LCAO MO theory

Pariser, R., /. Chem. Phys. 21, 568, "An improvement in the n-electron approximation in LCAO-MO theory."... 

The origin of cyclopropenone chemistry goes back to the successful preparation of stable derivatives of the cyclopropenium cation <5 3), the first member of a series of Huckel-aromatic monocyclic carbo-cations possessing a delocalized system of (4n + 2)-7r-electrons. This experimental confirmation of LCAO-MO theory stimulated efforts to prepare other species formally related to cyclopropenium cation by a simple resonance description of electron distribution, namely cyclopropenone 7 and methylene cyclopropene (triafulvene) 8 ... 

Lattice energies (continued) theory, 22 10-16 unit cell parameter, 22 11 Lawrencium, 31 4 LCAO-MO theory, 22 204 [L(CH,0)Cr(pdmg)Cu(Hj0)]2+, structure, 43 236-237... 

Some individual compounds have been studied using LCAO-MO theory in the Wolfsberg-Helmholz approximation (5). Although this method is somewhat more realistic and allows one to account for other properties (such as "charge-transfer bands, EPR, and NMR experiments) nevertheless, compared to the crystal field model it is much more laborious, it is only vahd for the individual case, and the choice of parameters in often rather arbitrary. 

In view of the complex nature of those molecules, it is not surprising that ab initio calculations are lacking. However, in the case of ferrocene with its very high D5d symmetry, attempts have been made using the simplified SCF-LCAO-MO theory of Roothaan (68). Yamazaki (69) reported on such calculations but so briefly that it is difficult to identify the ordering of the... 

Organometallic Ni—S metal cluster X-ray measurements, LCAO-MO theory 37 With S bonded to three Ni atoms, Ni—Ni bond length increases by 0.04 nm Ni—Ni interactions weaken due to longer Ni—Ni distance... 

They can be compared with the diatomic (covalent) bond order of Wiberg [52] formulated in the standard SCF-LCAO-MO theory,... 

Hartree-Fock-Roothaan SCF theory, using molecular orbitals expanded as linear combinations of atomic orbital basis functions (LCAO-MO theory)... 

For solids with more localized electrons, the LCAO approach is perhaps more suitable. Here, the starting point is the isolated atoms (for which it is assumed that the electron-wave functions are already known). In this respect, the approach is the extreme opposite of the free-electron picture. A periodic solid is constructed by bringing together a large number of isolated atoms in a maimer entirely analogous to the way one builds molecules with the LCAO approximation to MO (LCAO-MO) theory. The basic assumption is that overlap between atomic orbitals is small enough that the extra potential experienced by an electron in a solid can be treated as a perturbation to the potential in an atom. The extended- (Bloch) wave function is treated as a superposition of localized orbitals, centered at each atom ... 

For lattices with more than one atom per lattice point, combinations of Bloch sums have to be considered. In general, the LCAO approach requires that the result be the same number of MOs (COs in sohds) as the number of atomic orbitals (Bloch sums in sohds) with which was started. Thus, expressing the electron-wave functions in acrystaUine sohd as linear combinations of atomic orbitals (Bloch sums) is really the same approach used in the 1930s by Hund, Mulliken, Htickel, and others to construct MOs for discrete molecules (the LCAO-MO theory). 

In earlier chapters, it was seen how a qualitative energy-level diagram for the smallest repeating chemical point group, or lattice point (known to crystallographers as the basis, or asymmetric unit), can be used to approximate the relative placement of the energy bands in a solid at the center of the BZ. This is so because the LCAO-MO theory is equivalent to the LCAO band scheme, minus consideration of the lattice periodicity. The present chapter will investigate how the orbital interactions vary for different values of the wave vector over the BZ. 

Most chemists are well acquainted with LCAO-MO theory. The numbers of atomic orbitals, even in large molecules, however, are miniscule compared to a nonmolecular solid, where the entire crystal can be considered one giant molecule. In a crystal there are in the order of 10 atomic orbitals, which is, for all practical purposes, an infinite number. The principle difference between applying the LCAO approach to solids, versus molecules, is the number of orbitals involved. Fortunately, periodic boundary conditions allow us to smdy solids by evaluating the bonding between atoms associated with a single lattice point. Thus, the lattice point is to the solid-state scientist, what the molecule is to the chemist. 

Some conclusions can be drawn, if we restrict ourselves to LCAO-MO theory. The eigenfunctions, fm, are given by... 

The annulenes are that series of monocyclic polyolefins (C H ) containing a complete system of contiguous double bonds. While benzene (the best known member of this class of compounds) has been in evidence for some time it is only of late that interest in the higher members has become apparent. This interest has its origins in the LCAO-MO theory of re-elec-tron systems as formulated by E. Hiickel (in particular the "Hiickel rule relating aromatic stability to structure). Although the non-classical chemistry of the benzenoid hydrocarbons had previously been the subject of some conjecture, Httckel s theoretical studies provided the first satisfactory explanation of the peculiar stability of this class of compounds and, incidently, the elusiveness of cyclobutadiene. 

Huckel s rule (in its original form) stated that monocyclic polyenic molecules are aromatic only if their re-systems contain An + 2) re-el ec-trons, where n is an integer 1>. There have been many advances in LCAO-MO theory since Hiickel s original contributions (although the simplest approximation still bears his name, i.e. HMO), and today a more precise statement of the rule might read as follows. 

We shall now consider the electron configurations of each of the homo-nuclear diatomic molecules of the elements in the first short Period on the basis of the LCAO-MO theory outlined above. 

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