Spin-restricted state

Fig. 2. Relative energy differences between the spin states of [Fe2] 1, for each spin state optimized structures denoted opt as well as for single-point calculations on the optimized high-spin structure denoted as single point, where the energy of the closed-shell spin restricted state has arbitrarily been set to zero.

Here E is the valence-shell energy in the maximum spin-polarised state E is the valence-shell energy in the spin-restricted state. The correction obtained was added to the atom energy determined from the relativistic calculation. Thus, the calculations took into account, though not in a fully rigorous form, both the relativistic effects essential for core states and the spin-polarisation effects essential for valence states of free atoms. 

A spin projected result does not give the energy obtained by using a restricted open-shell calculation. This is because the unrestricted orbitals were optimized to describe the contaminated state, rather than the spin-projected state. In cases of very-high-spin contamination, the spin projection may fail, resulting in an increase in spin contamination. 

Let us further restrict ourselves to the case of K = 0 spin wave states, namely, those states that can be sampled usually by electromagnetic waves. Hence, then, it is necessary to have the character... 

Filatov, M., Shaik, S., 1999, Application of Spin-Restricted Open-Shell Kohn-Sham Method to Atomic and Molecular Multiplet States , J. Chem. Phys., 110, 116. 

This functional satisfy a variational principle [3,5] EVo o[p0, m0 < EVo>Bo[p, m. EVoJ3o [p0, hi0] denote the ground state energy with density p0(r), and magnetization m0(r) of a particular system characterized by the external fields (v0(r), Bt>(r)). One of the main differences between the spin-restricted and spin-polarized cases is that the one-to-one relation between the external potential and the density cannot be extrapolated to the set of quantities (v0(r), B0(r)) and (p0(r), m0(r)) [3]. 

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. 

Spin-restricted procedures, signified by an R prefix (e.g. RHF, RMP), constrain the a and (3 orbitals to be the same. As such, the resulting wavefunctions are eigenfunctions of the spin-squared operator (S2) that correspond to pure spin states (doublets, triplets, etc). The disadvantage of this approach is that it restricts the flexibility in the... 

Figure 1 shows the one-electron energies of iron(II) porphine in both spin-restricted and spin-unrestricted (different orbitals for different spins) form. The geometry used has a planar porphine with an Fe-N distance of 2.01 A and no axial ligands. This calculation assumes a high-spin quintet state with the configuration (dxy) ... 

It is interesting to note here that the value of the rate constant for DPC (14a), 1-NC (a-12), methoxycarbonylphenylcarbene (53), and FL (23) with O2 are 5 X 10 , (3.5 0.7) X 10 , 8.6 x 10 and (1.4 0.2) x 10 AT s-, respectively. The difference in the rate constants is not as large as that observed for the hydrogen atom abstraction rate constants for those carbenes (Table 9.9) and do not reflect the difference in the magnitude of AGst- The reason is probably because the rate constant of triplet carbenes is very fast and because the singlet states do not interact with triplet oxygen because of the spin restriction. 

Note that in Figs. 2 and 3 the prime indicates that converged BS orbitals were provided as initial guess, whereas the double prime denotes that converged high-spin orbitals were employed. The energy of the closed-shell restricted state denoted as S=R has been arbitrarily set to zero. 

MCSCF results for phenylnitrene using a complete active space formed from the six phenyl tt orbitals and the two nitrogen p orbitals and the eight electrons contained therein are presented in Table 14.1. Note that, because of symmetry and spin restrictions, only the 2 Al state must be determined as the second root of the MCSCF. The CAS results are quite... 

As noted earlier, there is a kinetic restriction on the reactivity of dioxygen, as the one-electron reduction to superoxide is unfavoured. This is demonstrated in the redox potentials shown in Figure 51. The triplet ground state of the molecule may also impose spin restrictions on its reactions. 

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