Page 9 - 71 the abc of dft_opt

1.2. WHAT IS A KOHN-SHAM CALCULATION? 17 18 CHAPTER 1. INTRODUCTION
distances are given in Bohr radii (a o = 0.529 ˚ A). In this example, the electrons are in a where Φ(r 1 , r 2 ) = φ 0 (r 1 )φ 0 (r 2 ). This is a much simpler set of equations to solve, since it
spin singlet, so that their spatial wavefunction Ψ(r 1 , r 2 ) is symmetric under interchange of only has 3 coordinates. Even with many electrons, say N, one would still need to solve only
r 1 and r 2 . Solution of Eq. (1.1) is complicated by the electrostatic repulsion between the a 3-D equation, and then occupy the first N/2 levels, as opposed to solving a 3N-coordinate
particles, which we denote as V ee . It couples the two coordinates together, making Eq. (1.1) Schr¨odinger equation. If we can get our non-interacting system to accurately ’mimic’ the
a complicated partial differential equation in 6 coordinates, and its exact solution can be quite true system, then we will have a computationally much more tractable problem to solve.
demanding. In Fig. 1.1 we plot the results of such a calculation for the total energy of the How do we get this mimicking? Traditionally, if we think of approximating the true wave-
molecule, E +1/R, the second term being the Coulomb repulsion of the nuclei. The position function by a non-interacting product of orbitals, and then minimize the energy, we find the
of the minimum is the equilibrium bond length, while the depth of the minimum, minus the Hartree-Fock equations, which yield an effective potential: 2
zero point vibrational energy, is the bond energy. More generally, the global energy minimum HF 1 1 n(r )
#
3 #
determines all the geometry of a molecule, or the lattice structure of a solid, as well as all the v S (r) = v ext (r) + 2 d r |r − r | . (1.6)
#
vibrations and rotations. But for larger systems with N electrons, the wavefunction depends
on all 3N coordinates of those electrons. The correction to the external potential mimics the effect of the second electron, in particular
(kcal/mol) 0 GGA as −1/r, reflecting an effective charge of Z − 1. Note that insertion of this potential into
screening the nuclei. For example, at large distances from the molecule, this potential decays
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exact
HF
LDA
Eq. (1.5) now yields a potential that depends on the electronic density, which in turn is
calculated from the solution to the equation. This is termed therefore a self-consistent set
E(H 2 )+1/R-E(2H) -100 of equations. An initial guess might be made for the potential, the eigenvalue problem is
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then solved, the density calculated, and a new potential found. These steps are repeated
until there is no change in the output from one cycle to the next – self-consistency has been
reached. Such a set of equations are often called self-consistent field (SCF) equations. In
Fig. 1.1, we plot the Hartree-Fock result and find that, although its minimum position is very
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1
0.8
0.6
0.4
this method, and traditional methods attempt to improve the wavefunction to get a better
R( ˚ A) 1.2 1.4 1.6 accurate, it underbinds the molecule significantly. This has been a well-known deficiency of
energy. The missing piece of energy is called the correlation energy.
Figure 1.1: The total energy of the H 2 molecule as a function of internuclear separation.
In a Kohn-Sham calculation, the basic steps are very much the same, but the logic is
entirely different. Imagine a pair of non-interacting electrons which have precisely the same
We note at this point that, with an exact ground-state wavefunction, it is easy to calculate
density n(r) as the physical system. This is the Kohn-Sham system, and using density
the probability density of the system:
functional methods, one can derive its potential v S (r) if one knows how the total energy
1 2
n(r) = 2 d r |Ψ(r, r )| . (1.4) E depends on the density. A single simple approximation for the unknown dependence of
3 #
#
the energy on the density can be applied to all electronic systems, and predicts both the
3
The probability density tells you that the probability of finding an electron in d r around r is energy and the self-consistent potential for the fictitious non-interacting electrons. In this
3
n(r)d r. For our H 2 molecule at equilibrium, this would look like the familiar two decaying view, the Kohn-Sham wavefunction of orbitals is not considered an approximation to the
exponentials centered over the nuceli, with an enhancement in between, where the chemical exact wavefunction. Rather it is a precisely-defined property of any electronic system, which
bond has formed. is determined uniquely by the density. To emphasize this point, consider our H 2 example in
Next, imagine a system of two non-interacting electrons in some potential, v S (r), chosen the united atom limit, i.e., He. In Fig. 1.2, a highly accurate many-body wavefunction for
somehow to mimic the true electronic system. Because the electrons are non-interacting, the He atom was calculated, and the density extracted. In the bottom of the figure, we plot
their coordinates decouple, and their wavefunction is a simple product of one-electron wave- both the physical external potential, −2/r, and the exact Kohn-Sham potential. 3 Two non-
functions, called orbitals, satisfying: interacting electrons sitting in this potential have precisely the same density as the interacting
1 2 2 For the well-informed, we note that the correction to the external potential consists of the Hartree potential, which is double that shown above,
2 3
− ∇ + v S (r) φ i (r) = & i φ i (r), (1.5) less the exchange potential, which in this case cancels exactly half the Hartree.
2 3 It is a simple exercise to extract the exact Kohn-Sham potential from the exact density, as in section ??.