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Here I refer to a particular book Molecular Quantum Mechanics by Peter W. Atkins and Ronald S. Friedman, but similar derivation could be found in many other texts.

So, when obtaining the explicit form of the Fock matrix elements for RHF formalism (p. 295 in 4th edition), authors go from

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to

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just by mentioning that $\psi_u$ is expanded as a linear combination of basis functions $\theta$.

The only way I could go from the first equation to the second one is by expanding the same spatial orbital $\psi_u$ on the left and on the right sides of integrand expression, i.e. before and after $1/r_{12}$, differently, $$ \psi_u = \sum_l c_{lu} \theta_{l} \quad \text{"on the left side"} \, , \\ \psi_u = \sum_m c_{mu} \theta_{m} \quad \text{"on the right side"} \, . $$

Is it true? And if yes, why on earth this should be done this way?

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This makes sense to me. $l$ and $m$ are two different summation indices. If you used the same index, say $l$, you would have only the $c^*_{lu}$ $c_{lu}$ terms, without the cross terms when $l\neq m$.

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  • $\begingroup$ Ugh! Yes, of course. Just was fooled by too much extra stuff on top of otherwise pretty simple expansion. Thanks! $\endgroup$
    – Wildcat
    Commented Jul 23, 2014 at 18:07
  • $\begingroup$ I know the feeling... ;-) $\endgroup$
    – Physico
    Commented Jul 23, 2014 at 18:10

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