Why do we multiply in the total wave function but add in the LCAO method?

Question

If to particles with wave functions $\psi_a$ and $\psi_b$ are „combined“ their total wave function is given by:

$$\psi(r_1,r_2)= A[\psi_a(r_1)\psi_b(r_2) \pm \psi_b(r_1)\psi_a(r_2)]$$

(+ for bosons and - for fermions, $A$ is a constant, here I‘m ignoring spin.)

But if we combine the wave functions of two atomic orbitals $\psi_c$ and $\psi_d$ it is done by a simple Linear Combination of Atomic Orbitals (LCAO):

$$\psi = C_1\psi_c + C_2\psi_d$$

Where the $C_1$, $C_2$ are constants.

These two processes of generating the total wave function out of existing ones feel to me as if they should be carried out in the same way. And I understand that we do it this way in the first case since we can’t tell the particles apart and the total wave function therefore has either to be symmetric or antisymmetric. So what is the justification that we can simply add them in the second LCAO case?

Answer 1

These formulae do different things. The first one takes 2 (or more) single-particle wavefunctions and gives you a multiple-particle wavefunction that can be interpreted as having one particle in one state and another particle in another state. The second one instead produces a single-particle wavefunction that can be interpreted as a single particle in a superposition of the states. Since they do different things, they look different too.

In quantum chemistry, the overall wavefunction of all the electrons together is often assumed to be of the first form (called a Slater determinant), combining $N$ single-electron spin-orbital wavefunctions. (Reduces to your formula for $N=2,$ replace the determinant with the permanent for bosons.) (In general, the wavefunction may be a linear superposition of these.)

$$\psi(\vec r_1,\ldots,\vec r_N)=\frac1{\sqrt{N!}}\begin{vmatrix}\psi_1(\vec r_1)&\ldots&\psi_1(\vec r_N)\\\vdots&\ddots&\vdots\\\psi_N(\vec r_1)&\ldots&\psi_N(\vec r_N)\end{vmatrix}$$

The problem of finding the overall wavefunction is now reduced to finding $N$ single-electron wavefunctions (the molecular (spin-)orbitals $\psi_n$). LCAO provides plausible candidates for these by combining atomic orbitals. The two types of combination work together to produce the final wavefunction. You cannot replace one with the other.