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Houben-Weyl Methoden der organischen Chemie vol.E17e by Wiberg K.B., et al. (eds.)

By Wiberg K.B., et al. (eds.)

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The molecular states are now characterized by the quantum number R for the absolute value of the total angular momentum along R. This coupling scheme is often referred to as Hund's case c c o ~ p l i n g . ~ ~ * ~ ~ c. 77cFor regions of space where the interaction energy and the fine structure atomic splitting are approximately equal, a transition region between (A,S) and (J,,Jb) couplings may exist. Here, instead of successively diagonalizing perturbations of different sizes, as in (a) and (b), essentially one must simultaneously diagonalize the perturbations (H'-Ho).

Also, (Q"),, and ( Q ~ ) , . , , are the expectation values of the multipole moments of the isolated atoms. "' Thus, Eb')(nonres) corresponds to the interaction energy between the permanent moments of the atoms or molecules. For 2" Fig. 2. Schematic diagram showing how hybridization of orbitals with different parity can produce a dipole moment in an atom. Consider a hydrogen 2s orbital, $2s, and a hydrogen 2 p , orbital, $ z p z , both centered on the nucleus n. The hybrid orbital, formed by adding $2s and $2pz.

33 However, if aV/ay is proportional to V , then Eq. (89) can be further simplified with the use of the hierarchy of perturbation '~~ equations of different orders and, in this special case, Y a r i ~ showed that the wave function through the nth order suffices to calculate dE/ay through the (2n I)st order. + B. The Electrostatic Hellmann-Feynman Theorem The electrostatic Hellmann-Feynman theorem' 2 3 9 124 provides a conceptually interesting way of determining the forces on the nuclei of an arbitrary molecular configuration.

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