Page 49 - 71 the abc of dft_opt

12.4. CORRELATION INEQUALITIES 97 98 CHAPTER 12. SCALING
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E LDA [n γ ] = −γd 0 d r n(r)/r s (r), vanishing linearly with γ, correctly. But most systems of This is the fundamental inequality of uniform scaling, as it tells us inequalities about how
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chemical interest are closer to the high-density limit, where LDA fails. correlation contributions scale.
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To see an example, apply Eq. (12.21) to n = n γ and write γ = 1/γ, to yield
Exercise 45 Extracting exchange
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If someone gives you an exchange-correlation functional E XC [n], define a procedure for F[n ] ≤ T[n !]/γ + V ee [n !]/γ . (12.22)
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extracting the exchange contribution.
But we can just drop the primes in this equation, and multiply through by γ, and add
As usual, one should not focus too much on the failings of the correlation energy alone. 2
Remember, exchange scales linearly, and dominates correlation. When we add both contri- (γ − γ)T[n] to both sides:
butions, we find Fig. 12.3. Clearly, the XC energy is best at γ = 1, and worsens as γ grows, γ T[n] + γV ee [n] ≤ T[n γ ]/γ + V ee [n γ ] + (γ − γ)T[n]. (12.23)
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-1 Now the left-hand-side equals the right-hand-side of Eq. (12.21), so we can combine the two
-2 He atom equations, to find
-3 (γ − 1) T[n γ ] ≤ γ (γ − 1) T[n]. (12.24)
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E XC [n γ ] -4 Note that for γ > 1, we can cancel γ − 1 from both sides, to find T[n γ ] < γ T[n]. As
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-7 we scale the system to high density, the physical kinetic energy grows less rapidly than the
-8 non-interacting kinetic energy. For γ < 1, the reverse is true, i.e.,
exact
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LDA
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-10 T[n γ ] ≤ γ T[n] γ > 1, T[n γ ] ≥ γ T[n] γ < 1. (12.25)
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γ Exercise 47 Scaling V ee
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Figure 12.3: Exchange-correlation energy of the He atom, both exactly and within LDA, as the density is squeezed.
because of the cancellation of errors. V ee [n γ ] ≥ γV ee [n] γ > 1, V ee [n γ ] ≤ γV ee [n] γ < 1. (12.26)
Exercise 46 Scaling LDA in Wigner approximation: These inequalities actually provide very tight bounds on these large numbers. But of
greater interest are the much smaller differences with Kohn-Sham values, i.e., the correlation
Using the correct exchange formula and the simple Wigner approximation for the correlation contributions. So
energy of the uniform gas, Eq. (8.18), and the simple exponential density for the He atom,
calculate the curves shown in Figs. 12.2 and 12.3. Exercise 48 Scaling E C
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12.4 Correlation inequalities E C [n γ ] ≥ γE C [n], γ > 1, E C [n γ ] ≤ γE C [n], γ < 1, (12.27)
This reasoning breaks down when correlation is included, because this involves energies eval- and
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uated on the physical wavefunction. As mentioned above, a key realization there is that the T C [n γ ] ≤ γ T C [n], γ > 1, T C [n γ ] ≥ γ T C [n], γ < 1. (12.28)
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scaled ground-state wavefunction is not the ground-state wavefunction of the scaled density,
because the physical wavefunction minimizes both T and V ee simultaneously. However, we Exercise 49 Scaling inequalities for LDA
can still use the variational principle to deduce an inequality. We write Give a one-line argument for why LDA must satisfy the scaling inequalities for correlation
@> ?@ @> ?@
@ ˆ ˆ @ @ ˆ ˆ @ energies.
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F[n γ ] = %Ψ[n γ ] @ T + V ee @ Ψ[n γ ]& ≤ %Φ γ [n] @ T + V ee @ Φ γ [n]& (12.20)
or In fact, for most systems we study, both E C and T C are relatively insensitive to scaling the
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F[n γ ] ≤ γ T[n] + γV ee [n] (12.21) density toward the high-density limit, so these inequalities are less useful.