CHEM545: Theory and practice of molecular electronic structure
Introduction: course overview, history of quantum chemistry.
Energy units. Energy scale relevant to chemistry.
Born-Oppenheimer approximation: Qualitative discussion.
Read introductory chapters from Szabo and John Pople's Nobel Lecture;
familiarize yourself with the course webpage and software infrastructure;
install IQmol; prepare questions about the course
infrastructure (due next Tuesday).
Born-Oppenheimer approximation: Derivation and discussion.
Physical meaning of the derivative terms (NaI example).
Consequences of the breakdown of
Born-Oppenheimer approximation (Laurie Butler example).
HW2: Analyze derivative
coupling terms (due September 6).
Introduction to IQmol and HPCC (bring your
laptop to class if possible).
Computational lab #1
assignment (due September 6).
Lecture 4. Quiz #1.
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: Orbitals and determinants. 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.
Read Head-Gordon's review.
Quiz #2. Understanding MO-LCAO framework. Review of atomic orbitals.
Bonding in H2+. Generalization
for many-electron molecules assuming independent electrons.
Qualitative MO-LCAO picture of
bonding, bond order in diatomic molecules.
Review: One-electron systems (atoms and molecules, MO-LCAO).
Determinants are eigenstates of separable Hamiltonians.
Ground and excited states on non-interacting electrons (Aufbau principle).
Pseudo-independent electrons - mean-field approximation (qualitative
discussion). Water example: MO-LCAO picture of bonding and
review of symmetry.
HW3: Properties of determinants, Slater rules, symmetry of two-electron integrals.
Computational lab #2:
Bonding and molecular orbitals of
Slater rules. Hartree-Fock energy expression: Coulomb and
exchange terms. Mean-field and self-interaction.
Review of Variational Principle.
Review of non-interacting electrons.
Hartree-Fock equations: Derivation using Variational Principle.
Canonical Hartree-Fock equations.
One-electron energies and total HF energy.
Canonical Hartree-Fock orbitals and
HW4: Koopmans theorem.
Computational lab #3:
Koopmans theorem and
ionized states of formaldehyde.
Review of Hartree-Fock equations. Koopmans theorem: In-depth discussion. Application of Koopmans theorem to water and uracil.
Hartree-Fock equations in MO-LCAO form: Definitions and discussion.
Electron density and density matrix. Matrix of the Fock operator in the
AO basis. How to solve Hartree-Fock equations?
HW5: Self-consistent procedure.
Review of HF equations.
One-electron basis sets. Hydrogen-like atom solutions and Slater type orbitals.
Cusp and asymptotic decay. Contracted Gaussian sets. N-zeta and
Pople split-vaence bases. Polarization and diffuse functions.
HW6: Contracted basis sets.
Hartree-Fock equations in MO-LCAO form: Review. How to solve them:
Self-consistent procedure. Choosing the guess:
CORE, SAD, READ, BASIS2 options. OCCUPIED and MOM keywords.
Formal attributes of the 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.
Why Hartree-Fock wave functions are too ionic -- the H2
Performance of Hartree-Fock method for energy differences: The good,
the bad, and the ugly. Isogyric and isodesmic reactions.
Spin functions and spin operators for one and two electrons.
Pauli matrices, Sz and S2 operators.
Different character of Sz and S2.
HW7: Isogyric and isodesmic reactions.
Review of homeworks.
H2 example: the structure of FCI matrix in minimal basis.
Spatial and spin parts of two-electron wave functions.
Low-spin and high-spin determinants.
Spin-operators acting on Slater determinants.
HW8: Calculate the expectation value of
S2 with a two-electron determinant and analyze the result.
Finish H2 example. How correlation can tune covalent versus ionic character. Analisys of different
wave functions (triplet, singlet, open-shell singlet). Spin-contamination and Lowdin dilemma.
UHF versus RHF solutions and bond-breaking. Discussion on electron correlation.
Lecture 18. MIDTERM.
Review of the midterm exam.
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. HW: Prepare one paragraph description of what sort of questions
are relevant to your research (e.g., thermochemistry, excited states) and example of molecules you work on.
Perturbation theory approach to electron correlation. MP2 theory.
Review of MP2 theory. Electron correlation the right way: Coupled-cluster theory.
Computational lab #4: IR and NMR spectra.
Lecture cancelled. Home work reading assignment: Read 2 papers about DFT theory
(A brief oberview of DFT and A recent review with extensive benchmarks).
Review of CI, MP2, and CC approaches for electron correlation. The gold standard: CCSD(T).
Density matrices: Introduction. Reduced DMs and calculation of expectation
values. OPDM and TPDM. Energy expression and N-representability problem.
Density Functional Theory: Wilson's proof of Hohenberg-Kohn theorems.
Representing density by Kohn-Sham determinant.
Kohn-Sham equations. Different approaches to exchange-correlation functional.
LDA, GGA, Hybrid
Long-range corrected functionals. 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. Excited-state analysis (MOs, Rydberg formula, etc).
Computational lab #5:
CIS calculations of formaldehyde.
"Single-Reference ab Initio Methods for the Calculation of
Excited States of Large Molecules".
Excited states: Correlated approaches.
Read: "Equation-of-motion coupled-cluster methods for open-shell and electronically excited species: The hitchhiker's guide to Fock space"
Open-shell species. Read
Quantum Chemistry of Open Shell Species .
Density matrices and wave function analysis. Partial charges (Mulliken, Lowdin, NBO). Natural Bond Orbital analysis.
NBO analysis of formaldehyde.