Diffuse and Polarization Functions

We would normally write the electronic ground state electron configuration of a carbon atom as ls-2s 2p-. Despite the intellectual activity that has gone into defining mythical valence states for carbon atoms in different bonding situations, no one would include a d-orbital in the description of ground state carbon. 

Carbon atoms in free space have spherical symmetry, but a carbon atom in a molecule is a quite different entity because its charge density may well distort from spherical symmetry. To take account of the finer points of this distortion, we very often need to include d, f,. .. atomic orbitals in the basis set. Such atomic orbitals are referred to as polarization functions because their inclusion would allow a free atom to take account of the polarization induced by an external electric field or by molecule formation. 1 mentioned polarization functions briefly in Section 9.3.1. 

Let s see what Pople and coworkers have to say about polarization functions. 

Self-Consistent Molecular Orbital Methods. XVn Geometries and Binding Energies of Second-Row Molecules. A Comparison of Three Basis Sets J. B. Collins, P. von R. Schleyer, J. S. Binkley and J. A. Pople The Journal of Chemical Physics 64 (1976) 5142-5151 

Three basis sets (minimal s-p, extended s-p and minimal s-p with d functions on the second row atoms) are used to calculate geometries and binding energies of 24 molecules containing second row atoms, d functions are found to be essential in the description of both properties for hypervalent molecules and to be important in the calculations of two-heavy-atom bond lengths even for molecules of normal valence. 

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G (nv= 1) valence sp sets, with diffuse and polarization functions on all atoms... 

In the G2 and G3 [10,11] theories, the Mpller-Plesset perturbation theories of the 2-nd and 4-th orders are used to estimate the consequences of extending orbital basis sets by including the diffuse and polarization functions. These attempts, however, do not allow one to eliminate a systematic error of about 6 millihartree per electronic pair, which, in the frame of the G2 and G3 theories, bears the pompous name of higher level correlation of unknown nature. These latter are parametrized in the form ... 

The calculated structures (at the QCISD level with triple- basis sets which include diffuse and polarization functions for Si, Ge and Sn, and ECPs for Pb)168 of D3h MII5 group 14 anions, which are all minima on the PES, are given in Figure 40. The M—H... 

From a more quantitative point of view, it is more difficult to achieve accurate computations for DPEs than for PAs of neutral substrates, because molecular anions are involved in the former case. However, when using second-order perturbation theory (MP2), a coupled-cluster theory (CCSD(T)) or a density functional theory (DFT/B3LYP), in conjunction with a moderate atomic basis set including a set of diffuse and polarization functions, such as the 6-3114—FG(d,p) or cc-aug-pVDZ sets, the resulting DPE errors appear to be fairly systematic. To some extent, the accuracy rests on a partial but uniform cancellation of errors between the acid and its conjugate base. Therefore, use of appropriate linear regressions between experimental and calculated values allows the DPEs for new members of the series to be evaluated within the chemical accuracy of 4=0.1 eV or 4=10 kJmoD. ... 

There is a general agreement that accurate quantum studies on hydrogen bonds require flexible basis sets, i.e., triple- with at least one set of diffuse and polarization functions (especially on H atoms it is surprisingly frequent to find in the literature calculations that exclude these functions from hydrogens) and treatment of electron correlation by means of either wave function-based or DFT-based methods [34-41]. Since these requirements have rendered useless much of the outdated theoretical material published before 1990, one can consider that the efforts to characterize hydrogen bonding from a quantum viewpoint are more or less 15 years old (see the historical comments in Ref. 5). 

SM calculations are broadly based on either the (i) Hartree-Fock method (ii) Post-Hartree-Fock methods like the Mpller-Plesset level of theory (MP), configuration interaction (Cl), complete active space self-consistent field (CASSCF), coupled cluster singles and doubles (CCSD) or (iii) methods based on DFT [24-27]. Since the inclusion of electron correlation is vital to obtain an accurate description of nearly all the calculated properties, it is desirable that SM calculations are carried out at either the second-order Mpller-Plesset (MP2) or the coupled cluster with single, double, and perturbative triple substitutions (CCSD(T)) levels using basis sets composed of both diffuse and polarization functions. 

Styszyriski et al. [31] studied the performance of BFs in relativistic HF and non-relativistic and relativistic electron correlation calculations for the BF, AIF, and GaF molecules. It was found that at both the non-relativistic and relativistic HF levels, the diffuse and polarization functions of an AC basis set can be effectively substituted by BFs unless the calculations with the AC basis itself yield near-HF-limit energies. However, in correlation calculations, the benefits of BFs were not obvious because when many BFs were used, the rate of convergence was usually rained. Nevertheless, when the number of BFs was relatively small, a considerable improvement could be observed in the correlation energy at practically no additional computational cost. 

The term basis set refers to the set of atom-centered mathematical functions chosen to describe atomic orbitals. These atomic orbitals are subsequently combined into molecular orbitals in the LCAO (see Linear Combination of Atomic Orbitals (LCAO)) approximation. Minimal basis sets, split-valence basis sets, and split-valence sets augmented with diffuse and polarization functions have all been used to examine the properties of hydrogen-bonded complexes. The weight of the evidence suggests that basis set which are not at least augmented split-valence basis sets are inadequate. 

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