Orbitals, angular dependence

Orbital Angular dependence function Orbital Angular dependence function... 

To incorporate the angular dependence of a basis function into Gaussian orbitals, either spherical haimonics or integer powers of the Cartesian coordinates have to be included. We shall discuss the latter case, in which a primitive basis function takes the form... 

Figure A1.2 Models schematically illustrating the angular dependence functions A ,(6, 0). There is no unique way of representing the angular dependence functions of all seven f orbitals. An alternative to the set shown is one f 3, three f i, fy i, and three fx(i2 3j,2), fj.(j,2 3x2), and fj(x2 3x2).

The dipole moment m depends on the total angular momentum, which may be written in terms of the orbital angular moment operator L and the total eleetron spin S. 

The dependence of the j3-deuterium effect on the spatial orientation of the isotopic bond with respect to the developing -orbital, on the a-carbon atom was elegantly demonstrated by Shiner and Humphrey (1963). This work will not be discussed in detail here suffice it to say the suggestion is made that j8-deuterium effects are better correlated by the postulate of hyperconjugation and its angular dependence than by the simple steric model (Shiner and Humphrey, 1963). 

Three-dimensional representations of the angular dependence of some of the atomic orbitals are shown in Fig. 1. The coordinate axes and angles are defined in Fig. 6-5. 

The g-value of a free electron is a scalar, ge = 2.00232. In a radical species, g becomes a matrix because of the admixture of orbital angular momentum into S through spin-orbit coupling. The components of the g-matrix thus differ from ge to the extent that p-, d-, or f-orbital character has been incorporated, and they differ from one another, depending on which p-, d-, or f-orbitals are involved. 

For diatomic molecules, there is coupling of spin and orbital angular momenta by a coupling scheme that is similar to the Russell-Saunders procedure described for atoms. When the electrons are in a specific molecular orbital, they have the same orbital angular momentum as designated by the m value. As in the case of atoms, the m value depends on the type of orbital. When the internuclear axis is the z-axis, the orbitals that form a bonds (which are symmetric around the internuclear axis) are the s, pz, and dzi orbitals. Those which form 7r bonds are the px, p, dlz, and dyi orbitals. The cip-y2 an(i dxy can overlap in a "sideways" fashion with one stacked above the other, and the bond would be a 8 bond. For these types of molecular orbitals, the corresponding m values are... 

The solutions of the angular dependent part are the spherical harmonics, Y, known to most chemists as the mathematical expressions describing shapes of (hydrogenic) atomic orbitals. It is noted that Y is defined only in terms of a central field and not for atoms in molecules. 

Important aspects of the interaction of strong laser fields with molecules can be missed in standard TOF experiments, most notably the population of electronically excited states. However, by studying vibrational excitation, the frequency and dephasing of the vibrational motion can be used to identify the electronic state undergoing the vibrational motion. In some cases, this turns out to be a ground state, and in others, an excited state. Once we have identified an excited state, we are left with the question of how and why the state was populated by the strong field. In one example above (the Ij A state discussed in Sect. 1.3.3), the excited state is formed by the removal of an inner orbital electron, in this case a iru electron. This correlates with the measured angular dependence for the ionization to this state. 

It is well known that the angular dependence of each Cartesian p orbital is simply a unit vector in the corresponding Cartesian direction (px = ux, etc.). We can therefore associate each p, with a Cartesian direction vector dh... 

Kettle and his co-workers (39—42) used a model rather similar to the AOM to discuss stereochemistry. A perturbation approach led to the proportionality of MO energies (relative to the unperturbed orbitals) to squared overlap integrals, as in the AOM. For systems where the valence shell orbitals are evenly occupied, the total stabilization energy shows no angular dependence, suggesting that steric forces determine the equilibrium geometry. 

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