Linear combination of atomic orbitals. See

The concepts of hybridisation and resonance are the cornerstones of VB theory. Unfortunately, they are often misunderstood and have consequently suffered from much unjust criticism. Hybridisation is not a phenomenon, nor a physical process. It is essentially a mathematical manipulation of atomic wave functions which is often necessary if we are to describe electron-pair bonds in terms of orbital overlap. This manipulation is justified by a theorem of quantum mechanics which states that, given a set of n respectable wave functions for a chemical system which turn out to be inconvenient or unsuitable, it is permissible to transform these into a new set of n functions which are linear combinations of the old ones, subject to the constraint that the functions are all mutually orthogonal, i.e. the overlap integral J p/ip dT between any pair of functions ip, and op, (i = j) is always zero. This theorem is exploited in a great many theoretical arguments it forms the basis for the construction of molecular orbitals as linear combinations of atomic orbitals (see below and Section 7.1). 

In the d o case of a singly-occupied metal d-orbital interacting with a doubly occupied ligand orbital (which in general will be a synnetry-adapted linear combination of atomic orbitals, see Table VI) the charge deformation can be written as (21) ... 

LCAO (linear combination of atomic orbitals) See orbital. 

Configuration Interaction (or electron correlation) adds to the single determinant of the Hartree-Fock wave function a linear combination of determinants that play the role of atomic orbitals. This is similar to constructing a molecular orbital as a linear combination of atomic orbitals. Like the LCAO approximation. Cl calculations determine the weighting of each determinant to produce the lowest energy ground state (see SCFTechnique on page 43). 

Most of the commonly used electronic-structure methods are based upon Hartree-Fock theory, with electron correlation sometimes included in various ways (Slater, 1974). Typically one begins with a many-electron wave function comprised of one or several Slater determinants and takes the one-electron wave functions to be molecular orbitals (MO s) in the form of linear combinations of atomic orbitals (LCAO s) (An alternative approach, the generalized valence-bond method (see, for example, Schultz and Messmer, 1986), has been used in a few cases but has not been widely applied to defect problems.)... 

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,... 

Hiickel MO calculations of the 7r-electron density for pyrazolo[3,4-d]-pyrimidine 290 reveal N-3 to be the most electron rich (69CJC1129). The same conclusion was reached with simple linear combination of atomic orbitals (LCAO) calculations (291). LCAO data for electron densities on pyrazolo[4,3-LCAO calculations exaggerate electronegativities of nitrogen atoms (see 293)... 

Called by Mulliken the LCAO ( linear combination of atomic orbitals ) form. For a critique of this type of approximation see R. S. Mulliken, J. Chem. Phys. 3, 375 (1935). It appears to have been first suggested by Len-nard-Jones. 

We will now take into account the hypothetical linear molecule, Li3. The valence electron cloud is spherical then, in the course of the linear combination of atomic orbitals, the three atomic valence electron clouds overlap to form one continuous distribution, and two distributions with nodes, that is, three MOs (see Figure 1.14). While the length of the chain is augmented, the number of electronic states, into which the atomic 2s state splits during the linear combination of atomic orbitals, increases. In this regard, the number of states equals the number of atoms. 

Many of the principles and techniques for calculations on atoms, described in section 6.2 of this chapter, can be applied to molecules. In atoms the electronic wave function was written as a determinant of one-electron atomic orbitals which contain the electrons these atomic orbitals could be represented by a range of different analytical expressions. We showed how the Hartree-Fock self-consistent-field methods could be applied to calculate the single determinantal best energy, and how configuration interaction calculations of the mixing of different determinantal wave functions could be performed to calculate the correlation energy. We will now see that these technques can be applied to the calculation of molecular wave functions, the atomic orbitals of section 6.2 being replaced by one-electron molecular orbitals, constructed as linear combinations of atomic orbitals (l.c.a.o. method). 

The discussion of Fig. 2-3 fits well with the LCAO description but the degree to which a solid is covalent or metallic is independent of which basis slates are used in the calculation. Most of the analy.sis of covalent solids (hat will be made here will be based upon linear combinations of atomic orbitals, but we also wish to understand them in terms of free-cleclron-like behavior. (These two extreme approaches are illustrated for cesium chloride in Fig. 2-2.) Frcc-electron-like behavior is treated in Chapter 18, where two physical parameters will be designated, one of which dominates in the covalent solid and one of which dominates in the metallic solid. It can be useful here to see how these parameters correspond to the concepts discussed so far. 

Orbitals) (CO) are expressed as a linear combination of atomic orbitals (LCAO) or similar local functions. There is an obvious parallelism with the traditional Molecular Orbital Theory see Molecular Orbital Theory) developed by chemists, in which the wavefimctions describing the motion of an electron in the molecule (i.e. the Molecular Orbitals) (MO) are also expressed as a LCAOs. As a matter of fact, the first orbital describing the motion of an electron in a polyatomic system was written by Bloch in 1928 and it was a CO. Hence, MO theory finds its roots in solid-state physics. 

Molecular Orbital Description. The molecular orbital description of the H bond has received surprisingly little attention. Pimentel considered HF2 qualitatively with molecular orbitals composed of linear combinations of atomic orbitals (1634). (See also 1529.) Using only fluorine p orbitals directed along the bond pA and Pb) and the hydrogen atom Is orbital (j), three molecular orbitals result. These are shown in the second column of Fig. 8-4. Since the H bond involves four electrons,... 

If H denotes the total Hamiltonian of the H2 molecule, CF found A as a function of the internuclear distance R by minimizing < H > with respect to the wave function (24). Their findings were (a) for R <. 6Requiiibrium, A = 1 (b) for R > 1.6 Requilibriums A falls quite rapidly to zero as the H2 molecule is stretched further. For A = 1, Eq. (24) is the molecular orbital wave function built as a linear combination of atomic orbitals, whereas for R > 1.6 ReqUiiibrium one sees that electrons quickly go back on to their own atoms . 

Also, not being atom centered orbitals, plane wave basis sets do not easily lead to chemical insight on the electronic structure of the system studied it is hard to describe the result of a plane wave calculation in a Linear Combination of Atomic Orbital (LCAO) framework, although it is the basis of many simplified, but qualitative, electronic structure models. We will see later that tools have been designed to construct such chemical insight when using plane wave basis sets. 

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