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An orthogonal plane wave with wave number $k$ is written as

$$ OPW_k=e^{ ik\cdot r}-\sum_\alpha \psi_\alpha(r) \int \psi^*_\alpha (r'') e^{ik\cdot r''} d\tau'',$$

where index $\alpha$ and $k$ stands for core states and conduction band states.

In the above equation first term is plan wave term and in second term we use two wave functions $\psi_\alpha(r)$ and $\psi^*_\alpha (r'')$ for core states. My question is why do we use two wave functions and why do subtract second term from the first term ?

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Just made some minor fixes to your formatting. – Michael Brown Mar 9 '13 at 3:25
Where is this equation coming from (what context)? And what is $\tau''$? Do you mean $r''$? – Michael Brown Mar 9 '13 at 3:28
I don't know. why do they use $ \tau^" $ i have got this problem while studying OPW method. Thank you @Michael Brown for your cooperation. – sky rain Mar 9 '13 at 4:05
I guess the $d\tau''$ in the integral should be a $dr''$ – Andre Holzner Mar 9 '13 at 7:41

$$ \psi_\alpha(r) \int \psi^*_\alpha (r'') e^{ik\cdot r''} dr'' $$

is the component of $\Psi_\alpha(r)$ not orthogonal to $e^{ik\cdot r}$.

So one is subtracting all components not orthogonal to any existing (basis) function $\Psi_\alpha(r)$ from the (newly added) plane wave $e^{ik\cdot r}$, similar to Gram Schmidt orthogonalization .

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