Linear Combination of Atomic Orbitals LCAO Method

4 LINEAR COMBINATION OF ATOMIC ORBITALS (LCAO) METHOD Algebraic approximation 

Usually we apply the self-consistent field approach with the LCAO method this is then the SCF LCAO MO. In the SCF LCAO MO method, each molecular orbital is presented as a linear combination of atomic orbitals Xs 

The approximation, in which the molecular orbitals are expressed as linear combinations of the atomic orbitals, is also called the algebraic approximation  

Curiosity these people liked to amuse themselves with little rhymes. Felix Bloch has translated a poem by Walter Huckel from German to English. It does not look like a 

Electronic Motion in the Mean Field Atoms and Molecules 

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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 ... 

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". ... 

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,... 

Much work has been reported on type B azapentalenes, and this can be classified according to the degree of sophistication of the calculations. Simple Hiickel linear combination of atomic orbitals (LCAO) methods have been widely used, though more recently more sophisticated Pariser-Parr-Pople (PPP) methods (using only re-electrons) and all-electron CNDO calculations have been reported. 

Of the various methods of approximating the correct molecular orbitals, we shall discuss only one- the linear combination of atomic orbitals (LCAO) method. We assume that we can approximate the correct molecular orbitals by combining the atomic orbitals of the atoms that form the molecule. The rationale is that most of the time the electrons will be nearer and hence controlled by oneor the other of the two nuclei, and when this is so, the molecular orbital should be very nearly the same as the atomic orbital for that atom. The basic process is the same as the one wc employed in constructing hybrid atomic orbitals except that now we are combining orbitals on different atoms to form new orbitals that are associated with the entire molecule. We... 

A crystalline solid can be considered as a huge, single molecule subsequently, the electronic wave functions of this giant molecule can be constructed with the help of the molecular orbital (MO) methodology [19]. That is, the electrons are introduced into crystal orbitals, which are extended along the entire crystal, where each crystal orbital can accommodate two electrons with opposite spins. A good approximation for the construction of a crystal MO is the linear combination of atomic orbitals (LCAO) method, where the MOs are constructed as a LCAO of the atoms composing the crystal [19]. 

As was emphasized before (cf. Chapter 3), a molecule is not simply a collection of its constituting atoms. Rather, it is a system of atomic nuclei and a common electron distribution. Nevertheless, in describing the electronic structure of a molecule, the most convenient way is to approximate the molecular electron distribution by the sum of atomic electron distributions. This approach is called the linear combination of atomic orbitals (LCAO) method. The orbitals produced by the LCAO procedure are called molecular orbitals (MOs). An important common property of the atomic and molecular orbitals is that both are one-electron wave functions. Combining a certain number of one-electron atomic orbitals yields the same number of one-electron molecular orbitals. Finally, the total molecular wave function is the... 

The electron shell of molecules is constructed by linear combination of atomic orbitals (LCAO method) of the participating atoms to form bonding ct and n molecular orbitals (MOs), non-bonding n (lone pairs of electrons) and anti-bonding... 

Molecular orbital theory originated from the theoretical work of German physicist Friederich Hund (1896-1997) and its apphcation to the interpretation of the spectra of diatomic molecules by American physical chemist Robert S. MuUiken (1896-1986) (Hund, 1926, 1927a, b Mulliken, 1926, 1928a, b, 1932). Inspired by the success of Heitler and London s approach, Finklestein and Horowitz introduced the linear combination of atomic orbitals (LCAO) method for approximating the MOs (Finkelstein and Horowitz, 1928). The British physicist John Edward Lennard-Jones (1894-1954) later suggested that only valence electrons need be treated as delocalized inner electrons could be considered as remaining in atomic orbitals (Lennard-Jones, 1929). 

There are a number of band-structure methods that make varying approximations in the solution of the Kohn-Sham equations. They are described in detail by Godwal et al. (1983) and Srivastava and Weaire (1987), and we shall discuss them only briefly. For each method, one must eon-struct Bloch functions delocalized by symmetry over all the unit cells of the solid. The methods may be conveniently divided into (1) pesudopo-tential methods, (2) linear combination of atomic orbital (LCAO) methods (3) muffin-tin methods, and (4) linear band-structure methods. The pseudopotential method is described in detail by Yin and Cohen (1982) the linear muffin-tin orbital method (LMTO) is described by Skriver (1984) the most advanced of the linear methods, the full-potential linearized augmented-plane-wave (FLAPW) method, is described by Jansen... 

It is well known that one of the standard approaches to Eq. (2.1) is the linear combination of atomic orbitals (LCAO) method it consists in expanding the states of the solid in linear combination of atomic (or molecular) orbitals of the composing atoms (or molecules). This method, when not applied in oversimplified form, provides an accurate description of core and valence bands in any type of crystal (metals, semiconductors, and insulators). Applied with some caution, the method also provides precious information on lowest lying conduction States, replacing whenever necessary atomic orbitals with appropriate localized orbitals. ... 

Within the tight-binding (TB) approach. Slater and Roster [64] described the linear combination of atomic orbitals (LCAO) method as an eflRcient scheme for calculation of the electronic structure of periodic solids. As this method is computationally much less demanding than other methods such as the plane-wave methods, it has been extensively employed to calculate electronic structures of various metals, semiconductors, clusters and a number of complex systems such as alloys and doped systems. The calculation of the electronic structure requires solving the Schrodinger equation with the TB Hamiltonian given by... 

The most interesting application for our purposes is to construct MOs by the linear combination of atomic orbitals (LCAO) method, where the variable parameters are the coefficients of the linear combination of some basic orbitals y 9 (Ritz method). It can be shown that, in this case, the best orbitals are obtained by solving the eigenvalue equation for matrix H ... 

For modeling purposes, the electronic structure of a tip is approximated by a linear combination of atomic orbital (LCAO) method. Cluster models of 10-20 atoms are utilized [78]. It has been found that the tunneling current is concentrated on a single apex atom, if the other front most atoms of a tip are not located on the same level. Hence, the apex atom of the tip matters (examples are in Fig. 10.18) [79]. 

Normally these molecular orbitals are obtained as expansions in a set of atom-centred basis functions (the linear combination of atomic orbitals (LCAO) method), m being the number of such functions. Recently, two-dimensional numerical integration methods have been developed to solve the MCSCF equations for linear molecules. The dimension m is then, in principle, infinite (practice, it is determined by the size of the grid used in the numerical integration). The molecular-orbital space is further divided into three subspaces the inactive, the active and the external orbitals. The inactive and active subspaces constitute the internal (occupied) orbital subspace, while the external orbitals are unoccupied. The CASSCF wavefunction is formed as a linear combination of configuration state functions (CSFs) generated from these orbitals in the following way. 

Tight binding (TB) and linear combination of atomic orbitals (LCAO) methods represent the more chemical approach to the problem of surface state calculations. They are basically fitting techniques, but, given a reasonable choice of parameters, they can add considerable detail to the basis provided by self-consistent calculations. The method, as applied to surfaces, was initiated by Hirabayashi [73] and developed into a useable form by Pandey and Phillips [74, 75]. 

The hydrogen molecule H2 is the simplest molecule which forms an electron-pair bond. Many calculations have been made for this molecule, which is a prototype for many other chemical bonds. One of the two basic quantum-mechanical treatments of the hydrogen molecule involves constructing a molecular orbital for the bond from a linear combination of atomic orbitals (LCAO method). The other involves constructing the molecular orbital as the product of wave functions for each of the two electrons forming the bond. Both of these methods will be outlined. 

In 1962, Sugano showed that the Seitz model (115) could be interpreted as a molecular orbital model (123), an interpretation that clarifies analysis of these systems. In this interpretation, the absorption bands observed in the TI(I) doped alkali halide system come from the electronic transition aigf a g) hu), but the excited states are still calculated assuming an ionic interaction between the metal and the hgand. Since the thallium-chlorine bond is actually largely covalent, Bramanti et al. (118) modified the approach and used a semiempirical molecular orbital (MO) calculation to describe the energy levels of T1(I) doped KCl. Molecular orbitals were constructed by the linear combination of atomic orbitals (LCAO) method from the 6s and 6p metal orbitals and the 3p chlorine orbitals. Initial calculations were conducted with the one-electron approximation the method was then expanded to include Coulomb and spin-orbit interactions. The results of Bramanti et al. were consistent with experimental... 

Any method of solution of Eq. [25] is specific of the kind of basis set used. In the remaining part of this chapter, we will always refer to the use of one-electron local basis sets within the linear combination of atomic orbitals (LCAO) method. Accordingly, nf AOs in the 0-cell are chosen and replicated in the other cells of the crystal to form the periodic component u r k) of nf Bloch functions. In particular, by denoting the p-th AO, with the origin at in the 0-cell, as x (r and the corresponding AO in a different cell, the g-cell, as x (r — r — g) or, equivalently, x (r — r ), the expression used for M (r k) consists of a linear combination of the equivalent AOs in all N cells of the crystal ... 

Since calculation of MO s from first principles is difficult, the usual approximate approach is the linear combination of atomic orbitals (LCAO) method. It seems reasonable that MO s of a molecule should resemble atomic orbitals (AO s) of the atoms of which the molecule is composed. From known shapes of AO s, one can approximate shapes of MO s. The linear combinations (additions and subtractions) of two atomic s orbitals to give two molecular orbitals are pictured in Figure 2.16. One MO results fi om addition of the parts of AO s that overlap, the other Irom their subtraction. 

The first term is important and the results are expressed in terms of the resonance integral, (3, via a Linear Combination of Atomic Orbitals (LCAO) method. These indices are a measure of the change in rr-energy of a transition state in an electrophilic reaction where the electrons become located (51) the transition systems are calculated with two electrons less and one or two atoms less than the parent. 

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