Density

Density matrices are not only nice for mathematical reasons, they also offer a stringent physical definition of orbitals. The point of this post is to show how you can create what is called "natural orbitals" from any kind of wave function. These things are also interesting with respect to orbital imaging (as introduced here and discussed for example here).

In order not to make any prior assumptions, I have to use general wave functions. In real life you would usually look at these things from the perspective of an orbital basis. And then the expressions would be simpler (maybe I will show that, too).

The one particle density matrix γ₁ of an n-particle wave function Ψ is defined as:



This quantity is a "convenient partial sum". It lets you evaluate all the properties of 1-particle operators without the need of the whole wave function. This operation (in a finite orbital basis) is carried out in many quantum chemical codes.

Because the particles are interchangable you only need one such density matrix. (An n-particle density matrix which is analogously defined, lets you evaluate the matrix element of any n-particle operator.)

First we have to define what a one particle operator is. I think the most straight forward (and hopefully correct) definition ist the following:
An operator A is said to act only on r₁ if

The idea is that you can do all the integrations with respect to the other variables ahead. You need some variable renaming to make sure that the operator only acts on the second function.

This contains already the density matrix



From the density matrix we can construct, what is called "natural orbitals" without the need of any a priory definition of an orbital basis.

Maybe it would be easier to understand it in the matrix representation but in the general case we can also define an operator of the following form.



We may also symmetrize it (again easier in cartesian coordinates; edit: the 1-particle density of a wave function is already symmetric). This symmetrized operator has clearly defined orthogonal 1-particle eigenfunctions, the so called "natural orbitals" (and occupation numbers as eigenvalues). The natural orbitals represent the 1-particle density matrix. And this one particle density matrix contains all the information needed for 1-particle operators (like e.g. the dipole). So natural orbitals are something kind of physcial in my opinion.

The eigenvalues of the density matrix are the occupations. In a Hartree-Fock wave function the natural orbitals correspond to the usual HF orbitals and the eigenvalue of occupied orbitals is one (zero for virtuals). The density matrix is idempotent (or equivalently a projection). In the general case there is no clear distinction between occupied and virtual orbitals and there is strictly spoken no such thing as a HOMO or LUMO. But then we notice that HF represents most wave functions quite well in "well behaved" cases. So you may still think about HOMOs and LUMOs even in a more strict sense.