Page 26 - 71 the abc of dft_opt

5.4. ATOMIC CONFIGURATIONS 51 52 CHAPTER 5. MANY ELECTRONS
We will see later how DFT resolves this paradox, by showing how an orbital calculation can this quantity for atoms. All atomic densities, when plotted as function of r with the nucleus
in principle yield the exact energetics. 25
Ar atom
20
5.4 Atomic configurations
4πr 2 n(r) 15
We have seen how the Hartree-Fock equations produce a self-consistent set of orbitals with
orbital energies. In many cases, the true wavefunction has a strong overlap with the HF 10
wavefunction, and so the behavior of the interacting system can be understood in terms of
5
the HF orbitals. We occupy the orbitals in order.
This is especially true for the atoms, and the positions of the HF orbitals largely determine 0
0 0.5 1 1.5 2 2.5 3
the strucutre of the periodic table. If we first ignore interaction, and consider the hydrogenic
levels, we get the basic idea. For Hydrogenic atoms (N = 1), there is an exact degeneracy r
between all orbitals of a given principal quantum number. For each n, there are n values of Figure 5.1: Radial density in Ar atom.
l, ranging from 0 to n − 1. For each l, there are 2l + 1 values of m l = −l, ..., 0, ..l. Thus
each closed shell contains at the origin, have been found to decay monotonically, although that’s never been generally
proven. Close to a nucleus, the external potential dominates in the Schr¨odinger equation,
n−1
- 2
g n = (2l + 1) = n(n − 1) + n = n (5.16) and causes a cusp in the density, whose size is proportional to the nuclear charge:
l=0
dn
2
Since each orbital can hold 2 electrons, the nth-shell contains 2n electrons. The first shell dr | r=0 = −2Z n(0) (5.17)
closes at N = 2, the next at 10, the next at 28, and so forth. for a nucleus of charge Z at the origin. This is Kato’s cusp condition, and we already saw
This scheme works up to Ar, but then fails to account for the transition metals. To examples of this in earlier exercises. It is also true in our one-dimensional examples with
understand these, we must consider the actual HF orbitals, which are not hydrogenic. Since delta-function external potentials.
2
the HF potentials are not Coulombic, the degeneracy w.r.t. different l-values is lifted, and Typically, spherical densities are plotted multiplied by the phase-space factor 4πr . This
we have & nl . As expected, for fixed n, the higher l, the less negative the energy. Thus the 2s means the area under the curve is precisely N. Furthermore, the different electronic shells
orbitals lie lower than 2p. This effect does not change the filling order described above. are easily visible. We take the Ar atom as an example. In Fig. 5.1, we plot its radial density.
But for Z between 7 and 20, the 4s orbital dips below the 3d. So it gets filled before the This integrates to 18 electrons. We find that the integral up to r = 0.13 contains 2 electrons,
2
3d, and, for example, the ground-state configuration of Ca is [Ar]4s . Then the 3d starts up to 0.25 contains 4, 0.722 contains 10, and 1.13 contains 12. These correspond to the 2
filling, and so the transition metals appear a little late. Even when the 3d orbital energy drops 1s electrons, 2 2s electrons, 6 2p electrons, and 2 3s electrons, respectively. Note that the
below the 4s, the total energy remains lower with the 4s filled. The situation is similar for peaks and dips in the radial density roughly correspond to these shells.
higher n. The decay at large distances is far more interesting. When one coordinate in a wavefunction
A list of the filled orbitals is called the electronic configuration of a system. This still does is taken to large distances from the nuclei, the N-electron ground-state wavefunction collapses
not determine the ground-state entirely, as many different angular momenta combinations, to the product of the square-root of the density times the (N −1)-electron wavefunction. This
both spin and orbital, (terms) can have the same configuration. Hund’s rules are used to means the square-root of the density satisfies a Schr¨odinger-like equation, whose eigenvalue
choose which one has the lowest energy. Our treatment will not require details beyond is the difference in energies between the two systems:
knowing the lowest configuration. % β
n(r) = Ar exp(−αr) (r → ∞) (5.18)
√
where α = 2I, and
5.5 Atomic densities
I = E(N − 1) − E(N) (5.19)
Since we will be constructing a formally exact theory of interacting quantum mechanics based is first ionization potential. In fact, the power can also be deduced, β = (Z − N + 1)/α − 1,
on the one-electron density, it seems appropriate here to discuss and show some pictures of and A is some constant. Thus, a useful and sensitive function of the density to plot for