CHEM545: Theory and practice of molecular electronic structure

1. Introduction. Energy units. An overview. Born-Oppenheimer approximation: Qualitative discussion. PES's: Concepts and definitions, relation to chemistry. HW1: read introductory chapters from Szabo and MHG review.
2. 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). HW2: Analyze derivative coupling terms by PT.
3. Valid N-electron wave functions. Slater determinants. Exact solution of the electronic Schroedinger equation: FCI/CBS. Factorial scaling of FCI and the need of approximations. Theoretical model chemistries.
4. Theoretical model chemistries: cont-d. Calibration of approximate methods. Different measures of errors. Scaling, variational properties, and size-consistency. HW3: Understanding orbitals and determinants (Szabo).
5. Understanding MO-LCAO framework. Review of atomic orbitals. Bonding in H₂⁺. Generalization for many-electron molecules assuming independent electrons. Qualitative discussion of Hartree-Fock model (pseudo-independent electrons). What is involved in setting up calculations?
6. Independent electrons and determinants. Pseudo-independent electrons and interpretation of molecular orbitals. Setting WebMO calculations. Water molecule: SCF calculations, MOs, symmetries of MOs and symmetry of many-electron wave functions. HW4: Formaldehyde: MOs and bonding, symmetry analysis, geometry optimization and frequencies using WebMO.
7. Formalism, notations, and matrix elements. Integrals and Slater rules. Hartree-Fock energy: derivation and analysis. Coulomb and exchange operators. Begin derivation of Hartree-Fock equations. HW5: permutational symmetry of two-electron integrals.
8. Hartree-Fock equations: Derivation using Variational Principle. Fock operator. Canonical Hartree-Fock equations. One-electron energies and total HF energy.
9. Hartree-Fock equations: Review. Canonical Hartree-Fock orbitals and Koopmans theorem. Examples: Assigning MO characters in formaldehyde and uracyl. Relation to photoelectron experiments. HW6: Koopmans theorem and formaldehyde.
10. Quiz #1. MO-LCAO Hartree-Fock equations: Definitions and discussion. Electron density and density matrix. Matrix of the Fock operator in the AO basis.
11. Quiz #2. MO-LCAO Hartree-Fock equations: Cont-d. Self-consistent procedure, choosing the guess and algorithm, etc. Scaling and resource requirements. HW7: Solving non-linear equation iteratively.
12. 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. HW8: Contraction schemes for Pople and general basis sets.
13. Review of basis sets and contraction schemes. Performance of Hartree-Fock theory: Equilibrium geometries and vibrational frequencies. Anharmonicities. Systematic and non-systematic errors, scaling. Convergence of the results with respect to one electron basis set.
14. Performance of Hartree-Fock theory for relative energies: Internal rotation and bond strength. Importance and magnitude of correlation: H₂ example. Dodging the bullet isogyric and isodesmic reactions. HW8: Calculating heats of formation using isodesmic pathways.
15. Midterm: All about Hartree-Fock method and basis sets.
16. Spin functions and spin operators for one and two electrons. Pauli matrices, S_z and S² operators. Different character of S_z and S².
17. Electron density, density matrix, and wave function analysis. Mulliken and Lowdin atomic charges. HW9: NBO calculations for formaldehyde. HW10: First computational assignment for the project.
18. Natural Bond Orbital analysis: Overview and example.
19. Two-electron spin functions: continued. Two-electron determinants: He atom example. Closed and open-shell determinants, high-spin triplet states, two-configurational low-spin functions. Spatial and spin parts of two-electron wave functions. Hartree-Fock solution of H₂ in the minimal basis set: fixed ratio of ionic and covalent configurations. Restricted and unrestricted HF and dissociation problem. HW11: Calculate expectation value of S² for a 2-electron determinant and analyze the result.
20. Back to the exact solution of SE: FCI, electron correlation and excited states. 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. Separation of spin and spatial parts in two-electron wave-functions and how spin determines symmetry of spatial wave function.
21. Intermediate normalization, correlation energy, and the structure of FCI matrix. Relative importance of highly excited determinants. Truncated CI models and their lack of size-extensivity.
22. Quiz #3. Electron correlation: MP2 theory: derivation and discussion. Scaling of MP2.
23. MP2 theory: cont-d. Basis sets for correlated calculations. Performance and limitations of of MP2 theory. Dynamical and non-dynamical correlation. Coupled-cluster method: Exponential ansatz.
24. Coupled-cluster methods: cont-d. Exponential versus linear ansatz and size-extensivity. Derivation of CCSD equations: Non-variational properties and the projection method. Inclusion of triple excitations explicitly and perturbatively. Scaling, performance, and limitations.
25. Density functional theory. Hohenberg-Kohn theorem and the existence of density functional. Physical basis of HKT (Wilson proof). Kohn-Sham equations. Semi-empirical nature of current DFT and known limitations.
26. Excited states: Koopmans and FCI description. Hierarchy of approximations: CIS, CIS(D), EOM-CCSD. Accuracy, cost, and limitations. Diazomethane example.
27. Some aspects of open-shell calculations. Spin-contamination, UHF versus ROHF and Lowdin dilemma. On spin-contamination within DFT. Spin-contamination in correlated wave functions.