Atomic 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 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). 

The bonding in the XeF2 molecule can be explained quite simply in terms of a 3-center, 4 electron bond that spans all three atoms in the molecule. The bonding in this molecular orbital description involves the filled 5pz orbital of Xe and the half-filled 2pz orbitals of the two F-atoms. The linear combination of these three atomic orbitals affords one bonding, one non-bonding and one anti-bonding orbital, as depicted below ... 

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. 

In the Pauiing-SIater theory, one desires the central wave functions to possess unilateral directional properties so as to be correlated with one particular attached atom. Hence the P-S central functions must have the same transformation properties as do those of the attached atoms before linear combinations of the latter are taken. Thus the problem of finding the linear combinations of the central orbitals which exhibit the proper directional properties is simply the reverse of finding the proper linear combinations of the attached orbitals in the MuIIiken procedure. The difference is only that in the P-S theory, the linear combinations are in the central rather than attached portion, and their construction corresponds to transformation from an irreducible representation to a... 

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... 

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...

Atomic orbitals can combine and overlap to give more complex standing waves. We can add and subtract their wave functions to give the wave functions of new orbitals. This process is called the linear combination of atomic orbitals (LCAO). The number of new orbitals generated always equals the number of starting orbitals. 

The n system is made up from the three pz orbitals on the carbon atoms. The linear combination of these orbitals takes the form of Equation 1.9, with three terms, creating a pattern of three molecular orbitals, tpi, ip2 and 3. In the allyl cation there are two electrons left to go into the n system after filling the a framework (and in the radical, three, and in the anion, four). 

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... 

Recall from Section 2.11(a) Polyatomic molecular orbitals that molecular orbitals are formed as linear combinations of atomic orbitals of the same symmetry and similar in energy. We find first a linear combination of atomic orbitals on peripheral atoms (in this case H) and the coefficient c, and then combine the combinations of appropriate symmetry with atomic orbitals on the central atom. The linear combinations of atomic orbitals of peripheral atoms (also called symmetry-adapted linear combinations—SALC) are given after Table RS5.1 in the Resource Section 5. Thus, we have to find linear combinations in Djh point group that belong to symmetry classes Al , Al", and E. We also have to keep in mind that H atom has only one s orbital. 

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. 

What we have just described has its counterpart in a mathematical treatment called the LCAO (linear combination of atomic orbitals) method. In the LCAO treatment, wave functions for the atomic orbitals are combined in a linear fashion (by addition or subtraction) in order to obtain new wave functions for the molecular orbitals. 

As described previously for the hydrogen molecule, molecular orbital (MO) theory takes the atomic orbitals of the atoms, and mathematically combines the wave functions that represent these atomic orbitals (using an approach known as the linear combination of atomic orbitals). This combination produces new molecular orbitals that describe the regions of space occupied by the bonding electrons. The number of new molecular orbitals formed is the same as the number of atomic orbitals combined. The wave functions that represent the new molecular orbitals can be used to calculate the energy of an electron in those molecular orbitals. 

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. 

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