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LHF

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]:


vxslat($\displaystyle \bf r$) = $\displaystyle {\frac{{n_s}}{{\rho({\bf r})}}}$  $\displaystyle \sum_{{a}}^{{\text{occ.}}}$$\displaystyle \underline{{u}}_{a}^{T}$$\displaystyle \underline{{\chi}}$($\displaystyle \bf r$)$\displaystyle \underline{{\chi}}^{T}_{}$($\displaystyle \bf r$)$\displaystyle \underline{{\underline{S}}}^{{-1}}_{}$$\displaystyle \underline{{\underline{K}}}$ $\displaystyle \underline{{u}}_{a}^{}$  . (16.8)

Here, the vector $ \underline{{\chi}}$($ \bf r$) contains the basis functions, $ \underline{{\underline{S}}}$ stands for the corresponding overlap matrix, the vector $ \underline{{u}}_{a}^{}$ collects the coefficients representing orbital a , and the matrix $ \underline{{\underline{K}}}$ represents the non-local exchange operator $ \hat{{v}}^{\text}_{}$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:

$\displaystyle \tilde{{\phi}}_{i}^{}$($\displaystyle \bf r$) = φi($\displaystyle \bf r_{0}^{}$)$\displaystyle \left(\vphantom{ \frac{\vert{\bf r\vert}}{\vert{\bf r}_0\vert}}\right.$$\displaystyle {\frac{{\vert{\bf r\vert}}}{{\vert{\bf r}_0\vert}}}$$\displaystyle \left.\vphantom{ \frac{\vert{\bf r\vert}}{\vert{\bf r}_0\vert}}\right)^{{{(Q+1)/\beta_i -1}}}_{{}}$e-βi(|$\scriptstyle \bf r$|-|$\scriptstyle \bf r_{0}$|)  , (16.9)

where $ \bf r_{0}^{}$ is the reference point (not in the asymptotic region), β = $ \sqrt{{-2\epsilon_i}}$ and Q is the molecular charge. A surface around the molecule is used to defined the points $ \bf r_{0}^{}$ .


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