For a molecule with N atoms, there are 3N Cartesian coordinates, so the gradient is a vector g (as above), with 3N terms (dV/dx_i).

The second derivatives have the form (d²V/dx_idx_j), involving each coordinate, and so make up a 3N´3N matrix, called the Hessian matrix, H.

For a multidimensional function, therefore, we require the inverse of the Hessian matrix:

x_k+1 = x_k – gH^–1

Features of the Newton-Raphson method

· Efficient for small molecules, converges quickly

· Approximation of quadratic surface poor, particularly far from minimum

· Calculation, inversion and storage of Hessian is computationally difficult for large molecules

· Can locate stationary points other than minima (TSs)

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