R-CCSD

The scalar relativistic (SR) corrections were calculated by the second-order Douglas-Kroll-Hess (DKH2) method [53-57] at the (U/R)CCSD(T) or MRCI level of theory in conjunction with the all-electron aug-cc-pVQZ-DK2 basis sets that had been recently developed for iodine [58]. The SR contributions, as computed here, account for the scalar relativistic effects on carbon as well as corrections for the PP approximation for iodine. Note, however, that the Stuttgart-Koln PPs that are used in this work include Breit corrections that are absent in the Douglas-Kroll-Hess approach [58]. 

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]. 

The CR-CCSD(T) method provides somewhat better results, when compared with the CR-CCSD[T] approach, so in the numerical examples described in the next section we focus on the results of the CR-CCSD(T) calculations, but we need the CR-CCSD[T] theory to understand the connection between the CR-CC methods and their higher-order QMMCC counterparts (see the discussion below). It is also useful to consider the renormalized CCSD[T] and CCSD(T) methods (the R-CCSD[T] and R-CCSD(T) approaches), which can be viewed as simplified variants of the CR-CCSD[T] and CR-CCSD(T) approaches in which the 0203(2) moments in the CR-CCSD[T] and CR-CCSD(T) formulas, Eqs. (41) and (39), are replaced by their lowest-order estimates, ( j j i3 (EjvT2)c )- The R-CCSD[T] and R-CCSD(T) energies are defined as follows [11-13,30,31,33,35,37] ... 

Although calculations of entire molecular PESs involving single bond breaking require using CR-CCSD[T] and CR-CCSD(T) methods rather than their simplified renormalized versions [11-13,30,31,33,35,37], these R-CCSD[T] and R-CCSD(T) approaches allow us to understand the relationship between the standard and completely renormalized CC approaches. 

The above analysis implies that the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) methods can be viewed as the MMCC-based extensions of the standard CCSD[T] and CCSD(T) approaches. Very similar extensions can be formulated for other noniterative CC approaches. In particular, we can use the MMCC formalism to renormalize the CCSD(TQf) method of Ref. [23], in which the correction due to the combined effect of triples and quadruples is added to the CCSD energy. The resulting completely renormalized CCSD(TQ) (CR-CCSD(TQ)) approaches are the examples of the MMCC(2,4) approximation, defined by Eq. (35). As in the case of the CR-CCSD[T] and CR-CCSD(T) methods, we use the MBPT(2)-like expressions to define the wave function o) in the CR-CCSD(TQ) energy formulas. Two variants of the CR-CCSD(TQ) method, labelled by the extra letters a and b , are particularly useful. The CR-CCSD(TQ),a and CR-CCSD(TQ),b energies will be defined as follows [11-13,30,31,33,35] ... 

In analogy to the R-CCSD[T] and R-CCSD(T) approaches and their standard CCSD[T] and CCSD(T) counterparts, it can be shown that the R-CCSD(TQ)-l,a scheme, obtained by simplifying the CR-CCSD(TQ),x (x=a,b) equations according to Eqs. (68) and (69), reduces to the factorized CCSD(TQf) approach of Kucharski and Bartlett [23], when the (TQl.a nominator in the R-CCSD(TQ)-l,a energy expression, Eq. (68), is replaced by 1. Indeed, the CCSD(TQf) energy can be given the following form ... 

The simple relationships between the renormalized and completely renormalized CCSD[T], CCSD(T), and CCSD(TQ) methods and their standard counterparts, discussed above, imply that computer costs of the R-CCSD[T], R-CCSD(T), CR-CCSD[T], CR-CCSD(T), R-CCSD(TQ)-n,x, and CR-CCSD(TQ),x (n = 1,2, x = a, b) calculations are essentially identical to the costs of the standard CCSD[T], CCSD(T), and CCSD(TQf) calculations. In analogy to the standard CCSD[T] and CCSD(T) methods, the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) approaches are procedures in the noniterative steps involving triples and procedures in the iterative CCSD steps. More specifically, the CR-CCSD[T] and CR-CCSD(T) approaches are twice as expensive as the standard CCSD[T] and CCSD(T) approaches in the steps involving noniterative triples corrections, whereas the costs of the R-CCSD[T] and R-CCSD(T) calculations are the same as the costs of the CCSD[T] and CCSD(T) calculations [77]. The memory and disk storage requirements characterizing the R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR-CCSD(T) methods are essentially identical to those characterizing the standard CCSD[T] and CCSD(T) approaches (see Ref. [77] for further details). In complete analogy to the noniterative triples corrections, the costs of the R-CCSD(TQ)-n,x calculations are identical to the costs of the CCSD(TQf) calculations (the CCSD(TQf) method... 

The Tables 5.4, 5.5, 5.6 show the calculation results of some electrical characteristics for H2,02, N2, CO2, CO, CN, HCl, HCN, NaCl, OH, NaH"", CH4, and H2O molecules, which are important for astrophysical and atmospheric problems. In the work [88] the calculations were carried out using the finite-field method at the (R) CCSD(T) level of theory with different aVXZ basis sets (X = Q, 5). For these cases, the amplitudes of the applied fields have been chosen as follows F = 0.0025 a.u., = 0.0001 a.u., FajSy = 0.00,001 a.u. and Fg,pys = 0.000001 a.u. Multipole moments up to 4th order are presented in Table 5.4. For comparison, in Table 5.4 the other literature data are also given. Table 5.5 presents the calculated and measured values (we have chosen the more reliable ones) of multipole polarizabilities. 

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