Quantum Chemistry: Restricted and Unrestricted Hartree-Fock for closed and open shell systems. Correlation energy and Post-Hartree Fock methods: configuration interaction and Density functional theory.
Statistical thermodynamics: phase space and statistical ensembles. Canonical partition function for ideal gases and chemical equilibria. Statistical thermodynamics of liquids. Structural properties of
complex system via Monte Carlo Simulations.
The course is divided into two modules, each providing 3 CFU. The first module is dedicated to Quantum Chemistry (QC) and the second to statistical Thermodynamics ( ST).
QC is expected to provide the basic theoretical background for the study and interpretation of the electronic structures in molecular systems. In particular, the student will get acquainted with the computational aspects of the most relevant ab initio methodologies.
TS is conceived to furnish the competencies and understanding aimed at bridging the gap between the macroscopic behavior of the chemical systems at equilibrium and their microscopic nature. In this framework, it is expected that the student familiarizes with the probabilistic approach in statistical thermodynamics (phase space, distribution function, ensemble averages, partition function).
Prerequisites
none
Teaching Methods
Class lectures.
At the end of the QC and ST module, DEMO computational applications are foreseen (simple QM calculations on small organic molecules and determination of the pair distribution functions in liquid water using Monte Carlo simulations)
Further information
Lecture notes didactic material are available online on the Professor's homepage (restricted access from unifi domain/proxy only for CFS students)
Type of Assessment
Oral Examination
Course program
QUANTITY CHEMISTRY MODULE
Elements of vector Algebra and matrices. Basis sets and operators. Hilbert Spaces, Bra and Ket formalism. An operator's matrix representation. Adjoint and Hermitian Operators.
Eigenvectors and eigenvalues. Variational Theorem. Reduction of the constrained minimization of the energy expectation value to an eigenvalue problem.
Born Oppenheimer Approximation. Electronic and nuclear Hamiltonian. Conditions for the diagonality of kinetic energy matrix and Born Oppenheimer surfaces.
Hartree-Fock's multielectronic wavefunctions: Slater's determinant and spin-orbitals. Deriving the Hartree-Fock canonical equations for N spin-orbitals.
Spin integration for closed shell systems in the restricted procedure. Derivation of the equations for closed-shell systems and RHF energy calculation in terms of Coulomb and Exchange operators.
Gaussian basis functions. Primitives and contractions. LCAO-MO Approach for RHF for Closed Shell Systems. Deriving the equations of Roothan-Hall. Example: molecular orbitals in the water.
Unrestricted Hartee-Fock (UHF). Eliminating the spin and deriving the Poble-Nebset equations for open-shell systems. Alpha and beta density matrices and spin densities.
Toward the combination of determinants: the hydrogen molecule and the problem of the dissociation. Analysis of the unrestricted solution. The unrestricted method in the MC-SCF variant.
Spin operators. Calculation of spin state for a single (RHF or UHF) Slater determinant. Examples of the restricted/Unrestricted HF determinants for the hydrogen molecule. Spin contamination and pure spin states.
Combinations of Slater determinants. Configuration Interaction (CI). The CI matrix and Condon-Slater rules.
Density matrices and formulation of Hartree-Fock theory in terms of density matrix. Hoenberg-Kohn's theorem.
The Kohn-Sham equations.
STATISTICAL THERMODYNAMICS MODULE
Many-particles systems. Phase space. First and second principle of the thermodynamics. Statistical Interpretation. Classic systems. Lagrangian and Hamiltonian in classical mechanics.
Distribution Functions, Deterministic Systems and Ergodic Conjecture. Gibbs ensemble averages and time averages .
Microcanonical Ensemble. Statistical Entropy and Thermodynamic Entropy. Entropy of a perfect gas. Suckur-Tetrode equation and the statistical mechanics derivation of the perfect gas law
Canonical ensemble. Helmholtz's molecular partition and free energy functions.
Energy fluctuations in canonical systems at equilibrium and constant volume heat capacity. Molecular partition functions and chemical equilibria in gases
Virial theorem in the canonical ensemble: Equipartition theorem and Equation of state in simple liquids.
Reduced distribution functions. The radial distribution function.
Role of the radial distribution function in liquids: g(r) and average internal energy, g(r) and equation of state, g(r) and the reversible work