What is the difference between product wave function and true wave function for many electron system I know expression for product wave function, and I know true wave function for many electron system can't be expressed.
But can someone tell me what is missing in the product wave function?
What is the approximation?
 A: The electronic Hamiltonian for a multi-electron atom (or molecule) generally contains three components. (When I write "electronic Hamiltonian", the Born-Oppenheimer approximation - which allows for separation of electronic and nuclear motion - is already implicitly invoked.)
$$\hat{H} = \sum_i{\hat{T}_i} + \sum_i\sum_a{\hat{V}^\mathrm{en}_{ia}} + \sum_i\sum_{j>i}{\hat{V}^\mathrm{ee}_{ij}}$$
where the first summation represents the kinetic energy ($T_i$ being the kinetic energy of the $i$-th electron), the second representing electron-nucleus attractions ($a$ is the summation index for nuclei - if the system is an atom then you can drop this summation), and the last representing electron-electron repulsions.
If we ignore the electron-electron repulsions entirely (or equivalently, treat them as a constant, which has no impact on the eigenfunctions but simply shifts the energy eigenvalues by a constant amount), then you can rearrange the terms and write
$$\hat{H} = \hat{H}_1 + \hat{H_2} + \cdots + \hat{H}_n$$
where $n$ is the total number of electrons and $\hat{H}_i$ is an operator that only acts on one electron at a time:
$$\hat{H}_i = \hat{T}_i + \sum_a \hat{V}^\mathrm{en}_{ia}$$
and it's relatively straightforward to show (using separation of variables) that the product function $\psi = \psi_1\psi_2\cdots\psi_n$ satisfies the Schrodinger equation
$$\hat{H}\psi = E\psi$$
with $\hat{H}_i \psi_i = E_i \psi_i$ and $E = E_1 + E_2 + \cdots + E_n$.
So, the thing that is "missing" is simply the proper treatment of electron-electron repulsions. Sometimes this is also referred to as "correlation", but this has the potential to be ambiguous, as the word correlation can be used to refer to different aspects of electron-electron repulsions.
