I know this from a quantum mechanics class:

$$\langle \psi_i | \psi_j\rangle = \delta_{ij}$$

But does it also apply for:

$$\langle \psi_i |V| \psi_j\rangle = \delta_{ij}$$

where $V$ is any arbitrary potential? That is, if I try to compute the expectation value of $V$ using 2 different states $\psi_i$ and $\psi_j$ I get $0$?