Potential energy surface second-order derivatives

The Newton-Raphson block diagonal method is a second order optimizer. It calculates both the first and second derivatives of potential energy with respect to Cartesian coordinates. These derivatives provide information about both the slope and curvature of the potential energy surface. Unlike a full Newton-Raph son method, the block diagonal algorithm calculates the second derivative matrix for one atom at a time, avoiding the second derivatives with respect to two atoms. 

Analytical gradient energy expressions have been reported for many of the standard models discussed in this book. Analytical second derivatives are also widely available. The main use of analytical gradient methods is to locate stationaiy points on potential energy surfaces. So, for example, in order to find an expression for the gradient of a closed-shell HF-LCAO wavefunction we might start with the electronic energy expression from Chapter 6,... 

It should be clear from our discussion of potential energy surfaces that we have to examine the gradient of the electron density and the matrix of second derivatives, in order to make progress. The gradient of the electron density P(r) is, in Cartesian coordinates,... 

A saddle point is a stationary point on the multidimensional potential energy surface. It is a stable point in all dimensions except one, where the second-order derivative of the potential is negative (see Appendix E). The classical energy threshold Eci or barrier height of the reaction corresponds to the electronic energy at the saddle point relative to the electronic energy of the reactants. 

A very important point is that, contrary to methods based on a Hartree-Fock zero-order wave function, those rooted in the Kohn-Sham approach appear equally reliable for closed- and open-shell systems across the periodic table. Coupling the reliability of the results with the speed of computations and the availability of analytical first and second derivatives paves the route for the characterization of the most significant parts of complex potential energy surfaces retaining the cleaness and ease of interpretation of a single determinant formalism. This is at the heart of more dynamically based models of physico-chemical properties and reactivity. 

For M of order 10 We,( is of order 0.1. Thus, typical vibrational and rotational energies are of order 10 and lO"", respectively, compared to electronic energies. The Djj R) correction to the potential-energy surface Sj R) is of order lO-", similar to a rotational spacing. The mixing of electronic states i j is proportional to the squares of the first and second derivative couplings, i.e., 10 and 10", respectively. This provides... 

Hessian matrix A matrix of second-order partial derivatives that determine curvature used in calculations to test for minima on a potential energy surface. 

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