Hybrid electron functions

In order to derive the tetrahedral hybrid electron functions we shall follow the method of Pauling and show that the linear combination of one s and three p wave functions give maximum stability of the bond. The linear combination has the form  

We shall attempt to determine the values of the coefficients b, r, and d which permit the greatest possible overlap of the resulting orbitals with the orbital of another atom. Let us suppose that the bond is formed in the direction of the z axis, so that the linear combination of the orbitals may be limited to 

The wave functions js and ijjp arc also normalized and mutually orthogonal  

We are concerned here only with the angular dependence of the wave function and we may assume that the radial parts of the wave functions are so closely similar that their differences can be neglected. If we then regard 1/2 Vtt as part of the normalization constant and put/(r) = 1 we obtain  

For the particular case of the direction along the z axis i.e. when 6 = 0) 

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The electronic structure calculations were carried out using the hybrid density functional method B3LYP [15] as implemented in the GAUSSIAN-94 package [16], in conjunction with the Stevens-Basch-Krauss (SBK) [17] effective core potential (ECP) (a relativistic ECP for Zr atom) and the standard 4-31G, CEP-31 and (8s8p6d/4s4p3d) basis sets for the H, (C, P and N), and Zr atoms, respectively. 

B3LYP Model. A Hybrid Density Functional Model which improves on the Local Density Model by accounting explicitly for non-uniformity in electron distributions, and which also incorporates the Exchange Energy from the Hartree-Fock Model. The B3LYP model involves three adjustable parameters. 

All electron calculations were carried out with the DFT program suite Turbomole (152,153). The clusters were treated as open-shell systems in the unrestricted Kohn-Sham framework. For the calculations we used the Becke-Perdew exchange-correlation functional dubbed BP86 (154,155) and the hybrid B3LYP functional (156,157). For BP86 we invoked the resolution-of-the-iden-tity (RI) approximation as implemented in Turbomole. For all atoms included in our models we employed Ahlrichs valence triple-C TZVP basis set with polarization functions on all atoms (158). If not noted otherwise, initial guess orbitals were obtained by extended Hiickel theory. Local spin analyses were performed with our local Turbomole version, where either Lowdin (131) or Mulliken (132) pseudo-projection operators were employed. Broken-symmetry determinants were obtained with our restrained optimization tool (136). Pictures of molecular structures were created with Pymol (159). 

Note that, because electron correlation often stabilizes delocalized electronic structures over localized ones, HF theory tends to be inaccurate for such systems in the opposite direction from DFT, and thus, again, hybrid ACM functionals tend to show improved performance by an offsetting of errors. 

There is no unequivocal answer to the question as to which is the better method. Calculations by the VB method are likely to be more reliable than those by the MO method, but in practice are much more difficult to carry out. For many-electron molecules the MO procedure is simpler to visualize because we combine atomic orbitals into molecular orbitals and then populate the lower-energy orbitals with electrons. In the VB method, atomic orbitals are occupied, but the electrons of different atoms are paired to form bonds, a process that requires explicit consideration of many-electron wave functions. To put it another way, it is easier to visualize a system of molecular orbitals containing N electrons than it is to visualize a hybrid wave function of N electrons. 

Towards fabrication of SWNT-based molecular electronic devices, two methods have been used to assemble the 03-SWNTs on functionalized SAMs of OPEs, as shown in Figure 5.10. The first, termed chemical assembly , is based on a condensation reaction between the carboxylic acid functionalities of O3-SWNTs and the amine functionalities of SAMs to form amides. The results show that O3 -SWNTs coat the amino-terminated SAM with a high degree of surface coverage. The second method is based on physical adsorption via layer-by-layer (LBL) deposition with bridging of metal cations, i.e., Fe3+ on carboxylate terminated SAMs or Cu2+ on thiol-terminated SAMs. The oxidatively shortened 03 -SWNTs are shown to be perpendicular to the surface with random adsorption of longer tubes. The patterned nanotube assemblies may be useful in hybridized electronic devices, where device functions can be modified by the orientation and stacking of SWNTs, and the properties of the SAM. 

In order to overcome the limitations of currently available empirical force field param-eterizations, we performed Car-Parrinello (CP) Molecular Dynamic simulations [36]. In the framework of DFT, the Car-Parrinello method is well recognized as a powerful tool to investigate the dynamical behaviour of chemical systems. This method is based on an extended Lagrangian MD scheme, where the potential energy surface is evaluated at the DFT level and both the electronic and nuclear degrees of freedom are propagated as dynamical variables. Moreover, the implementation of such MD scheme with localized basis sets for expanding the electronic wavefunctions has provided the chance to perform effective and reliable simulations of liquid systems with more accurate hybrid density functionals and nonperiodic boundary conditions [37]. Here we present the results of the CPMD/QM/PCM approach for the three nitroxide derivatives sketched above details on computational parameters can be found in specific papers [13]. 

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