CHEM545: Theory and practice of molecular electronic structure (2006)

1. Introduction. Energy units. Ab initio methods: An overview. Born-Oppenheimer approximation: Qualitative discussion. PES's: Concepts and definitions, relation to chemistry.
2. Introduction. Energy units. Born-Oppenheimer approximation: Derivation and discussion. Physical meaning of derivative terms (NaI example). Consequences of the breakdown of Born-Oppenheimer approximation (Laurie Butler example, NO dimer).
3. Valid N-electron wave functions. Slater determinants. Exact solution of electronic Schroedinger equation: FCI/CBS. Factorial scaling of FCI and the need of approximations. Theoretical model chemistries.
4. Calibration of approximate methods. Different measures of errors. Independent electrons and MC-LCAO approach.
5. MO-LCAO: 1-electron systems. H₂⁺ in the minimal basis set example. Generalization to larger bases. H₂ with non-interacting electrons. Qualitative discussion of Hartree-Fock model. Performing calculations using WebMO interface: water molecule. What is involved in setting up calculations. Orbitals, HF wave function, electronic configuration. Assigning orbital character and Lewis structures.
6. Water molecule and symmetries of MOs. Character tables and symmetry operations. Orbital symmetries and symmetries of many electron wave functions: H₂ example. Closed and open-shell wave functions: H₂O vs H₂O⁺. Determinants and Slater rules: Overlap between determinants. Notations for matrix elements.
7. Brief review of notations (determinants, the Hamiltonian, and integrals). Matrix elements of one and two particle operators: Simple example using non-antisymmetrized two-electron functions. Slater rules. Hartree-Fock energy and the meaning of two-electron integrals. Coulomb and exchange operator. Derivation of Hartree-Fock equations using Variational Principle and Lagrange multipliers.
8. Quiz. Review of Hartree-Fock energy expression and analysis of Hartree-Fock equations. Canonical form of Hartree-Fock equation and Koopmans theorem. WebMO/Q-Chem example: Ground and electronically excited states of the water cation using Koopmans theorem. IP by delta E versus Koopmans.
9. Canonical Hartree-Fock equations and Koopmans theorem. Relation to non-interacting electrons and excited states. MO-LCAO representation and Roothan equations. Matrix form of Fock operator. Hartree-Fock density and density matrix. How to solve Hartree-Fock equations: Self-consistent procedure. Different guess orbitals/densities and convergence procedures (Q-Chem manual).
10. One-electron basis sets. Hydrogen-like atom solutions and Slater type orbitals. Cusp and asymptotic decay. Contracted Gaussian sets. N-zeta. Polarization and diffuse functions. Contraction schemes and number of basis functions versus number of primitives in Pople's split-valence bases.
11. Review of bases, some examples of when polarization/n-zeta are important. Connection to calculations: memory requirements, determining most time consuming step (Fock matrix build), scaling. Tricks of the trade. Limitations of HF: Analysis of H₂ wave function and breaking of the HF model in the case of bond-breaking.
12. Performance of Hartree-Fock model. Equilibrium structures and frequencies (harmonic versus anharmonic): accuracy, systematic trends, and empirical scaling factors. Energy differences and error cancellations. Bond dissociation energies and correlation. Tricks: isogyric/isodesmic reactions.
13. Performance of Hartree-Fock model: Example of isodesmic scheme of calculating heat of isomerization reaction. Spin in non-relativistic electronic structure theory. Spin operators for one and two electrons: Pauli matrices, S_z and S². Valid two electron spin functions.
14. Spin and determinants: H₂ example. Open-shell determinants and expectation values of S². Separation of spin and spatial parts in two-electron wave-functions and how spin determines symmetry of spatial wave function.
15. Midterm: All about Hartree-Fock and molecular orbital theory.
16. Back to the exact solution of SE: FCI, electron correlation and excited states. Structure of FCI matrix. Spin- and spatial symmetry and block-diagonal structure of the Hamiltonian matrix. Intermediate normalization. Correlation energy. H₂ example: FCI in the minimal basis set. Complete N-electron basis set and symmetries of the determinants. Excited states and the correlated ground state. Variational flexibility of FCI solution in terms of ionic versus covalent terms in the wave function.
17. Comments on midterm problems. Review of FCI matrix, intermediate normalization, and correlation energy. Consequences of electron correlation (using the H₂ example). Back to H₂: Excited states. How can we compute them? Hartree-Fock calculations for the lowest state in each symmetry/spin-symmetry block for single-configurational states, as M_s=1 triplet. Two-determinantal character of open-shell singlets: HF is not appropriate. Configuration Interaction Singles: The simplest excited state method. CIS wave function and CIS eigen problem. Formaldehyde example: CIS calculations using WebMO, orbital analysis, assigning symmetry and MO character.
18. CIS and excited states, formaldehyde example. Symmetry and transition dipole: Selection rules for one-electron transitions. Open-shell triplet vs singlet and connection to the homework. Valence and Rydberg excited states. Rydberg states versus ionized states. Vertical vs adiabatic excitation energies, caveat of excited state optimization. Performance and limitations of CIS.
19. Including electron correlation. Truncated CI and size-extensivity. Perturbation theory: MP2 method. Scaling and performance overview. Basis sets for correlated calculations.
20. Limitations of MP2: H₂ example. Dynamical and non-dynamical correlation. Beyond MP2: coupled-cluster ansatz and size-extensivity.
21. Coupled-cluster methods. CCSD ansatz and equations. Scaling and accuracy. CCSD(T) and CCSDT. Limitations of CCSD: bond-breaking example, unbalanced treatment of two configurations.
22. Hierarchy of approximations for correlation treatment: A review. Basis set requirements for correlated calculations. Slow convergence and two-electron cusp. Correlation-consistent bases, cc-pVnZ vs cc-cVnz. Freezing the core. Extrapolation and energy additivity schemes. G2/G3 theories.
23. Density Functional Theory. Hohenberg-Kohn theorems. Kohn-Sham equations. LDA and GGA functionals.
24. Hierarchy of methods for excited states. EOM-CCSD wave function, equations, and similarity-transformed Hamiltonian. Accuracy, cost, and limitations. Energy additivity tricks. Spin- and spatial symmetry.
25. Excited states: cont-d. Size-extensivity of CIS, CISD, and EOM. Be and Be-Ne example. Balance in electronic structure calculations: Singlet-triplet gap in formaldehyde calculated by energy differences and by excited state methods. Some challenges of open-shells (spin-contamination, UHF vs ROHF).
26. Multi-configurational wave functions: Examples (bond-breaking, ethylene twisting, and diradicals). Multi-reference family of methods. CASSCF and beyond. Active space selection: minimal versus full valence. Alternative approaches: EOM spin-flip method for bond-breaking.
27. Time to put it all together: Students' presentations. Great job, everyone!