RHF theory

An initial equilibrium structure is obtained at the Hartree-Fock (HF) level with the 6-31G(d) basis [47]. Spin-restricted (RHF) theory is used for singlet states and spin-unrestricted Hartree-Fock theory (UHF) for others. The HF/6-31G(d) equilibrium structure is used to calculate harmonic frequencies, which are then scaled by a factor of 0.8929 to take account of known deficiencies at this level [48], These frequencies are used to evaluate the zero-point energy Ezpe and thermal effects. 

The following questions deal with a specific application of ab initio RHF theory to ethylene (C2H4). 

Figure 6.12 The anti-SN2 reaction of chloride with allyl chloride. While RHF theory does well with the reactant/product geometries, it significantly overestimates the C-Cl bond lengths in the C2 symmetric TS structure based on calculations at more reliable levels of theory...

In order to improve on the RHF values, electron correlation is included in the ab initio method by using MP2. Comparing the results for the 6-31 lG(d,p) basis set for all the diatomics, electron correlation results in a sizable correction to the RHF results differences of 8.7 ppm for A1H, 8.0 ppm for A1F, 3.7 ppm for A1C1 and 7.4 ppm for A1NC are noted. In the case of A1H, the isotropic value of the aluminum shielding calculated at the MP2 level is greater than the result from RHF theory however, for the fluoride, chloride and isocyanide, the opposite trend holds. Gauss et al. (42) also observed this trend for the hydride, fluoride and chloride. 

An initial equilibrium structure is obtained by geometry optimization at the Hartree-Fock (HF) level with the 6-31G(d) basis.68 69 Spin-restricted Hartree-Fock (RHF) theory is used for singlet states and spin-unrestricted Hartree-Fock theory (UHF) for others. 

More generally, properties such as vibrational frequencies and reaction energies, which depend on the form of the potential curve, are better predicted with CASSCF theory than with RHF theory, provided the active orbital space has been properly defined. In Table 1, we compare the N2 full-valence CASSCF and RHF results for Z)e, re, coe, and v in various correlation-consistent basis sets. At the CASSCF level, De is in much better agreement with the experiment than at the RHF level. Moreover, the CASSCF re and coe are both close to the experimental values, although the RHF errors are slightly overcorrected because the CASSCF method overemphasizes the role of the antibonding orbitals. 

The Roothaan-Hall equations are not applicable to open-shell systems, which contain one or more unpaired electrons. Radicals are, by definition, open-shell systems as are some ground-state molecules such as NO and 02. Two approaches have been devised to treat open-shell systems. The first of these is spin-restricted Hartree-Fock (RHF) theory, which uses combinations of singly and doubly occupied molecular orbitals. The closed-shell approach that we have developed thus far is a special case of RHF theory. The doubly occupied orbitals use the same spatial functions for electrons of both a and spin. The orbital expansion Equation (2.144) is employed together with the variational method to derive the optimal values of the coefficients. The alternative approach is the spin-unrestricted Hartree-Fock (UHF) theory of Pople and Nesbet [Pople and Nesbet 1954], which uses two distinct sets of molecular orbitals one for electrons of a spin and the other for electrons of / spin. Two Fock matrices are involved, one for each type of spin, with elements as follows ... 

Quinet and Champagne have made an ah initio study of the vibrational contributions to y in acetylene, ethylene and ethane, using RHF theory with MP2 corrections and including first order anharmonicity contributions. There is a general increase in the effect in going fi om sp to sp hybridized carbons. Vibrational effects account for 10-20% of the intensity-dependent refiactive index (calculated at zero frequency) and about 50% of the Kerr constant for acetylene. 

RHF) theories, respectively. For most systems RHF is adequate, such that the "R" is usually... 

There are two alternatives to RHF theory, both of which allow a transition state to be calculated. In Unrestricted Hartree... 

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