measure of the quality of a dissociation curve is the nonparallelity error (NPE), the absolute difference between the maximum and minimum deviations from the FCI energy. For MRMP the NPE is 4mE (6-3IG), 9mE, (cc-pVDZ), whereas for CASSCF/ L-CTD the NPE is 5 mE , (6-3IG), 6 mE , (cc-pVDZ), showing that the CASSCF/L-CTD provides a quantitative description of the bond breaking with a nonparallelity error competitive with that of MRMP.

In both the HF molecule and BH molecules, the HF/L-CTSD and HF/L-CTD theories give energies comparable to CCSD at the equilibrium geometry. However, as the bond is stretched, they both display significantly increased errors, typical of a single-reference theory. At stretched geometries, the errors of the HF/L-CTSD method are worse than those of CCSD and comparable to those of the linearized CCSD theory. 

The perturbative analysis in Section II.D showed that single-reference L-CTSD is exact through third order in the fluctuation potential, much like L-CCSD theory, and our results are consistent with this analysis. This suggests that one of the things we... 

The 1 isomer is difficult to detect and identify by IR spectroscopy, since the IR spectrum only contains one weak line. In contrast, there are two transitions of intermediate strength in the Raman spectrum, and consequently Raman spectroscopy can be used for identification without the need of isotopic labeling. An alternative approach for detection is LIF spectroscopy. Excited state calculations, using linear and quadratic CCSD theory, indicates that the... 

Zeta with polarization functions for the second-row atoms and Triple-Zeta for the metal atoms. Highly correlated methods such as the CCSD theory are even more demanding in term of basis sets quality and this is one of the limiting steps for further applications in the field of transition metal complexes electronic spectroscopy [12]. 

We shall provide an overview of the applications that have been made over the period being review which demonstrate the many-body Brillouin-Wigner approach for each of these methods. By using Brillouin-Wigner methods, any problems associated with intruder states can be avoided. A posteriori corrections can be introduced to remove terms which scale in a non linear fashion with particle number. We shall not, for example, consider in any detail hybrid methods such as the widely used ccsd(t) which employs ccsd theory together with a perturbative estimate of the triple excitation component of the correlation energy. 

The determinant 0) plays key role in the CC theory. It is called the reference determinant and in the standard version of the CC theory this is the determinant used to generate all necessary excitations in the wave function. For this reason the standard CC theory is called the single reference theory. Also, in the standard implementation of the theory, the T operator includes only single and double excitations from 0) (the CCSD theory),... 

The configuration coefficients of those higher excitations are products of amplitudes of lower level excitations. For example, the main contributions of the triple excitations in the CCSD theory are given as products of amplitudes of single excitations and double excitations, tf (see Eq. 3.6), and the main contributions of the quadruple excitations are given as products of two double-excitation amplitudes, (see Eq. 3.7). 

Multireference State-Specific Generalization of 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 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... 

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... 

Our semiempirical r-electron CCSD theory is based on the standard CCSD scheme. The CCSD wave function has the form ... 

First comparison of 2nd hyperpolarizability obtained in CCSD/PPP approximations were performed in Ref. [73]. The results of ar-electron CCSD calculations and especially cue-CCSD calculations in comparison with experimental data [12] are presented in the Table 3.2. The Table 3.2 results reveals quite similar calculated values (r-CCSD and cue-CCSD) with experimental data. A detailed comparison of the results for different variants of CCSD theory with FCI values were performed in Ref. [31]. 

Systematic improvement of the cue(()-CCSD theory with Z up to / = 6 is still not enough to give results coinciding with the cue-CCSD results. However, unlike the MP2 approach, improvement of the quality of the wave-function with / leads to more accurate values for both polarizability and 2nd hyperpolarizability. Also, as a rule, the lower level of theory gives a lower bound for the calculated optical property. 

As noted in the Sect. 3.3.2, the polyacenes are characterized by a more complex structure of the wave function, and therefore, to adequately describe this structure a higher level of theory is needed. It is expected that for the polycyclic aromatic hydrocarbons discussed in this section the selection of an appropriate correlation radius is a very important aspect of the calculation. To study the effect of the level of accounting for the electron correlation effects for polyacenes, we have calculated the polarizability and 2nd hyperpolarizabUity values for different levels of the cue-CCSD theory. In Figs. 3.14 and 3.15 the dependencies of the specific values of these properties on the number of the r-electrons are shown. 

L. Visscher, K. G. DyaU, T. J. Lee. Kramers-Restricted Closed-SheU CCSD Theory. Int. J. Quantum Chem. Quantum Chem. Symp., 29 (1995) 411 19. 

Here we have to note that eq. (4.48) only reduces to eq. (4.49) in the case when the cluster amplitudes are fully converged. We see that the cancellation of disconnected contributions in the bwccd theory is achieved iteratively and, furthermore, exact cancellation is achieved by the full convergence of cluster amplitudes. It can therefore be concluded that bwccd theory is fully equivalent to the standard ccd approximation, which exploits the linked cluster theorem. Similar arguments have been used to demonstrate the equivalency of bwccsd and ccsd theories [9]. The extension of these arguments to other coupled cluster approximations, such as bwccsdt and ccsDT theories, is straightforward. 

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