Theory and practice of molecular electronic structure (2010)
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.
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.
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.
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
download verbose output, view it by iQSee. Q/A discussion.
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.
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.
HW4: MO-LCAO picture of bonding:
formaldehyde example (computational).
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.
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.
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.
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.
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.
HW7: Solving a non-linear equation iteratively.
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
HW8: Contraction schemes for Pople and
general basis sets.
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.
HW9: First computational assignment for the
How to run Q-Chem on the HPCC cluster.
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
HW10: Using bond separation reactions for
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, Sz and S2 operators.
Different character of Sz and S2.
Midterm: All about Hartree-Fock method and basis sets.
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.
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.
Density Functional Theory. Hohenberg-Kohn theorems. Kohn-Sham equations.
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
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.
Excited states: cont-d. Symmetry, spin, and character of excited states.
Rydberg and valence states. Rydberg formula. Diazomethane example.
Using electron density for wave function analysis.
Mulliken and Lowdin atomic charges.
Natural Bond Orbital analysis: An overview and the formaldehyde
Using dipole moments to assess the quality of partial charges.
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.
Quiz #4. MP2 theory: derivation and discussion.
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.
Coupled-cluster and equation-of-motion methods.