The orthogonalized plane waves

Question

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 ?

Answer 1

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

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 .