Basis sets, diffuse triple-zeta

As already anticipated, one of advantages of using TD-DFT is that this method does not exhibit any dramatic dependence on the size of the basis set. For valence transitions, many studies indicate that a medium-size basis set (valence double-zeta or triple-zeta adding polarization and diffuse functions) provides VEE close to convergence [53-58]. 

Even larger basis sets are now practical for many systems. Such basis sets add multiple polarization functions per atom to the triple zeta basis set. For example, the 6-31G(2d) basis set adds two d functions per heavy atom instead of just one, while the 6-311++G(3df,3pd) basis set contains three sets of valence region functions, diffuse functions on both heavy atoms and hydrogens, and multiple polarization functions 3 d functions and 1 f function on heavy atoms and 3 p functions and 1 d function on hydrogen atoms. Such basis sets are useful for describing the interactions between... 

Barone also introduces two new basis sets, EPR-Il and EPR-llI. These are optimized for the calculation of hyperfine coupling constants by density functional methods. EPR-Il is a double zeta basis set with a single set of polarization functions and an enhanced s part. EPR-III is a triple zeta set including diffuse functions, double d polarization functions and a single set off functions. 

The starting point is our previously performed calculations [3] using the Huzinaga basis set [20] (9s) for Be and (4s) for H, triple-zeta contracted, supplemented by the three 2p orbitals proposed for Be by Ahlrichs and Taylor [21] with exponents equal to 1.2, 0.3 and 0.05 respectively. This initial basis set, noted I, includes one s-type bond-function the exponent of which is equal to 0.5647. Several sets of diffuse orbitals have then been added to this basis I. Their corresponding exponents were determined by downward extrapolation from the valence basis set, using the Raffenetti [22] and Ahlrichs [21] procedure. Three supplementary basis sets noted II, III and IV containing respectively one, two and three... 

Details of the extended triple zeta basis set used can be found in previous papers [7,8]. It contains 86 cartesian Gaussian functions with several d- and f-type polarisation functions and s,p diffuse functions. All cartesian components of the d- and f-type polarization functions were used. Cl wave functions were obtained with the MELDF suite of programs [9]. Second order perturbation theory was employed to select the most energetically double excitations, since these are typically too numerous to otherwise handle. All single excitations, which are known to be important for describing certain one-electron properties, were automatically included. Excitations were permitted among all electrons and the full range of virtuals. 

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. 

The ECP basis sets include basis functions only for the outermost one or two shells, whereas the remaining inner core electrons are replaced by an effective core or pseudopotential. The ECP basis keyword consists of a source identifier (such as LANL for Los Alamos National Laboratory ), the number of outer shells retained (1 or 2), and a conventional label for the number of sets for each shell (MB, DZ, TZ,...). For example, LANL1MB denotes the minimal LANL basis with minimal basis functions for the outermost shell only, whereas LANL2DZ is the set with double-zeta functions for each of the two outermost shells. The ECP basis set employed throughout Chapter 4 (denoted LACV3P in Jaguar terminology) is also of Los Alamos type, but with full triple-zeta valence flexibility and polarization and diffuse functions on all atoms (comparable to the 6-311+- -G++ all-electron basis used elsewhere in this book). 

Ab initio density functional theory calculations were also carried out on the CH2=XH(A) and CH(A)=XH2 series of molecules. The basis set used was the CEP-TZDP+ described previously26 and is more extensive than the DZP basis set used in the CAS(4,4)-OVB calculations. In TZDP+ the valence electron wave function is expanded in a triple-zeta sp set of functions plus a double set of polarization d-type functions plus a set of diffuse sp-type functions. The B3LYP exchange-correlation functional20 as defined in the Gaussian 94 program set35 was used in all the DFT calculations. 

With the addition of polarization functions and/or diffuse functions to the basis sets, the Pople notation can become rather cumbersome. For example, the 6-311++G(3df,2pd) set has a single zeta core and triple zeta valence shell, diffuse functions for all the atoms. Regarding polarized functions, there are three sets of d functions and one set of f functions on the non-hydrogens and two sets of p functions and one set of d orbitals on the hydrogens. 

A more sequential approach to the analysis of the systematic error of ab initio methods has been proposed in [14]. The same set of molecules as in [10] has been analyzed there. For this set the series of calculations using the basis sets aug-cc-pVxZ containing both polarization and diffuse functions with the number of exponents x in their respective radial parts up to x = 6 (single zeta x = 1, double zeta - DZ -x = 2, triple zeta - TZ - x = 3, etc.) and with the account of correlation effects in the range of methods from MP2 up to CCSD(T) had been performed and then fitted to the formulae [15-18] ... 

The AO basis set should be of triple zeta or better quality polarization functions are important and additional basis functions with very tight and diffuse exponents are helpful. A good compromise between accuracy and cost is provided by the AO basis set given by Chipman [5], enlarged by an additional tight s function. 

FCI energies of the ground state and several excited states (3 12+, 2 ll, and 2 2A states) were obtained by Olsen et al. [66] in 1989 using a DZP basis set augmented with diffuse functions. These data have been used as tests for a wide variety of EOM/FR-CC methods, including CCSD [20, 24], CCSDT-la [44], CC3 [45], CCSDT-3 [46], and CCSDt [52], Later Hirata et al. [49] obtained FCI results with the 6-31G basis set. Shiozaki et al. [57] have obtained FCI results with the augmented correlation-consistent polarized valence double-zeta (cc-pVDZ) and valence triple-zeta (aug-cc-pVTZ) sets. 

All calculations were performed with the Gaussian 98 suite of programs [4] using the hybrid DFT/HF B3LYP method. The all-electron 6-311+G(2df) basis set was used for H, O, Si, and Ge, whereas the valence triple zeta pseudopotential basis set of Stoll et al. [5] was used for Sn. The Sn basis set was augmented with diffuse function exponents one-third in size of the outermost valence exponents, the two-membered d-polarization set of Huzinaga, [6] and f(QSn = 0.286. This basis set combination will be called (v)TZ throughout this paper. 

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