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In the LHF implementation the exchange potential in Eq. (16.3) is computed on each grid-point and numerically
integrated to obtain orbital basis-sets matrix elements. In this case the DFT grid is needed but no auxiliary basis-set is required.
The Slater potential can be computed numerically on each grid point (as in Eq. 16.3) or using a basis-set expansion as[156]:

Here, the vector
**()**
contains the basis
functions,
stands
for the corresponding overlap matrix, the vector
collects the coefficients representing orbital *a*
,
and the matrix
represents the non-local exchange
operator
*NLx*
in the
basis set.
While the numerical Slater is quite expensive but exact, the basis set method is very fast but its accuracy
depends on the completeness of the basis set.
Concerning the correction term, Eq. (16.3) shows that it depends on the exchange potential itself.
Thus an iterative procedure is required in each self-consistent step: this is done using the conjugate-gradient method.

Concerning conditions (16.4) and (16.5), both are satisfied
in the present implementation.
KS occupied orbitals are asymptotically continued[164] on the asymptotic grid point **r** according to:

() = *φ*_{i}()*e*^{-βi(||-||)} , |
(16.9) |

where
is the reference point (not in the asymptotic region),
*β* =
and *Q*
is the molecular charge.
A surface around the molecule is used to defined the points
.

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TURBOMOLE M