Open-shell restricted HF

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... 

Another way to computationally treat unpaired electrons is to employ restricted open-shell HF (ROHF) theory. Here, we encounter another pit-fall. It is an artifact called symmetry breaking (97). Whereas ROHF wave functions are pure spin states, the ROHF wave function may not retain the symmetry of the molecule. Suppose a molecule has C2V symmetry. The wave function should have the same symmetry, e.g., the orbital lobes on either side of the symmetry plane should be identical. However, with symmetry breaking, the two sides are not equal. The unsymmetrical ROHF wave function may even give lower energy than a physically correct (symmetrical)... 

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... 

The common way to treat free radicals is with the unrestricted HF method or UHF method. In this method, we employ separate spatial orbitals forthe a and the )3 electrons, giving two sets of MOs, one for a and one for electrons. Less commonly, free radicals are treated by the restricted open-shell HF or ROHF method, in which electrons occupy MO s in pairs as in the RHF method, except forthe unpaired electron(s). The theoretical treatment of open-shell species is discussed in [1,10, l(k, /)], in particular, compare the performance of the UHF and ROHF methods. 

For open-shell molecules, spin contamination can be a problem, as mentioned earlier. DFT and coupled-cluster theory are resistant to spin contamination and may be helpful. One may also choose spin-restricted open-shell HF (ROHF) as a starting point, instead of UHF, unless dissociation behavior is important. ROHF has no spin contamination. 

We shall follow the unrestricted Hartree-Fock (UHF) formalism for obtaining the restricted open-shell HF (ROHF) functions to derive the Hartree-Fock equations for excited states. For the sake of simplicity, we restrict our attention to the first excited state. The problem can be described as ... 

For a closed-shell system, the spin-up and spin-down Fock operators are equal and the spin-orbitals are obtained in pairs of equal shape and energy. Instead of dividing the set of orbitals into spin-up and spin-down orbitals, it is also possible to pursue a division into closed-shell and open-shell orbitals. This leads to the restricted open-shell HF (ROHF) method [23-25]. The formalism for this method is slightly more involved than the UHF formalism, but the general ideas are identical. The ROHF wavefunction is an eigenfunction of the total spin squared (5 ) operator, while the UHF wavefunction does not have this feature. The energy of the UHF wavefunction, on the other hand, is lower than that of the ROHF wavefunc-... 

These two issues recently initiated interest in the development of alternative open-shell CC schemes. " " A first suggestion was to start from a restricted open-shell HF (ROHF) instead of a UHF reference function. Since the ROHF wavefunction is already a spin eigenfunction, it can be shown that energies obtained from a spin-orbital based CC treatment correspond to so-called spin-projected energies ... 

MP methods have been developed for both spin-restricted HF (RHF), spin-unrestricted HF (UHF), and restricted open-shell HF (ROHF) wavefunctions to investigate both closed- and open-shell systems (see Table 2). For the calculation of electron systems with multireference character such as biradicals various multireference state (MRS) MP methods have been developed (Table 2). All these methods describe atoms, molecules, and reaction systems in the gas phase. However, many chemical reactions take place in solution phases. For this purpose, MP methods are available that start from a solvent corrected wavefunction where mostly polarizable continuum models are used (see Self-consistent Reaction Field Methods). 

For the s elements a comparably large number of polarization functions is needed, even for HF and DFT at DZV or TZV level, if one wants to stay within the above error limits. The subsequent p orbitals are energetically in reach and thus often partly occupied, for the heavier s elements this holds also for the d orbitals and for the 6s and 7s elements additionally for the f orbitals. The respective functions are determined best in restricted open-shell HF calculations with the respective occupations, e.g s p for the optimization of p functions of Mg. Thus, for error-consistency in the case of s elements, comparably large basis sets are needed, for instance for the triple zeta basis for Mg two diffuse p and two diffuse d sets and additionally one steep d set for the polarization of the 2p shell. For Ba apart from a diffuse p and a (4,1 d set, a (4 f set had to be added even to the double zeta basis. These functions are partly occupied, for instance in case of BaOby ca. 0.1 electrons. Neglecting the f set changes the equilibrium distance for BaO (at DFT level) by ca. 9 pm and the bond energy by ca. 50 kJ/mol, which is not tolerable at all. 

Both HF and DFT calculations can be performed. Supported DFT functionals include LDA, gradient-corrected, and hybrid functionals. Spin-restricted, unrestricted, and restricted open-shell calculations can be performed. The basis functions used by Crystal are Bloch functions formed from GTO atomic basis functions. Both all-electron and core potential basis sets can be used. 

To distinguish between closed-shell and open-shell configurations (and determinants), one may generally include a prefix to specify whether the starting HF wavefunction is of restricted closed-shell (R), restricted open-shell (RO), or unrestricted (U) form. (The restricted forms are total S2 spin eigenfunctions, but the unrestricted form need not be.) Thus, the abbreviations RHF, ROHF, and UHF refer to the spin-restricted closed-shell, spin-restricted open-shell, and unrestricted HF methods, respectively. 

Perturbative approximation methods are usually based on the Mpller-Plesset (MP) perturbation theory for correcting the HF wavefunction. Energetic corrections may be calculated to second (MP2), third (MP3), or higher order. As usual, the open- versus closed-shell character of the wavefunction can be specified by an appropriate prefix, such as ROMP2 or UMP2 for restricted open-shell or unrestricted MP2, respectively. 

To elaborate point (i), we chose thiophene oligomers as model systems [41]. In the present context, it is convenient to characterize the geometry with a single relevant parameter, the inter-ring C-C bond length this bond shortens in radical-cations in agreement with the valence-bond scheme shown in Figure 1.7. The results of ab initio and semiempirical HF spin-restricted open-shell self-consistent field... 

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