RHF/SCF calculations

Figure 8 Torsional diagram based on an RHF-SCF calculation for [(rf -NC4H4)-Co(C2B9Huj] and diazaferrocene. The sketches correspond to the distinct ring rotamers...

The diagonalization of the one-particle density matrix P on the Cl level yields the natural orbitals (NOs) and the occupation numbers of the molecule. In an RHF-SCF calculation, all occupation numbers are either 0 or 2, which means that an MO is either doubly occupied by 2 electrons or not occupied. If we invoke a Cl calculation, the mixing of configurations results in fractional occupation numbers of the NOs. Still in a normal molecule, such as ethane, water, ammonia, etc., the occupation numbers for the ground state will be close to 0 or 2. The occupation numbers of at least two NOs of excited states usually deviate from the values 0 and 2. In a biradicaloid structure occupation numbers will change and typically two of them will be close to 1. However, this method cannot characterize zwitterions since occupation numbers should be close to 0 and 2, respectively, in this case. The approach by Jug and Poredda yields both zwitterionic and diradical character of species, based on the valence criterion of Gopinathan and Jug. The valence is defined according to equation (5), which can be reformulated as equation (6). 

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... 

For systems with unpaired electrons, it is not possible to use the RHF method as is. Often, an unrestricted SCF calculation (UHF) is performed. In an unrestricted calculation, there are two complete sets of orbitals one for the alpha electrons and one for the beta electrons. These two sets of orbitals use the same set of basis functions but different molecular orbital coefficients. 

The method of calculating wavefunctions and energies that has been described in this chapter applies to closed-shell, ground-state molecules. The Slater determinant we started with (Eq. 5.12) applies to molecules in which the electrons are fed pairwise into the MO s, starting with the lowest-energy MO this is in contrast to free radicals, which have one or more unpaired electrons, or to electronically excited molecules, in which an electron has been promoted to a higher-level MO (e.g. Fig. 5.9, neutral triplet). The Hartree-Fock method outlined here is based on closed-shell Slater determinants and is called the restricted Hartree-Fock method or RHF method restricted means that the electrons of a spin are forced to occupy (restricted to) the same spatial orbitals as those of jl spin inspection of Eq. 5.12 shows that we do not have a set of a spatial orbitals and a set of [l spatial orbitals. If unqualified, a Hartree-Fock (i.e. an SCF) calculation means an RHF calculation. 

All calculations were carried out within the approximation of intermediate neglect of differential overlap (37-42) (INDO-RHF-SCF) which includes parameterization for transition metals. A restricted open-shell formalism, developed by Zerner et al. (37,38), was employed to prevent spin contamination and to make the quantitative evaluation of the relative spin state energies possible. This method has been used successfully to study simple transition metal complexes like [FeCl ]" (42), [CuCl ]2" ( ), and ferrocene ( ) as well as larger and more complicated systems like model oxyheme (61) and carbonylheme ( ) and model oxyhorseradish peroxidase ( ) complexes. 

Turning now to comparisons of CCP, MMB, and CAT, the relative energies of the germane doublet, quartet, and sextet spin states have been calculated using the same INDO-RHF-SCF method as for the model complexes and the results are presented in Tables V and VI. 

Special interest has been focused on the structure and reactivity of thiirane derivatives bearing exocyclic double bonds, methylenethiirane (or allene episulfide), thiiranone, and thiiranimine. The molecular structure of allene episulfide has been optimized by Closed-Shell SCF (CS-SCF) calculation using restricted Hartree-Fock (RHF) built into the GAUSSIAN 80 program, with the... 

The two different isomers were modelled through rigid [Rh(PH3)4] fragments. Calculations at the RHF-SCF level were made through use of pseudopotentials for the rhodium and phosphorus atoms. A double- basis set was used for Rh and active hydrogen atoms, whereas a minimal basis set was employed for phosphines. The relative positions of the hydrogen atoms was completely optimized. 

Introductory descriptions of Hartree-Fock calculations [often using Rootaan s self-consistent field (SCF) method] focus on singlet systems for which all electron spins are paired. By assuming that the calculation is restricted to two electrons per occupied orbital, the computation can be done more efficiently. This is often referred to as a spin-restricted Hartree-Fock calculation or RHF. 

There is more than one solution to the SCF equations for the system, and the calculation procedure converges to a solution which is not the minimum (often a saddle point in wavefunction space). This indicates an RHF-to-RHF or UHF-to-UHF instability, depending on the wavefunction type. 

In fact, it turns out that the orbitals resulting from SCF or valence MCSCF calculations in molecules ean be described in extremely simple terms by comparing them with the RHF orbitals of the separated atoms. 

If we except the Density Functional Theory and Coupled Clusters treatments (see, for example, reference [1] and references therein), the Configuration Interaction (Cl) and the Many-Body-Perturbation-Theory (MBPT) [2] approaches are the most widely-used methods to deal with the correlation problem in computational chemistry. The MBPT approach based on an HF-SCF (Hartree-Fock Self-Consistent Field) single reference taking RHF (Restricted Hartree-Fock) [3] or UHF (Unrestricted Hartree-Fock ) orbitals [4-6] has been particularly developed, at various order of perturbation n, leading to the widespread MPw or UMPw treatments when a Moller-Plesset (MP) partition of the electronic Hamiltonian is considered [7]. The implementation of such methods in various codes and the large distribution of some of them as black boxes make the MPn theories a common way for the non-specialist to tentatively include, with more or less relevancy, correlation effects in the calculations. 

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