In summary, by solving the electronic Schrodinger equation at a variety of geometries and searching for geometries where the gradient vanishes and the Hessian matrix has all positive eigenvalues, one can find stable structures of molecules(and ions). The Schrodinger equation is a necessary aspect of this process because the movement of the electrons is governed by this equation rather than by Newtonian classical equations. The information gained after carrying out such a geometry optimization process include(1) all of the interatomic distances and internal angles needed to specify the equilibrium geometry(Rae) and(2)the total electronic energy E at this particular geometry It is also possible to extract much more information from these calculations. For example, by multiplying elements of the Hessian matrix(oE/OR, aR, by the inverse quare roots of the atomic masses of the atoms labeled a and b, one forms the mass- eighted Hessian(m, m)(aE/aR, OR,)whose non-zero eigenvalues give the harmonic vibrational frequencies(o, of the molecule. The eigenvectors (Rka of the mass wieghted Hessian mantrix give the relative displacements in coordinates r a that accompany vibration in the k normal mode (i.e, they describe the normal mode motions). Details about how these harmonic vibrational frequencies and normal modes are obtained were discussed earlier in the Background Material B. Molecular Change-reactions, isomerization, interactions Changes in bonding 88 In summary, by solving the electronic Schrödinger equation at a variety of geometries and searching for geometries where the gradient vanishes and the Hessian matrix has all positive eigenvalues, one can find stable structures of molecules (and ions). The Schrödinger equation is a necessary aspect of this process because the movement of the electrons is governed by this equation rather than by Newtonian classical equations. The information gained after carrying out such a geometry optimization process include (1) all of the interatomic distances and internal angles needed to specify the equilibrium geometry {Raeq} and (2) the total electronic energy E at this particular geometry. It is also possible to extract much more information from these calculations. For example, by multiplying elements of the Hessian matrix (¶ 2E/¶Ra¶Rb ) by the inverse square roots of the atomic masses of the atoms labeled a and b, one forms the mass￾weighted Hessian (ma mb ) -1/2 (¶ 2E/¶Ra¶Rb ) whose non-zero eigenvalues give the harmonic vibrational frequencies {wk} of the molecule. The eigenvectors {Rk,a} of the mass￾wieghted Hessian mantrix give the relative displacements in coordinates Rka that accompany vibration in the kth normal mode (i.e., they describe the normal mode motions). Details about how these harmonic vibrational frequencies and normal modes are obtained were discussed earlier in the Background Material. B. Molecular Change- reactions, isomerization, interactions 1. Changes in bonding