Correlation-consistent basis sets cardinal number

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. 

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 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. 

Fig. 8.16. Errors (in mEh) relative to the basis-set limit in the Hartree-Fock energy (dotted line) and in the valence CISD correlation energy (full line) for the ground state of the carbon atom calculated using correlation-consistent basis sets cc-pVXZ. On the left, we have plotted the errors on a linear scale as a function of the cardinal number X on the right, we have plotted the errors on a logarithmic scale.

The polarized core-valence correlation-consistent basis sets for first-row atoms are listed in Table 8.13. We note that the same pattern is followed for the core-valence functions of the first-row atoms as for the valence functions of helium in Table 8.11. The number of ftinctions in the core-valence sets may therefore be calculated from the cardinal number as... 

Fig. 8.17. Gaussian exponents of the carbon correlation-consistent basis sets of cardinal numbers 2-5 on a logarithmic scale with tight functions to the left and diffuse functions to the right. The exponents of the valence-correlating orbitals present in the cc-pVXZ root sets are located in the middle and are plotted using larger dots, the exponents of the core-correlating orbitals are located on the left, and the diffuse functions of the augmented aug-cc-p(C)VXZ sets are located on the right. For each angular momentum, we have plotted the exponents for cardinal numbers X = 2 at the bottom and X = 5 at the top.

The first point to note about the correlation-consistent basis sets in Table 8.16 is that the convergence is in all cases uniform and systematic - for the energies, for the bond distances, and for the bond angle. Scrutiny of the table reveals that, with each increment in the cardinal number, all errors are reduced by a factor of at least 3 or 4. Clearly, the correlation-consistent basis sets provide a convenient framework for the quantitative study of molecular systems at the Hartree-Fock level. We also note that the results for the cc-pVXZ and cc-pCVXZ basis sets are very similar. Apparently, the molecular core orbitals are quite atom-like and unpolarized by chemical bonding. In Hartree-Fock calculations, therefore, the use of the smaller valence cc-pVXZ sets is recommended. 

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]. 

For the helium atom, the correlation-consistent sets cc-pVYZ have the same composition as the orbitals of the principal expansion with N = X. Identifying the cardinal number X with N in (8.4.1), we then obtain the following simple expression for the error in the cc-pVYZ basis ... 

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