CHEM545: Theory and practice of molecular electronic structure (2010)

1. Introduction: course overview, history of quantum chemistry. Energy units. Energy scale relevant to chemistry. Born-Oppenheimer approximation: Qualitative discussion. PESs: Concepts and definitions, relation to chemistry. HW1: read introductory chapters from Szabo and MHG review. Lecture slides.
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). HW2: Analyze derivative coupling terms by PT. Lecture slides.
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. HW3: Orbitals and determinants. Lecture slides.
4. Using WebMO and iQSee. WebMO: build ethylene and methanol, run SCF/STO-3G, look at the orbitals. Check symmetry ("Clean-up" tool). iQSee: prepare input for methanol SCF/STO-3G, import to WebMO, run QChem, download verbose output, view it by iQSee. Q/A discussion.
5. Theoretical model chemistries: cont-d. Review of one- and many-electron bases and the respective approximations. Calibration of approximate methods. Different measures of errors. Scaling, variational properties, and size-consistency. Lecture slides.
6. 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). Qualitative MO-LCAO picture of bonding, bond order in diatomic molecules. From diatomics to nucleobases: bonding in ionized dimers of nucleobases. Lecture slides. HW4: MO-LCAO picture of bonding: formaldehyde example (computational).
7. Review: Determinants are eigenstates of separable Hamiltonians. Ground and excited states on non-interacting electrons (Aufbau principle). Slater rules and matrix elements. Hartree-Fock energy expression: Coulomb and exchange operators. HW5: Symmetry of two-electron integrals.
8. Quiz #1 (Slater rules and integrals notations). Review of Variational Principle. Geometrical interpretation of VP. Hartree-Fock equations: Derivation using Variational Principle. Fock operator. Canonical Hartree-Fock equations. One-electron energies and total HF energy. Lecture slides.
9. Hartree-Fock equations: Review. Canonical Hartree-Fock orbitals and Koopmans theorem. Review of symmetry. Examples: Assigning MO characters in water and uracil. Relation to photoelectron experiments. Benzene dimer example. HW6: Koopmans theorem and formaldehyde, symmetry of the electronic states of the formaldehyde cation. Lecture slides.
10. Hartree-Fock equations in MO-LCAO form: Definitions and discussion. Electron density and density matrix. Matrix of the Fock operator in the AO basis. Self-consistent procedure. Choosing the guess: CORE, SAD, READ, BASIS2 options. OCCUPIED and MOM keywords.
11. Quiz #2 (Hartree-Fock equations in matrix form, MO-LCAO). Review of HF equations. One-electron basis sets. Hydrogen-like atom solutions and Slater type orbitals. Cusp and asymptotic decay. Contracted Gaussian sets. Lecture slides. HW7: Solving a non-linear equation iteratively.
12. One-electron basis sets: cont-d. 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. Basis set size and cost/scaling of SCF calculations. Lecture slides. HW8: Contraction schemes for Pople and general basis sets.
13. One-electron basis sets: Review and QA session. Cartesian versus pure angular momentum. Quiz #3 (one-electron basis sets, computational scaling of HF method). Formal attributes of HF model (variational, size-extensive, etc). Review calibration, systematic and non-systematic errors. Accuracy of HF for molecular structures and vibrational frequencies (discuss harmonic versus anharmonic frequencies), systematic errors, using scaling factors. Lecture slides. HW9: First computational assignment for the project. How to run Q-Chem on the HPCC cluster.
14. Performance of Hartree-Fock method for energy differences: The good, the bad, and the ugly. Isogyric and isodesmic reactions. Why Hartree-Fock wave functions are too ionic -- the H₂ example. HW10: Using bond separation reactions for accurate thermochemistry.
15. H₂ example: the structure of FCI matrix in minimal basis, review of point group symmetry. Analysis of FCI: How electron correlation recovers correct character of the wave function. Spin functions and spin operators for one and two electrons. Pauli matrices, S_z and S² operators. Different character of S_z and S².
16. Midterm: All about Hartree-Fock method and basis sets.
17. Midterm review. Spin functions and spin operators for one and two electrons: cont-d. Spatial and spin parts of two-electron wave functions. Low-spin and high-spin determinants. Spin-operators acting on Slater determinants. Spin-contamination, UHF, RHF, and ROHF. HW11: Calculate the expectation value of S₂ with a two-electron determinant and analyze the result.
18. Electron density and density matrices. Density matrix and calculation of observables. One- and two- particle DMs. Energy expression and N-representability problem. DM and wave function analysis: partial charges and dipole moments.
19. Density Functional Theory. Hohenberg-Kohn theorems. Kohn-Sham equations.
20. Different approaches to exchange-correlation functional. LDA, GGA, Hybrid functionals. Long-range corrected functional. Empirical dispersion correction. Numerical aspects of KS-DFT and performance of modern functionals.
21. Excited states: What are they? Koopmans and FCI description. Conceptual methodological problems: Limitation of VP and open-shell (two-determinantal) character. The simplest model: CIS. Assigning the character of excited states. HW12: CIS calculations of formaldehyde. Notes on the final project preparation.
22. Excited states: cont-d. Symmetry, spin, and character of excited states. Rydberg and valence states. Rydberg formula. Diazomethane example. Size-intensivity. Lecture slides.
23. Using electron density for wave function analysis. Mulliken and Lowdin atomic charges. Natural Bond Orbital analysis: An overview and the formaldehyde example (WebMO). Using dipole moments to assess the quality of partial charges.
24. Consequences of electron correlation. Dynamical and non-dynamical correlation. Intermediate normalization, correlation energy, and the structure of FCI matrix. Relative importance of excited determinants. Truncated CI models and their lack of size-extensivity. Lecture slides.
25. Quiz #4. MP2 theory: derivation and discussion.
26. MP2-cont-d. Scaling of MP2. Basis sets for correlated calculations. Using frozen-core in correlated methods. Performance and limitations of MP2 theory. Coupled-cluster methods. Exponential ansatz and size-extensivity. Coupled-cluster equations: projection method.
27. Coupled-cluster and equation-of-motion methods. Lecture slides.
28. Project presentations.
29. Project presentations.