Hartree-Fock by varying the coefficients of a wave function expansion

In deriving the Hartree-Fock equations, we will expand the single-particle functions in a known basis and vary the coefficients, that is, the new single-particle wave function is written as a linear expansion in terms of a fixed chosen orthogonal basis (for example the well-known harmonic oscillator functions or the hydrogen-like functions etc). We define our new Hartree-Fock single-particle basis by performing a unitary transformation on our previous basis (labelled with greek indices) as $$$$\psi_p^{HF} = \sum_{\lambda} C_{p\lambda}\phi_{\lambda}. \tag{13}$$$$ In this case we vary the coefficients $$C_{p\lambda}$$. If the basis has infinitely many solutions, we need to truncate the above sum. We assume that the basis $$\phi_{\lambda}$$ is orthogonal. A unitary transformation keeps the orthogonality, as discussed in exercise 1 below.