Localized molecular orbitals lone-pair

Molecular orbitals are not unique. The same exact wave function could be expressed an infinite number of ways with different, but equivalent orbitals. Two commonly used sets of orbitals are localized orbitals and symmetry-adapted orbitals (also called canonical orbitals). Localized orbitals are sometimes used because they look very much like a chemist s qualitative models of molecular bonds, lone-pair electrons, core electrons, and the like. Symmetry-adapted orbitals are more commonly used because they allow the calculation to be executed much more quickly for high-symmetry molecules. Localized orbitals can give the fastest calculations for very large molecules without symmetry due to many long-distance interactions becoming negligible. 

The electrostatic energy is calculated using the distributed multipolar expansion introduced by Stone [39,40], with the expansion carried out through octopoles. The expansion centers are taken to be the atom centers and the bond midpoints. So, for water, there are five expansion points (three at the atom centers and two at the O-H bond midpoints), while in benzene there are 24 expansion points. The induction or polarization term is represented by the interaction of the induced dipole on one fragment with the static multipolar field on another fragment, expressed in terms of the distributed localized molecular orbital (LMO) dipole polarizabilities. That is, the number of polarizability points is equal to the number of bonds and lone pairs in the molecule. One can opt to include inner shells as well, but this is usually not useful. The induced dipoles are iterated to self-consistency, so some many body effects are included. 

All lone pair orbitals have a node between the two atoms and, hence, have a slightly antibonding character. This destabilizing effect of the lone pair localized molecular orbitals corresponds to the nonbonded repulsions between lone pair atomic orbitals in the valence bond theory. In the MO theory all bonding and antibonding resonance effects can be described as sums of contributions from orthogonal molecular orbitals. Hence, the nonbonded repulsions appear here as intra-orbital antibonding effects in contrast to the valence-bond description. 

Percent of localization of the methanide lone pair on the carbon atom according to the calculated NLMO (natural localized molecular orbital). 

Despite the quantitative victory of molecular orbital (MO) theory, much of our qualitative understanding of electronic structure is still couched in terms of local bonds and lone pairs, that are key conceptual elements of the valence bond (VB) picture. VB theory is essentially the quantum chemical formulation of the Lewis concept of the chemical bond [1,2]. Thus, a chemical bond involves spin-pairing of electrons which occupy valence atomic orbitals or hybrids of adjacent atoms that are bonded in the Lewis structure. In this manner, each term of a VB wave function corresponds to a specific chemical structure, and the isomorphism of the theoretical elements with the chemical elements creates an intimate relationship between the abstract theory and the nature of the... 

The last equation has been used to analyze occupied-unoccupied localized molecular orbital pair contributions for excitations in chiral metal complexes and metallahelicenes [260, 261], as well as in chiral organic acids derived from amino acids by substitution of the amino group with —OH and —F [170]. The analyses in terms of canonical MOs and LMOs may be considered complementary tools, with the canonical MO analysis generally leading to fewer contributions since the canonical MOs are well adapted to describe electronic excitations. The analysis in terms of LMOs allows one to focus on selected chemist s orbitals of interest, such as contributions to excitations from a given lone pair or localized n orbital, or from metal-centered orbitals, which can also be very useful. 

It is interesting to compare the implications expressed by these subrules with the depiction of some localized molecular orbitals in Figure 3-41 [97], The lone pair of electrons occupies more space than do the bonding pairs in the vicinity of the central atom. Also, a bond to a more electronegative ligand such as fluorine occupies less space in... 

Figure 3-41. Localized molecular orbitals represented by contour lines denoting electron densities of 0.02, 0.04, 0.06, etc. electron/bohr from theoretical calculations for the S-H, S-F, and S=0 bonds and the lone pair on sulfur [98],...

Fig. 7.9. Contour plots of the localized molecular orbitals in the (H2SiO)2 molecule for la) the Si-0 bond (b) the Si-H bond (c) and (d) the lone pair orbitals (after Kudo and Nagase, 1985 reproduced with the publisher s permission).

If the Coulomb interaction between electrons of different pairs is ignored, each localized bond and lone pair contribute independently to the total energy, which implies a perfect additivity of bond energies. In the independent-particle model, the localized bond function can be visualized as a two-center molecular orbital occupied by two electrons. Nevertheless, it is possible to reproduce deviations from additivity rules within this scheme, for instance, by taking into account overlap (for a review, see e.g. 2>). 

Localized molecular orbitals (LMOs) are certain combinations of delocalized molecular orbitals (eigenfunctions of the Hamiltonian) such that charge density is concentrated in particular regions of the molecule. Individual LMOs can therefore be identified with bond orbitals between a pair of atoms, lone pair orbitals on isolated atoms, and inner shell atomic orbitals. LMO methods have recently been introduced as a means of calculating ROA spectra without the necessity for extracting para-... 

It is also interesting to use molecular orbitals localized in core, lone pairs, and bond regions, rather than fully delocalized canonical orbitals. A good choice of the remaining n(n - l)/2 Lagrangian multipliers should lead to those localized orbitals, but such an a priori choice is hardly feasible. Most often, localized orbitals are determined by applying an adequate unitary transformation on previously obtained canonical orbitals ... 

NBOs were conceived as a chemist s basis set since they correspond closely to the picture of localized bonds and lone pairs as basic units of molecular structure. The procedure for obtaining them starts from the one-particle density matrix, from which a set of orthonormal NBOs is obtained. The transformation from canonical orbitals to NBOs includes several steps that are not described here, but it is important to stress that the main contribution to the total density matrix comes from a set of one- and two-centred occupied orbitals, w, the former being identified either as core orbitals or lone pairs and the latter as <7 or tt bonds. The transformation also yields unoccupied orbitals, ijj, that are identified as cr or tt antibonds (<7 or tt ) or extra-valence-shell... 

Fig. 8.30. A shock therapy. The localized molecular orbitals of water molecule, (a) An orbital, which is strougly localized on the oxygen nucleus and nearly spherical (practically Is of the oxygen atom), (b) and (c) are the OH bond orbitals, one for each OH bond, (d) The first of the two lone parr orbitals is a hybrid with its axis within the molecular plane, (e) The second of the two lone pair orbitals (orthogonal to the plane of the molecule) is a pure 2p orbital of oxygen. It is clear that a sum of the last two orbitals leads to a hybrid, which has to go up off the plane. On the other hand, a subtraction of these orbitals brings a twin hytaid but down from the plane. In this way, we recover (in a fully correct way) the equivalent lone pairs of water molecules seen in textbooks.

The effect of the overlap of electron lone pairs on the Fermi contact part of NMR nuclear spin-spin couplings has been analysed by Malkina et by the use of visualization of spin-spin coupling pathways by real-space funetions. The authors also applied the decomposition of the coupling deformation density into contributions from localized molecular orbitals... 

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