Page 30 - 71 the abc of dft_opt

6.2. HOHENBERG-KOHN THEOREMS 59 60 CHAPTER 6. DENSITY FUNCTIONAL THEORY
from the variational principle, where the search is over all normalized positive densities. Any negative of the external potential (up to a constant). Note that it would be marvellous if
such wavefunction can then be fed into Eq. (6.7) to construct the unique corresponding we could find an adequate approximation to F for our purposes, so that we could solve Eq.
potential. We denote the minimizing wavefunction in Eq. (6.8) by Ψ[n]. This gives us a (6.13) directly. It would yield a single integrodifferential equation to be solved, probably
verbal definition of the ground-state wavefunction. The exact ground-state wavefunction of by a self-consistent procedure, for the density, which could then be normalized and inserted
density n(r) is that wavefunction that yields n(r) and has minimizes T + V ee . We may back into the functional E[n], to recover the ground-state energy. In the next section, we
also define the exact kinetic energy functional as will examine the original crude attempt to do this (Thomas-Fermi theory), and find that,
although the overall trends are sound, the accuracy is insufficient for modern chemistry and
@ @
T[n] = %Ψ[n] @ T @ Ψ[n]&, (6.10) HF
@
@
@ ˆ@
materials science. Note also that insertion of F [n] will yield an equation for the density
and the exact electron-electron repulsion functional as equivalent to the orbital HF equation.
@ @ An important formal question is that of v-representability. The original HK theorem was
@ @
V ee [n] = %Ψ[n] @ V ee@ Ψ[n]&. (6.11) proven only for densities that were ground-state densities of some interacting electronic prob-
@ˆ @
The second Hohenberg-Kohn theorem states that the functional F[n] is universal, i.e., lem. The constrained search formulation extended this to any densities extracted from a
it is the same functional for all electronic structure problems. This is evident from Eq. single wavefunction, and this domain has been further extended to densities which result
from ensembles of wavefunctions. The interested reader is referred to Dreizler and Gross for
(6.8), which contains no mention of the external potential. To understand the content of
this statement, consider some of our previous problems. Recall our orbital treatment of a thorough discussion.
2
one-electron problems. The kinetic energy functional, T[φ] = 1 2 ! dx |φ (x)| , is the same
#
functional for all one-electron problems. When we evaluate the kinetic energy for a given 6.3 Thomas-Fermi
trial orbital, it is the same for that orbital, regardless of the particular problem being solved.
Similarly, when we treated the two electron case within the Hartree-Fock approximation, we In this section, we briefly discuss the first density functional theory (1927), its successes, and
approximated its limitations. Note that it predates Hartree-Fock by three years.
In the Thomas-Fermi theory, F[n] is approximated by the local approximation for the
2
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1
1
1
3
3
F[n] ≈ F HF [n] = 1 d r |∇n| + 1 d r d r n(r)n(r ) . (6.12) (non-interacting) kinetic energy of a uniform gas, plus the Hartree energy
3 #
8 n 4 |r − r | 1 1 1 #
#
3
3
3 #
Then minimization of the energy functional in Eq. (6.9) with F HF [n] yields precisely the F TF [n] = A S d rn 5/3 (r) + 1 d r d r n(r)n(r ) . (6.14)
2 |r − r | #
Hartree-Fock equations for the two-electron problem. The important point is that this single
formula for F HF [n] is all that is needed for any two-electron Hartree-Fock problem. The Several points need to be clarified. First, these expressions were developed for a spin-
second Hohenberg-Kohn theorem states that there is a single F[n] which is exact for all unpolarized system, i.e., one with equal numbers of up and down spin electrons, in a spin-
independent external potential. Second, in the kinetic energy, we saw in Chapter ?? how the
electronic problems.
power of n can be deduced by dimensional analysis, while the coefficient is chosen to agree
2 2/3
Exercise 25 Errors in Hartree-Fock functional with that of a uniform gas, yielding A S = (3/10)(3π ) . We will derive these numbers in
Comment, as fully as you can, on the errors in F HF [n] relative to the exact F[n] for two detail in Chapter 8. They can also be derived by classical arguments applied to the electronic
electrons. fluid.
The last part of the Hohenberg-Kohn theorem is the Euler-Lagrange equation for the Insertion of this approximate F into the Euler-Lagrange equation yields the Thomas-Fermi
energy. We wish to mininize E[n] for a given v ext (r) keeping the particle number fixed. We equation:
5 1 n(r )
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therefore minimize E[n] − µN, as in chapter ??, and find the Euler-Lagrange equation: A S n 2/3 (r) + d r + v ext (r) = µ (6.15)
3 #
3 |r − r | #
δF
+ v ext (r) = µ. (6.13) We will focus on its solution for spherical atoms, which can easily be achieved numerically with
δn(r)
a simple ordinary differential equation solver. We can see immediately some problems. As
We can identify the constant µ as the chemical potential of the system, since µ = ∂E/∂N. r → 0, because of the singular Coulomb external potential, the density is singular, n → 1/r 3/2
2
The exact density is such that it makes the functional derivative of F exactly equal to the (although still integrable, because of the phase-space factor 4πr ). Again, as r → ∞,