Say I have an arbitrary molecule in the Born-Oppenheimer approximation, and furthermore say that I can approximate the molecule as having only one active electron. What methods exist to calculate the density of states as a function of the energy of an electron in the continuum (that is, with positive energy)?
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The continuum states are different in several aspects.
First, there are a countably infinite number of bound molecular states vs an uncountably infinite number of states in any finite range of the continuum. Thus, the ratio of total number of bound states over the number of states in even a small range at the beginning of the continuum is effectively zero. So in any definition of density of states in which the continuum is finite, below the first ionization potential the density is zero.
Second, the continuum states are free and therefore at infinity the energy eigen-states will resemble plane waves. Since these states cover all space, the potential from the molecule has a negligible effect. Consider calculating $\langle \phi| V |\phi\rangle$ where $V$ is the potential due to the molecular nuclei and bound electrons. Since $\phi$ is spread over all space, this is similar to asking what if when calculating $\langle\phi|\phi\rangle$ we only integrated over a finite region. Compared to the infinite size of space, this is truly negligible.
Therefore the density of states in the continuum after the first ionization energy (and before the second one) can be approximated by: $$D(E) \propto \sqrt{E-E_0}$$ where $E_0$ is the ionization edge.
The x-ray absorption of molecules, or x-ray photoelectron spectroscopy (XPS), is probably the closest you'll have to experimental measurements which you can try to work backwards to check any density of state calculations. Here is a graph of x-ray absorption of common gas atoms/molecules which shows the strong effects of the ionization edge. Note that the edges there are showing when a new atomic/molecular energy level is now able to reach the continuum, so this is demonstrating the strong edge between continuum vs bound states.
Here is a paper looking at XPS for solids and trying to work backwards to the bound state levels. They need to know the density of states in the continuum to do this, and comment:
"... the final state electrons are ~ 1250 eV into the continuum and the lattice potential affects them very little. Therefore, the appropriate final state density will be proportional simply to $\epsilon^{1/2}$" (where $\epsilon$ is the energy of the free electron)
Beyond this initial approximation $\sqrt{E-E_0}$, there will also be structure in the density of states due to combinatorics of arranging energy in the bound states. Further combinatorics and tradeoffs of energy between the free electrons occurs at the second ionization energy, and so on.
The main point is that the density of states in the continuum should be completely determined by calculating the ionization energies, and the energy levels of the bound states of the remaining ion. The energy levels of the free electron and ion can be considered separately.
Update: Here is an x-ray absorption overview, which also shows some fine structure within ~ 50eV of an edge. Absorption is mostly dominated by single transitions, and the more electrons which change energy level usually the more suppressed the matrix element, so this type of structure in the absorption spectrum is likely showing more about the bound state density than the final state.
John
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I also was looking for this. I guess there are many method among them
- Gilat-Raubenheimer method for k - space integration
- tetrahedron method
- special point method
BMS
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hamza hebal
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