Potensial Energy Surface

Pertemuan V

Definition

• • • Femtosecond spectroscopy experiments show that molecules vibrate in many different directions until an energetically accessible reaction path is found. In order to examine these ideas computationally, the entire potential energy surface (PES) or an approximation to it must be computed. A PES is either a table of data or an analytic function, which gives the energy for any location of the nuclei comprising a chemical system

The full PES for the hypothetical molecule ABC requires four dimensions to display (3N − 6 = 3 coordinate degrees of freedom plus one dimension for energy). The three-dimensional plot (left) represents a hyperslice through the full PES showing the energy as a function of two coordinate dimensions, the AB and BC bond lengths, while taking a fixed value for the angle ABC (a typical choice might be the value characterizing the global minimum on the full PES). A further slice of this surface (right) now gives the energy as a function of a single dimension, the AB bond length, where the BC bond length is now also treated as frozen (again at the equilibrium value for the global minimum)

Points on a potential energy surface.

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PROPERTIES OF POTENTIAL ENERGY SURFACES

Once a PES has been computed, it can be analyzed to determine quite a bit of information about the chemical system. The PES is the most complete description of all the conformers, isomers, and energetically accessible motions of a system.

One of the most common reasons for doing a PES computation is to subsequently study reaction dynamics. The vibrational properties of the molecule can also be obtained from the PES.

COMPUTING POTENTIAL ENERGY SURFACES

• • • Computing a complete PES for a molecule with N atoms requires computing energies for geometries on a grid of points in 3N-6 dimensional space. This is extremely CPU-intensive because it requires computing a set of X points in each dimension, resulting in X 3N-6 single point computations. Because of this, PES's are typically only computed for systems with a fairly small number of atoms

• • • • • Some software packages have an automated procedure for computing all the points on a PES, but a number of technical problems commonly arise. Some programs have a function that halts the execution when two nuclei are too close together; this function must be disabled. SCF procedures often exhibit convergence problems far from equilibrium. At different points on the PES, the molecule may have different symmetries. This often results in errors when using software packages that use molecular symmetry to reduce computation time.

The use of symmetry by a program can often be turned off.

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FITTING PES RESULTS TO ANALYTIC EQUATIONS

The criteria on fitting PES results to analytic equations have been broken down into a list of 10 specific items. Below is the list as given by Schatz: 1. The analytic function should accurately characterize the asymptotic reactant and product molecules.

2. It should have the correct symmetry properties of the system.

3. It should represent the true potential accurately in the interaction regions for which experimental or nonempirical theoretical data are available.

4. It should behave in a physically reasonable manner in those parts of the interaction regions for which no experimental or theoretical data are available.

5. It should smoothly connect the asymptotic and interaction regions in a physically reasonable way.

6. The interpolating function and its derivatives should have as simple an algebraic form as possible consistent with the desired goodness of fit.

7. It should require as small a number of data points as possible to achieve an accurate fit.

8. It should converge to the true surface as more data become available.

9. It should indicate where it is most meaningful to compute the data points.

10. It should have a minimal amount of “ad hoc” or “patched up” character.