next up previous
Next: About this document ... Up: lecture_9 Previous: Hartree-Fock theory

Probing energy levels and the Koopmans approximation

Quantum theory successfully predicts the shell structure of atoms and can explain the observed features of the periodic table (see end of chapter 5). The shell structure and the ionization energies of atoms can be probed by a technique known as photoelectron spectroscopy.



The technique works very much like the photoelectric effect in metals except that we use incident EM radiation (usually X-rays or UV rays) to knock electrons out of individuals atoms, thereby leading to measures of the ionization energies. In this method, the ionization energy IE plays the same role as the work function of a metal.



Suppose a photon of energy $h\nu$ strikes an atom or cation in order to probe one of the ionization energies $IE_i$. The electron will be ejected from the atom or ion and have a residual kinetic energy $p^2/2m_e$. Thus, by energy conservation

\begin{displaymath}
h\nu = IE_i + {p^2 \over 2m_e}
\end{displaymath}

Given the experimental numbers, we can compare the measured ionization energies to the orbital energies generated in a HF calculation. We find a very interesting result. If the energies are ordered according to $\varepsilon_1 < \varepsilon_2 < \cdots < \varepsilon_M$, then the first ionization energy $IE_1$ is approximately equal to the negative of the highest HF energy $-\varepsilon_M$:

\begin{displaymath}
IE_1 \approx -\varepsilon_M
\end{displaymath}

Similarly, the second ionization energy has the property

\begin{displaymath}
IE_2 \approx -\varepsilon_{M-1}
\end{displaymath}

Finally, the $M$th ionization energy $IE_M$ satisfies

\begin{displaymath}
IE_M \approx -\varepsilon_1
\end{displaymath}

These relations are known as Koopmans's approximation.



The essential approximations made in these relations are, firstly, the approximations inherent in the HF theory. But beyond this, it is also assumed that the energies do not change much as the electrons are sequentially stripped out of the atom. Obviously, these energies should change, as the screening and repulsion effects of the other electrons is reduced, and the electrons are drawn closer to the positively charged nucleus, but the effects are small. This type of approximation is known as the frozen core approximation.


next up previous
Next: About this document ... Up: lecture_9 Previous: Hartree-Fock theory
Mark E. Tuckerman 2011-11-03