Atomic orbitals linear combination model

The implementation of such a model mostly depends on the choice of the atomic orbitals. Linear combinations of Slater Type Orbitals arc natural and moreover allow a good description of one-center matrix elements even at large intemuclear distances. However, a complete analytical calculation of the two-center integrals cannot be performed due to the ETF, and time consuming numerical integrations [6, 7] are required (demanding typically 90% of the total CPU time). 

To make the picture of 7T-electrons more intelligible the model of linear combinations of single electron atomic orbitals to molecular orbitals is helpful (Fig. 14). In this model one concentrates only on the outermost electrons or valence orbitals. Starting from the atomic wavefunctions the s, px and py atomic orbitals are combined in the (x,y)-plane to sp and sp2 orbitals. These sp and sp2 orbitals of the different atoms combine to molecular orbitals, building the molecular structure framework in the (x,y)-plane. The electrons in these molecular orbitals are called a-electrons and their wavefunctions are symmetric perpendicular to the (x,y)-plane extending only over two neighboring atoms. 

Fig. 3 Gouterman s four-orbital linear combination of atomic orbital model...

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. 

In binary semiconductors having a sphalerite structure (the structure of zinc blende), the valence band in the tight-binding MO model is a combination of sp -hybrid orbitals, linear combinations of the s and three p orbitals on the metal atom that are directed toward the neighboring nonmetal atoms and form bonding combinations with atomic orbitals (s and p) on them. Similarly the empty conduction band is then built up of the antibonding combinations of the sp -hybrids on the metal atoms directed away from the bonded neighbor toward the interstices of the lattice. 

To improve our model we note that s- and /7-orbitals are waves of electron density centered on the nucleus of an atom. We imagine that the four orbitals interfere with one another and produce new patterns where they intersect, like waves in water. Where the wavefunctions are all positive or all negative, the amplitudes are increased by this interference where the wavefunctions have opposite signs, the overall amplitude is reduced and might even be canceled completely. As a result, the interference between the atomic orbitals results in new patterns. These new patterns are called hybrid orbitals. Each of the four hybrid orbitals, designated bn, is formed from a linear combinations of the four atomic orbitals ... 

The bent-bond model can be expressed in orbital terms by assuming that the two components of the double bond are formed from sp3 hybrids on the carbon atoms (Figure 3.19) That this model and the ct-tt model are alternative and approximate, but equivalent, descriptions of the same total electron density distribution can be shown by converting one into the other by taking linear combinations of the orbitals, as shown in Figure 3.20. But neither form of the orbital model can predict the observed deviations from the ideal angles of 109° and 120°. 

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. 

The simple orbital basis expansion method which is used in the implementation of most models of molecular electronic structure consists of expanding each R as a linear combination of determinants of a set of (usually) atom-centred functions of one or two standard forms. In particular most qualitative and semi-quantitative theories restrict the terms in this expansion to consist of the (approximate) occupied atomic orbitals of the constituent atoms of the molecule. There are two types of symmetry constraint implicit in this technique. 

Molecular Orbital Theory Model. Oxygen and hydrogen atoms in H2O are held together by a covalent bond. According to the quantum molecular orbital theory of covalent bonding between atoms, electrons in molecules occupy molecular orbitals that are described, using quantum mechanical language, by a linear combination of... 

Feynman model. The Feynman approach, or LCAO (hnear combination of atomic orbitals) method, assumes that a wavefunction of valence electrons i// in a metal is a linear combination of atomic functions ... 

Basis Functions. Functions usually centered on atoms which are linearly combined to make up the set of Molecular Orbitals. Except for Semi-Empirical Models where basis functions are Slater type, basis functions are Gaussian type. 

Various theoretical methods and approaches have been used to model properties and reactivities of metalloporphyrins. They range from the early use of qualitative molecular orbital diagrams (24,25), linear combination of atomic orbitals to yield molecular orbitals (LCAO-MO) calculations (26-30), molecular mechanics (31,32) and semi-empirical methods (33-35), and self-consistent field method (SCF) calculations (36-43) to the methods commonly used nowadays (molecular dynamic simulations (31,44,45), density functional theory (DFT) (35,46-49), Moller-Plesset perturbation theory ( ) (50-53), configuration interaction (Cl) (35,42,54-56), coupled cluster (CC) (57,58), and CASSCF/CASPT2 (59-63)). 

Molecular-orbital theory treats molecule formation from the separated atoms as arising from the interaction of the separate atomic orbitals to form new orbitals (molecular orbitals) which embrace the complete framework of the molecule. The ground state of the molecule is then one in which the electrons are assigned to the orbitals of lowest energy and are subject to the Pauli exclusion principle. Excited states are obtained by promoting an electron from a filled molecular orbital to an orbital which is normally empty in the ground state. The form of the molecular orbitals depends upon our model of molecule formation, but we shall describe (and use in detail in Sec. IV) only the most common, viz., the linear combination of atomic orbitals approximation. 

The obvious deficiency of crystal-field theory is that it does not properly take into account the effect of the ligand electrons. To do this a molecular-orbital (MO) model is used in which the individual electron orbitals become a linear combination of the atomic orbitals (LCAO) belonging to the various atoms. Before going into the general problem, it is instructive to consider the simple three-electron example in which a metal atom with one ligand atom whose orbital contains two electrons. Two MO s are formed from the two atomic orbitals... 

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