Page 52 - 71 the abc of dft_opt

104 CHAPTER 13. ADIABATIC CONNECTION
ˆ
Furthermore, if λ simply multiplies V , then
dE λ λ ˆ λ
= % φ | V | φ &, (13.6)
dλ
Chapter 13 so that 1 1
λ ˆ
λ
E = E λ=0 + dλ % φ | V | φ &. (13.7)
0
Adiabatic connection This is often called the Hellmann-Feynman theorem.
Exercise 57 Hellmann-Feynman theorem
Show that the 1-d H-atom and 1-d harmonic oscillator satisfy the Hellman-Feynman theorem.
1
In this chapter, we introduce an apparently new formal device, the coupling constant of How does the approximate solution using a basis set do?
the interaction for DFT. This differs from that of regular many-body theory. But we see
In modern density functional theory, one of the most important relationships is between
at the end that this is very simply related to scaling.
the coupling constant and coordinate scaling. We can see this for these simple 1-d problems.
The Schr¨odinger equation at coupling constant λ is
13.1 One electron > ?
ˆ
ˆ
λ
λ λ
T + λV (x) φ (x) = E φ (x) (13.8)
ˆ
Introduce a parameter in H, say λ. Then all eigenstates and eigenvalues depend on λ, i.e.,
Eq. (1.9) becomes: If we replace x by γx everywhere, we find
=
ˆ λ λ
λ λ
H φ = ε φ . (13.1) < 1 T + λV (γx) φ (γx) = E φ (γx) (13.9)
ˆ
ˆ
λ λ
λ
γ 2
Most commonly,
ˆ
ˆ
ˆ λ
ˆ
H = T + λV , 0 < λ < ∞. (13.2) Furthermore, if V is homogeneous of degree p,
> ?
2
ˆ
2 p ˆ
λ
λ λ
A λ in front of the potential is often called the coupling constant, and then all eigenvalues T + λγ γ V (x) φ (γx) = γ E φ (γx) (13.10)
and eigenfunctions depend on λ. For example, for the harmonic oscillator of force constant
2
2
k = ω , the ground-state wavefunction is φ(x) = (ω/π) 1/4 exp(−ωx /2). Since V (x) = Thus, if we choose λ γ p+2 = 1, then Eq. (13.10) is just the normal Schr¨odinger equation,
√
2 2 λ λ
kx /2, λV (x) = λkx /2, i.e., the factor of λ multiplies k. Then ω → λω, and so and we can identify φ (γx), which is equal (up to normalization) to φ (x) with φ(x). If we
γ
scale both of these by 1/γ, we find
√  1/4
λω √ 2
λ
λ
φ (x) =   exp(− λωx /2) (harmonic oscillator). (13.3) φ (x) = φ 1/γ (x) = φ 1/(p+2)(x) (13.11)
π λ
For a λ-dependent Hamiltonian, we can differentiate the energy with respect to λ. Be- In this case, changing coupling constant by λ is equivalent to scaling by λ 1/4 , but this
cause the eigenstates are variational extrema, the contributions due to differentiating the relation depends on the details of the potential.
wavefunctions with respect to λ vanish, leaving
Exercise 58 λ-dependence
ˆ λ
dE λ λ ∂H λ Show that, for a homogeneous potential of degree p,
= % φ | | φ &. (13.4)
dλ ∂λ λ 2
E = λ p+2 E (13.12)
Thus
ˆ λ
1 1 ∂H Exercise 59 λ-dependence of 1-d H
λ
λ
E = E λ=1 = E λ=0 + dλ % φ | | φ &. (13.5)
0 ∂λ Find the coupling-constant dependence for the wavefunction and energy for the 1-d H-atom,
1 2
V (x) = −δ(x), and for the 1-d harmonic oscillator, V (x) = x .
1 c !2000 by Kieron Burke. All rights reserved. 2
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