You need to know this material if you wish to understand most of what this text offers,so I urge you to read the background Material if your education to date has not yet d to it Let us now return to the discussion of how theory deals with molecular structure. We assume that we know the energy e(R D at various locations of the nuclei. In some cases, we denote this energy V(R )and in others we use E(R,) because, within the Born Oppenheimer approximation, the electronic energy e serves as the potential v for the molecule's vibrational motions. As discussed in the backgound material. one can then perform a search for the lowest energy structure(. g. by finding where the gradient vector vanishes aE/aR =0 and where the second derivative or Hessian matrix (aE/ar aR,has no negative eigenvalues). By finding such a local-minimum in the energy landscape, theory is able to determine a stable structure of such a molecule. The word stable is used to describe these structures not because they are lower in energy than all other possible arrangements of the atoms but because the curvatures, as given in terms of eigenvalues of the Hessian matrix(OE/OR aR), are positive at this particular geometry. The procedures by which minima on the energy landscape are found may involve simply testing whether the energy decreases or increases as each geometrical coordinate is varied by a small amount. Alternatively, if the gradients aE/aR are known at a particular geometry, one can perform searches directed"downhill"along the negative of the gradient itself. By taking a small"step"along such a direction, one can move to a new geometry that is lower in energy. If not only the gradients aE/aR, but also the second derivatives(OE/aR,aR ) are known at some geometry, one can make a more"intelligent'5 You need to know this material if you wish to understand most of what this text offers, so I urge you to read the Background Material if your education to date has not yet adequately been exposed to it. Let us now return to the discussion of how theory deals with molecular structure. We assume that we know the energy E({Ra}) at various locations {Ra} of the nuclei. In some cases, we denote this energy V(Ra ) and in others we use E(Ra ) because, within the Born￾Oppenheimer approximation, the electronic energy E serves as the potential V for the molecule’s vibrational motions. As discussed in the Backgound Material, one can then perform a search for the lowest energy structure (e.g., by finding where the gradient vector vanishes ¶E/¶Ra = 0 and where the second derivative or Hessian matrix (¶ 2E/¶Ra¶Rb ) has no negative eigenvalues). By finding such a local-minimum in the energy landscape, theory is able to determine a stable structure of such a molecule. The word stable is used to describe these structures not because they are lower in energy than all other possible arrangements of the atoms but because the curvatures, as given in terms of eigenvalues of the Hessian matrix (¶ 2E/¶Ra¶Ra ), are positive at this particular geometry. The procedures by which minima on the energy landscape are found may involve simply testing whether the energy decreases or increases as each geometrical coordinate is varied by a small amount. Alternatively, if the gradients ¶E/¶Ra are known at a particular geometry, one can perform searches directed “downhill” along the negative of the gradient itself. By taking a small “step” along such a direction, one can move to a new geometry that is lower in energy. If not only the gradients ¶E/¶Ra but also the second derivatives (¶ 2E/¶Ra¶Ra ) are known at some geometry, one can make a more “intelligent