### N.16 A single Slater determinant is not exact

The simplest example that illustrates the problem with representing a general wave function by a single Slater determinant is to try to write a general two-variable function as a Slater determinant of two functions and . You would write

A general function cannot be written as a combination of the same two functions and at every value of . However well chosen the two functions are.

In fact, for a general antisymmetric function , a single Slater determinant can get right at only two nontrivial values and . (Nontrivial here means that functions and should not just be multiples of each other.) Just take and . You might object that in general, you have

where , , , and are some constants. (They are or values at or , to be precise). But if you plug these two expressions into the Slater determinant formed with and and multiply out, you get the Slater determinant formed with and within a constant, so it makes no difference.

If you add a second Slater determinant, you can get right at two more values and . Just take the second Slater determinant's functions to be and , where is the deviation between the true function and what the first Slater determinant gives. Keep adding Slater determinants to get more and more -​values right. Since there are infinitely many -​values to get right, you will in general need infinitely many determinants.

You might object that maybe the deviation from the single Slater determinant must be zero for some reason. But you can use the same ideas to explicitly construct functions that show that this is untrue. Just select two arbitrary but different functions and and form a Slater determinant. Now choose two locations and so that and are not in the same ratio to each other. Then add additional Slater determinants whose functions you choose so that they are zero at and . The so constructed function is different from just the first Slater determinant. However, if you try to describe this by a single determinant, then it could only be the first determinant since that is the only single determinant that gets and right. So a single determinant cannot get right.