Orbital connections

Fig. 4a Preferred configuration of electron spins in the a orbital connecting a hydrogen atom to an sp2-hybridized carbon atom bearing unpaired 7t spin density, b Molecular n orbital consisting of two carbon pz orbitals and an H2 group orbital generated by hyperconjugative interaction of an, fp2-hybridized C atom bearing unpaired spin with a CH2—R group...

C. A bonding -orbital connects two atoms in a separate region from the straight line between them. 

For zirconia without anion impurity, orbital connected with hydrogen is absent in the forbidden gap and appear in the conduction band (Figs. 2-6 on the left side). Thus, hydrogen on surface of zirconia must be shallow donor (i.e. with one donor electron on the each hydrogen). 

For zirconia with hydroxyl group of bridge type and anion boron, orbital connected with boron appear in the forbidden gap. Thus, zirconia with anion boron and hydrogen on surface can be acceptor or deep donor. 

In this product, only the terminal projection of GT(a>) is needed. For example, if the left terminal has l and the right terminal has r orbitals connecting the bridge with the electrodes, Gr(co) has the rank n = l + r. Consequently, the electron propagator will have the same rank ... 

We consider a multireference Cl (MRCI) wave function calculated from a set of MCSCF orbitals. The Cl reference state is denoted by CI> at X0 and by CI(g)> at the displaced geometry X0 + g. In addition to the reorthonormalization part, the MRCI orbital connection contains the MCSCF orbital... 

The electronic gradient /(1) (which has no orbital part) has the same structure as the configuration part of the MCSCF electronic gradient [Eq. (84)] and may be constructed in the configuration basis, requiring /, SU), /<0>), and k(1 I(0) in the MO basis. (The k(1),/(0> integrals are needed since the orbital connection includes the MCSCF orbital reoptimization effects.)... 

Since HF orbitals are used, the orbital connection and thus H(n) contain reorthonormalization as well as reoptimization contributions. The HF orbital response parameters are obtained by solving Eqs. (64)-(66). 

Ignoring the field dependence of the dipole operator (MCSCF orbital connections are field independent) we write... 

The other situation where long-range coupling can be expected is when the orbitals connecting the two coupling nuclei (X and Y) are forced by a rigid molecular architecture to adopt a W relationship ... 

The degree of freedom (q,p) has a resonant term Vcosq. There exist unstable fixed points ( = 7i,p = 0), and the separatrix orbits connecting them. The separatrix orbits of the nonlinear resonance are given by the following ... 

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