Basis sets, correlation-consistent

The primary disadvantage of ANO basis sets is that a very large number of primitive GTOs are necessary for converging towards the basis set limit. Dunning and coworkers have proposed a somewhat smaller set of primitives that yields comparable results 

Several different sizes of cc basis sets are available in terms of final number of contracted functions. These are known by their acronyms cc-pVDZ, cc-pVTZ, cc-pVQZ, cc-pV5Z and cc-pV6Z correlation consistent polarized Valence Double/Triple/Quadru-ple/Quintuple/Sextuple Zeta). The composition in terms of contracted and primitive (for the s- and p-part) functions is shown in Table 5.3. Note that each step up in terms of quality increases each type of basis function by one, and adds a new type of higher 

Basis Hydrogen First row elements Second row elements  

The polarization consistent basis sets again employ an energetic criterion for determining the importance of each type of basis function. The level of polarization beyond the isolated atom is indicated by a value after the acronym, i.e. a pc-0 basis set is 

The main advantage of the ANO and cc basis sets is the ability to generate a sequence of basis sets which converges toward the basis set limit. For example, from a series of 

A+B L -fl/2) have also been used. The theoretical assumption underlying an inverse power dependence is that the basis set is saturated in the radial part (e.g. the cc-pVTZ ba.sis is complete in the s-, p-, d- and f-function spaces). This is not the case for the correlation consistent basis sets, even for the cc-pV6Z basis the errors due to insuficient numbers of s- to i-functions is comparable to that from neglect of functions with angular moment higher than i-functions. 

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There are several types of basis functions listed below. Over the past several decades, most basis sets have been optimized to describe individual atoms at the EIF level of theory. These basis sets work very well, although not optimally, for other types of calculations. The atomic natural orbital, ANO, basis sets use primitive exponents from older EIF basis sets with coefficients obtained from the natural orbitals of correlated atom calculations to give a basis that is a bit better for correlated calculations. The correlation-consistent basis sets have been completely optimized for use with correlated calculations. Compared to ANO basis sets, correlation consistent sets give a comparable accuracy with significantly fewer primitives and thus require less CPU time. 

Several basis schemes are used for very-high-accuracy calculations. The highest-accuracy HF calculations use numerical basis sets, usually a cubic spline method. For high-accuracy correlated calculations with an optimal amount of computing effort, correlation-consistent basis sets have mostly replaced ANO... 

Bauschlicker ANO Available for Sc through Cu (20.vl5/il0r/6/4 ). cc—pVnZ [n = D, T, Q, 5,6) Correlation-consistent basis sets that always include polarization functions. Atoms FI through Ar are available. The 6Z set goes up to Ne only. The various sets describe FI with from i2s p) to [5sAp id2f g) primitives. The Ar atoms is described by from [As pld) to ils6pAd2>f2g h) primitives. One to four diffuse functions are denoted by... 

Rappe, Smedley and Goddard (1981) Stevens, Basch and Krauss (1984) Used for ECP (effective core potentitil) calculations Dunning s correlation consistent basis sets (double, triple, quadmple, quintuple and sextuple zeta respectively). Used for correlation ctilculations Woon and Dunning (1993)... 

Electron correlation studies demand basis sets that are capable of very high accuracy, and the 6-31IG set I used for the examples above is not truly adequate. A number of basis sets have been carefully designed for correlation studies, for example the correlation consistent basis sets of Dunning. These go by the acronyms cc-pVDZ, cc-pVTZ, cc-pVQZ, cc-pV5Z and cc-pV6Z (double, triple, quadruple, quintuple and sextuple-zeta respectively). They include polarization functions by definition, and (for example) the cc-pV6Z set consists of 8. 6p, 4d, 3f, 2g and Ih basis functions. 

We need to look at the convergence as a function of basis set and amount of electron correlation (Figure 4.2). For the former we will use the correlation consistent basis sets of double, triple, quadruple, quintuple and, when possible, sextuple quality (Section 5.4.5), while the sensitivity to electron correlation will be sampled by the HF, MP2 and CCSD(T) methods (Sections 3.2, 4.8 and 4.9). Table 11.1 shows how the geometry changes as a function of basis set at the HF level of theory. 

Peterson, K.A. and Puzzarini, C. (2005) Systematically convergent basis sets for transition metals. II. Pseudopotential-based correlation consistent basis sets for the group 11 (Cu, Ag, Au) and 12 (Zn, Cd, Hg) elements. Theoretical Chemistry Accounts, 114, 283-296. 

Raymond, K. S., Wheeler, R. A., 1999, Compatibility of Correlation-Consistent Basis Sets With a Hybrid Hartree-Fock/Density Functional Method , J. Comput. Chem., 20, 207. 

Four basis sets were examined BSl and BS3 are based on the Couty-Hall modification of the Hay and Wadt ECP, and BS2 and BS4 are based on the Stuttgart ECP. Two basis sets, BSl and BS2, are used to optimize the geometries of species in the OA reaction, [CpIr(PH3)(CH3)]++ CH4 [CpIr(PH3)(H)(CH3)2]+, at the B3LYP level, while the other basis sets, BS3 and BS4, are used only to calculate energies at the previously optimized B3LYP/BS1 geometries. BSl is double-zeta with polarization functions on every atom except the metal atom. BS2 is triple-zeta with polarization on metal and double-zeta correlation consistent basis set (with polarization functions) on other atoms. BS3 is similar to BSl but is triple-zeta with polarization on the metal. BS4 is similar to BS2 but is triple-zeta with polarization on the C and H that are involved in the reaction. The basis set details are described in the Computational Details section at the end of this chapter. 

The quality of quantum-chemical calculations depends not only on the chosen n-electron model but also critically on the flexibility of the one-electron basis set in terms of which the MOs are expanded. Obviously, it is possible to choose basis sets in many different ways. For highly accurate, systematic studies of molecular systems, it becomes important to have a well-defined procedure for generating a sequence of basis sets of increasing flexibility. A popular hierarchy of basis functions are the correlation-consistent basis sets of Dunning and coworkers [15-17], We shall use two varieties of these sets the cc-pVXZ (correlation-consistent polarized-valence X-tuple-zeta) and cc-pCVXZ (correlation-consistent polarized core-valence X-tuple-zeta) basis sets see Table 1.1. 

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. 

Table 1.1 Correlation-consistent basis sets for first-row atoms.

There are two possible solutions to this problem. We may either modify our ansatz for the wavefunction, including terms that depend explicitly on the interelectronic coordinates [26-30], or we may take advantage of the smooth convergence of the correlation-consistent basis sets to extrapolate to the basis-set limit [6, 31-39], In our work, we have considered both approaches as we shall see, they are fully consistent with each other and with the available experimental data. With these techniques, the accurate calculation of AEs is achieved at a much lower cost than with the brute-force approach described in the present section. 

The prerequisites for high accuracy are coupled-cluster calculations with the inclusion of connected triples [e.g., CCSD(T)], either in conjunction with R12 theory or with correlation-consistent basis sets of at least quadruple-zeta quality followed by extrapolation. In addition, harmonic vibrational corrections must always be included. For small molecules, such as those contained in Table 1.11, such calculations have errors of the order of a few kJ/mol. To reduce the error below 1 kJ/mol, connected quadruples must be taken into account, together with anhar-monic vibrational and first-order relativistic corrections. In practice, the approximate treatment of connected triples in the CCSD(T) model introduces an error (relative to CCSDT) that often tends to cancel the... 

Table 4-4 Convergence of the (Zmax + 5) 3 extrapolated cc-pVnZ correlation-consistent basis set MP2 correlation energies (Eh) to the MP2-R12 limit see Eq. (6.2).

We have recently employed the Dunning correlation-consistent basis sets for our pair natural orbital CBS extrapolation algorithm, Eqs. (2.1) and (2.2) [50]. The results produced a substantial improvement over the raw second-order energies, but were inferior to the (lmax + ) 3 extrapolations listed in Table 4.4. The residual underestimation of... 

Table 7 Convergence of the scaled PNO extrapolated CBS/cc-pVnZ, correlation-consistent basis set higher-order [i.e. CCSD(T)-MP2] correlation energies (Eh) to the CCSD(T)-R12 limit.

The Cauchy moments are derived and implemented for the approximate triples model CC3 with the proper N scaling (where N denotes the number of basis functions). The Cauchy moments are calculated for the Ne, Ar, and Kr atoms using the hierarchy of the coupled-cluster models CCS, CC2, CCSD, CC3 and a large correlation-consistent basis sets augmented with diffuse functions. A detailed investigation of the one- and A-electron errors shows that the CC3 results have the accuracy comparable to the experimental results. 

In the next section, we recapitulate the derivation of the Cauchy moment expressions for CC wavefunction models and give the CC3-specific formulas we also outline an efficient implementation of the CCS Cauchy moments. Section 3 contains computational details. In Section 4, we report the Cauchy moments calculated for the Ne, Ar, and Kr gases using the CCS, CC2, CCSD, CCS hierarchy and correlation-consistent basis sets augmented with diffuse functions. In particular, we consider the issues of one- and A-electron convergence and compare with the Cauchy moments obtained from the DOSD approach and other experiments. 

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. 

It is noticeable that correlation-consistent basis sets are not able to accurately reproduce the bond distances and energies of the excited states in II symmetry, which may be attributed to the relatively inadequate p-space. The problem was eliminated in the case of H-73 basis set. The polarization set that was purposefully optimized to correlate with the 2p orhitals of hydrogen atoms greatly improved the bond lengths and excitation energies, and reduced the errors to be within 100 cmof exact values. 

Correlation Consistent Basis Sets with Relativistic Effective Core Potentials The Transition Metal Elements Y and Hg... 

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