ROHF methods

A separate HF-Xa ealeulation is therefore needed in order to caleulate eaeh ionization energy. What we do is to plaee half an electron in the orbital from which the electron is supposedly ionized and re-do the HF-Xa ealeulation. The hypothetical state with a fractional electron is sometimes called an Xa transition state, a phrase borrowed from ehemieal kinetics. We treat the transition state by UHF or ROHF methods aceording to personal preferenee. 

Agreement with experiment is not particularly good, and the ROHF method cannot give regions of negative spin density, since the spin density is just the sum of the squares of the partially occupied orbitals in this model. Corresponding calculations with the UHF method give Table 18.4. 

Figure 3. Molecular-orbital diagrams as obtained by the ROHF method. Dashed lines indicate MOs dominated by the metal d-orbitals, the solid lines stand for doubly occupied or virtual ligand orbitals. Orbitals which are close in energy are presented as degenerate the average deviation from degeneracy is approximately 0.01 a.u. In the case of a septet state (S=3), the singly occupied open-shell orbitals come from a separate Fock operator and their orbital energies do not relate to ionization potentials as do the doubly occupied MOs (i.e. Koopmann s approximation). For these reasons, the open-shell orbitals appear well below the doubly occupied metal orbitals. Doubly occupying these gives rise to excited states, see text.

Figure 3 The numbers at each site in the top half (above the dotted line connecting the extreme atoms to the left and right of the diagram) are the numbers of classical structures which can be constructed with hydrogen (muonium) attached to the position indicated and the unpaired electron at the indicated site. The corresponding numbers in the bottom half are the spin densities in atomic units from UHFAA calculations on the fully optimised geometry of CeoMu using an ST0-3G basis set within the ROHF method.

All of the ab initio results are collected in Tables 2 and 3. The tables differ only in the theoretical method used. Table 2 used the ROHF method and Table 3 used the UHF method using optimised geometries from Table 2. 

Apart from type 62, which is only slowly convergent to the optimised geometry, the other centres are well described by the ROHF method. Polyhedral views of the three type a structures are shown in Fig. 6. These all illustrate the change of hybridisation at the point of muonium attachment and at the adjacent carbon atom where the unpaired electron is effectively localised as expected from addition to an alkene. The bi and c defects (Fig. 7) are quite different. The expected hybridisation change to sp is clearly present for the atom bonded to muonium, but other significant distortions are not obvious. This is consistent with the prediction from resonance theory (Fig. 8) that the unpaired electron for these structures is delocalised over a large number of centres. 

It is possible to construct a HF method for open-shell molecules that does maintain the proper spin symmetry. It is known as the restricted open-shell HF (ROHF) method. Rather than dividing the electrons into spin-up and spin-down classes, the ROHF method partitions the electrons into closed- and open-shell. In the easiest case of the high-spin wavefunction ( op = — electrons in op... 

Both objects are much less complicated than the total A -particle wavefunction itself, since they only depend on three spatial variables. The electron density is manifestly positive (or zero) everywhere in space while the spin-density can be positive or negative. If, by convention, there are more spin-up than spin-down electrons, the positive part of the spin-density will prevail and there will usually be only small regions of negative spin-density that arise from spin-polarization. This spin-polarization is physically important and is already included in the UHF method but not in the ROHF method that, by construction, can only describe the... 

For Eq. (9.35) to be useful the density matrix employed must be accurate. In particular, localization of excess spin must be well predicted. ROHF methods leave something to be desired in this regard. Since all doubly occupied orbitals at the ROHF level are spatially identical, they make no contribution to P only singly occupied orbitals contribute. As discussed in Section 6.3.3, this can lead to the incorrect prediction of a zero h.f.s. for all atoms in the nodal plane(s) of the singly occupied orbital(s), since their interaction with the unpaired spin(s) arises from spin polarization. In metal complexes as well, the importance of spin polarization compared to tire simple analysis of orbital amplitude for singly occupied molecular orbitals (SOMOs) has been emphasized (Braden and Tyler 1998). 

The common way to treat free radicals is with the unrestricted Hartree-Fock method or UHF method. In this method, we employ separate spatial orbitals for the oc and the jl electrons, giving two sets of MO s, one for oc and one for fj electrons. Less commonly, free radicals are treated by the restricted open-shell Hartree-Fock or ROHF method, in which electrons occupy MO s in pairs as in the RHF method, except for the unpaired electron(s). The theoretical treatment of open-shell species is discussed in various places in references [1] and in [12]. 

The electronic structure methods are based primarily on two basic approximations (1) Born-Oppenheimer approximation that separates the nuclear motion from the electronic motion, and (2) Independent Particle approximation that allows one to describe the total electronic wavefunction in the form of one electron wavefunc-tions i.e. a Slater determinant [26], Together with electron spin, this is known as the Hartree-Fock (HF) approximation. The HF method can be of three types restricted Hartree-Fock (RHF), unrestricted Hartree-Fock (UHF) and restricted open Hartree-Fock (ROHF). In the RHF method, which is used for the singlet spin system, the same orbital spatial function is used for both electronic spins (a and (3). In the UHF method, electrons with a and (3 spins have different orbital spatial functions. However, this kind of wavefunction treatment yields an error known as spin contamination. In the case of ROHF method, for an open shell system paired electron spins have the same orbital spatial function. One of the shortcomings of the HF method is neglect of explicit electron correlation. Electron correlation is mainly caused by the instantaneous interaction between electrons which is not treated in an explicit way in the HF method. Therefore, several physical phenomena can not be explained using the HF method, for example, the dissociation of molecules. The deficiency of the HF method (RHF) at the dissociation limit of molecules can be partly overcome in the UHF method. However, for a satisfactory result, a method with electron correlation is necessary. 

The allyl radical is of particular theoretical interest as a small molecule which exhibits the phenomenon of doublet instability, or symmetry breaking. As a consequence, the restricted open-shell HF (ROHF) method fails to reproduce the C2v equilibrium structure predicted by experimental smdies [74]. One must, therefore, resort to unrestricted (UHF or UKS) or multiconfigurational (MCSCF) methods. The results obtained using different functionals are reported in table 4. As for the methyl radical, a good agreement is found between the... 

Fig. 1 Vibronic coupling constant (solid line) and energy gradient (dotted line) of hydrogen molecule anion using the ROHF method with the basis set (a) 6-3IG and (b) 6-3IG with their first derivatives. The electronic energy (dashed line) is shown against the displacement from the equilibrium geometry of the neutral molecule...

Typical structures are specified in Table 1 which uses the labelling of carbon atoms in Cjo defined in Fig. 1. The restricted open-shell Hartree-Fock (ROHF) method was used in all geometry optimizations using a minimal basis set of orbitals (STO-3G) [13]. These calculations are therefore exploratory in nature. Here we have chosen to use the standard ab initio ROHF method since it is well-known that the UHF method (as used in the PRDDO approximation [9]) does not give wave functions which are eigenstates of the total spin operator S. The effect of spin contamination on molecular properties is uncertain, particularly if the contamination is high (the... 

The electronic states and the relevant matrix elements have been determined using the Restricted Open-shell Hartree-Fock (ROHF) method, [337] followed by a configuration interaction (Cl) calculation with double excitations. The active space is hmited to 10 molecular orbitals (MO), consisting of 2 occupied, 1 singly-occupied, and 7 unoccupied MOs. Excitations to the higher MOs are neglected. The total munber of configuration state functions (CSFs) in the active space amounts to 479. 

ROHF means a single-determinant wavefunction with maximal spin projection that is automatically eigenfunctions of S with the maximal spin projecton value S = ris/2. So, for the ROHF method projection on a pure spin state is not required. The space sjonmetry of the Hamiltonian in the ROHF method remains the same as in the RHF method, i.e. coincides with the space symmetry of nuclei configuration. The double-occupancy constraint allows the ROHF approach to obtain solutions that are eigenfunctions of the total spin operator. The molecular orbitals diagram for the ROHF half-closed shell is given in Fig. 4.1, (left). 

The ROHF LCAO method for crystals differs from the UHF method in the equations defining the CO. Let n = na + n electrons per primitive unit cell be considered (ria > nfj). For the Ufj CO the closed-shell RHF LCAO equations are solved, the CO of ria + rifj electrons with a spin are found from the equation analogous to (4.57). In the ROHF method the total density P(i m) and spin density P P "(iim) matrices are defined as... 

The UHF method many-electron wavefunction is the eigenfunction of the spin projection operator Sz with zero eigenvalue, the ROHF method many-electron function... 

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