Computational chemistry gradient methods

By the early 1980s, the first reviews on DFT appeared [107-110], underlining its great potential for applications to atoms and molecules. Also the first publications about analytic gradients for the Kohn-Sham method appeared in the literature [111,112]. This clearly moved the Kohn-Sham method into the computational chemistry arena. Still it took almost a decade more before stable geometry optimization with analytic gradients... 

Since optimization problems in computational chemistry tend to have many variables, essentially all commonly used methods assume that at least the first derivative of the function with respect to all variables, the gradient g, can be calculated analytically (i.e. directly, and not as a numerical differentiation by stepping the variables). Some methods also assume that the second derivative matrix, the Hessian H, can be calculated. 

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