Correlation-consistent functions

Krishnan R, Binkley J S, Seeger R and Popie J A 1980 Self-consistent molecular orbital methods XX. A basis set for correlated wave functions J. Chem. Phys. 72 650-4... 

An extension of this last notation is aug—cc—pVDZ. The aug denotes that this is an augmented basis (diffuse functions are included). The cc denotes that this is a correlation-consistent basis, meaning that the functions were optimized for best performance with correlated calculations. The p denotes... 

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. 

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... 

Some of the basis sets discussed here are used more often than others. The STO—3G set is the most widely used minimal basis set. The Pople sets, particularly, 3—21G, 6—31G, and 6—311G, with the extra functions described previously are widely used for quantitative results, particularly for organic molecules. The correlation consistent sets have been most widely used in recent years for high-accuracy calculations. The CBS and G2 methods are becoming popular for very-high-accuracy results. The Wachters and Hay sets are popular for transition metals. The core potential sets, particularly Hay-Wadt, LANL2DZ, Dolg, and SBKJC, are used for heavy elements, Rb and heavier. 

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. 

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. 

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. 

Stuttgart pseudopotential for Au with a uncontracted (lls/10p/7d/5f) valence basis set and a Dunning augmented correlation consistent valence triple-zeta sets (aug-cc-pVTZ) for both C and N, but with the most diffuse f function removed, was used. 

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. 

Dunning has developed a series of correlation-consistent polarized valence n-zeta basis sets (denoted cc-pVnZ ) in which polarization functions are systematically added to all atoms with each increase in n. (Corresponding diffuse sets are also added for each n if the prefix aug- is included.) These sets are optimized for use in correlated calculations and are chosen to insure a smooth and rapid (exponential-like) convergence pattern with increasing n. For example, the keyword label aug-cc-pVDZ denotes a valence double-zeta set with polarization and diffuse functions on all atoms (approximately equivalent to the 6-311++G set), whereas aug-cc-pVQZ is the corresponding quadruple-zeta basis which includes (3d2flg,2pld) polarization sets. 

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. 

For the related [CpIr(PH3)(CH3)]+ system, four basis sets were used. Basis set one (BS1) is the same as the ones described above for Ir and P, but the C and H are described as D95. Basis set two (BS2) is the Stuttgart relativistic, small core ECP basis set (49) augmented with a polarization function for Ir, and Dunning s correlation consistent double-zeta basis set with polarization function (50) for P, C and H. Basis set three (BS3) is the same as BS1 except the d-orbital of Ir was described by further splitting into triple-zeta (111) from a previous double-zeta (21) description and augmented with a f-polarization function (51). Basis set four (BS4) is the same as BS2 for Ir, P, and most of the C and H, but the C and H atoms involved in the oxidative addition were described with Dunning s correlation consistent triple-zeta basis set with polarization. 

This model is consistent with (6.67), and can be seen as a multi-variate version of the IEM model. The role of the second term (eC 1) is simply to compensate for the additional diffusion term in (6.91). Note that, like with the flamelet model and the conditional-moment closure discussed in Chapter 5, in the FP model the conditional joint scalar dissipation rates ( ap ip) must be provided by the user. Since these functions have many independent variables, and can be time-dependent due to the effects of transport and chemistry, specifying appropriate functional forms for general applications will be non-trivial. However, in specific cases where the scalar fields are perfectly correlated, appropriate functional forms can be readily established. We will return to this question with specific examples below. 

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. 

A variety of extrapolation algorithms have been applied to the sequences generated by the correlation-consistent cc-pVnZ basis sets [12, 51-55], Dunning and his colleagues had initially suggested fitting their calculations to an exponentially decaying function [12, 51, 52],... 

Krishnan R, Binkley JS, Seeger R, Pople JA(1980) Self-consistent molecular-orbital methods. 20. Basis set for correlated wave-functions. J Chem Phys 72 650-654... 

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. 

The Cauchy moments of Kr (Table 4) have been calculated using the largest currently available correlation-consistent basis augmented with diffuse functions - d-aug-cc-pV5Z basis. As for the Ar atom, the CCS and CC2 results overestimate the DOSD values for smaller k and underestimate for larger k. The CCSD model behaves in the same manner. The CC3 model systematically... 

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