Energy minimisation methods derivative

In order to use a derivative minimisation method it is obviously necessary to be able to calculate the derivatives of fhe energy wifh respecf to the variables (i.e. the Cartesian or interna] coordinates, as appropriate). Derivatives may be obtained either analytically or numerically. The use of analytical derivatives is preferable as fhey are exact, and because they can be calculated more quickly if only numerical derivatives are available then it may be more effective to use a non-derivative minimisation algorithm. The problems of calculating analytical derivatives with quantum mechanics and molecular mechanics were discussed in Sections 3.4.3 and 4.16, respectively. 

Quantum mechanical calculations are restricted to systems with relatively small numbers of atoms, and so storing the Hessian matrix is not a problem. As the energy calculation is often the most time-consuming part of the calculation, it is desirable that the minimisation method chosen takes as few steps as possible to reach the minimum. For many levels of quantum mechanics theory analytical first derivatives are available. However, analytical second derivatives are only available for a few levels of theory and can be expensive to compute. The quasi-Newton methods are thus particularly popular for quantum mechanical calculations. 

To optimise the geometry, the energy must be expressed as a function of atomic displacements. This yields the partial derivatives crucial to automatic minimisation algorithms. The expressions for the total energy derivatives with respect to atomic displacements are quite complex for ab initio and semi-empirical methods but trivial for empirical schemes like Molecular Mechanics (MM). Virtually all modern computer codes provide extensive, efficient facilities for determining ground state molecular geometries. 

In practice, of course, the surface is only quadratic to a first approximation and so a number of steps will be required, at each of which the Hessian matrix must be calculated and inverted. The Hessian matrix of second derivatives must be positive definite in a Newton-Raphson minimisation. A positive definite matrix is one for which all the eigenvalues are positive. When the Hessian matrix is not positive definite then the Newton-Raphson method moves to points (e.g. saddle points) where the energy increases. In addition, far from a mimmum the harmonic approximation is not appropriate and the minimisation can become unstable. One solution to this problem is to use a more robust method to get near to the minimum (i.e. where the Hessian is positive definite) before applying the Newton-Raphson method. 

I is the identity matrix. The six first derivatives of the energy with respect to the strain components e, measure the forces acting on the unit cell. When combined with the atomic coordinates we get a matrix with 3N - - 6 dimensions. At a minimum not only should there be no force on any of the atoms but the forces on the unit cell should also be zero. Application of a standard iterative minimisation procedure such as the Davidon-Fletcher-Powell method will optimise all these degrees of freedom to give a strain-free final structure. In such procedures a reasonably accurate estimate of the initial inverse Hessian matrix is usually required to ensure that the changes in the atomic positions and in the cell dimensions are matched. 

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