Numerical Hessian for a plannar molecule with n atoms

n atoms:(x1, y1, 0), (x2, y2, 0), (x3, y3, 0), ...

A total of (1 + 9*n^2) energy calculations need to be performed:

1: at equilibrium geometry.
n*1: for 1z-1z 2z-2z,3z-3z,....
[n*(n-1)/2]*2: for 1z-2z, 1z-3z, 1z-4z, 2z-3z, 2z-4z, 3z-4z.
[(2n)^2]*4/2: for the second derivatives with respect to x(s) and y(s)

1 + n + n*(n-1) + 8*n^2 = 1 + 9*n^2

For example, Numerical Hessian for a plannar molecule with 3 atoms

3 atoms:(x1, y1, 0),(x2, y2, 0),(x3, y3, 0)

82 energy calculations need to be performed:

1: at equilibrium geometry.
3: for 1z-1z 2z-2z,3z-3z
3*2: for 1z-2z, 1z-3z, 2z-3z.
2*6^2: for the second derivatives with respect to x(s) and y(s)

1 + 3 + 6 + 72 = 82 (verified by an AM1 method=fullnum Hessian calculation on HOCl)

For a planar molecule such as trans-HNOH with 4 atoms, 1+9*4^2=145 energy calculations need to be carried out to obtain the numerical Hessian.

 $force temp(1) = 1100.00, 1200.00, 1300.00, 1400.00, 1500.00,
                  1600.00, 1700.00, 1800.00, 1900.00, 2000.00
  method=fullnum projct=.t.
 $end
 $contrl maxit=200 mult=2 scftyp=uhf runtyp=hessian coord=unique $end
 $basis gbasis=am1 $end
 $system timlim=1000000 mwords=80 $end
 $data
 HNOH
 Cs

 O           8.0   0.0094634591  -0.0934516366   0.0000000000
 N           7.0   1.2729078083   0.1027207551   0.0000000000
 H           1.0  -0.4112060327   0.7952578868   0.0000000000
 H           1.0   1.7288347653  -0.8045270053   0.0000000000
 $end

For a non-planar  molecule with 4 atoms, see the post on H2NO.