Orbitals LCAO method linear combination

There are a variety of methods for the computation of the MOs that interact in the transition states of [4+2]-cycloadditions. The LCAO method (linear combination of atomic orbitals) is often employed, and the basic idea is as follows. The MOs of the -systems of alkenes, conjugated polyenes, or conjugated polyenyl cations, radicals, or anions all are built by so-called linear combinations of 2p AOs. In a somewhat casual formulation, one might say that the MOs of these -systems are constructed with the help of the 2pz AOs. These AOs are centered at the positions of the n C atoms that are part of the -system. LCAO computations describe a conjugated -electron system that extends over n s/ 2-hybridized C atoms by way of n Ji-type MOs. 

The quantum mechanics methods in HyperChem differ in how they approximate the Schrodinger equation and how they compute potential energy. The ab initio method expands molecular orbitals into a linear combination of atomic orbitals (LCAO) and does not introduce any further approximation. 

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... 

This is the classical and general LCAO approximation (linear combination of atomic orbitals) of the molecular orbital method. 

To apply the variational method, we first need to have a trial function. When the system under study is a molecule composed of n atoms, the trial function, which is to become a molecular orbital, is a linear combination of atomic orbitals (LCAO) ... 

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... 

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