Download Electron Atoms: Hamiltonian, Ground State, Correlation, Ionization, and Koopman's Theorem and more Exercises Physical Chemistry in PDF only on Docsity!
Many Electron Atoms
1. Please see the PowerPoint slides and class notes for answers to these qualitative questions. 2. (a)
(b)
^ ^ ^
3. Be: Z = 4, Ground State Configuration: 1s
2 2s
2
(a) Hamiltonian
H H KE H Electron ( / Nucleus Attraction ) H Electron / Electron Re pulsion
SI Units 2 2 2 2 2 2 2 2 1 1 2 2 3 3 4 4
2 2 2 2
0 1 0 2 0 3 0 4 2 2 2 2 2 2
0 12 0 13 0 14 0 23 0 24 0 34
H r r r r m m m m
e e e e
r r r r
e e e e e e
r r r r r r
Atomic Units
2 2 2 2 1 1 2 2 3 3 4 4
1 2 3 4
12 13 14 23 24 34
H r r r r
r r r r
r r r r r r
(b) Ground State Slater Determinant
1 1 1 1
2 2 2 2
3 3 3 3
4 4 4 4
1 1 (2)^ 1 (2)^ 2 (2)^ 2 (2)
24 1 (3)^ 1 (3)^ 2 (3)^ 2 (3)
s s s s
s s s s
s s s s
s s s s
4. (a) Correlation Energy of Lithium
EHF 7.432 au
3
exp 1 2 3 1
exact n n
E E IE IE IE IE eV
au eV au eV
E corr Eexact E (^) HF au au
kJ au kJ mol kJ mol au
(b) Ionization Energy and Electron Affinity
(i) Ionization Energy (IE 1 ): Li Li
**+
-
IE 1 (HF) = EHF (Li
) - EHF (Li) = -7.236 au - (-7.432 au) = 0.196 au x 27.21 eV/au = 5.33 eV
The experimental First Ionization Energy of Li is IE 1 (exp) = 5.39 eV
A more accurate QM treatment which includes correlation energy would yield a result
very near to experiment.
(ii) Electron Affinity (EA): Li + e
- Li -
EA(HF) = EHF (Li
- ) - EHF (Li) = -7.427 au - (-7.432 au) = +0.005 au x 27.21 eV/au
= +0.14 eV
5. Koopman's Theorem: The Ionization Energy is:
IE 1 - HOMO = -(-0.616 au) = +0.616 au x 2625 kJ/mol / 1 eV = +1617 kJ/mol
vs. IE 1 (exp) = +1314 kJ/mol.
The reasons for the deviation from experiment are:
(i) Koopman's Theorem neglects relaxation of the remaining n-1 electrons in the ion.
(ii) Differences in the Correlation Energy in the ion and neutral atom are neglected.
