Linear combination of atomic orbitals LCAO model

Using the same atomic orbital twice in constructing molecular orbitals in the Linear Combination of Atomic Orbitals (LCAO) model. 

A complementary approach is the Tensor Surface Harmonic Theory [19], based on the linear combination of atomic orbitals (LCAO) model, which explicitly incorporates the atomic positions. A set of atomic cores on the surface of a sphere are considered, and a basis set of s atomic orbitals used. If only these s orbitals are used, then the results are identical to the spherical jellium model. The three most stable orbitals are respectively 1,3 and 5 fold degenerate, leading to closed shells at 2, 8 and 18 electrons. 

Hydrogen normally exists as a diatomic molecule, H2, with a covalent bond connecting the two hydrogen atoms. Section 3.3 established that the orbitals of a molecule are different from those of an individual atom. Is it possible to use the hydrogen atom and its atomic orbitals to make predictions about the orbitals in a molecule In one model, the atomic orbitals of the atom are mixed and combining two such atoms may lead to a molecule. This idea of mixing atomic orbitals to form a molecular orbital is called the linear combination of atomic orbital (LCAO) model, and in some cases it helps predict the relative energy of these molecular orbitals. 

In Eq. (2.30), F is the Fock operator and Hcore is the Hamiltonian describing the motion of an electron in the field of the spatially fixed atomic nuclei. The operators and K. are operators that introduce the effects of electrons in the other occupied MOs. Hence, when i = j, J( (= K.) is the potential from the other electron that occupies the same MO, i ff IC is termed the exchange potential and does not have a simple functional form as it describes the effect of wavefunction asymmetry on the correlation of electrons with identical spin. Although simple in form, Eq. (2.29) (which is obtained after relatively complex mathematical analysis) represents a system of differential equations that are impractical to solve for systems of any interest to biochemists. Furthermore, the orbital solutions do not allow a simple association of molecular properties with individual atoms, which is the model most useful to experimental chemists and biochemists. A series of soluble linear equations, however, can be derived by assuming that the MOs can be expressed as a linear combination of atomic orbitals (LCAO)44 ... 

Ab-initio (nonempirical, from first principles ) methods also use the HF-SCF model but includes all electrons and uses minimal approximation. Basis sets of functions based on linear combinations of atomic orbitals (LCAO) increase in complexity from the simplest (STO-3G) to more complex (3-21G( )) to extended basis sets (6-311 + G ) for the most accurate (and most time-consuming) results. Treat systems up to 50 atoms. 

This highly successful qualitative model parallels the most convenient quantum mechanical approach to molecular orbitals the method of linear combination of atomic orbitals (LCAO). We have assumed that the shapes and dispositions of bond orbitals are related, in a simple way to the shapes and dispositions of atomic orbitals. The LCAO method makes the same assumption mathematically to... 

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. 

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]. 

Spin properties are notoriously difficult to calculate accurately57. Here, we are actually calculating spin populations, with their intrinsic uncertainties, and not the directly observed hyperfine interactions. On the other hand, analyses of the hyperfine interactions in the ESR spectra to give experimental atomic orbital occupancies for the radical electron are based on a simplistic, rigid linear combination of atomic orbitals (LCAO)-MO model with the reference electron-nuclear coupling parameters taken from the free atom. No allowance is made for radial or angular polarization of the atomic orbitals in the molecular environment. Thus agreement at these levels between calculated and experimental values can only be qualitative, at best. 

The cluster model of HAp/methyl acetate interface was shown in Fig.2 overlap population analysis was applied to this model. Using Monte Carlo method, 300 sampling points were put around each atom in the cluster. Molecular orbitals in the cluster were constructed by a linear combination of atomic orbitals (LCAO). Atomic orbitals used in this model were ls-2p for C, ls-2p for O, Is for H, ls-3d for P and ls-4p for Ca, which were numerically calculated for atomic Hartree-Fock method. Overlap population was evaluated by Mulliken s population analysis. 

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... 

The Linear Combination of Atomic Orbitals (LCAO) approximation is fundamental to many of our current models of chemistry. Both the vast majority of the calculational programs that we use, be they ab initioy density functional, semiempirical molecular orbital, or even some sophisticated force-fields, and our qualitative understanding of chemistry are based on the concept that the orbitals of a given molecule can be built from the orbitals of the constituent atoms. We feel comfortable with the Ji-HOMO (Highest Occupied Molecular Orbital) of ethylene depicted as a combination of two carbon p-orbitals, as shown in Fig. 2.1, although this is not a very accurate description of the electron density of this Molecular Orbital (MO). The use of the Jt-Atomic Orbitals (AOs), however, makes it easier to understand both the characteristics of the MO itself and the transformations that it can undergo during reactions. 

The tight-binding model is an approach to the electronic band structure from the atomic borderline case. It describes the electronic states starting from the limit of an isolated atom. It is assumed that the Fourier transform of the Bloch function can be approximated by the linear combination of atomic orbitals (LCAO). Thus, the band structure of solids is investigated starting from the Hamiltonian of an isolated atom centered at each lattice site of the crystal lattice. 

In this model, each hydrogen atom has a Is atomic orbital and, as the two atoms approach, molecular orbitals (MOs) are formed as a linear combination of atomic orbitals (LCAO). The coordinate system is shown in Fig. 2.5. The first atomic orbital, is centered on atom 1 at R. The value of 01 at a point in space r is 0i(r) or, since its value depends on the distance from its origin, we sometimes write = 0i(r - Rj). The second atomic orbital is centered on atom 2 at R2, i.e., 02 = 02( 2)- Th exact Is orbital of a hydrogen atom centered at R has the form... 

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