next up previous contents index
Next: Time-Dependent Fields Up: Hartree-Fock for Molecules Previous: Hartree-Fock for Molecules   Contents   Index

Roothan Equations

The equations Eq. (V.4.8) are called Roothan equations (they are usually written for spin-independent Fock-operator \bgroup\color{col1}$ {\mathcal F}$\egroup. We summarize the situation so far: We can then write the Roothan equations as
$\displaystyle \underline{\underline{\mathcal F}}^{\uparrow\uparrow}\underline{\underline{C}}$ $\displaystyle =$ $\displaystyle \underline{\underline{S}} \underline{\underline{C}} \underline {\underline{\varepsilon}}^{\uparrow}$ (4.8)
$\displaystyle \underline{\underline{\mathcal F}}^{\downarrow\downarrow}\underline{\underline{C}}$ $\displaystyle =$ $\displaystyle \underline{\underline{S}} \underline{\underline{C}} \underline{\underline{\varepsilon}}^{\downarrow},$ (4.9)

where \bgroup\color{col1}$ \underline{\underline{\varepsilon}}$\egroup are diagonal matrices for the energies \bgroup\color{col1}$ \varepsilon_k$\egroup. Now these look like simultaneous linear equations but of course they are not, because the Fock-operator depends on the coefficients \bgroup\color{col1}$ c_{kl}$\egroup that we try to determine: recall \bgroup\color{col1}$ {\mathcal F} \equiv \hat{H}^0 + \hat{J}-\hat{K}$\egroup with
$\displaystyle \hat{J}^{\uparrow\uparrow}_{l'l}$ $\displaystyle \equiv$ $\displaystyle \sum_{i=1}^{2N}\langle l'\nu_i \vert U \vert\nu_i l\rangle$ (4.10)
$\displaystyle \hat{K}^{\uparrow\uparrow}_{l'l}$ $\displaystyle \equiv$ $\displaystyle \sum_{i=1}^{2N}\langle l'\nu_i \vert U \vert l \nu_i\rangle
\langle \sigma_i\vert\uparrow\rangle
,$  

where we first considered spin up. The \bgroup\color{col1}$ i$\egroup-sum runs over spin-orbitals, i.e. AOs including the spin. We now assume \bgroup\color{col1}$ U$\egroup to be spin-independent.
$\displaystyle \hat{J}_{l'l}$ $\displaystyle \equiv$ $\displaystyle \sum_{\sigma=0,1}\sum_{k=1}^N\langle l' \nu_{2k-\sigma} \vert U \...
...igma} l\rangle
= 2 \sum_{k=1}^N \langle l' \psi_k \vert U \vert \psi_k l\rangle$  
$\displaystyle \hat{K}_{l'l}$ $\displaystyle \equiv$ $\displaystyle \sum_{\sigma=0,1}\sum_{k=1}^N
\langle l' \nu_{2k-\sigma} \vert U ...
...uparrow\rangle
=\sum_{k=1}^N \langle l' \psi_k \vert U \vert l \psi_k \rangle
,$  

where \bgroup\color{col1}$ \sigma=0$\egroup corresponds to spin down and \bgroup\color{col1}$ \sigma=1$\egroup corresponds to spin up, and the orbital part of the spin-orbital \bgroup\color{col1}$ \nu_{2k-\sigma}$\egroup is \bgroup\color{col1}$ \psi_k$\egroup by definition. Now everything is expressed in terms of orbitals only and the spin has just led to the factor of two in front of the direct term!

We now use the LCAO expansion \bgroup\color{col1}$ \psi_{k}=\sum_m c_{mk}\psi_m$\egroup and thus obtain

$\displaystyle \hat{J}_{l'l}$ $\displaystyle =$ $\displaystyle 2 \sum_{k=1}^N \sum_{m,m'=1}^M c_{m'k}^*c_{mk}
\langle l' m' \ver...
...\vert m l\rangle =
\sum_{m,m'=1}^MP_{m'm}\langle l' m' \vert U \vert m l\rangle$  
$\displaystyle \hat{K}_{l'l}$ $\displaystyle =$ $\displaystyle \phantom{2}\sum_{k=1}^N \sum_{m,m'=1}^M c_{m'k}^*c_{mk} \langle l...
...ngle
=\frac{1}{2}\sum_{m,m'=1}^M P_{m'm}\langle l' m' \vert U \vert l m \rangle$  
$\displaystyle P_{m'm}$ $\displaystyle \equiv$ $\displaystyle 2\sum_{k=1}^N c_{m'k}^*c_{mk},$   populations (4.11)

where the populations depend on the \bgroup\color{col1}$ c$\egroup's: it is them who are responsible for the non-linearity (self-consistent character) of the Roothan equations. Summarizing, the matrix elements of the Fock operator are given by
$\displaystyle {\mathcal F}^{\uparrow\uparrow}_{l'l}$ $\displaystyle =$ $\displaystyle \left(\hat{H^0}\right)^{\uparrow\uparrow}_{l'l}$ (4.12)
  $\displaystyle +$ $\displaystyle \sum_{m,m'=1}^M P_{m'm}[\{c_{ij}\}] \left( \langle l' m' \vert U \vert m l\rangle
-\frac{1}{2} \langle l' m' \vert U \vert l m \rangle\right).$  


next up previous contents index
Next: Time-Dependent Fields Up: Hartree-Fock for Molecules Previous: Hartree-Fock for Molecules   Contents   Index
Tobias Brandes 2005-04-26