Page 50 - 71 the abc of dft_opt

12.5. VIRIAL THEOREM 99 100 CHAPTER 12. SCALING
12.5 Virial theorem or, swapping γ to 1/γ in the derivative,
d
Any eigenstate wavefunction extremizes the expectation value of the Hamiltonian. Thus any 2T[n] + V ee [n] = | γ=1 (T[n γ ] + V ee [n γ ]) . (12.35)
small variations in such a wavefunction lead only to second order changes, i.e., dγ
2
E[φ sol + δφ] = E[φ sol ] + O(δφ ). (12.29) This then is the generalization of Eq. (12.29) for density functional theory: Because Ψ[n]
ˆ
minimizes F, variations induced by scaling (but keeping the density fixed) have zero derivative
ˆ
Assume φ is the ground-state wavefunction of H. Then φ γ is not, except at γ = 1. But around the solution, leading to Eq. (12.35).
ˆ
small variations in γ near 1 lead only to second order changes in % H &, so
d 12.6 Kinetic correlation energy
ˆ
| γ=1 % φ γ | H | φ γ & = 0. (12.30)
dγ
As usual, the more interesting statement comes from the KS quantities. We apply Eq. (12.35)
−2
But, from Eq. (12.4), dT[φ γ ]/dγ = 2γT[φ], and from Eq. (12.5), dV [φ γ ]/dγ = −γ % x dV & to both the interacting and Kohn-Sham systems, and subtract, to find:
dx
at γ = 1, yielding
dV dE XC [n γ ]
2T = % x &. (12.31) E XC [n] + T C [n] = dγ | γ=1 (12.36)
dx
The right-hand side is called the virial of the potential. So this must be satisfied by any Exercise 51 Virial expressed as scaling derivative
solution to the Schr¨odinger equation, and can be used to test how accurately an approximate Prove Eq. (12.36).
solution satisfies it.
This statement is trivially true for Hartree and Exchange energies, so we subtract off the
Note that this is true for any eigenstate and even for any approximate solution, once that exchange contribution, and write:
solution is a variational extremum over a class of wave-functions which admits scaling, i.e., a
class in which coordinate scaling of any member of the class yields another member of that T C [n] = dE C [n γ ] | − E C [n] (12.37)
class. Linear combinations of functions do not generally form a class which admits scaling. dγ γ=1
This is an extremely useful and powerful statement. It tells us how, if we’re given a functional
Exercise 50 Using the virial theorem
Of the exercises we have done so far, which approximate solutions satisfy the virial theorem for the correlation energy, we can extract the kinetic contribution simply by scaling.
Eq. (12.31) exactly, and which do not? What can you deduce in the two different cases? Exercise 52 LDA kinetic correlation energy:
We now derive equivalent formulae for the density functional case. We will first generalize unif
Eq. (12.29), and then Eq. (12.31). As always, instead of considering the ground-state energy Assuming the correlation energy density of a uniform gas is known, e C (r s ), give formulas
unif
unif
ˆ
ˆ
ˆ
itself, we consider just the construction of the universal functional F[n]. Write F = T + V ee . for both t C (r s ) and u C (r s ).
Then Exercise 53 Scaling to find correlation energies:
ˆ
F[n] = %Ψ[n]|F|Ψ[n]&, (12.32)
and, if we insert any other wavefunction yielding the density n(r), we must get a higher 1. By applying Eq. (12.37) to n γ (r), show
number. Consider especially the wavefunctions Ψ γ [n 1/γ ]. These yield the same density, but
are not the minimizing wavefunctions. Then dE C [n γ ]
T C [n γ ] = γ − E C [n γ ] (12.38)
d dγ
ˆ
| γ=1 %Ψ γ [n 1/γ ]|F|Ψ γ [n 1/γ ]& = 0. (12.33)
dγ 2. Next, consider Eq. (12.38) as a first-order differential equation in γ for E C [n γ ]. Show
But expectation values of operators scale very simply when the wavefunction is scaled, so that
1 ∞ dγ #
that E C [n γ ] = −γ #2 T C [n γ !], (12.39)
d ( 2 ) γ γ
| γ=1 γ T[n 1/γ ] + γV ee [n 1/γ ] = 0, (12.34)
dγ using the fact that E C [n γ ] vanishes as γ → 0.