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##### Lecture 9. Many-Electron Atoms

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**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**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)**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.)**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)**Electron has “intrinsic spin” angular momentum, which**has nothing to do with orbital angular momentum in an atom.**spin**space**Electrons are indistinguishable.** Probability doesn’t change.**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.**Ground state of Helium**Slate determinants provide convenient way to antisymmetrize many-electron wave functions.**= 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)**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!**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 !**Example:**Particle in a box ground state**=**= 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**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)**= 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)**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 **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.**HF equation (one-electron equation)**Any one electron sees only the spatially averaged position of all other electrons.(Electron correlation ignored)**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).**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**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**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)