Hybridization matrix, hybrid orbitals

It is usually of interest to express the atomic orbitals as functions of the hybrid orbitals. As the transformation matrix is orthogonal, its inverse is... 

By analyzing the density matrix composition of planar and 3D structures of seven atom clusters (II and IV of Fig 1), calculated using scalar relativistic pseudo-potential at the GGA theory level, Fernandez and coworkers conclude that the planarity of An clusters is driven by the hybridization of the half-filled 6s orbital with the fully occupied 5d 2 orbital, which is favored by relativistic effects. Thus, the three valence electrons in the orbitals 6s and 5d 2, form a sticky-waist cylinder , where the cylinder is due to the almost filled s + d 2 hybrid, and the sticky-waist is due to the nearly half-filled s — d 2 hybrid orbital. 

The form of P may be greatly simplified since the matrix element (19) is roughly equal to the matrix element of those orbitals within the two interacting configurations that differ in one-electron occupancy (Salem et al., 1975 Epiotis, 1978 Epiotis and Shaik, 1978a). In the present case the relevant orbitals are hybrid orbitals on R and X respectively. Thus the new relation for p becomes (20). Since this term is strongly dependent on the... 

We saw in 11-3 that a set of equivalent tetrahedral hybrid orbitals could also be constructed from a set of s, d, d s, and d, orbitals. Mathematical expressions for these hybrids (sd1) can be obtained from eqns (11-5.14) by changing p, to d, p to d., and pf to d, . That these are the correct changes can be deduced from the fact that the operators Oc>, 0Cjb, and Oc acting on the column matrix... 

Recalling from Section 3.4.1, we use the s orbital and the pz orbital to form two equivalent hybrid orbitals, one pointing in the +z direction and the other in the —z direction. These two orbitals are called sp hybrids, since they are formed by one s and one p orbital. The wavefunctions of the sp hybrid orbitals are given by eqs. (3.4.6) and (3.4.7). In matrix form the wavefunction are... 

As introduced in Chapter 3, for AX systems, the hybrid orbitals are the linear combinations of atomic orbitals on central atom A that point toward the X atoms. In addition, the construction of the sp" hybrids was demonstrated. In this section, we will consider hybrids that have d-orbital contributions, as well as the relationship between the hybrid orbital coefficient matrix and that of the molecular orbitals, all from the viewpoint of group theory. 

The third step of the NBO algorithm is the calculation of the natural hybrid orbitals (NHOs), which form the two-center (sometimes three-center) NBOs. The density matrix in the NAO basis is first partitioned into one-center and two-center subblocks. Each one-center block is searched for NAOs, which have an occupancy >1.90e. These one-center orbitals... 

In the case of hydrocarbons, the calculation still comprises five parameters, namely two Coulomb integrals ac and h, two bonds, resonance integrals /Sec and /Sqh and one resonance integral for two orbitals centered on the same carbon Ice- As usual, the interaction terms between non-bonded atoms are neglected. The parameters a and /3 are the matrix elements of a non-specified effective Hamiltonian with respect to the sp3 or sp2 hybrid orbitals of carbon and the Is orbitals of hydrogens. For the a bonds of conjugated hydrocarbons 4S>, the following set of values has been used... 

The (assumed localised) orbitals are assumed to be constructed by pairwise addition of pairs of hybrid orbitals the e matrix can be transformed to a tri-diagonal form to reflect this model of the electronic structure. 

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