1. Lecture 6. Many-Electron Atoms. Pt.4. Physical significance of Hartree-Fock solutions: Electron correlation, Aufbau principle, Koopmans’ theorem & Periodic trends References Ratner Ch. 9.5-, Engel Ch. 10.5-, Pilar Ch. 10 Modern Quantum Chemistry, Ostlund & Szabo (1982) Ch. 3.3 Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005), Ch.7 Computational Chemistry, Lewars (2003), Ch. 5 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

2. Electron-electron repulsion Indistinguishability Helium Atom First (1 nucleus + 2 electrons) (Review) We cannot solve this Schrödinger equation analytically. (Two electrons are not separable nor independent any more.)  A series of approximations will be introduced. 1. Electron-electron repulsion (correlation) The 1/r 12 term removes the spherical symmetry in He. ~H atom electron at r 1 ~H atom electron at r 2 newly introduced : Correlated, coupled

3. Hartree-Fock equation (One-electron equation) spherically symmetric & -Two-electron repulsion operator (1/r ij ) is replaced by one-electron operator V HF (i), which takes it into account in an “average” way. -Any one electron sees only the spatially averaged position of all other electrons. -V HF (i) is spherically symmetric. - (Instantaneous, dynamic) electron correlation is ignored. -Spherical harmonics (s, p, d, …) are valid angular-part eigenfunctions (as for H-like atoms). -Radial-part eigenfunctions of H-like atoms are not valid any more. optimized V eff includes

4. A single Slater determinant never corresponds to the exact wave function. E HF > E 0 (the exact ground state energy) Correlation energy: a measure of error introduced through the HF scheme E C = E 0  E HF (< 0) –Dynamical correlation –Non-dynamical (static) correlation Post-Hartree-Fock method (We’ll see later.) –Møller-Plesset perturbation: MP2, MP4, … –Configuration interaction: CISD, QCISD, CCSD, QCISD(T), … –Multi-configuration self-consistent-field method: MCSCF, CAFSCF, … Electron Correlation (P.-O. Löwdin, 1955) Ref) F. Jensen, Introduction to Computational Chemistry, 2 nd ed., Ch. 4

5. Solution of HF-SCF equation gives

6. Solution of HF-SCF equation: Z-  (measure of shielding) 00.31 1.722.09 8.49 8.69 2.42 2.58 2.78 2.86 3.15 3.17 3.51 3.55 3.87 3.90 4.24 8.88 8.93 9.10 9.71 9.36 10.11 9.73 10.52 9.93 10.88 10.24 11.24 more shielded less shielded

7. Solution of HF-SCF equation: Effective nuclear charge (Z-  is a measure of shielding.) higher energy, bigger radiuslower energy, smaller radius

8. www.periodictable.com/Properties/A/AtomicRadius.v.wt.html Source: www.chemix-chemistry-software.com/school/periodic_table/atomic-radius-elements.html larger smaller

9. As well as the total energy, one also obtains a set of orbital energies. Remove an electron from occupied orbital a. Orbital energy = Approximate ionization energy Physical significance of orbital energies (  i ): Koopmans’ theorem (T. C. Koopmans, 1934) Physica, 1, 104 Ostlund/Szabo Ch.3.3

10. length energy Atomic orbital energy levels & Ionization energy of H-like atoms Total energy eigenvalues are negative by convention. (Bound states) depend only on the principal quantum number. 1 Ry Minimum energy required to remove an electron from the ground state IE (1 Ry for H)    atomic units

11. Koopmans’ theorem: Validation from experiments

12. Hartree-Fock orbital energies  i & Aufbau principle degenerate For H-like atoms ” ” Hartree-Fock orbital energies  i depend on both the principal quantum number (n) and the angular quantum number (l). Within a shell of principal quantum number n,  ns   np   nd   nf  …   

13. Aufbau (Building-up) principle for transition metals 10.3

14. Aufbau (Building-up) principle for transition metals

15. Electronegativity (~ IE + EA) ~Lowest Unoccupied AO/MO (LUMO) ~Highest Occupied AO/MO (HOMO) small high low or deep small large Na  + Cl +    NaCl    Na + + Cl 

16. Periodic trends of many-electron atoms

17. Periodic trends of many-electron atoms: Electronegativity http://www.periodictable.com/Properties/A/Electronegativity.bt.wt.html

18. Periodic trends of many-electron atoms: 1 st ionization energy http://www.periodictable.com/Properties/A/IonizationEnergies.bt.wt.html

19. Periodic trends of many-electron atoms: Electron affinity http://www.periodictable.com/Properties/A/ElectronAffinity.bt.wt.html

20. Periodic trends of many-electron atoms: “Atomic” radius http://www.periodictable.com/Properties/A/AtomicRadius.bt.wt.html