Open shell systems triplet

Unrestricted monodeterminantal treatments using different orbitals for different spins for open-shell systems (free radicals, triplet states, etc.) [41,42],... 

A somewhat modified MO LCAO scheme, without restriction on the identity of spin orbitals (p and

unrestricted Hartree-Fock (UHF) method and is usually used to treat open-shell systems (free radicals, triplet states, etc.). Electron correlation is partially taken into account in this method, and therfore it can be expected to be more efficient than the RHF method when applied to calculate potential energy surfaces of chemical rearrangements whose intermediate or final stages may involve the formation of free- or bi-radical structures. The potentialities of the UHF method are now under active study in organic reaction calculations. Also, it is successfully coming into use in chemisorption computations (6). 

Semi-empirical methods (Section 3.9) sometimes employ the so-called half-electron THE) method for desenbing open-shell systems, such as doublets and triplets. In thrs odel a doublet state is <... 

Modern implementations of the MP2-F12 method combine the CABS approximation ]20] with robust density fitting techniques [21, 22] and local approaches ]23]. The coefficients are usually constrained at the values predetermined from the cusp conditions, as one half for singlet pairs and one quarter for triplet pairs in the spin-adapted formalism [24, 25]. The MP2-F12 methods have been extended to treat open-shell systems with unrestricted [26, 27, 28], restricted [29, 30] and multireference [29] formalisms. 

In FIF calculations on open-shell molecules, a complication arises because in addition to doubly occupied and virtual MOs, there are now also singly occupied MOs (one in the case of radicals and two in the case of triplet diradicals). In a first attempt to construct a ground state wavefunction Pq for such a molecule, a Slater determinant may be set up that contains the appropriate number of singly occupied MOs and leaves all the others occupied by two electrons of opposite spin. This type of wavefunction is called restricted open-shell Hartree-Fock (ROHF) restricted because the electrons of opposite spin in the closed shells are constrained to occupy MOs / that are spatially identical for electrons of a ( up ) and p ( down ) spin. A procedure to variationally optimize such a wavefunction for open-shell systems has been devised. ... 

Thus, in case of a closed-shell system the delocalization index between two domains equals the difference between the singlet-coupled and 1/3 of the triplet-coupled pairs formed between the domains (so-called effective pairon population" ). However, such straightforward relationship is not valid for open-shell systems. 

In Hartree-Fock theory, complications arise only for open-shell systems, where the active-active rotations are in some cases redundant, in other cases nonredundant For instance, for open-shell states constructed by distributing two electrons between two orbitals, we found in Section 10.1.2 that the active-active rotations are redundant for the triplet state but nonredundant for the singlet state. We can easily imagine that the situation becomes even more complicated in the MCSCF case, where the wave function is generated by optimizing simultaneously the orbital-rotation parameters and a (potentially) laige number of Cl coefficients. Fortunately, for the more common MCSCF models such as those based on the CAS and RAS concepts, the question of redundancies is simple and unexpected redundancies will only rarely arise. 

The theory had never been tested on a logical model system. Let us consider in detail one representative case, the superimposable stacking of the two benzene rings, one from each triplet diphenylcarbene molecule. These are considered to represent idealized modes of dimeric interaction of the aromatic ring parts of open-shell molecules in ordered molecular assemblies like crystals, liquid crystals and membranes. 

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