ROHF approximation

The bold (normal) notation of symmetry is used for the calculations performed in the UHF (RHF or ROHF) approximation. 

One of the most important qualitative results is probably the effect of orbital degeneracy on the multielectronic states of the P4 molecule in the reduced forms (see Figs 1,4, and 5). In the ROHF approximation the degenerated many-electronic wave functions of the P4 ground states can be presented as follows ... 

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.

All the calculations of F2 are carried out with a simple basis set of double-zeta polarization type, the standard 6-31G(d) basis set, and are performed at a fixed interatomic distance of 1.44 A, which is approximately the optimized distance for a full Cl calculation in this basis set. Only the corresponding orbitals are referred to as the active orbitals , while the orbitals representing the lone pairs, so-called spectator orbitals , remain doubly occupied in all calculations. A common point to the various VB methods we use, except the VBCI method, is that at the dissociation limit, the methods converge to two F fragments at the restricted-open-shell Hartree Fock (ROHF) level. 

A curious effect, prone to appear in near degeneracy situations, is the artifactual symmetry breaking of the electronic wave function [27]. This effect happens when the electronic wave function is unable to reflect the nuclear framework symmetry of the molecule. In principle, an approximate electronic wave function will break symmetry due to the lack of some kind of non-dynamical correlation. A typical example of this case is the allyl radical, which has C2v point group symmetry. If one removes the spatial and spin constraints of its ROHF wave function, a lower energy symmetry broken (Cs) solution is obtained. However, if one performs a simple CASSCF or a SCVB [28] calculation in the valence pi space, the symmetry breaking disappears. On the other hand, from the classical VB point of view, the bonding of the allyl radical is represented as a superposition of two resonant structures. 

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. 

In Table 3, the MP2, MP3, EN2 and EN3 energies for the lowest 1Hi state of CH2 and NHj are compared with the SCF (ROHF), UGA-CC [at both linear and quadratic levels of the first interacting space (is) approximation, referred to as L-CCSD(is) and CCSD(is), respectively], and various Cl [including full Cl (FCI)] results, considering both DZ and DZP basis sets. (For full SD space CCSD results and other limited Cl results, see Table 2 of (13)). These results indicate that MP2 underestimates the exact correlation energy by about 12-15%, yielding 84.8 and 88.5% of the correlation energy for the 1H/ state of CH2 and NH, respectively, with... 

The geometry of butadiene in the different electronic states has been computed at the ROHF/MP2 level of approximation using 6-31G basis sets. Excitation energies have been computed with the CASSCF/CASPT2 method,... 

In 2013, Li et al. [15] used the ROHF-UCCSD(T)-F12 method with the aug-cc-pVTZ basis set and a frozen-core (FC) approximation to study the characteristics of the three X - H20 (X = F, Cl, Br) complexes, but no other stationary points in the potential surface were reported. Since the reaction rate constants and their temperature dependence are sensitive to the accuracy of the potential energy surface, we wiU, in the present paper, adopt high-level ab initio coupled-cluster methods along with correlation-consistent cc-pV5Z-PP basis sets to study all stationary points on the potential surface for the Br -I- H2O HBr -I- OH reaction. 

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 restricted open-shell Hartree-Fock (ROHF) and the unrestricted Hartree Fock Method ( UHF) approximations permit, however, open-shell systems to be described, while maintaining the simplicity of the single-determinant approximation. This is made at the stage of self-consistent electronic-structure calculations. Afterwards, the obtained spin-orbitals can be used to get the correct total spin many-determinant wavefunction and to calculate the corresponding electron energy. 

The results obtained in post-HF methods for solids refer mainly to the energy of the ground state but do not provide the correlated density matrix. The latter is calculated for sohds in the one-determinant approximation. The density matrix calculated for crystals in RHF or ROHF one-determinant methods describes the many-electron state with the fixed total spin (zero in RHF or defined by the maximal possible spin projection in ROHF). Meanwhile, the UHF one-determinant approximation formally corresponds to the mixture of many-electron states with the different total spin allowed for the fixed total spin projection. Therefore, one can expect that the UHF approach partly takes into account the electron correlation. In particular, of interest is the question to what extent UHF method may account for correlation effects on the chemical bonding in transition-metal oxides. An answer to this question can be obtained in the framework of the molecular-crystalline approach, proposed in [577] to evaluate the correlation corrections in the study of chemical bonding in crystals. 

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