The standard quadrature grid

Customarily, in developing a quadrature rule to approximate the integral 

For the radial integration, we have adopted the Euler-Maclaurin formula proposed by Murray et al. [47], in which the radial points and weights are given by 

To illustrate the effect of orientational dependence of the energy, Table 3 lists the S-null/6-31G(d) energy of the Ci molecule HOOF in three different orientations. Orientation I is the standard orientation, orientation II is obtained from the standard orientation by a rotation of 20° about the x-axis, and orientation in is obtained from the standard orientation by successive rotations of 20° about the x-axis and 30° about the y-axis. The energies are listed for two grids SG-1 and (20,50), which has 1000 points per atom, more typical of the coarser grids 

Total energies (hartrees) of HOOF in various orientations 

When grid orientation is not properly taken into account, loss of rotational invariance manifests itself in derivative calculations as well, often with more pronounced effects. In particular, calculated harmonic frequencies of low-lying vibrational modes can be adversely affected, as we have demonstrated elsewhere [55]. An example of this will be given later. 

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