LCAO method

Application of the simple molecular orbital theory to cyclobutadiene (cf Exercise 4-6) leads to prediction of four one-electron energy levels a + 2p, a, a, and a 2p. Use of Hund s rule leads to the following electronic configuration for the four tt electrons  

The calculated DE, is zero. Therefore simple molecular orbital theory predicts (apparently correctly) that cyclobutadiene should have no resonance energy and, if Hund s rule is applicable, a lowest diradical (triplet) state. 

The further prediction that the dipositive ion (2tt electrons) corresponding to cyclobutadiene should have paired electrons (singlet state) and a resonance energy comparable. to benzene has not as yet been confirmed. 

The latter case is particularly important because all odd alternant hydrocarbon radicals turn out to have a nonbonding molecular orbital. 

Although the electron distribution resulting from an electron in any one of these orbitals does not have four—fold symmetry, this symmetry does obtain for an electron distributed equally 

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Molecules. The electronic configurations of molecules can be built up by direct addition of atomic orbitals (LCAO method) or by considering molecular orbitals which occupy all of the space around the atoms of the molecule (molecular orbital method). 

LCAO method A method of calculation of molecular orbitals based upon the concept that the molecular orbital can be expressed as a linear combination of the atomic orbitals. 

The accuracy of most TB schemes is rather low, although some implementations may reach the accuracy of more advanced self-consistent LCAO methods (for examples of the latter see [18,19 and 20]). However, the advantages of TB are that it is fast, provides at least approximate electronic properties and can be used for quite large systems (e.g., thousands of atoms), unlike some of the more accurate condensed matter methods. TB results can also be used as input to detennine other properties (e.g., photoemission spectra) for which high accuracy is not essential. 

Slater P C and Koster G F 1954 Simplified LCAO method for the periodic potential problem Phys. Rev. 94 1498-524... 

The basis of constructing the MOs is the linear combination of atomic orbitals (LCAO) method. This takes account of the fact that, in the region close to a nucleus, the MO wave function resembles an AO wave function for the atom of which the nucleus is a part. It is reasonable, then, to express an MO wave function 1/ as a linear combination of AO wave functions Xi on both nuclei ... 

For a homonuclear diatomic molecule with nuclei labelled 1 and 2 the LCAO method gives the MO wave function... 

The Hiickel MO method is based on the LCAO method for diatomic molecules discussed in Section 7.2.1. Extension of the LCAO method to polyatomic molecules gives a secular determinant of the general type... 

Notice that 1 haven t made any mention of the LCAO procedure Hartree produced numerical tables of radial functions. The atomic problem is quite different from the molecular one because of the high symmetry of atoms. The theory of atomic structure is simplified (or complicated, according to your viewpoint) by angular momentum considerations. The Hartree-Fock limit can be easily reached by numerical integration of the HF equations, and it is not necessary to invoke the LCAO method. 

A determination of the dissociation constants of the compounds reveals that 5-azauracil (pi a = 6.73) is practically of the same acidity as 6-azauracil and considerably more acidic than uracil, A fundamental difference between 5-azauracil, on the one hand, and 6-azauracil and uracil, on the other, lies in the low stability of 5-azauracil toward acid and especially to alkaline hydrolysis. This fact appears to be in agreement with the differences in electron densities of these substances computed by the simple MO-LCAO method. ... 

If the basic set is chosen to consist of atomic orbitals, this relation forms the fundament for the MO-LCAO method in molecular and crystal theory. In its SCF form this approach was first used by Coulson (1938), and later it has been systematized by Roothaan (1951). More details about the SCF results within molecular theory will be given later in a special section. 

The energy spectrum of the dodecaborides (YB,2, YbB,2, LuBj2> computed by the MO-linear combination of atomic orbitals (LCAO) method shows that the direction of the electron transfer is uncertain". ... 

One of the most popular of the semi-empirical LCAO methods is that of Hfickel. It is applicable to planar molecules which have n -electron systems. The delocalization of these systems, as treated by this method, has particular chemical significance. The traditional application is to the benzene molecule. Historically, different structures of this molecule were suggested by K6kuI6+ and by Dewar that are described in virtually all textbooks of organic chemistry. These structures represented the first efforts to represent the delocalization of the 3r orbitals in such systems. In the present context the delocalization can be better specified with the use of the method of Hilckel. 

A given application of the LCAO method is characterized by a set of integrals. For a diatomic molecule they are... 

This example of the LCAO method, as applied to diatomic molecules, is perfectly general. However, it is simpler for hoinonuclear diatomics, for which Haa = Hbb- Then, Eq. (134) becomes... 

Linear combination of atomic orbitals (LCAO) method, 16 736 Linear condensation, in silanol polycondensation, 22 557-558 Linear congruential generator (LCG), 26 1002-1003 Linear copolymers, 7 610t Linear density, 19 742 of fibers, 11 166, 182 Linear dielectrics, 11 91 Linear elastic fracture mechanics (LEFM), 1 509-510 16 184 20 350 Linear ethoxylates, 23 537 Linear ethylene copolymers, 20 179-180 Linear-flow reactor (LFR) polymerization process, 23 394, 395, 396 Linear free energy relationship (LFER) methods, 16 753, 754 Linear higher a-olefins, 20 429 Linear internal olefins (LIOs), 17 724 Linear ion traps, 15 662 Linear kinetics, 9 612 Linear low density polyethylene (LLDPE), 10 596 17 724-725 20 179-211 24 267, 268. See also LLDPE entries a-olefin content in, 20 185-186 analytical and test methods for,... 

Spin-Spin Coupling.—The relative roles of different coupling mechanisms in organophosphorus compounds have been studied theoretically.66 The spin-spin interaction constants for methylphosphine have been calculated by the MO-LCAO method.67 Unresolved isotropic coupling to boron caused broadening of the vinyl... 

MO LCAO methods, 34 136 Molecular-beam surface scattering, 26 26, 27 Molecular Cage, 34 226 Molecular design in cyclodextrin, 32 427 Molecular dynamics diffusion in zeolites, 42 2, 4-6 argon, 42 20... 

Equation 1.132 has the property of never giving an energy that is lower than the true energy resonance principle cf Pauling, 1960). This property allows us to assign to the MO wave functions that are obtained by linear combination of the AO functions of the separate atoms (Linear Combination of Atomic Orbitals, or LCAO, method), by progressive adjustment of the combinatory parameters, up to achievement of the lowest energy. 

Hybrid orbitals may be considered as perfectioned AOs, adopted in the calculation of localized MOs in polyatomic molecules, with the LCAO method (cf. section 1.17.1). In the case of hybrid orbitals sp, the four linear combinations of s and p orbitals (Tbi, Tc2, Te, Te ) that lead to tetrahedral symmetry are... 

These properties of the d-shell chromophore (group) prove the necessity of the localized description of d-electrons of transition metal atom in TMCs with explicit account for effects of electron correlations in it. Incidentally, during the time of QC development (more than three quarters of century) there was a period when two directions based on two different approximate descriptions of electronic structure of molecular systems coexisted. This reproduced division of chemistry itself to organic and inorganic and took into account specificity of the molecules related to these classical fields. The organic QC was then limited by the Hiickel method, the elementary version of the HFR MO LCAO method. The description of inorganic compounds — mainly TMCs,— within the QC of that time was based on the crystal field... 

The application of the method to bulk systems is most easily introduced by first considering a lattice of atoms with overlapping s orbitals, 3, and corresponding free atomic energy levels, Es. Generalizing the LCAO method... 

Comparison of Ionization Potentials of Ferrocene Calculated by the SCF-A Transition State Procedure and the SCF-LCAO Method... 

SCF-X-SW transition state (191) (eV) Hartree-Fock SCF-LCAO method (58) (eV) Experimental (eV) ... 

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