CHEM01200604005 A. K. Pathak - Homi Bhabha National Institute

Newton –Raphson (NR) method expand the true function (energy) to second order
around the current point x
0
.
1
f ( x) f( x ) g( x x ) H( x x )
2
2
=
0
+ −
0
+ −
0
(1)
where, g is the gradient (force) and H is the Hessian. Requiring the force to be zero (in
eq. 1) produces the following step,
( − ) = − (2)
−1
x x0
H g
In the co-ordinate system (x) where the Hessian is diagonal, the NR step may be written
as
f
Δ x = ∑ Δx Δ x =−
(3)
' ' ' i
i, i
i
εi
where
f
i
is the projection of the gradient (force) along the Hessian eigenvector with
eigen value ε
i
.
As the real function contains terms beyond the second order, the NR formula may
be used iteratively for approaching towards the stationary point. Near minimum, all the
Hessian eigenvalues are positive and the step direction is opposite to the gradient
direction. If, one Hessian eigenvalue is negative, the step in this direction will be along
the gradient and thus increase the function value. In this circumstances, the optimization
may end up at a stationary point, which, having one negative Hessian eigenvalue, first
order saddle point. If the stationary point has n negative Hessian eigenvalues, it is called
n th order saddle point in the potential energy surface. So, in general the NR method will
attempt to converge on the nearest stationary point (either minimum or saddle point).
Other problem is the use of inverse Hessian for control of the step size. If one of the
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