Say we have a system of 2 electrons, an ansatz for the wavefunction is then a Slater determinant consisting of two orbitals. $$ \left|\Psi_{Slater}\right>= \frac{1}{\sqrt{2!}}\left(\left|\phi_1\phi_2\right>-\left|\phi_2\phi_1\right>\right) $$
How is this related to the basis expansion ansatz: $$ \left|\Psi\right>= \sum_i c_i \left|\phi_i\right> $$
The basis expansion formula would give me a function of 1 coordinate x_1 if i projected it on the position space, while the slater determinant ansatz would give me a function of two coordinates x_1, x_2. x_i stands for a set of x,y,z coordinates. Spin is neglected in this example. I understand that for the basis set expansion in an orthonormal basis, one can obtain the coefficients by the inner product $$ \left<\phi_i|\Psi \right>=c_i $$
but how can i use this on a slater determinant ? I only understand how this works if everything relies on one coordinate x_i, then i can calculate this: $$ \left<\phi_i|\Psi \right>=\int\phi_i(x)\Psi(x)dx=c_i $$
But how does this work if the wavefunction is multidimensional like the slater determinant wavefunction ? $$ \left<\phi_i|\Psi_{Slater} \right>=\int\phi_i(x_1)\Psi_{Slater}(x_1,x_2)dx= c_i(x_2)? $$ This expression makes no sense, since the remaining object would still be a function of $$x_2$$
To my knowledge only basis functions of one coordinate are used in calculations, like 3D Gaussians. Assuming that i have a wavefunction given in slater determinant form, how would i find the coefficients of the basis functions ?
I am afraid i am mixing up things that don't belong together, clarifications would be welcome.