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 , 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

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

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

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

Solution of HF-SCF equation gives

Solution of HF-SCF equation: Z-  (measure of shielding) more shielded less shielded

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

Source: larger smaller

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

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

Koopmans’ theorem: Validation from experiments

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  …   

Aufbau (Building-up) principle for transition metals 10.3

Aufbau (Building-up) principle for transition metals

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 

Periodic trends of many-electron atoms

Periodic trends of many-electron atoms: Electronegativity

Periodic trends of many-electron atoms: 1 st ionization energy

Periodic trends of many-electron atoms: Electron affinity

Periodic trends of many-electron atoms: “Atomic” radius