Steps in Energy Minimization

It is useful to describe the generation of a new density matrix from the previous matrix by unitary transformation  

Every unitary matrix U can be represented by an exponential function of an anti-Hermitian matrix 

In this notation, the exponential parametrization of the new density matrix becomes 

The density matrix (and thus the Flartree-Fock energy) can now be seen as functions of the parameter A and the requirement for an energy minimum becomes 

If one is not yet at a minimum, one can for example use the method of steepest descent to arrive at the optimum density matrix. Inserting the explicit dependence of Py on Al , e.g., in the form of the Taylor expansion of Eq. [124] around A]] = 0, one obtains for the direction of steepest descent  

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Angle bending is the next step in energy minimization. It occurs when strain release by rotation of substituents is not enough to accommodate all the steric strain or when rotation is not possible, as with spherical substituents. Quantitative results are scarce an interesting example is the methylation of a series of 3,5-dialkylimidazo(l,2-a)pyridines (81T83 81T91) (Scheme 114). 

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