Local minimum, 339------------------------Localized Molecular Orbitals

Linear dependence of basis functions, 164 Linear response function, 261 Linear scaling techniques, 80 Linear Synchronous Transit (LST) optimization method, 328 Local Density Approximation (LDA), Local Spin Density Approximation (LSDA) functionals, 182 Local minimum, 339 Localized Molecular Orbitals (LMO), 227 Localized orbital methods, 227 Localized Orbital/local oRiGin (LORG), for calculating magnetic properties, 252 Locally Updated Planes (LUP) optimization method, 330... 

An error function depending on parameters. Only minima are of interest, and the global minimum is usually (but not always) desired. This may for example be determination of parameters in a force field, a set of atomic charges, or a set of localized Molecular Orbitals. 

The nonorthogonal LMOs were determined in the same manner as in the previous subsection. The atomic orbitals used for the determination are two Is orbitals of the hydrogen atoms and 2p(o) orbitals of the halogen atom. All the overlaps between the atomic orbital (AO) and the nonorthogonal LMO are greater than 0.9 (0.9004 at minimum). The molecular orbitals are therefore well localized. 

Figure 2. Minimum atomic localization energy in benzenoid hydrocarbons plotted against energy of the highest occupied molecular orbital (HOMO).

Usually, a covalent bond between two atoms is formed by two electrons, one from each atom. These electrons tend to be partly localized in the region between the two nuclei. If the orbitals of these electrons are I l and I 2, the molecular orbital of the bonded atoms must be their linear combination, symmetric Rb = R + Ri and antisymmetric = I l - I l- As illustrated in Fig. 2.2, the former orbital has a minimum of energy at certain distance and generally has lower energy than the... 

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