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Lecture 9. Many-Electron Atoms
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  1. Lecture 9. Many-Electron Atoms References • Engel, Ch. 10 • Ratner & Schatz, Ch. 7-9 • 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

  2. Helium First (1 nucleus + 2 electrons) • Electron-electron repulsion • Indistinguishability newly introduced Electron-electron repulsion ~H atom electron at r1 ~H atom electron at r2 : Correlated r12 term removes spherical symmetry in He. Cannot solve the Schrödinger equation analytically (not separable/independent any more)

  3. Many-electron (many-body) wave function To first approximation electrons are treated independently. ~H atom orbital An N-electron wave function is approximated by a product ofN one-electron wave functions (orbitals). (Hartree product) Orbital Approximation or Hartree ApproximationSingle-particle approach or One-body approach Does not mean that electrons do not sense each other.(We’ll see later.)

  4. Hartree Approximation (1928)Single-Particle Approach Nobel lecture (Walter Kohn; 1998) Electronic structure of matter • Impossible to search through • all acceptable N-electron • wavefunctions. • Let’s define a suitable subset. • N-electron wavefunction • is approximated by • a product ofN one-electron • wavefunctions. (Hartree product)

  5. Electron has “intrinsic spin” angular momentum, which has nothing to do with orbital angular momentum in an atom.

  6. spin space

  7. Electrons are indistinguishable.  Probability doesn’t change.

  8. Antisymmetry of electrons (fermions) Electrons are fermion (spin ½).  antisymmetric wavefunction Quantum postulate 6: Wave functions describing a many-electron system must change sign (be antisymmetric) under the exchange of any two electrons.

  9. Ground state of Helium Slate determinants provide convenient way to antisymmetrize many-electron wave functions.

  10. Excited state of Helium

  11. = 0 Slater determinant and Pauli exclusion principle • A determinant changes sign when two rows (or columns) are exchanged. •  Exchanging two electrons leads to a change in sign of the wave function. • A determinant with two identical rows (or columns) is equal to zero. •  No two electrons can occupy the same state. “Pauli’s exclusion principle” “antisymmetric” = 0 4 quantum numbers (space and spin)

  12. N-electron wave function: Slater determinant • N-electron wave function is approximated by • a product ofN one-electronwave functions (hartree product). • It should be antisymmetrized. but not antisymmetric!

  13. Ground state of Lithium

  14. 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 Lower E (closer to E0) Minimize E[] by changing !

  15. Example: Particle in a box ground state

  16. = = Approximation to solve the Schrödinger equation using the variational principle • Nuclei positions/charges & number of electrons in the molecule • Set up the Hamiltonian operator • Solve the Schrödinger equation for wave function , but how? • Once  is known, properties are obtained by applying operators • No exact solution of the Schrödinger eq for atoms/molecules (>H) • Any guessed trial is an upper bound to the true ground state E. • Minimize the functional E[] by searching through all acceptable • N-electron wave functions

  17. Hartree Approximation (1928)Single-Particle Approach Nobel lecture (Walter Kohn; 1998) Electronic structure of matter • Impossible to search through • all acceptable N-electron • wavefunctions. • Let’s define a suitable subset. • N-electron wavefunction • is approximated by • a product ofN one-electron • wavefunctions. (Hartree product)

  18. = ij Hartree-Fock (HF) Approximation • Restrict the search for the minimum E[] to a subset of , which • is all antisymmetric products of N spin orbitals (Slater determinant) • Use the variational principle to find the best Slater determinant • (which yields the lowest energy) by varying spin orbitals (orthonormal)

  19. Assume that electrons are uncorrelated. • Use Slater determinant for many-electron wave function • Each  has variational parameters (to change to minimize E) • including effective nuclear charge 

  20. Constrained Minimization of EHF[SD]

  21. and Hartree-Fock (HF) Equation (one-electron equation) • Fock operator: “effective” one-electron operator • two-electron repulsion operator (1/rij) replaced by one-electron operator VHF(i) • by taking it into account in “average” way Two-electron repulsion cannot be separated exactly into one-electron terms. By imposing the separability, the Molecular Orbital Approximation inevitably involves an incorrect treatment of the way in which the electrons interact with each other.

  22. HF equation (one-electron equation) Any one electron sees only the spatially averaged position of all other electrons.(Electron correlation ignored)

  23. Self-Consistent Field (HF-SCF) Method • Fock operator depends on the solution. • HF is not a regular eigenvalue problem that can be solved in a closed form. • Start with a guessed set of orbitals • Solve HF equation • Use the resulting new set of orbitals • in the next iteration and so on • Until the input and output orbitals • differ by less than a preset threshold • (i.e. converged).

  24. Solution of HF-SCF equation gives

  25. Hartree-Fock (HF) Energy

  26. Hartree-Fock (HF) Energy: Integrals

  27. Two-electron interactions (Vee) • Coulomb integral Jij (local) • Coulombic repulsion between electron 1 in orbital i and electron 2 in orbital j • Exchange integral Kij (non-local) only for electrons of like spins • No immediate classical interpretation; entirely due to antisymmetry of fermions > 0, i.e., a destabilization

  28. Aufbau (Building-up) Principle

  29. Koopman’s Theorem • 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

  30. Koopman’s Theorem: Examples

  31. Electron Correlation • A single Slater determinant never corresponds to the exact wavefunction. • EHF > E0 (the exact ground state energy) • Correlation energy: a measure of error introduced through the HF scheme • EC = E0- EHF (< 0) • Dynamical correlation • Non-dynamical (static) correlation • Post-Hartree-Fock method • Møller-Plesset perturbation: MP2, MP4 • Configuration interaction: CISD, QCISD, CCSD, QCISD(T)

  32. The lowest energy term for p 2 and d 6

  33. Restricted vs. Unrestricted HF