Determinant matrix elements

Photo Credit: Samuel Monnier

Determinant matrix elements

In quantum chemistry we often want to do integrals like this:

This integral is generically called a “matrix element”, and specifically it is a “determinant matrix element”.

The integration in this matrix element is over electronic coordinates The functions \( \Phi_K = \phi\sub{k_1}(\B{x_1})\ldots\phi(\B{x}\sub{N}) \) are Hartree orbital products. \( M \) is some kind of operator — perhaps a function of one or more of those coordinates, or derivatives of those coordinates — which importantly is symmetric in the electron coordinates, since all electrons are indistinguishable. (Read here and here to know more).

Now you can see that the determinant matrix element above involves products of two antisymmetrizers \( \mathscr{A} \). The parentheses make clear that these antisymmetrizers only “operate” on the electron coordinates in the functions \( \Phi_K \) immediately to their right.

Why are (determinant) matrix elements important?

The reason for that matrix elements are important is they allows to convert a partial differential equation, such as the Schrodinger equation, into a matrix equation involving numbers. Matrix equations involving numbers are much easier to solve on computers.

The reason for that determinant matrix elements are important is that a determinant is the simplest reasonable model wavefunction for a group of electrons.

Clearly, we need to be able to work out or simplify this matrix element in order to do numerical calculations.

Matrix elements between determinant wavefunctions

Luckily we can use previous results showing that the antisymmetrizeris hermitian, and that a product of two antisymmetrizers is idempotent i.e.

To make further progress we need to make certain assumptions about the operator \( M \).