CHEM545: Theory and practice of molecular electronic structure
- Introduction: course overview, history of
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, John Pople's Nobel Lecture,
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.
Overview of software and hardware tools for the class.
Note: Updated materials about course resources are posted in the
HW3: Learning software and hardware.
Orbitals and determinants.
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.
Review: Factorial scaling of the exact solution of SE (FCI)
and the need of approximations.
Theoretical model chemistries. 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.
Quiz 1. 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 #2 (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.
Hartree-Fock equations: Review. Canonical Hartree-Fock equations.
One-electron energies and total HF energy.
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 #3 (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.
One-electron basis sets: Review and QA session. Cartesian versus
pure angular momentum.
Quiz #4 (one-electron basis sets, computational scaling of HF method).
Formal attributes of HF model (variational, size-extensive, etc).
Accuracy of HF for molecular structures and vibrational
frequencies (discuss harmonic versus anharmonic frequencies),
systematic errors, using scaling factors.
How to run Q-Chem on the HPCC cluster: Updated instructions.
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
HW8: Basis sets and using bond separation
reactions for accurate thermochemistry.
Lecture 13: Midterm! All about Hartree-Fock theory and basis
H2 example: the structure of FCI matrix in minimal basis, review
of point group symmetry.
Spin functions and spin operators for one and two electrons.
Pauli matrices, Sz and S2 operators.
Different character of Sz and S2. Spatial and spin parts of two-electron wave functions.
Low-spin and high-spin determinants.
Spin-operators acting on Slater determinants.
HW9: Calculate the expectation value of
S2 with a two-electron determinant and analyze the result.
Project ideas are due 10/16!
Review of spin functions. Analysis of determinants for minimal basis
Electron density and density matrices. Density matrix and calculation
of observables. One- and two- particle DMs.
Energy expression and N-representability problem.
Midterm review. DM and wave function analysis: partial charges and dipole
moments. NBO analysis.
HW10: NBO calculations for formaldehyde and first computational assignment for the project.
Density Functional Theory. Hohenberg-Kohn theorems. Kohn-Sham equations. LDA and GGA.
Different approaches to exchange-correlation functional. LDA, GGA, Hybrid
functionals. Long-range corrected functionals. Empirical dispersion
correction. Numerical aspects of KS-DFT and performance of modern
functionals (see recent review).
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.
HW11: CIS calculations of formaldehyde.
Quiz #5 (singly excited determinants, Be example, CIS).
Excited states: cont-d. Symmetry, spin, and character of excited states.
Rydberg and valence states. Rydberg formula. Diazomethane example.
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 #6 (structure of FCI matrix).
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.
Review of electron correlation. MP2 and CC methods: overview and
discussion. Correlated methods for excited states:
Equation-of-motion coupled-cluster methods.
Excited states: time-dependent derivation. TD-HF and TD-DFT.
Excited states: Review and discussion of CIS and TD-DFT. SIE and excited states.
Lecture slides. Suggested reading:
Dreuw and Head-Gordon review.
Excited states: Review. Wave-function analysis (transition densities,
difference densities, attachment-detachment densities).
Project presentations: Great job, everyone!
Special topic: Multireference-methods. If you must...