Lecture 4. Many-Electron Atoms. Pt.2
Variation Principle &
Application to the He Ground State
References
•
•
•
•
•
Ratner Ch. 7.2-7.4, Engel Ch. 10.4, Pilar Ch. 6.1-6.4
Modern Quantum Chemistry, Ostlund & Szabo (1982) Ch. 1
Quantum Chemistry, McQuarrie (1983) Ch. 7-8
Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005), Ch.7
Computational Chemistry, Lewars (2003), Ch.4
• A Brief Review of Elementary Quantum Chemistry
http://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.html
http://vergil.chemistry.gatech.edu/notes/hf-intro/hf-intro.html
Solution of the Schrödinger equation for He atom
electron
correlation
~H
atom
electron
~H atom electron
h(1)
h(2)
at r1
at r2
Initial set up for many-electron (many-body) wave function
Hartree product of
1-electron wave function
(~ H atom orbital) (1928)
asymmetric
Introduction of spin
Antisymmetry
Slater
determinant
(1929)
How do they look like?
 1s(H)
Atomic Units (a.u.)
?
Perturbation method?
(Read Ratner Ch. 5.4, 7.2 & Appendix B)
Variation method?
(Read Ratner Ch. 7.3, Engel 10.4 & Prove the theorem)
Apply these methods to the Hartree product first
and then to the Slater determinant
(almost transparent, because the Hamiltonian does not
include any spin operator)
Evaluate the energy of the Hartree product first.
1)
2)
(-0.5Eh for H)
(experimentally determined; IE1 + IE2)
IE1=?, IE2=? (quiz)
Evaluate the energy of the Hartree product first.
1)
2)
(-0.5Eh for H)
(experimentally determined; IE1 + IE2)
IE1=0.904, IE2=2.000
Variational theorem and Variational method
If you know the exact (true) total energy eigenfunction
True ground state energy
For any approximate (trial) ground state wave function
=
Better trial function
=E
true ground state
Lower E (closer to E0)
Minimize E[] by changing !
Proof: Variational theorem
Variation method: an example
h2

8mL
We cannot do better with this (limited) trial wave function.
Example:
Particle in a box
ground state
Variation method: Hartree product for He atom
?
smaller
larger
Source: www.chemix-chemistry-software.com/school/periodic_table/atomic-radius-elements.html
www.periodictable.com/Properties/A/AtomicRadius.v.wt.html
Variation method: Better wave function for He atom
Variation method: Better wave function for He atom