Page 25 - 71 the abc of dft_opt

5.3. CORRELATION 49 50 CHAPTER 5. MANY ELECTRONS

atom Z E HF E E C Exercise 21 Atomic energies
H 1 -0.5 -0.5 0
He 2 -2.862 -2.904 -0.042 This exercise uses the numbers in Table 5.3.
Li 3 -7.433 -7.478 -0.045 1. By plotting ln(−E) versus ln(Z), find the dependence of the total energy on Z for large
Be 4 -14.573 -14.667 -0.094
Z.
B 5 -24.529 -24.654 -0.125
C 6 -37.689 -37.845 -0.156 2. Repeat for the correlation energy.
N 7 -54.401 -54.589 -0.188 3. Plot the correlation energy (a) across the first row and (b) down the last column of the
O 8 -74.809 -75.067 -0.258 periodic table. Comment on the results.
F 9 -99.409 -99.733 -0.324
Ne 10 -128.547 -128.937 -0.39 But we strive for chemical accuracy in our approximate solutions, i.e., errors of less than
2 mH per bond. Another important point to note is that in fact, we often do not need total
Ar 18 -526.817 -527.539 -0.722
Kr 36 -2752.055 -2753.94 -1.89 energies to this level of accuracy, but rather only energy differences, e.g., between a molecule
Xe 54 -7232.138 -7235.23 -3.09 and its separated constituent atoms, in which the correlation contribution might be a much
Rn 86 -22866.745 -22872.5 -5.74 larger fraction, and in which compensating errors might occur in the separate calculation of
the molecule and the atoms. An example of this is the core electrons, those electrons in closed
Table 5.1: Total energies in HF, exactly, and correlation energies across first row and for noble gas atoms.
subshells with energies below the valence electrons, which are often relatively unchanged in
a chemical reaction. A large energy error in their contribution is irrelevant, as it cancels out
To find the orbitals in Hartree-Fock, we must minimize the energy as a functional of
each orbital φ i (r), subject to the constraint of orthonormal orbitals. Doing this, using the of energy diffferences.
Quantum chemistry has developed many interesting ways in which to calculate the corre-
techniques of chapter 2 yields the Hartree-Fock equations
lation energy. These can mostly be divided into two major types: perturbation theory, and
< =
1 2 wavefunction calculations. The first is usually in the form of Moller-Plesset perturbation the-
− ∇ + v ext (r) + v H (r) φ i (r) + f F,i [{φ i }](r) = & i φ(r) (5.14)
2 ory, which treats the Hartree-Fock solution as the starting point, and performs perturbation
where the last contribution to the potential is due to the Fock operator, and is defined by theory in the Coulomb interaction. These calculations are non-variational, and may produce
energies below the ground-state energy. Perhaps the most common type of wavefunction
∗
#
#
N 1 φ (r ) φ i (r )
- j
3 #
f F,i [{φ i }](r) = − d r φ j (r). (5.15) calculation is configuration interaction (CI). A trial wavefunction is formed as a linear com-
#
j |r − r | bintation of products of HF orbitals, including excited orbitals, and the energy minimized.
Note that this odd-looking animal is orbital-dependent, i.e., it is a different function of r for In electronic structure calculations of weakly-correlated solids, most often density functional
each occupied orbital. Also, for just one electron, the Hartree and Fock terms in the potential methods are used. Green’s function methods are often applied to strongly-correlated systems.
cancel, as they should. In order to discuss the pro’s and con’s of HF calculations, we must
Exercise 22 Correlation energy
first discuss the errors it makes.
Show that the correlation energy is never positive. When is it zero?
An interesting paradox to note in chemistry is that most modern chemists think of reac-
5.3 Correlation
tivity in terms of frontier orbitals, i.e., the HOMO (highest occupied molecular orbital) and
the LUMO (lowest unoccupied), and their energetic separation, as in Fig. 1.3. In the wave-
The definition of correlation energy remains the same for N electrons as for two: It is the
function approach, these are entirely constructs of the HF approximation, which makes errors
error made by a Hartree-Fock calculation. Table 5.3 lists a few correlation energies for
atoms. We see that correlation energies are a very small (but utterly vital) fraction of the of typically 0.2 Hartrees in binding energies. Thus, although this approximation obviously
contains basic chemical information, needed for insight into chemical reactivity, the resulting
total energy of systems. They are usually about 20-40 mH/electron, a result we will derive
later. thermochemistry is pretty bad. On the other hand, to obtain better energetics, one adds
many more terms to the approximate wavefunction, losing the orbital description completely.