Energy difference gradient vector

Section III then introduces the various approximate energy expressions that are used to determine the wavefunction corrections within each iteration of the MCSCF optimization procedure. Although many of these approximate energy expressions are defined in terms of the same set of intermediate quantities (i.e. the gradient vector and Hessian matrix elements), these expressions have some important formal differences. These formal differences result in MCSCF methods that have qualitatively different convergence characteristics. 

If the electric field gradient (EFG) is non-zero, for instance due to a non-cubic valence electron distribution and/or non-cubic lattice site symmetry, electric quadrupole interaction as visualized by the precession of the quadrupole moment vector about the field gradient axis sets in and splits the degenerate I = 3/2 level into two substates with magnetic spin quantum numbers mj = 3/2 and 1/2 (Fig. 2.4). The energy difference between the two substates A q is observed in the spectrum as the separation between the two resonance fines. These two resonance lines in the spectrum refer to the two transitions between the two substates of the... 

Two approaches to this equation have been employed. (/) The scalar product is formed between the differential vector equation of motion and the vector velocity and the resulting equation is integrated (1). This is the most rigorous approach and for laminar flow yields an expHcit equation for AF in terms of the velocity gradients within the system. (2) The overall energy balance is manipulated by asserting that the local irreversible dissipation of energy is measured by the difference ... 

Equations 9.3a and 9.3b give the energy of the upper and lower part of the cone (f/ and f/fi). In Eqs 9.3a and 9.3b, the first term represents Q in Eq. 9.2, while the expression under the square root sign corresponds to Tin Eq. 9.2. is the reference energy at the apex of the cone. The remaining qnantities in these two equations are energy derivatives. The quantity in Eq. 9.3g is the gradient difference vector, while the qnantity in Eq. 9.3h... 

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