Effective valence basis-sets

Unlike semiempirical methods that are formulated to completely neglect the core electrons, ah initio methods must represent all the electrons in some manner. However, for heavy atoms it is desirable to reduce the amount of computation necessary. This is done by replacing the core electrons and their basis functions in the wave function by a potential term in the Hamiltonian. These are called core potentials, elfective core potentials (ECP), or relativistic effective core potentials (RECP). Core potentials must be used along with a valence basis set that was created to accompany them. As well as reducing the computation time, core potentials can include the effects of the relativistic mass defect and spin coupling terms that are significant near the nuclei of heavy atoms. This is often the method of choice for heavy atoms, Rb and up. 

The most common way of including relativistic effects in a calculation is by using relativisticly parameterized effective core potentials (RECP). These core potentials are included in the calculation as an additional term in the Hamiltonian. Core potentials must be used with the valence basis set that was created for use with that particular core potential. Core potentials are created by htting a potential function to the electron density distribution from an accurate relativistic calculation for the atom. A calculation using core potentials does not have any relativistic terms, but the effect of relativity on the core electrons is included. 

Initially, the level of theory that provides accurate geometries and bond energies of TM compounds, yet allows calculations on medium-sized molecules to be performed with reasonable time and CPU resources, had to be determined. Systematic investigations of effective core potentials (ECPs) with different valence basis sets led us to propose a standard level of theory for calculations on TM elements, namely ECPs with valence basis sets of a DZP quality [9, 10]. The small-core ECPs by Hay and Wadt [11] has been chosen, where the original valence basis sets (55/5/N) were decontracted to (441/2111/N-11) withN = 5,4, and 3, for the first-, second-, and third-row TM elements, respectively. The ECPs of the second and third TM rows include scalar relativistic effects while the first-row ECPs are nonrelativistic [11], For main-group elements, either 6-31G(d) [12-16] all electron basis set or, for the heavier elements, ECPs with equivalent (31/31/1) valence basis sets [17] have been employed. This combination has become our standard basis set II, which is used in a majority of our calculations [18]. 

At B3LYP/6-311G(2d,p), pseudo-relativistic effective core potential and a (31/31/1) valence basis set were used for Si, Ge, Sn, Pb, from Ref 40. 

Another series of composite computational methods, Weizmann-n (Wn), with n = 1-4, have been recently proposed by Martin and co-workers W1 and W2 in 1999 and W3 and W4 in 2004. These models are particularly accurate for thermochemical calculations and they aim at approximating the CBS limit at the CCSD(T) level of theory. In all Wn methods, the core-valence correlations, spin-orbit couplings, and relativistic effects are explicitly included. Note that in G2, for instance, the single-points are performed with the frozen core (FC) approximation, which was discussed in the previous section. In other words, there is no core-valence effect in the G2 theory. Meanwhile, in G3, the corevalence correlation is calculated at the MP2 level with a valence basis set. In the Wn methods, the core-valence correlation is done at the more advanced CCSD(T) level with a specially designed core-valence basis set. 

The geometry optimizations have been carried out at the MP2 level of theory17 using effective core potentials (ECPs)18 for the heavier elements. Hydrogen and the first and second row elements B, C, N, O, Na, Mg, Al and Si were described by standard all electron 6-31G(d) basis sets.19 For tungsten we used the relativistic ECP developed by Hay and Wadt and the corresponding (441/2111/21) split-valence basis set.20 A pseudopotential with a (31/31/1) valence basis set was used for Cl, Ga, In and Tl.21 This basis set combination is our standard basis set II.22... 

A fluorine substituent shortens the adjacent C—C bonds and lengthens the opposite bond in the cyclopropane ring. The lengthening of the C—C bond opposite to a CF2 group is remarkable (cf 98 and 104 in Table 9), but the mean C—C distance is smaller even in these molecules than in cyclopropane (1). In the other fluoro derivatives, all C—C bonds are shorter than in 1. The C—C bonds appear to be longer, the C—F bonds shorter in a cis isomer than in the trans isomer , and the same applies to C—C and C—F bonds in cis compared to trans CHF—CHF moieties within the same molecule. Ab initio calculations with a 4-2IG split valence basis set reflect the above trends quite well, except for the distinction of effects in the cis and trans forms ". The additivity of substituent effects, which... 

Two methods are mainly responsible for the breakthrough in the application of quantum chemical methods to heavy atom molecules. One method consists of pseudopotentials, which are also called effective core potentials (ECPs). Although ECPs have been known for a long time, their application was not widespread in the theoretical community which focused more on all-electron methods. Two reviews which appeared in 1996 showed that well-defined ECPs with standard valence basis sets give results whose accuracy is hardly hampered by the replacement of the core electrons with parameterized mathematical functions" . ECPs not only significantly reduce the computer time of the calculations compared with all-electron methods, they also make it possible to treat relativistic effects in an approximate way which turned out to be sufficiently accurate for most chemical studies. Thus, ECPs are a very powerful and effective method to handle both theoretical problems which are posed by heavy atoms, i.e. the large number of electrons and relativistic effects. 

Topiol, S. and Osman, R., On the use of minimal valence basis sets with the coreless Hartree-Fock effective potential, J. Chem. Phys. 73, 5191-5196 (1980). 

Del Bene, J. E., and Shavitt, I., Basis-set effects on computed acid-base interaction energies using the Dunning correlation-consistent polarized split-valence basis sets, J. Mol. Struct. (Theo-chem) 307, 27-34 (1994). 

Throughout this chapter, unless specified otherwise, STO valence basis sets of triple-C (TZ) quality were used for the actinides Th-Am (54), while valence basis sets used for C and H were of the quality of double-, with 3type polarization functions (DZP) for C and H, respectively. In order to test the effects of different basis sets, we also used at several places the TZ basis sets with d- and /-type or p- and rf-type polarization functions (TZ2P) for C and H. The exponents of the polarization functions were taken from the standard basis sets library of ADF, i.e., (3d) = 2.20, C(4/) = 3.30 for C, and C (2p) = 1.25, C(3rf) = 2.50 for H. 

In the calculations based on effective potentials the core electrons are replaced by an effective potential that is fitted to the solution of atomic relativistic calculations and only valence electrons are explicitly handled in the quantum chemical calculation. This approach is in line with the chemist s view that mainly valence electrons of an element determine its chemical behaviour. Several libraries of relativistic Effective Core Potentials (ECP) using the frozen-core approximation with associated optimised valence basis sets are available nowadays to perform efficient electronic structure calculations on large molecular systems. Among them the pseudo-potential methods [13-20] handling valence node less pseudo-orbitals and the model potentials such as AIMP (ab initio Model Potential) [21-24] dealing with node-showing valence orbitals are very popular for transition metal calculations. This economical method is very efficient for the study of electronic spectroscopy in transition metal complexes [25, 26], especially in third-row transition metal complexes. 

Table 5 Effect of Basis Set on Valence Topological Population ... 

Obviously, the first consideration one must address when performing population analysis is the reliability of the wavefunction at hand. Whereas this topic is well beyond the scope of this review, a quick scan of the populations reported in this chapter indicates that basis set size and elearon correlation play an important role in defining the electronic distribution. For most compounds, a split-valence basis set augmented with polarization functions on the heavy atoms is the smallest basis set that produces reasonably consistent populations, regardless of the population method employed. One should further keep in mind the effects of electron correlation. Generally correlation contracts the core electrons, lengthens bonds, and reduces the ionic component of bonds. 

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