Open shell configuration

The tr E and the tr y terms vary for the different states in a very similar way to the FCI terms. The values obtained with the IP and MPS approximations for the term tr Q, for the ground and third state show a similar behaviour to those of the FCI calculation. However while the tr (2 D.) FCI value is higher in the second state (which has a dominant open shell configuration) than in the other states the opposite happens to the IP and MPS results. 

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. 

In the present work we present the generalisation of the theory to the case of K interacting fragments one of which may be described by an open shell configuration. This extension implies a drastic modification of the procedure which is here reported in full detail. 

Let us study a system constituted of K fragments Qi. ..flAr We assume that the first (K-1) fragments are closed-shell systems described by doubly occupied orbitals. The last fragment Ok has an open-shell configuration, consisting of singly occupied molecular orbitals of parallel spin. Obviously, the theory here developed can easily be restricted to the case of K closed-shell fragments. 

The UV-visible (vis) spectra of many organic radical ions show significant bath-ochromic shifts relative to their precursors. The open-shell configurations of singly occupied bonding (radical cations) or antibonding orbitals (radical anions) introduce new electronic transitions of lower energies, in the visible or near-IR (cf. Fig. 6.5). ... 

We now consider the theoretical calculation of excited-state wave functions. This is more difficult than ground-state calculations because we are dealing with open-shell configurations. The Hartree-Fock equations for a state of an open-shell configuration have a more complicated form than for closed shells, and there exist close to a dozen different approaches to excited-state Hartree-Fock calculations. As noted earlier, the Hartree-Fock wave function for a closed-shell state is a single determinant, but for open-shell states, we may have to take a linear combination of a few Slater determinants to get a Hartree-Fock function that is an eigenfunction of S and Sz and has the correct spatial symmetry. 

Before the effective hamiltonian can be used in actual calculations some means must be found for expressing the terms Gcore [equation (33)] and the projection operator terms in equations (31) or (34) in a form which is convenient for computing matrix elements this is the subject of parameterization, which is dealt with in Section 3. Two other formal problems remain at this level. Firstly there is the need to modify equation (29) and, as a result, equations (31) and (34) if the atomic calculations on the separate atoms are of the open shell kind as is usually the case. In order not to bias the later molecular calculation the core operators and projection terms can be derived for some average of all the possible open-shell configurations,25 although care should be exercised in the choice of the hamiltonian for which the... 

Despite their name, rare earth elements , lanthanides are in fact not especially rare each is more common in the earth s crust than silver, gold or platinum. They possess characteristic 4/ open-shell configurations and exhibit... 

The UHF formalism becomes inconvenient for open-shell configurations of atoms or molecules with point-group symmetry. Unless specific restrictions are imposed, the self-consistent occupied orbitals fall into sets that are nearly but not quite transformable into each other by operations of the symmetry group. By imposing equivalence and symmetry restrictions, these sets become symmetry-adapted basis states for irreducible representations of the symmetry group. This makes it possible to construct symmetry-adapted /V-clcctron functions, as described in Section 4.4. The constraints in general invalidate the theorems of Brillouin and Koopmans. This restricted theory (RHF) is described in detail for atoms by Hartree [163] and by Froese Fischer [130],... 

When using a spin-free hamiltonian operator, the spin functions are introduced as mulplicative factors, yielding the spin-orbitals. There are two spin-orbitals per orbital. An electronic configuration is defined by the occupancies of the spin-orbitals. Open-shell configurations are those in which not all the orbitals are doubly occupied. 

The total, antisymmetric function for a closed-shell configuration is expressed as a Slater determinant built-up from the spin-orbitals. In the case of open-shell configurations, a linear combination of Slater determinants may be needed in order to obtain a function with the same symmetry and multiplicity characteristics as the state under consideration. 

In the case of open-shell configurations, such as those corresponding, for example, to the lowest singlet and triplet excited states, one can use the SCF formulation of Birss and Fraga (1963) and Fraga and Birss (1964). The use of virtual orbitals provides, however, a simpler way of determining approximate functions for excited states. This approximation has been used in the calculations reported and reviewed here. 

If the shell structure is not conserved, the number of matrix elements to be included in the energy expression will be increased considerably and yield nonequivalent + and — orbitals for open-shell configurations. When the variational procedure is applied to the energy, one obtains equations that are essentially the same as the nonrelativistic equations [Eqs. (13)—(16) of Ref. 47]. Since all the deviations from the conventional SCF equations are included in Eqs. (44)-(49), the SCF equations for the TCMSs are omitted. 

Table 3 shows how the intensity of ionisation from an open-shell configuration is divided between states of the two possible spin multiplicities see Eq. (24). 

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