HF calculations results

To demonstrate the phenomenon, HF calculated results [20] for the ground-state 4s orbital as well as for the excited 3d, 4d and 5d orbitals (resulting due to 3p rid transitions) of free Ca and Ca in Ca C60 are depicted in Figure 25. The calculations for the encaged Ca atom were performed in the framework of the A-potential model. 

We present in Table I results from calculations on bond energies, bond distances and vibrational frequencies for the simple MH hydrides of the coinage triad M=Cu,Ag,and Au as well as the isoelectronic series MH+ with M=Zn,Cd,and Hg.Table I contains experimental data (lH) as well as results from non-relativistic (JLl) and relativistic (4.) Hartree-Fock-Slater (HFS) calculations. Results from a similar set of calculations on the metal-dimers M2 (M=Cu,Ag,and Au) as well as the dications (M=Zn,Cd,and Hg) are presented in Table II. 

Table 3. Theoretical bond lengths and vibrational frequencies for ZnCln " compared with experimental data (in parentheses). Values in bold are density functional results from Collins (1997) calculated using the Perdew (1986) exchange-correlation functional. Values in brackets are HF calculated results.

Figure 5. Molecular orbital energy difference A612 and 2//12 for the l

Semiempirical calculations are set up with the same general structure as a HF calculation in that they have a Hamiltonian and a wave function. Within this framework, certain pieces of information are approximated or completely omitted. Usually, the core electrons are not included in the calculation and only a minimal basis set is used. Also, some of the two-electron integrals are omitted. In order to correct for the errors introduced by omitting part of the calculation, the method is parameterized. Parameters to estimate the omitted values are obtained by fitting the results to experimental data or ah initio calculations. Often, these parameters replace some of the integrals that are excluded. 

The Onsager model describes the system as a molecule with a multipole moment inside of a spherical cavity surrounded by a continuum dielectric. In some programs, only a dipole moment is used so the calculation fails for molecules with a zero dipole moment. Results with the Onsager model and HF calculations are usually qualitatively correct. The accuracy increases significantly with the use of MP2 or hybrid DFT functionals. This is not the most accurate method available, but it is stable and fast. This makes the Onsager model a viable alternative when PCM calculations fail. 

At the other end of the spectrum are the ab initio ( from first principles ) methods, such as the calculations already discussed for H2 in Chapter 4. I am not trying to imply that these calculations are correct in any strict sense, although we would hope that the results would bear some relation to reality. An ab initio HF calculation of the potential energy curve for a diatomic Aj will generally give incorrect dissociation products, and so cannot possibly be right in the absolute sense. The phrase ab initio simply means that we have started with a certain Hamiltonian and a set of basis functions, and then done all the intermediate calculations with full rigour and no appeal to experiment. 

The parameterization of MNDO/AM1/PM3 is performed by adjusting the constants involved in the different methods so that the results of HF calculations fit experimental data as closely as possible. This is in a sense wrong. We know that the HF method cannot give the correct result, even in the limit of an infinite basis set and without approximations. The HF results lack electron correlation, as will be discussed in Chapter 4, but the experimental data of course include such effects. This may be viewed as an advantage, the electron correlation effects are implicitly taken into account in the parameterization, and we need not perform complicated calculations to improve deficiencies in fhe HF procedure. However, it becomes problematic when the HF wave function cannot describe the system even qualitatively correctly, as for example with biradicals and excited states. Additional flexibility can be introduced in the trial wave function by adding more Slater determinants, for example by means of a Cl procedure (see Chapter 4 for details). But electron cori elation is then taken into account twice, once in the parameterization at the HF level, and once explicitly by the Cl calculation. 

The HF error depends only on the size of the basis set. The energy, however, behaves asymptotically as exp(—L),L being the highest angular momentum in the basis set, i.e. already, with a basis set of TZ(2df) (4s3p2dlf) quality the results are quite stable. Combined witii the fact that an HF calculation is the least expensive ab initio method, this means that tire HF error is not the limiting factor. 

If not otherwise stated the four-component Dirac method was used. The Hartree-Fock (HF) calculations are numerical and contain Breit and QED corrections (self-energy and vacuum polarization). For Au and Rg, the Fock-space coupled cluster (CC) results are taken from Kaldor and co-workers [4, 90], which contains the Breit term in the low-frequency limit. For Cu and Ag, Douglas-Kroll scalar relativistic CCSD(T) results are used from Sadlej and co-workers [6]. Experimental values are from Refs. [91, 92]. 

Tomas et al. [281] have calculated the tautomeric equilibrium of 1,2,3-benzotriazole in the gas phase and compared their results to experimental data [282] derived from ultraviolet spectroscopy. Experiment suggests that 35 is about 4 kcal/mol more stable than 34 this result is consistent with calculations [281] at the MP2/6-31G level, which predict 35 to be 2.5 kcal/mol more stable than 34. The same level of theory predicts 33 to be 5.0 kcal/mol more stable than 32 in the parent triazole system. Although experimental data are available indicating 35 to be the dominant tautomer in CDCf and d6-dimethyl sulfoxide solutions [279, 283], this equilibrium does not appear to have been the subject of any modeling, continuum or otherwise. It may prove to be somewhat challenging, however. Tomas et al. point out that correlation effects favor 35 by about 5 kcal/mol at the MP2 level AMI, PM3, and HF calculations with moderate basis sets all predict... 

The solid-state molecular dipole moment has been evaluated in a recent HF calculation on the crystal of urea. The result of 23.5-10 30 Cm, compared with... 

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