Computable Error Bounds in the Finite-Element Method

In order to provide computable error bounds in the finite-elementmethod, attention is mainly focused on specifying the constantsin convergence results. In the present note we demonstrate adifferent approach, namely a local-interpolation error-estimationtechnique that does not separate powers of the discretizationparameter h. It provides a bound for the relative error overarbitrary triangular grids which, in particular, characterizesthe influence of irregular grids.     

Electronic states and transitions of the TeX (X = C1, Br,I) radicals

Ab initio multireference configuration interaction calculations including spin-orbit coupling have been carried out for the first time for valence electronic states of the TeX (X = Cl, Br, I) radicals and compared with the results for the isovalent TeF and IO systems obtained earlier at a similar level of theoretical treatment. The calculated spectroscopic constants are in good agreement with experimental data in the rare cases when the latter are available. It is shown that the X² II(σ²π⁴π?³) ground state bonding becomes consistently weaker down the TeX group (calc. D_e, = 25480cm^?1 for TeF, 12 100cm^?1 for Tel) due to the more covalent character of bonding in the heavier radicals. The first excited state, A ⁴Σ^? (π?→ σ?), is calculated to be bound in all systems. It is split into Ω 1/2 and 3/2 components, with regular ordering in the Franck-Condon region, opposite to that of the ground X²II state. For larger internuclear distances, the A₁ ⁴Σ^? _1/2 state undergoes an avoided crossing with X₂ ²II_1/2, which causes a shoulder in the X₂ potential curve and also leads to a crossing between the A₁, and A₂ curves and large distinctions in their vibrational frequencies. The π? → σ? B²Σ^?, C²δ, and 1 ²Σ⁺ states are calculated to lie next in energy. They are all bound in the lightest of the TeX radicals, TeF, but successively lose their bonding character down the group. In contrast to oxygen monohalides, the 2²II(σ²π³ π?⁴) state has a repulsive potential curve. Electric dipole transition moments and radiative lifetimes have also been calculated for the low lying bound states in all systems. Most of them are found to be quite weak. The A^1,2 → X^1,2 spectra are dominated by parallel contributions, with the A₂ → X₁ being the strongest one. The T values for this transition are quite similar and lie in the 17–30 μs range. Radiative lifetime values for the B → X^1,2 transitions demonstrate very irregular behaviour for various, TeX radicals, due to strong admixture of A⁴Σ^? character to the X^1,2 states near the B²Σ^? potential minimum. The A^{1,2 4}Σ^? _1/2,3/2 and B²Σ^? _1/2 states of TeX (X = Cl, Br, I) still await their experimental observation.