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