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.
PESs: Concepts and definitions, relation to chemistry.
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).
Review of important quantum mechanical concepts. Matrices, eigenproblems,
and variational principle.
- Introduction to IQmol and HPCC (bring your
laptop to class if possible). Computational lab #1
assignment. Introduction to
IQmol. Remote submission
with IQmol: Server setup.
Review of important quantum mechanical concepts: Perturbation theory.
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
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.
Read Head-Gordon's review.
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 #2. 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.
Computational lab #2:
Bonding and molecular orbitals of
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
Slater rules and matrix elements.
HW4: Symmetry of two-electron integrals.
Review: Slater rules. Hartree-Fock energy expression: Coulomb and
exchange terms. Mean-field and self-interaction.
Review of Variational Principle. Geometrical interpretation of VP.
Hartree-Fock equations: Derivation using Variational Principle.
Hartree-Fock equations: Finish derivation.
Canonical Hartree-Fock equations.
One-electron energies and total HF energy.
Canonical Hartree-Fock orbitals and
Koopmans theorem. Example: MOs and IPs of water.
HW5: Koopmans theorem.
Computational lab #3:
Koopmans theorem and
ionized states of formaldehyde.
Review symmetry. MO-LCAO and Koopmans theorem: Examples (water, uracyl).
Hartree-Fock equations in MO-LCAO form: Definitions and discussion.
Electron density and density matrix. Matrix of the Fock operator in the
HW6: 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.
HW7: 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.
HW8: Isogyric and isodesmic reactions.
Lecture 15. Review session, Q/A.
Density matrices: Introduction. Reduced DMs and calculation of expectation
values. OPDM and TPDM. Properties of OPDM (trace of OPDM is equal to the number of electrons). Examples: Hartree-Fock and FCI OPDM for HF in the MO basis
(using H2 as an example).
Density matrices and wave function analysis. Partial charges (Mulliken, Lowdin, NBO). Natural Bond Orbital analysis.
Computational lab #4:
NBO analysis of formaldehyde.
Review of the midterm results.
Spin functions and spin operators for one and two electrons.
Pauli matrices, Sz and S2 operators.
Different character of Sz and S2.
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.
HW9: 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).
UHF versus RHF solutions and bond-breaking. Discussion on electron correlation. Introduction of intermediate normalization.
Intermediate normalization, correlation energy, and the structure of FCI matrix. Relative importance of excited
determinants. Truncated CI models and their lack of size-extensivity.
MP2 and CC theory: Qualitative discussion. Lecture slides.
Quiz #8. CC and MP2 theory.
Quiz #9 Density Functional Theory. Hohenberg-Kohn theorems.
Representing density by Kohn-Sham determinant.
Quiz #10. 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
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. Excited-state analysis (MOs, Rydberg formula, etc). Lecture slides.
Computational lab #5:
CIS calculations of formaldehyde.
Excited states: Correlated approaches. EOM-CC for electronically excited
and open-shell species.