Basis Set Considerations

It is useful to define first die different basis sets of wave functions used in calculating analytic dipole moment derivatives and infrared intensities diat will be discussed in the present section. These are described in Table 7.1. 

Yamaguchi et al. [182] have applied the equations derived by diem for analytic simultaneous evaluation of vibrational frequencies and intensities for closed-shell, open-shell imrestricted and open-shell restricted Hartree-Fock wave functions for a number of molecules using basis sets of different complexity. Their results illustrate very clearly the basis set dependence of calculated vibrational parameters. Some of their results are represented in Table 7.2. Several molecular quantities are included since the comparisons for the accuracy of predicted values are quite interesting. Even larger basis have been used by Amos in consistent analytic derivative calculations of harmonic frequencies, infrared and Raman intensities for H2O, NH3 and CH4 as test molecules [175,189]. The results for H2O, HF, CO, NH3, CH4 and C2H2 of Yamaguchi et al. [182] and Amos [175, 189] are summarized in Tables 7.3 and 7.4. 

The dipole mrunent of HCN is very well predicted almost independoitly on the basis set used, hitensities of the infirared active modes of CO2 are overestimated by neatly 150-200% for all basis sets. HCN intensities are quite well predicted for extmided basis sets, such as TZ+2P or 6-311++G(3d,3p), except for the C N stretching mode which is very weak in die erqterimoital spectrum. 

Vibrational wavenumbers are overestimated by approximately 5-7% at even very extended basis set RHF/SCF calculations. There is no clear correlation between the size of basis sets and agreonait with raqieriment. Predicted infrared intensities for water are quite disappointing, differing very significandy from the measured values. Particularly 

Symbol contraction (First row atoms ) contraction (Hydrogen) 

---

Next, difficulties encountered in ab initio molecular dynamics simulations were discussed. Topics covered were massive parallelization to address computer time problems, basis set considerations, density functionals and van der Waals interactions, relativistic corrections, and new integration schemes. Several simulation techniques used to gain chemical insight were summarized. Enhanced sampling methods, metadynamics and other methods to explore free energy surfaces, reaction pathways and transition states were covered. Simulation of spectra (IR, NMR, EXAFS) from ab initio molecular dynamics simulations was the subject of the remaining paragraphs. 

Table 4 shows that the effect of electron correlation inclusion on methyl barriers is in general less important than the basis set considerations described in Section 4.1. For example, there is nearly 50 cm" (15%) flattening of the calculated acetaldehyde barrier from the HF one (calculated at HF geometry) by the MP4(SDTQ) correction at the modest MP2 correlation level the decrease is 26 cm". The HF reference calculation at MP2 geometry (Table 5) reveals that... 

Tie first consideration is that the total wavefunction and the molecular properties calculated rom it should be the same when a transformed basis set is used. We have already encoun-ered this requirement in our discussion of the transformation of the Roothaan-Hall quations to an orthogonal set. To reiterate suppose a molecular orbital is written as a inear combination of atomic orbitals ... 

Calculations at the 6-31G and 6-31G level provide, in many cases, quantitative results considerably superior to those at the lower STO-3G and 3-21G levels. Even these basis sets, however, have deficiencies that can only be remedied by going to triple zeta (6-31IG basis sets in HyperChem) or quadruple zeta, adding more than one set of polarization functions, adding f-type functions to heavy atoms and d-type functions to hydrogen, improving the basis function descriptions of inner shell electrons, etc. As technology improves, it will be possible to use more and more accurate basis sets. 

In practice only low orders of perturbation dreory can be carried out, and it is often observed that the HF and MP2 results differ considerably, the MP3 result moves back towards the HF and the MP4 moves away again. For well-behaved systems tlte correct answer is normally somewhere between the MP3 and MP4 results. MP2 typically overshoots the correlation effect, but often gives a better answer than MP3, at least if medium sized basis sets are used. Just as the first term involving doubles (MP2) tends to overestimate the correlation effect, it is often observed that MP4 overestimates the effect of the singles and triples contributions, since they enter the series for the first time at fourth order. 

Even-tempered basis sets have the same ratio between exponents over the whole range. From chemical considerations it is usually preferable to cover the valence region better than the core region. This may be achieved by well-tempered basis sets. The idea is similar to the even-tempered basis sets, tire exponents are generated by a suitable formula containing only a few parameters to be optimized. The exponents in a well-tempered basis of size M are generated as ... 

Figure 11.1 shows the bond dissociation curve at the HF level with the STO-3G, 3-21G, 6-31G(d,p), cc-pVDZ and cc-pVQZ basis sets. The total energy drops considerably upon going from the STO-3G to the 3-21G and again to the 6-3IG(d,p) basis. This is primarily due to the improved description of the oxygen Is-orbital. The two different... 

The ab initio methods used by most investigators include Hartree-Fock (FFF) and Density Functional Theory (DFT) [6, 7]. An ab initio method typically uses one of many basis sets for the solution of a particular problem. These basis sets are discussed in considerable detail in references [1] and [8]. DFT is based on the proof that the ground state electronic energy is determined completely by the electron density [9]. Thus, there is a direct relationship between electron density and the energy of a system. DFT calculations are extremely popular, as they provide reliable molecular structures and are considerably faster than FFF methods where correlation corrections (MP2) are included. Although intermolecular interactions in ion-pairs are dominated by dispersion interactions, DFT (B3LYP) theory lacks this term [10-14]. FFowever, DFT theory is quite successful in representing molecular structure, which is usually a primary concern. 

Table 11 summarizes the relative conformation stabilities of various sulfmyl carbanions, based on the H/D exchange rates of the corresponding sulfmyl compounds 36-39. The results are in good agreement with the order of stabilities obtained from the MO calculations using the 3-21G basis set. This is remarkable, since the calculation did not take into consideration the solvent effect, despite the strong unsymmetrical solvation on the a-sulfmyl carbanion. 

First consider the dipole operator (O = r). The matrix elements on rhs of eq. 17 are thus just the dipole transition moments, and the commutator becomes C = -ip. As the exact solution (complete basis set limit) to the RPA is under consideration, we may use eq. 10 to obtain... 

According to earlier considerations, the effects of m-substituents have been considered generally to be of the sigma zero type. Swain and Lupton have questioned this conclusion. We have proceeded in the following manner to obtain evidence on the issue. From all the known data, 11 sets of meta substituent reaction series data were found which meet the minimum substituent criterion for basis sets. These reactions are listed in Table XI, with the pT .Pr. ... 

---