Page 51 - 71 the abc of dft_opt

12.7. POTENTIAL 101 102 CHAPTER 12. SCALING
3. Show Exercise 56 Virial for LDA:
dE C [n γ ]
U C [n γ ] = 2E C [n γ ] − γ (12.40) Show that the XC virial theorem is satisfied in an LDA calculation.
dγ
and 1 ∞ dγ # 12.8 Questions
E C [n γ ] = γ 2 U C [n γ !]. (12.41)
γ γ #3
1. What is the relation between scaling and changing Z for the H-atom?
4. Combine Eqs. (12.40) and (12.39) to find differential and integral relations between U C
and T C . 2. Is the virial theorem satisfied for the ground-state of the problem V (x) = − exp(−|x|)?
Exercise 54 Scaling of potentials: 3. For a homogeneous potential of degree p, e.g., p=-1 for the delta function well, we know
E = −T. Can we find the ground-state by simply maximizing T?
If v C [n](r) = δE C [n]/n(r), and I define E C [n](γ) = E C [n γ ], how is v C [n](γ, r) = δE C [n](γ)/n(r)
4. What is the exact kinetic energy density functional for one electron in one-dimension?
related to v C [n](r)?
How does it scale?
Exercise 55 Scaling LDA in Wigner approximation: loc
5. Does T [n] from chapter ?? satisfy the correct scaling relation? (Recall question 1 from
S
chapter ??).
Within the simple Wigner approximation for the correlation energy of the uniform gas,
Eq. (8.18), deduce both the potential and kinetic contributions, u unif (r S ) and t unif (r S ), 6. If Ψ is the ground-state for an interacting electronic system of potential v ext (r), of what
C C
respectively. Then check all 6 relations of the previous exercise. is Ψ γ a ground-state?
12.7 Potential

After our earlier warm-up exercises, it is trivial to derive the virial theorem for the exchange-
correlation potential. For an arbitrary system, the virial theorem for the ground-state yields:
N @ @
- @ @
2T = %Ψ @ r i · ∇ i V @ Ψ&, (12.42)
@
ˆ @
i=1
no matter what the external potential. For our problems, the electron-electron repulsion is
homogeneous of order -1, and so
1
3
2T + V ee = d r n(r)r · ∇v ext (r). (12.43)
This theorem applies equally to the non-interacting and interacting systems, and by subtract-
ing the difference, we find
1
3
E XC [n] + T C [n] = − d r n(r) r · ∇v XC (r). (12.44)
Since we can either turn off the coupling constant or scale the density toward the high-density
limit, this applies separately to the exchange contributions to both sides, and the remaining
correlation contributions:
1 3
E C [n] + T C [n] = − d r n(r) r · ∇v C (r). (12.45)
For almost any approximate functional, once a Kohn-Sham calculation has been cycled to
self-consistency, Eq. (12.44) will be satisfied. It is thus a good test of the convergence of
such calculations, but is rarely performed in practice.