Page 23 - 71 the abc of dft_opt

4.4. QUESTIONS 45 46 CHAPTER 4. TWO ELECTRONS

fixed Z adjustable Z HF exact
E -3 -3.0625 -3.0833 -3.154
% err -5 -3 -2 0
Table 4.2: Approximate Energies for 1d He.

Exercise 19 Ions on the verge of ionization

1. Using the simple exponential approximation, calculate the critical Z at which 1d He
becomes unstable.
2. Repeat using the exact HF result. Is there an alternative method of estimating Z c in this
case? Which is better?
3. The exact energy is accurately reproduced by
1 < 2 β =
E = −2Z + Z − α + (4.21)
2 Z
where α = 3/4 − 4/(3π) = 0.3255868 and β = 0.0342. Find the critical value of Z.
4. Calculate the ratio of E C to E X at Z c .

4.4 Questions

All the questions below are conceptual.
1. Is the HF estimate of the ionization potential for 1-d He an overestimate or underesti-
mate?
2. Consider the approximate HF calculation given in section 4.2. Comment on what it does
right, and what it does wrong. Suggest a simple improvement.
3. Which is bigger, the kinetic energy of the true wavefunction or that of the HF wavefunc-
tion for 1d He?