Applications of Energy Minimisation

The molecular mechanics or quantum mechanics energy at an energy minimum corresponds to a hypothetical, motionless state at OK. Experimental measurements are made on molecules at a finite temperature when the molecules undergo translational, rotational and vibration motion. To compare the theoretical and experimental results it is 

If all translational and rotational modes are fully accessible in accordance with the equipartition theorem, then Utrans(T) and U ,t(T) are both equal to per molecule (except that U i T) equals k T for a linear molecule) k is Boltzmann s constant. However, the vibrational energy levels are often only partially excited at room temperature. The vibrational contribution to the internal energy at a temperature T thus requires knowledge of the actual vibrational frequencies. The vibrational contribution equals the difference in the vibrational enthalpy at the temperature T and at 0 K and is given by  

J5 Normal modes of water Experirrwntal and (calculated) frequencies are shown. Theoretical frequencies calculated using a 6-3JG basis set 

We next solve the secular equation F —1 = 0 to obtain the eigenvalues and eigenvectors of the matrix F. This step is usually performed using matrix diagonalisation, as outlined in Section 1.10.3. If the Hessian is defined in terms of Cartesian coordinates then six of these eigenvalues will be zero as they correspond to translational and rotational motion of the entire system. The frequency of each normal mode is then calculated from the eigenvalues using the relationship  

We next calculate the first and then the second derivatives of the potential energy with respect to the three coordmates i, 2 and 3  

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