Linear-combination-of-atomic-orbitals LCAO approach

However, solving such an equation for a solid is something of a tall order because exact solutions have not yet been found for small molecules and even a small crystal could well contain of the order of lO atoms. An approximation often used for smaller molecules is that combining atomic wave functions can form the molecular wave functions. This linear combination of atomic orbitals (LCAO) approach can also be applied to solids. 

Linear-Combination-of-Atomic-Orbitals (LCAO) Approach used in MO-Theory... 

A compromise between computational effort and accuracy can be readied within an approximate KS-DFT scheme - a density functional based tight-binding (DFTB) method. This approximation assumes very strong atomic potential so that different atomic orbitals barely overlap. It can be considered the analog of the standard linear combination of atomic orbital (LCAO) approach with an LCAO representation of the KS orbitals. Like the LCAO approach, the atomic locations can be specified arbitrarily, so the method can be applied to nonperiodic (noncrystalline) materials. The fundamental step of the DFTB... 

The ansatz Eq. (2.41) is the Linear Combination of Atomic Orbitals (LCAO) approach. The coefficients a are the parameters of variation when minimizing the functional E[ ]. The set of functions belonging to a single atom 0 I j 1,..., rik) is called atomic basis set. 

These concepts are used in the simpler linear combination of atomic orbitals (LCAO) approach. We construct a molecular orbital by starting with the wavefunctions of isolated atoms and take a linear combination of these wavefunctions to describe the state of the electron in the molecule. Consider, for example, a diatomic molecule consisting of atoms A and B. For sufficiently large distance between them, the molecular wavefunction can be written as i/f+ = i/ a + [Pg.40]

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.)... 

In the present approach, the KS orbitals are expanded in a set of functions related to atomic orbitals (Linear Combination of Atomic Orbitals, LCAO). These functions usually are optimized in atomic calculations. In our implementation a basis set of contracted Gaussians VF/ is used. The basis set is in general a truncated (finite) basis set reasonably selected . 

The MO theory differs greatly from the VB approach and the basic MO theory is an extension of the atomic structure theory to molecular regime. MOs are delocalized over the nuclear framework and have led to equations, which are computationally tractable. At the heart of the MO approach lies the linear combination of atomic orbitals (LCAO) formahsm... 

In the MO approach molecular orbitals are expressed as a linear combination of atomic orbitals (LCAO) atomic orbitals (AO), in return, are determined from the approximate numerical solution of the electronic Schrodinger equation for each of the parent atoms in the molecule. This is the reason why hydrogen-atom-like wavefunctions continue to be so important in quantum mechanics. Mathematically, MO-LCAO means that the wave-functions of the molecule containing N atoms can be expressed as... 

Band Theory of Metals, Three approaches predict the electronic band structure of metals. The first approach (Kronig-Penney), the periodic potential method, starts with free electrons and then considers nearly bound electrons. The second (Ziman) takes into account Bragg reflection as a strong disturbance in the propagation of electrons. The third approach (Feynman) starts with completely bound electrons to atoms and then considers a linear combination of atomic orbitals (LCAOs). 

Molecular orbitals are characterized by energies and amplitudes expressing the distribution of electron density over the nuclear framework (1-3). In the linear combination of atomic orbital (LCAO) approximation, the latter are expressed in terms of AO coefficients which in turn can be processed using the Mulliken approach into atomic and overlap populations. These in turn are related to relative charge distribution and atom-atom bonding interactions. Although in principle all occupied MOs are required to describe an observable molecular property, in fact certain aspects of structure and reactivity correlate rather well with the nature of selected filled and unfilled MOs. In particular, the properties of the highest occupied MO (HOMO) and lowest unoccupied MO (LUMO) permit the rationalization of trends in structural and reaction properties (28). A qualitative predictor of stability or, alternatively, a predictor of electron... 

So far the discussion has concerned only the shapes and energies of atomic orbitals (AOs). Organic chemists really need to look at the orbitals for whole molecules. One way to construct such molecular orbitals (MOs) is to combine the atomic orbitals of the atoms that make up the molecule. This approach is known as the Linear Combination of Atomic Orbitals (LCAO). 

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... 

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). 

The procedure is called the linear combination of atomic orbitals (LCAO) approximation and can be used for molecules of any size. H2+ is a special case in that a wavefimction can be found that will solve the Schrodinger equation exactly, yet the MO approach will be used so that molecular orbitals can be derived. The simplest trial function for the H2+ system is written ... 

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