Cardinal number basis sets

As can be seen from the table, the number of AOs increases rapidly with the cardinal number X. Thus, with each increment in the cardinal number, a new shell of valence AOs is added to the cc-pVXZ set since the number of AOs added in each step is proportional to X2, the total number (Nbas) of AOs in a correlation-consistent basis set is proportional to X3. The core-valence sets cc-pCVXZ contain additional AOs for the correlation of the core electrons. As we shall see later, the hierarchy of correlation-consistent basis sets provides a very systematic description of molecular electronic systems, enabling us to develop a useful extrapolation technique for molecular energies. 

This equation contains two unknowns and we can thus extrapolate to the basis-set limit from two separate calculations with different cardinal numbers X and Y. This gives us the following simple expression for the energy at the basis-set limit [32, 33] ... 

Dunning s correlation consistent basis sets cc-pVAZ [27] augmented with diffuse functions [28] were used in the calculations. We considered cardinal numbers X—D, T, Q, 5, 6 and single (s), double (d), triple (t), and quadruple (q) augmentations. The orbitals were not allowed to relax in the coupled cluster response calculations. 

The Cauchy moments of Ne at the CCSD/q-aug-cc-pV5Z level were found in Ref. [4] to be converged within 1 % compared to the basis-set limit result. We have calculated the Cauchy moments also for the X—6 cardinal number. From the results in Table 1 it appears that the Cauchy moments at this level are significantly less than 1 % from the basis-set limit result. 

Table 3. The f em symmetric double dissociation of water (into 2H(ls 2S) + 0(2p4 3P) cut (ii)). The H-O-H angle is kept fixed at its equilibrium value taken from Ref. [139] (ae = 104.501 degree). R is an O-H distance and Re = 0.95785 A is the equilibrium value of R [139]. All energies E (in cm-1) are reported as E - E(Re, ae), where E(Re, ae) are the corresponding values of E at the equilibrium geometry. X is a cardinal number defining the aug-cc-pCVXZ basis sets used in the calculations. In all CC calculations, all electrons were correlated.

Table 5. The differences between CC/CR-CC energies, calculated relative to their equilibrium values (the CC/CR-CC E - E(Re, ore)] values in Table 3) and the corresponding MRCI(Q) relative energies (the MRCI(Q) [E — E(Re, ae)] values in Table 1) for the dissociation of a single O-H bond in water (into H(l.s 2S) 4- OH(X 2 n) cut (i)). X is a cardinal number defining the aug-cc-pCVXZ basis sets used in the calculations.

At the Hartree-Fock level the hyperpolarizabilities usually increase if the augmentation level and also if the cardinal number X are increased. For the correlated contribution to "yn (0) the convergence pattern is dominated by different effects At the CCSD level an increase of "yn (0) with the cardinal number beyond T is only found for the lower augmentation levels. In particular for molecules we observe, as illustrated in Table S for N2 and CH4, a monotonic decrease of Are second hyperpolarizability when the correlation treatment is improved in the series X = T, Q, 5, etc. The results for X = T typically overestimate the correlation contribution to yn (0) by a few percent. Many correlated hyperpolarizability calculations in the literature were performed with basis sets of triple- or similar quality and basis set convergence was often only explored with respect to augmentation with diffuse functions. From the above observations one may conclude that many of these studies obtained too large results for 7 (0). 

A fundamental characteristic of the FPA is the dual extrapolation to the one-and n-particle electronic-structure limits. The process leading to these limits can be described as follows (a) use families of basis sets, such as the correlation-consistent (aug-)cc-p(wC)VnZ sets [51,52], which systematically approach completeness through an increase in the cardinal number n (b) apply lower levels of theory with extended [53] basis sets (typically direct Hartree-Fock (HF) [54] and second-order Moller-Plesset (MP2) [55] computations) (c) use higher-order valence correlation treatments [CCSD(T), CCSDTQ(P), even FCI] [5,56] with the largest possible basis sets and (d) lay out a two-dimensional extrapolation grid based on the assumed additivity of correlation increments followed by suitable extrapolations. FPA assumes that the higher-order correlation increments show diminishing basis set dependence. Focal-point [2,49,50,57-62] and numerous other theoretical studies have shown that even in systems without particularly heavy elements, account must also be taken for core correlation and relativistic phenomena, as well as for (partial) breakdown of the BO approximation, i.e., inclusion of the DBOC correction [28-33]. 

A cardinal number group / with the sequence of regional tops assigned to it, for example, clockwise or counter-clockwise, will be called a basis(set) W(v, w, I) of the length 1. In the one of the / tops of a group we have I-1 variants of a motion into the rest of the tops in the next top the numbers of motion variants into the remanding ones is equal to 1-2, etc. A total number of different sequences of the round of / tops is equal to (M).. Just as the opposite sequence of a counter-clockwise round corresponds to every sequence of a clockwise round, the total number of different sequences of the / tops is equal to (/-I) /2. However, in this ratio a correction should be made for the case when 1=2. So, the number N(y, w, 1) of the basis(sets) lV(v, m , /) in the groups V(v, 1 is determined by the expression... 

In accordance with (2.51) and (2.52) every cardinal number groups 1=2 and 1=3 have a one basis(set), the cardinal number group 1=4 has three basis(sets), represented in Figure 2.5. The cardinal number group 1=5 has 12 basis(sets). 

The correlation consistent basis sets contain a systematically increasing amount of polarization functions not only with respect to the number of functions but also to the highest angular momentum quantum number, which is always just one value smaller than the cardinal number X. This series of basis sets is therefore well suited for and frequently used in the calculation of polarizabilities and hyperpolarizabilities. Often, one can observe a monotonic convergence of the results, which offer the possibility to extrapolate to a complete basis set limit. However, these basis sets quickly become very large with increasing cardinal number X. 

The number of basis functions grows with the third power of the cardinal number X in the series of cc-pVXZ basis sets. For first-row atoms, for example, the number of basis functions is ... 

In Figure 15.8 we have, for the calculated dipole moments in the aug-cc-pVXZ basis sets, plotted the mean errors, the standard deviations, the mean absolute errors and the maximum absolute errors. The plots indicate that the calculated dipole moments depend in a systematic mann - on the cardinal number and the correlation treatment In general, the dipole moments arc reduced as we improve the correlation treatment. Indeed, with the exceptions of CO and HNC, the dipole moment is always reduced as we go from Hartree-Fock to CCSD and then on to CCSD(T) - see Table 15.11. The MP2 dipole moments are less systematic but are usually slightly smaller than the CCSD numbers. At the aug-cc-pVQZ level, the mean absolute errors are 0.17 D for the Hartree-Fock model, 0.05 D for the MP2 model, 0.04 D for the CCSD model and 0.01 D for... 

From Figure 15.8, we see that the calculated dipole moments are more sensitive to the choice of the A -electron model than to the choice of cardinal number for the basis set. Even in the smallest basis, the CCSD(T) model is considerably more accurate than the CCSD model. Thus, for the CCSD(T) model, the smallest basis (aug-cc-pVDZ) gives a mean error of —0.006 D and the largest basis (aug-cc-pVQZ) a mean error of -1-0.006 D, the corresponding mean absolute errors being 0.02 and 0.008 D. In comparison, for the CCSD model, the mean errors are 0.02 and 0.03 D at the aug-cc-pVDZ and aug-cc-pVQZ levels, respectively, and the mean absolute errors 0.04 D in both cases. Whereas, at the CCSD and CCSD(T) levels, the dipole moment increases with the cardinal number, at the Hartree-Fock level, the change is in the opposite direction. The performance of the MP2 model appears to be less systematic. 

For the calculation of dipole moments, the aug-cc-pVXZ basis sets should be used for a flexible description of the outer valence region. In general, however, the quality of the calculations depends more on the correlation treatment than on the cardinal number, the aug-cc-pVTZ basis being sufficient for most applications. Whereas the Hartree-Fock model is typically in error by 0.1-0.2 D, the introduction of correlation at the MP2 and CCSD levels reduces the errors to about 0.05 and 0.04 D, respectively. The CX SD(T) errors are small - typically smaller than 0.01 D. At the coupled-cluster level, the dipole moment usually increases with the cardinal number at the Hartree-Fock level, the change is typically in the opposite direction but small. 

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