66835 - Properties of Molecular Matter

Learning outcomes

The course will develop the knowledge necessary to bridge molecular, surface and solid state properties and will investigate also dynamical aspects of matter.

Course contents

Part 1 (F. Zerbetto): The concept of probability stochastic and Bayesian probabilities. Examples and applications. Link with the concept of multiplicity.
Further examples and applications of probability conditioning.  Introduction to probability distributions.
Probability distributions: binomial and multinomial distributions, applications and examples of chemical interest, mean value and variance, their meaning in chemistry and physics.
Calculation of the first and second moment for some observables. The case of the expectation value of cos (theta) and cos ^ 2 (theta). Equipartition of energy. The Stirling's approximation.
Random walk, the discrete model generates a Gaussian function. Lagrange multipliers, introduction and simple applications.
Boltzmann equation, application to ideal gases: equation of state, pressure balance between different containers.
The Boltzmann distribution with maximization of entropy, its modification in the presence of physical constraints, examples and applications.
Free energy and its meaning, Boltzmann distribution of the free energy, partition functions, their applications, internal energy and entropy in terms of partition functions.
Practical examples of partition functions, mean values ​​and thermodynamic functions. Brief introduction to the unit'of measurement.
Use of units of measurement in simple examples and for simple applications. Calculation of the translational partition function, practical examples.
Partition function of rotational and vibrational degrees of freedom, applications and examples. The chemical potential from partition functions. The theory of the activated complex.
Introduction to the model of the disordered lattice. Vapor pressure, cavitation energy, surface tension, interfacial tension.
Entropy, energy, free energy and chemical potential of a two-component system with the model of the disordered lattice, the Bragg-Williams or mean field model.
Entropy, internal energy, free energy, chemical potential for ternary systems; standard potential and significance of activity with the model of the disordered lattice, the binodal curve and its analytical expression. The spinodal curve, its expression and simple applications; critical point as the third derivative of the free energy. Introduction to equilibrium models in statistical thermodynamics; isosbestic point; cooperative and non-cooperative transitions. Model for non-cooperative equilibrium, totally cooperative cases, and first neighbors or Zimm-Bragg cooperation model. Cooperative model with degeneration. Nucleation and crystallization. Ising model. Introduction to the Langmuir model with the disordered lattice model. Michaelis-Menten and its significance in materials science. Active transport and passive transport by means of a carrier through an interface. Sabatier's principle. Binding polynomials. Multiple binding by means of the binding polynomials, Scatchard plot, Hill plot; micelles.
Formation of multilayer and the BET model. Pauling's cooperativity model. Binding with relaxation; model of multiple binding with excluded volume.
Introduction to the physical chemistry of polymers. Entropy and internal energy according to Flory-Huggins, comparison with the case of non-polymeric solutions.
Introduction to Dissipative Particle Dynamics, comparison with the model of disordered lattice, computer simulations and practical examples.
Flory-Huggins: free energy, chemical potential, miscibility of polymers, the partition coefficient, dependence on the length of the chain.
Flory-Huggins parameter at the critical point,  <r> <r^2>, Kuhn's model, random walk and probability of cyclization.
Polymer radius; elasticity in one dimension; elasticity of many chains for generic deformations. Entropy, internal energy and free energy as a function of the density of the polymer: theta conditions," poor "and" good " solvents.

Part 2 (F. Negri): 

Program: The knowledge and understanding of intra-molecular properties and intermolecular interactions is fundamental for the design of new molecular materials. The modeling of these properties requires a critical choice and use of computational tools based on quantum-mechanics or classical mechanics The course therefore presents a panorama of the tools offered by computational chemistry for modeling charge and energy transport properties in conjugated molecular materials.    

 

 

Content.

1.     Molecular materials: organic semiconductors.

a)     P and n doping of organic and inorganic semiconductors: effect of doping on orbital levels of organic chromophores.

b) Comparison between HOMO-LUMO levels and work function of metal electrodes. Charge injection and transport inside the semiconductor.

c)     Band and hopping regimes for charge transport in organic semiconductors. Polaronic charge carriers .

d)    Excitonic interaction. Frenkel states and charge transfer states.

e)     Rate charge transfer described by Marcus equation: electronic coupling and electron-phonon coupling.

f)     Intramolecular reorganization energy: adiabatic potential method and determination of Huang-Rhys factors to estimate intramolecular reorganization energies.

g)     Direct and indirect methods for the calculation of electronic and excitonic couplings.

h)    Anisotropy of charge transport and effects induced by thermal disorder.

 

 

2.     Methods for the characterization of potential energy surfaces.  

a)     Topology of potential energy hypersurfaces. Minima, maxima, saddle points, transition states.

b)    Introduction to algorithms for geometry optimization.

c)     First order methods, steepest descent. Second order methods, Newton-Raphson.

d)    Cartesian coordinates and internal coordinates.

 

3.     Computational methods for the evaluation of intra- and inter- molecular properties with quantum-chemistry tools.

a)     Poly-electronic wave-functions: Hartree product and Slater determinant.

b)    Derivation of the total energy expression for H2 a simple polyelectronic molecule.

c)     Core integrals, Coulomb and Exchange integrals. Mono-eletronic and bi-electronic integrals determining the energy expression.

d)    Derivation of HF equations.

e)     HF equations in matrix form on atomic basis: Roothan Hall formulation.

f)     Orbital basis sets based on Gaussian wavefunctions. Polarization and diffuse functions. Pople notation.

 

4.     Quantum-chemical methods: beyond the Hartree Fock.

a)     Koopmans's theorem, ionization potential, electronic affinity.

b)    Unrestricted HF.

c)     Electronic correlation. Coulomb and Fermi hole.

d)    Methods based on density functional theory (DFT).

e)     Kohn-Sham equations. Local density, gradient corrected, and hybrid functionals. Introduction to long range corrected functionals and to the TDDFT method.

f)     Other methods introducing electron correlation with the variation approach: configuration interaction (CI), full CI, truncated CI, CIS, CID, CISD. Brillouin's theorem. Introduction to MCSCF, definition of CASSCF, choice of active space. Introduction to MR-CI and CASPT2.

g)     Methods introducing electronic correlation with the perturbation approach: MP2. The concept of  size consistency in quantum-chemical methods.

 

5.     Molecular dynamics and empirical force fields.

a)     Statistical thermodynamics: ensemble definition and phases' space.

b)    Time step discretization and equation of motion integration in molecular dynamics. Example: derivation of the Verlet algorithm.

c)     Molecular dynamics in ensembles different from the NVE using thermal baths.

d)    Comparison between molecular dynamics and Monte Carlo simulations as regard their application to flexible chemical systems.

e)     Initial conditions. Periodic boundary conditions and calculation of intermolecular interactions with the approach of the minimum image convention and using a cutoff radius.

f)     Equilibration. Trajectory analysis and temporal average calculation. Radial distribution function for gases, liquids and solids. Root mean square displacement and relation to the diffusion coefficient.

g)     Transport phenomena. Derivation of the diffusion equation or Fick's second law. Brownian motion and Gaussian probability. Derivation of the Einstein equation for the diffusion.

h)    Molecular mechanics empirical force fields. Fundamental terms and their general functional form. Multipolar expansion and Coulomb equation for electrostatic interactions. Atom type concept.

Auto-correlation functions. Brief introduction to kinetic Monte Carlo simulations for the investigation of transport properties.

Readings/Bibliography

K.A. Dill, S. Bromberg, Molecular Driving Forces, Garland Science

A. Leach, “Molecular Modelling Principles and Applications”, Prentice Hall, 2001.

Lectures notes

Teaching methods

This course is formed by frontal lectures accompanied by problem solution and sessions in the computational laboratories. During laboratory sessions the student applies the notions acquired by following the frontal lectures and learns how to choose and use the tools of chemical modeling applied to the study of chemical properties and inter-molecular interactions.

Assessment methods

Written examination with a number of questions and problems. The time allocated for the written examination is 2 hours for each module. The total score exceeds 30/30. Module 1 also requires a viva.

The final mark of the course “ Proprietà di materiali molecolari” is calculated as the weighted average on the basis of the credit numbers of the two modules: Module 1, Prof. Zerbetto, and Module 2, Prof. Negri.  

Teaching tools

PC, projector, Power Point presentations, blackboard

Office hours

See the website of Francesco Zerbetto

See the website of Fabrizia Negri