Open-shell electronic states

In open-shell electronic states, the orbitals are not all doubly occupied, and the preceding procedure is not applicable. However, if the wave function can be written as a single Slater determinant, one can use a modified procedure to obtain energy-localized MOs here also. The procedure is to deal with the a spin-orbitals and the jS spin-orbitals separately, using two different unitary transformation matrices Ba and B in (2.85). 

In section 7.4.2 we dealt with perhaps the simplest contribution to the effective Hamiltonian for a particular electronic state, that of the rotational kinetic energy. We now turn our attention to contributions which are only slightly more complicated, the so-called fine structure interactions involving the electron spin angular momentum S. Obviously for S to be non-zero, the molecule must be in an open shell electronic state with a general multiplicity (2S +1). 

In a classic paper, Frosch and Foley [117] derived an elFective Hamiltonian to describe the magnetic nuclear hyperfine interactions of a diatomic molecule in an open shell electronic state. Their Hamiltonian was expressed in the following form ... 

The CAS(RAS)SCF method is one of the best methods to study excited states and photochemical processes because it can in a balanced way treat closed and open shell electronic states of varying complexity and also of different spin, which is necessary in studies of intersystem crossing. However, calculations on excited states is often more complicated than those for a well defined ground state. Preferably,... 

The symmetric and antisymmetric squares have special prominence in molecular spectroscopy as they give information about some of the simplest open-shell electronic states. A closed-shell configuration has a totally symmetric space function, arising from multiplication of all occupied orbital symmetries, one per electron. The required antisymmetry of the space/spin wavefunction as a whole is satisfied by the exchange-antisymmetric spin function, which returns Fq as the term symbol. In open-shell molecules belonging to a group without... 

A radical is defined to be a molecule in an open shell electronic state. It is often, although not necessarily, very reactive and short-lived in a laboratory environment. Several new species have been studied since the publication of the previous supplement, although the number for which microwave transition frequencies have been measured is still quite small. Many of the new observations have been made by radio astronomers who now have access to frequencies up to 350 GHz. Experiments employing double resonance techniques (simultaneous irradiation with microwaves and either infrared or visible radiation) have also made a contribution to the development of the field. The information about linear molecules, in 2, 2, and states, is contained in section 3.2.1. The non-linear radicals, almost all of which are triatomic, are presented in 3.2.2 (Non-Unear triatomic) and 3.2.3 (Non-linear larger molecules). 

The Hamiltonian used to describe the microwave spectra of non-linear molecules in open-shell electronic states can be written as the sum of several separate terms ... 

However, in octahedral Cr(III) complexes, Cr3+ is a d3 metal ion with three electrons in the HOMO orbitals ttM (t ). Thus the complexes of Cr(III) have an open-shell ground-state configuration, which on excitation produces quartet and doublet states (Figure 1.13). 

One important feature of carbenes is the presence of two nonbonding electrons and two available orbitals, which are nominally located on the carbon atom. In bent carbenes, the two orbitals have different energies and are often denoted as a and -ir, with a being the in-plane orbital and -ir the out-of-plane orbital. Within this simple picture, four electronic states can be envisioned, which are depicted in Fig. la. Singlets and S3 have the two electrons in the same orbital (a and -IT, respectively) and are often characterized as closed-shell singlets. On the other hand, singlet S2 has the same open-shell electronic occupation (arr) as the triplet (T) state. 

The situation here is completely analogous to that obtained in the restricted open HE theory (ROHF). The states are not eigenfunctions of S, except when all the open-shell electrons have parallel spins (A p = 0 or = 0). This result is a consequence of the expansion (22) used to obtain Eq. (27). Actually, the spin decomposition, Eq. (22), for does not conserve in general the total spin S. However, we can form appropriate linear combinations of two-electron spin functions cri Sj, 8 81,52) that are simultaneously eigenfunctions of and S, and achieve a correct spin decomposition of the 2-RDM [85] ... 

Calculations predict that the lowest state of PN has an open-shell electronic configuration." " The Salem-Rowland Rule for ISC promoted by spin-orbit coupling (SOC) predicts that singlet to triplet relaxation will have its maximum rate when the singlet state is closed-shell. This is the case with diaryl carbenes where the absolute rate constants of ISC are in the order of Michl has recently pointed out the importance of donor-... 

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