### 19.2 Implementation

Both the OEP-EXX and LHF methods can be used in spin–restricted closed–shell and
spin–unrestricted open–shell ground state calculations. Both OEP-EXX and LHF are parallelized
in the OpenMP mode.

#### 19.2.1 OEP-EXX

In the present implementation the OEP-EXX local potential is expanded as [223]:

| (19.6) |

where g_{p} are gaussian functions, representing a new type of auxiliary basis-set (see directory
xbasen). Inserting Eq. (19.6) into Eq. (19.2) a matrix equation is easily obtained for the coefficient
c_{p}. Actually, not all the coefficients c_{p} are independent each other as there are other
two conditions to be satisfied: the HOMO condition, see Eq. (19.4), and the Charge
condition

| (19.7) |

which ensures that v_{x}^{EXX}(r) approaches -1∕r in the asymptotic region. Actually Eq. (19.6)
violates the condition (19.5) on the HOMO nodal surfaces (such condition cannot be achieve in
any simple basis-set expansion).

Note that for the computation of the final KS Hamiltonian, only orbital basis-set matrix elements
of v_{x}^{EXX} are required, which can be easily computes as three-index Coulomb integrals. Thus the
present OEP-EXX implementation is grid-free, like Hartree-Fock, but in contrast to all other
XC-functionals.

#### 19.2.2 LHF

In the LHF implementation the exchange potential in Eq. (19.3) 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. 19.3) or using a basis-set expansion
as [219]:

Here, the vector χ(r) contains the basis functions, S stands for the corresponding overlap matrix,
the vector u_{a} collects the coefficients representing orbital a, and the matrix K represents the
non-local exchange operator _{x}^{NL} 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. (19.3) 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 (19.4) and (19.5), both are satisfied in the present implementation. KS
occupied orbitals are asymptotically continued [227] on the asymptotic grid point r according
to:

| (19.9) |

where r_{0} 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 r_{0}.