Basis set employed

It has been noted in Hay s paper that the occupations for the d1, d4, d6, and d9 states are in principle arbitrary. This does not strictly hold true for density functional applications because of the above-mentioned dependence of the energy on the shape of the occupied orbitals. The density generated from occupying the dz2 differs from the one obtained from placing the electron in, e. g the d orbital. Feeding an approximate density functional with these two unequal densities may lead to non-identical energies (cf. Figure 5-2). In most practical applications, however, the errors introduced in this way should be much smaller than those caused by other limitations of the functional or basis set employed. 

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). 

Although there is no strict relationship between the basis sets developed for, and used in, conventional ah initio calculations and those applicable in DFT, the basis sets employed in molecular DFT calculations are usually the same or highly similar to those. For most practical purposes, a standard valence double-zeta plus polarization basis set (e.g. the Pople basis set 6-31G(d,p) [29] and similar) provides sufficiently accurate geometries and energetics when employed in combination with one of the more accurate functionals (B3LYP, PBEO, PW91). A somewhat sweeping statement is that the accuracy usually lies mid-way between that of M P2 and that of the CCSD(T) or G2 conventional wave-function methods. 

The ab initio SCF-MO calculations reported in this work were performed using the Gaussian 70 series of programs26. The basis sets employed are the STO-3G, STO-4G and 4-31G basis sets developed by Pople and co-workers. 

We performed CBS-4M single point energy calculations at these stationary points. The barrier height for the first proton transfer and the relative energy of the first dienolate are quite sensitive to the level of theory and basis set employed (Table 4.10 and Fig. 4.14). The initial Asp" (i.e. formate ion) carries the full negative charge in... 

It is found that the basis sets employed are well converged even though no rotational London orbitals are used in the calculations. The correlation effects are rather small, in the range of 1.5-3.5%, and for all but methane the correlation corrections are negative. 

The McLean-Chandler triple- basis set employed for aluminum was (12s9p/6s5p), expanded with two sets of polarization functions with exponents = 0.8 and 0.2. ... 

Thus, to the best of our knowledge, there is a lack of embedding cluster studies on the yttrium ceramics where with a sufficient precision both aspects of the ECM were taken in account. In the study [44], we attempted to fill such a gap and carried out the electronic structure calculations of the YBa2Cu307 ceramics at the Moller-Plesset level with a self-consistent account of the infinite crystal surrounding to the quantum cluster. The Gaussian basis set employed (6-31IG) was larger than those used in previous cluster calculations [16,20,22,29]. 

Comparable results are obtained for the NH3-HCI complex. SCF geometrical parameters (rnci and rnci) vary considerably with the basis set employed, while the MCSCF-Ml and SCF-Ml results are more stable (see Table 9). 

The C3V symmetry for the complex is confirmed. The order of magnitude of the binding energy confirms the importance ofthe BSSE (see also reference 16). Again, SCF energies show a greater dependence on the basis set employed when compared with the MCSCF-Ml and SCF-Ml ones (see Table 10) it is to be noted that SCF-Ml interaction energies are sufficiently close to the estimated HF limit of-5.5 kcal/mol[23]. 

Computed H-Cl stretching force constants of the complex, see Table 11, show a minor dependence on the basis set employed when the SCF-MI and MCSCF-MI procedures are applied. Moreover, the MCSCF-MI and SCF-MI results are very similar. The reliability of the SCF-MI and MCSCF-MI techniques is thus further demonstrated. 

Btiilding on atomic studies using even-tempered basis sets, universal basis sets and systematic sequences of even-tempered basis sets, recent work has shown that molecular basis sets can be systematically developed until the error associated with basis set truncation is less that some required tolerance. The approach has been applied first to diatomic molecules within the Hartree-Fock formalism[12] [13] [14] [15] [16] [17] where finite difference[18] [19] [20] [21] and finite element[22] [23] [24] [25] calculations provide benchmarks against which the results of finite basis set studies can be measured and then to polyatomic molecules and in calculations which take account of electron correlation effects by means of second order perturbation theory. The basis sets employed in these calculations are even-tempered and distributed, that is they contain functions centred not only on the atomic nuclei but also on the midpoints of the line segments between these nuclei and at other points. Functions centred on the bond centres were found to be very effective in approaching the Hartree-Fock limit but somewhat less effective in recovering correlation effects. 

The theory of upper and lower bounds is used in order to get estimates of the resolvent in the one-electron as well as in the many-electron case. The issue of completeness of the basis sets employed is central. 

Rizzo et al reported in refs. (17,18) results of nuclear shieldings and their polarizabilities for the same compounds but including correlation effects. By inspection of those values, we conclude that correlation affects dramatically both, nuclear magnetic shieldings and their polarizabilities for the CO compound but it is not so much important for H2O and CH4 molecules. In the case of the H2O molecule, the basis sets employed here must be improved since it is evident that they are not suitable enough to provide results of the same accuracy as those taken from the literature. 

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