Linear combination LCAO

LCAO (linear combination of atomic orbitals) refers to construction of a wave function from atomic basis functions LDA (local density approximation) approximation used in some of the more approximate DFT methods... 

Each MO is expanded in terms of the basis functions, conventionally called atomic orbitals (MO=LCAO, Linear Combination of Atomic Orbitals), although they are generally not solutions to the atomic HF problem. 

The raw output of a molecular structure calculation is a list of the coefficients of the atomic orbitals in each LCAO (linear combination of atomic orbitals) molecular orbital and the energies of the orbitals. The software commonly calculates dipole moments too. Various graphical representations are used to simplify the interpretation of the coefficients. Thus, a typical graphical representation of a molecular orbital uses stylized shapes (spheres for s-orbitals, for instance) to represent the basis set and then scales their size to indicate the value of the coefficient in the LCAO. Different signs of the wavefunctions are typically represented by different colors. The total electron density at any point (the sum of the squares of the occupied wavefunctions evaluated at that point) is commonly represented by an isodensity surface, a surface of constant total electron density. 

The Hartree-Fock orbitals are expanded in an infinite series of known basis functions. For instance, in diatomic molecules, certain two-center functions of elliptic coordinates are employed. In practice, a limited number of appropriate atomic orbitals (AO) is adopted as the basis. Such an approach has been developed by Roothaan 10>. In this case the Hartree-Fock differential equations are replaced by a set of nonlinear simultaneous equations in which the limited number of AO coefficients in the linear combinations are unknown variables. The orbital energies and the AO coefficients are obtained by solving the Fock-Roothaan secular equations by an iterative method. This is the procedure of the Roothaan LCAO (linear-combination-of-atomic-orbitals) SCF (self-consistent-field) method. 

When the Hartree-Fock method is applied to molecules, molecular orbitals are used instead of atomic orbitals. To construct the molecular orbitals, one widely used approximation is LCAO (linear combinations of atomic orbitals). According to molecular orbital theory, the total wave function of the system is written as a combination of molecular orbitals, spin functions describing electrons in terms of spin j(a) or — j p). 

Mulliken s population analysis is rooted in the LCAO (linear combination of atomic orbitals) formulation it is not directly applicable to other types of wavefunctions. With Cr i representing the coefficient of the rth type of atomic orbital (li, 2s, etc.) of atom k in the ith molecular orbital, we describe the latter by... 

The electronic states of a diatomic molecule such as H2 are approximately equal to linear combinations of atomic orbitals. For example, the groimd state is approximately proportional to the sum of Is orbitals from the two atoms IsA + Isb- An excited state is approximately proportional to the difference IsA — Isb- Although these LCAO (Linear Combination of Atomic Orbitals) wave functions are not quantitatively correct representations of the true wave functions, their shape, and hence their symmetry, is correct. 

In the simple version of the MO LCAO (linear combination of atomic orbitals) theory, the quantities Hjk and 8jk, defined by Eqs. (3-5), are treated as empirical parameters, thus there is no need for detailed information concerning the character of the atomic functions [Pg.3]

For diatomic (and linear polyatomic) molecules, the basis set usually consists of several STOs centered on each atom. Thus the MOs are expressed as LCAOs—linear combinations of atomic orbitals. In a mini- ia/-basis-set calculation, only inner-shell and valence-shell STOs are used. Thus a minimal basis calculation of HF would use as basis functions l.sH, 1 F, 2jf, 2/ ctf, 2p7TF, 2pTTF, where 2poF is a fluorine 2p AO along the internuclear (z) axis (i.e., a 2pz AO), and the 2pir and 2pir AOs are... 

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. 

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

This is the reason covalent bonding is referred sometimes as the product of LCAO (Linear Combination of Atomic Orbital). Since each covalent bond uses two and only two electrons, any excess (more than the number of electrons required for covalent bond) valence electrons will automatically become free electrons. This is because their atomic... 

The main idea of the method is to represent the wave function of a particle as a linear combination of some known localized states ipa(r, a), where a denote the set of quantum numbers, and a is the spin index (for example, atomic orbitals, in this particular case the method is called LCAO - linear combination of atomic orbitals)... 

LCAO (Linear Combination of Atomic Orbitals) It is simple and qualitative approximation which can explain the formation of molecular orbitals by combination of atomic orbitals. 

Ditchfield, R. Self-consistent perturbation theory of diamagnetism. I. A gage-invariant LCAO(linear combination of atomic orbitals) method for NMR chemical shifts, Mol. Phys. 1974, 27, 789-807. 

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