### 18.2 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.

#### 18.2.1 OEP-EXX

In the present implementation the OEP-EXX local potential is expanded as [172]:

| (18.6) |

where g_{p} are gaussian functions, representing a new type of auxiliary basis-set (see
directory xbasen). Inserting Eq. (18.6) into Eq. (18.2) 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. (18.4), and
the charge condition

| (18.7) |

which ensures that v_{x}^{EXX}(r) approaches -1∕r in the asymptotic region. Actually Eq.
(18.6) violates the condition (18.5) 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.

#### 18.2.2 LHF

In the LHF implementation the exchange potential in Eq. (18.3) 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. 18.3) or using a basis-set expansion
as [168]:

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. (18.3) 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 (18.4) and (18.5), both are satisfied in the present implementation.
KS occupied orbitals are asymptotically continued [176] on the asymptotic grid point r
according to:

| (18.9) |

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}.