Basis Sets and Effective Potentials

Effective potentials also depend on the type of basis set used, hi atomic orbital calculations, they are sometimes referred to as frozen-core potentials. In most cases, only the highest-energy s, p and d electrons are included in the calculation. In plane-wave calculations, effective potentials are known as pseudopotentials They come in different varieties soft or ultrasoft pseudopotentials need only a relatively low energy cut-off as they involve a larger atomic core.  

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Since there is an element of arbitrariness in carrying out a DFT calculation, namely in the choice of functional, one may argue that DFT is not a truly ab initio method. However, DFT is based on an exact formalism, and once a functional has been chosen, all that enters a calculation are the atomic numbers of the atoms in the system (apart from the choice of basis set and effective potentials, see next section). Therefore, in this respect DFT methods are certainly ab initio. Nevertheless, in spite of the impressive results that can be obtained with DFT-GGA methods, a major drawback of DFT is that there is no systematic way to improve the calculation such that the exact result is approached, something which is formally (though usually not practically) possible in wave-function-based methods. 

Kaupert, Heydtmann and Thiel"2 calculated the vibrational spectrum of monohalo-genated 1 at the HF level using the 6-31 G(d) basis set and effective core potentials with DZ + P basis sets for Cl, Br and I. Reduction from Z)3h to Cs symmetry leads to considerable coupling between modes (exceptions C—H stretching and CH2-deformation modes) of 1. Vibrational frequencies that are influenced by the halogen substituent are shifted to lower values with increasing mass of the halogen. 

Martins JBL, Andres J, Longo E, Taft CA (1995) A theoretical-study of (1010) and (0001) ZnO surfaces—molecular cluster model, basis-set and effective core potential dependence. Theochem-J Mol Struc 330 301-306... 

Let us consider the principal results of these calculations in the case of lanthanum trichloride, for which the most complete ab initio and density fimctional theory (DFT) calculations (both with various basis set modifications) are available. The obtained molecular parameters strongly depend on the valence basis sets and effective relativistic core potentials. Nevertheless, ab initio calculations tend to predict a planar structure, whereas DFT calculations favor pyramidal configurations. The V2 frequencies obtained in all calculations are too low compared with those found experimentally. This implies a low barrier to inversion and large thermal geometry fluctuations. In view of such uncertainty in the results of quantum-chemical calculations, Hargittai (2000) was very careful in her estimates. She tends to favor the planar geometry of lanthanide chlorides, although does not rule out the possibility that they are pyramidal. 

Optimize these three molecules at the Hartree-Fock level, using the LANL2DZ basis set, LANL2DZ is a double-zeta basis set containing effective core potential (ECP) representations of electrons near the nuclei for post-third row atoms. Compare the Cr(CO)5 results with those we obtained in Chapter 3. Then compare the structures of the three systems to one another, and characterize the effect of changing the central atom on the overall molecular structure. 

It is not possible to use normal AO basis sets in relativistic calculations The relativistic contraction of the inner shells makes it necessary to design new basis sets to account for this effect. Specially designed basis sets have therefore been constructed using the DKH Flamiltonian. These basis sets are of the atomic natural orbital (ANO) type and are constructed such that semi-core electrons can also be correlated. They have been given the name ANO-RCC (relativistic with core correlation) and cover all atoms of the Periodic Table.36-38 They have been used in most applications presented in this review. ANO-RCC are all-electron basis sets. Deep core orbitals are described by a minimal basis set and are kept frozen in the wave function calculations. The extra cost compared with using effective core potentials (ECPs) is therefore limited. ECPs, however, have been used in some studies, and more details will be given in connection with the specific application. The ANO-RCC basis sets can be downloaded from the home page of the MOLCAS quantum chemistry software (http //www.teokem.lu.se/molcas). 

C. S. Callam, S. J. Singer, T. F. Fowary, and C. M. Hadad, Computational analysis of the potential energy surfaces of glycerol in the gas and aqueous phases Effects of level of theory, basis set, and solvation on strongly intramolecularly hydrogen bonded systems. J. Am. Chem. Soc. 123, 11743 11754 (2001). 

The wave functions are expended in a plane wave basis set, and the effective potential of ions is described by ultrasoft pseudo potential. The generalized gradient approximation (GGA)-PW91, and local gradient-corrected exchange-correlation functional (LDA)-CAPZ are used for the exchange-correlation functional. 

For the very highest accuracy, the effect of at least core-valence correlation should be explored. This must be accompanied by some serious effort to extend the basis so that core-correlating functions are included. Using valence-optimized basis sets and including core correlation is not only a waste of computer time, but a potential source of problems, as it can substantially increase BSSE. This point is not well appreciated the prevailing view appears to be that no harm can come of correlating the core when the basis set is inadequate. This is not so. 

Calculated as described in Ref. 31 at the experimental temperature in the gas phase unless noted. The LANL2DZ basis set with effective core potentials was used for rhodium and iridium. 

We shall compare the potential curves obtained with the two different methods. Second order perturbation theory (CASPT2) has been used to estimate the remaining correlation effects in the FCI calculation with the smaller number of orbitals. This approach will be described in detail below. The spectroscopic constants are presented in Table 5-2. As can be seen, the two results are almost identical. The results are obviously far from experiment because of the small basis set used but that is not relevant to the present discussion. With the smaller number of orbitals we can now perform much more advanced calculations using larger basis sets and approach the experimental values. As an illustration, such a result is also given in the table. 

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