Orbital linear combination

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 present article is an attempt to review those studies of pyridinelike heterocycles (mono-azines) and, to a lesser extent, their analogues and derivatives that have interpreted the behavior and estimated various physico-chemical properties of the compounds by the use of data calculated by the simplest version of the MO LCAO (molecular orbital, linear combination of atomic orbitals) method (both molecular orbital energies and expansion coefficients). In this review, attention is focused upon the use of the simple method because it has been applied to quite extensive sets of compounds and to the calculation of the most diverse properties. On the other hand, many fewer compounds and physico-chemical properties have been investigated by the more sophisticated methods. Such studies are referred to without being discussed in detail. In a couple of years, we believe, the extent of the applications of such methods will also be wide enough to warrant a detailed review. 

The energies of the (/-orbitals for the system , , i = 1-5, are then obtained by diagonalization of the real symmetrical matrix Zfy, i = dzi...dy2. The real (/-orbital linear combinations which correspond to these energies are then obtained by substituting the solutions, into the sets of simultaneous equations derived from the secular determinant. 

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). 

A very simple procedure was proposed by Del Re 87 for representing the <7 bonds in saturated compounds. Its essential features are as follows, (a) Treat each bond as a two-electron problem, (b) Describe each electron in the bond by a molecular orbital linear combination of... 

The notation concerns are easily overcome by the following simple construct bearing the name of second quantization formalism.21 Let us consider the space of wave functions of all possible numbers of electrons and complement it by a wave function of no electrons and call the latter the vacuum state vac). This is obviously the direct sum of subspaces each corresponding to a specific number of electrons. It is called the Fock space. The Slater determinants eq. (1.137) entering the expansion eq. (1.138) of the exact wave function are uniquely characterized by subsets of spin-orbitals K = k,, k2,..., fc/v which are occupied (filled) in each given Slater determinant. The states in the list are the vectors in the carrier space of spin-orbitals (linear combinations of the functions of the (pk (x) = ma (r, s) basis. We can think about the linear combinations of all Slater determinants, may be of different numbers of electrons, as elements of the Fock space spanned by the basis states including the vacuum one. 

Up to this point nothing changes. The next assumption extends the above treatment of atoms to molecules. Within it the molecular orbitals - linear combinations of the atomic core orbitals with zero overlap - are taken to be the molecular core orbitals and are assumed to be filled. This allows one to write... 

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

In order to solve the Kohn-Sham equations (Eqn. (2)) we used the molecular orbital-linear combination of atomic orbitals (MO-LCAO) approach. The molecular wave functions 0j are expanded the symmetry adapted orbitals Xj) which are also expanded in terms of the atomic orbitals... 

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. 

In order to further describe the molecular wavefunctions or the molecular orbitals. Linear Combinations of Atomic Orbitals (LCAO) are normally used (LCAO method). Such a method of solution is possible since the directional dependence of the spherical-harmonic functions for the atomic orbitals can be used. The Pauli principle can be applied to the single-electron molecular orbitals and by filling the states with the available electrons the molecular electron configurations are attained. Coupling of the angular momenta of the open shell then gives rise to molecular terms. 

SCF-MO- Self-consistent field-molecular orbital-linear combination of atomic LCAO orbitals... 

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