Definition of linear response kernel in terms of wavefunctions (<PERSON>/<PERSON>)
I'm trying to understand the derivation of the linear response kernel in Parr/Yang's "Density-functional theory of atoms and molecules". First some background information: We look at a system of $N$ electrons which are described by the Hamiltonian
\begin{equation} H_\text{el}= -\frac{1}{2}\sum_i \nabla_i^2 + \sum_{i<j} \frac{1}{|\boldsymbol r_i - \boldsymbol r_j|} + \sum_i v(\boldsymbol r_i). \end{equation}
Imagine the potential $v$ to be the Coulomb potential due to the nuclei in a molecule. 346
We call the $k$-th eigenstate to this Hamiltonian $\Psi_k^{(0)} = \Psi_k^{(0)}(\boldsymbol x_1 \dots \boldsymbol x_N)$ (the (0) indicating that these are the "unperturbed" states, since we will introduce a perturbation soon). The $\boldsymbol x_i := (\boldsymbol r_i, m_{s, i})$ are collective spatial and spin degrees of freedom.
The electron density associated with the wavefunction $\Psi_k^{(0)}$ is defined as
\begin{equation} \rho_k(\boldsymbol r_1) = N \int \Psi_k^{(0)*}\Psi_k^{(0)} dm_{s, 1} d\boldsymbol x_2 \dots d\boldsymbol x_N. \end{equation}
Now we introduce a perturbation $\Delta V = \sum_i \Delta v(\boldsymbol r_i)$ to the Hamiltonian, which we assume to be small such that we can express our perturbed wavefunction by usual first-order time-independent perturbation theory: $\Psi_k = \Psi_k^{(0)}+\Psi_k^{(1)}$ (the first order correction being calculated via the well-known formula).
Parr/<PERSON> then come (on page 15) to the following expression for the change of the electron density due to the perturbation:
\begin{align} \Delta \rho_k(\boldsymbol r_1) &:= N \int \left(\Psi_k^\Psi_k - \Psi_k^{(0)} \Psi_k^{(0)} \right) dm_{s, 1} d\boldsymbol x_2 \dots d\boldsymbol x_N \ &= 2N \Re\left[ \sum_{j\neq k} \frac{\langle \Psi_j^{(0)}| \Delta V | \Psi_k^{(0)}\rangle}{E_k^{(0)} - E_j^{(0)}} \int \Psi_k^{(0)*}\Psi_j^{(0)} dm_{s, 1} d\boldsymbol x_2 \dots d\boldsymbol x_N \right]. \end{align}
Up to here it is quite easy (simply ignore terms quadratic in the first-order correction). 526