Variation of electronic energy levels $E_{nl}$ of a many-electron with $n$ and $l$

Question

Within the central field approximation, the electronic energy levels $E_{nl}$ depend on both $n$ and $l$.

1. For a given value of $n$, $E_{nl}$ increases with increasing value of $l$.

2. For a given value of $l$, $E_{nl}$ increases with increasing value of $n$.

I can understand (1) because increasing $l$ will increase the repulsive centrifugal term, and hence raise the energy. In other words, electrons orbitals with lower values of $l$ penetrate closer to the nucleus so that the effect of screening is reduced and hence have lower energy. But how do we explain (2)? For a fixed $l$, why does $E_{nl}$ increase with increasing $n$? Unlike the hydrogen atom, we do not have an analytical formula for $E_{nl}$ in general and I do not understand the physical reasoning given in page 359, second paragraph.

Answer 1

At fixed $l$, for each $n$ above the label of first electronic level (ground state), the corresponding eigenstate must be orthogonal to the previous ones. It is this constraint which forces the wavefunction to give larger weight on distances in average larger than the previous $n$ states, thus justifying a larger energy.

More formally, looking at the solution of a Schrödinger-like equation as a variational problem, the ground state is obtained by minimizing $$ \left<\psi_0 \left| H \right| \psi_0 \right > - \lambda (\left<\psi_0 \left|\right. \psi_0 \right > -1 ) $$ with respect to the state $\left|\psi_0 \right >$, $\lambda$ being the Lagrange multiplier which takes care of the normalization constraint and which corresponds to the energy. The extremum condition with respect to $\left|\psi_0 \right >$ provides the time independent Schrödinger-like equation $$ \left. \left. H \right| \psi_0 \right >= \lambda \left|\psi_0 \right > $$ I am using the term "Schrödinger-like" because the same considerations apply to a pure Schrödinger equation but also to non-linear equations corresponding to all kind of self-consistent field approsimations, including the Kohn-Sham equations coming from Density Functional Theory for the electronic states.

The next-above-the-minimum-energy state will be the solution of a variational problem for a new state $\left|\psi_1 \right >$, where an additional constraint is added, thus an additional term proportional to the scalar product $\left<\psi_1 \left|\right. \psi_0 \right >$ appears. The resulting Schrödinger-like equation will contain a term proportional to $\left|\psi_0 \right >$, i.e. the ground state wavefunction acts as an additional interaction term which, in coordinate representation, forces the state $\left|\psi_1 \right >$ to have a nodal surface and to assign a higher weight to larger distances. By construction, this state must have an energy larger than the ground state. The analysis of the effect of the extra term allows to rationalize the origin of such larger energy.

Answer 2

Sketch the effective potential energy curve $U_l(r)$ at fixed $l$, in the central field approximation. By this I mean that $U_l(r)$ includes the electrostatic effects from nucleus and the average over the electrons, and it also includes the centrifugal term $l(l+1)/r^2$ from rotational part of kinetic energy. Then as you go up in $n$ at given $l$ you are going up the energy levels in this well. Admittedly the well changes shape a bit as you go, but this gives a good physical insight I think. As $n$ increases there is an increase in both kinetic and potential energy. The former is illustrated by the increased number of nodes and associated $d^2 \psi/dr^2$; the latter is illustrated by the wavefunction extending further in the well.