Analysis of Search Direction Methods

The Hessian matrix G is real and symmetric. It can then be resolved for analysis in terms of its eigenvalues and its eigenvectors w as 

When cos 0 approaches zero, the step and the gradient become 

In terms of the normal mode analysis the Taylor series expansion for the energy can be rewritten as 

then there will be no displacements in these directions whether is zero or not. Since the energy is invariant to translations and rotations, the center of coordinates and moments about these coordinates will be conserved. In the former case it can be shown that X = 0 in the case of rotation the corresponding X may not equal zero except at the extreme point, but the corresponding e. = 0. Since the energy is also invariant to symmetry operations that leave the Hamiltonian invariant, the gradient vector will have nonvanishing components only along totally symmetric modes, A (see [43] for a more detailed proof). 

Since motion along totally symmetric modes cannot change the point group symmetry, this symmetry is preserved throughout the optimization. The exception is at the extreme points themselves, where a higher symmetry might be obtained. 

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