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In semi-empirical quantum chemistry, one frequently encounters the so called zero differential overlap approximation

$$\langle \mu \nu | \lambda \sigma \rangle = \delta_{\mu\nu}\delta_{\lambda\sigma} \langle \mu \mu | \lambda \lambda \rangle .$$

Why is it rather not written as

$$\langle \mu \nu | \lambda \sigma \rangle = \delta_{\mu\nu}\delta_{\lambda\sigma} \langle \mu \nu | \lambda \sigma \rangle = \langle \mu \mu | \lambda \lambda \rangle $$

since on the right hand side of the first equation there are no $\nu$ nor $\sigma$ contained anymore. So either all four variables plus the Kronecker Deltas (middle expression of second equation), or only the "remaining" variables after evaluation of the Kronecker deltas (last expression of second equation).

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    $\begingroup$ I think there is a summation which can't be seen in this notation. You have tensors of rank 2 there. $\endgroup$
    – Andreas K.
    Mar 13, 2013 at 12:12
  • $\begingroup$ Can you expand on your comment a bit? I dont understand where do you suspect a summation. $\endgroup$
    – TMOTTM
    Mar 13, 2013 at 13:39
  • $\begingroup$ Something like this $\langle \mu \nu | \lambda \sigma \rangle =\langle \chi^{\lambda},\chi_{\mu}\rangle \langle \chi^{\sigma},\chi_{\nu}\rangle=\langle \chi^{\lambda},\chi_{\mu} \rangle \langle \delta_{\lambda}^{\sigma}\chi^{\lambda},\delta_{\nu}^{\mu}\chi_{\mu} \rangle=\delta_{\lambda}^{\sigma}\delta_{\nu}^{\mu}\langle \chi^{\lambda},\chi_{\mu} \rangle \langle \chi^{\lambda}, \chi_{\mu}\rangle$ $\endgroup$
    – Andreas K.
    Mar 13, 2013 at 17:47
  • $\begingroup$ So you have the deltas acting only on the second inner product (or the first, but only on one of them). Actually it would be better to change the indexes to $\lambda '$ and $\mu '$ in the second inner product in order to not confuse them. But there is no problem... $\endgroup$
    – Andreas K.
    Mar 13, 2013 at 17:53

1 Answer 1

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$\mu$ and $\nu$ are dummy variables, they can take any values from $1$ to $N$. As such, you cannot evaluate the Kronecker delta, before specifying $\mu$ and $\nu$ are. For example, $\delta_{14} = 0$, and $\delta_{NN} = 1$, but that is only because you have been given what $\mu$ and $\nu$ are.

So the first line still contains $\nu$ and $\sigma$! What the equation is saying is that, when $\mu = \nu$, and $\lambda = \sigma$, for any $\mu, \lambda = 1, \cdots, N$, then $\langle \mu \mu | \lambda \lambda \rangle = \langle \mu \mu | \lambda \lambda \rangle $, which is obviously true (though it doesn't tell you what the numerical value is). But if any one of those conditions is not true then $\langle \mu \nu | \lambda \sigma \rangle = 0$.

Now what you wrote doesn't make sense. If $\langle \mu \nu | \lambda \sigma \rangle = \delta_{\mu \nu} \delta_{\lambda \sigma} \langle \mu \nu | \lambda \sigma \rangle $ then $1 = \delta_{\mu \nu} \delta_{\lambda \sigma}$ (if $\langle \mu \nu | \lambda \sigma \rangle \neq 0$). But this is obviously a false statement, since if say $\mu = 1, \nu = 2$ then we have $ 1 = 0$.

The bottom line is that $\mu, \nu, \lambda, \sigma$ are dummy variables, and you cannot evaluate the Kronecker delta without being given what the two indices are.

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  • $\begingroup$ I get the mechanism of the Kronecker delta and now I also get how to read the first equation. So in the integral of the right hand side in the first equation, the second $\mu$ is implicitly the $\nu$, which I plug in for the $\nu$ in the first Kronecker delta. $\endgroup$
    – TMOTTM
    Mar 13, 2013 at 11:31

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