In the spin averaged case,

(26) |

If *h*_{k} is a Gaussian then the integrals are proportional to . To a first approximation, since
only varies slowly with *n* we can approximate this to
, where

This approximation can be improved through some mathematical tricks,
since we know that a highly accurate form for is given
by where *s*=0.30917 (see Eq. 2.5.19).

If we define

(27) |

we can now do a limited expansion of this to give an
approximation for *f*(*s*),

(28) |

This step is one of the largest approximations within our method, leading to errors of up to 10%; however in practise its contribution to the total energy is minute and so the approximation is not unreasonable.

To find *f*(2) we need to know the second moment of , which
can be determined analytically as

(29) |

Finally substituting all of these new terms back in, we get an expression for :