Potential energy surface first-order derivatives

The steepest descent method is a first order minimizer. It uses the first derivative of the potential energy with respect to the Cartesian coordinates. The method moves down the steepest slope of the interatomic forces on the potential energy surface. The descent is accomplished by adding an increment to the coordinates in the direction of the negative gradient of the potential energy, or the force. 

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. 

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... 

The first approach is two-determinant CC theory, described above. Analytical derivatives have been implemented for the TD-CCSD method in ACES II, and these can be used to search excited state potential energy surfaces and to compute first-order properties of excited states. 

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