Abstracts of the CTTC School 2016
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General Introduction
Eduard Matito
Universidad País Vasco
Introduction to Chemical Bonding Analysis
In this course we will introduce a few concepts that will be used in the subsequent four Chemical Bonding courses.
- Density and Density Matrices.
- The Atom in a Molecule. Atomic partitions.
- Three-dimensional partitions
- The Quantum Theory of Atoms in Molecules (QTAIM) partition.
- Fuzzy partitions. The Topological Fuzzy Voronoi Cells (TFVC)
- Hilbert space partitions.
- Mulliken's partition.
- Three-dimensional partitions
- Population Analysis
- Electron Sharing Indices.
We will follow the next notes [1] and slides [2] (slides will be updated in due time).
Andreas Savin
CNRS and Sorbonne Universités, UPMC Univ Paris 06
Density Functional Theory
- Foundations of density functional theory
- The Hohenberg-Kohn theorem
- Accurate functionals
- Response functions
- Methods used in the frame of density theory
- The Kohn-Sham method
- Functionals for the kinetic, exchange, correlation energies
- Scaling
- Tools for constructing approximations, and extending density functional theory
- Adiabaitc connection
- Hybrids
- Approximations for density functionals
- LDA
- Semi-local approximations
- Hybrids
- Combination with Hartree-Fock
- Combination with multi-reference methods
- Methods for excited states
- Main limitations of approximations
- Systematic improvement
- Treatment of degeneracy and size-consistency
- Methods related to density functional theories
- Local potentials
- Random phase approximation
- Judging approximations
- Concepts from statistics
- Benchmarks and their limits
Tomás Rocha
Universidad Nacional Autónoma de México
Introduction to Method Development: Second Quantization
- Fock space: creation and annihilation operators.
- Fundamental anticommutation relationships.
- Representation of operators in second quantization.
- Rank reduction: some useful commutators and anticommutators.
- Singlet tensor operators.
- Use of second quantization in coupled cluster theory
- Coupled cluster energy.
- Connected coupled cluster equations.
José L. Gázquez
Universidad Autónoma Metropolitana-Iztapalapa, México
Chemical reactivity in density functional theory
The Taylor series expansion of the energy as a function of the number of electrons and the external potential, around an isolated chemical species, has been the basis of the density functional theory of chemical reactivity. The response functions that appear through this approach describe the inherent or intrinsic chemical reactivity, which may be used to infer the behavior of a molecule when it interacts with different families of reagents. In this presentation we will revise the concepts that arise from the derivatives of the energy with respect to the number of electrons, basically, the chemical potential and the chemical hardness, and the derivatives of the electronic density with respect to the number of electrons, basically, the Fukui function and the dual descriptor. We will also analyze the chemical potential equalization principle, the maximum hardness principle and the hard and soft acids and bases principle, and their importance to describe chemical interactions. Additionally, we will make use of the Hohenberg-Kohn-Mermin formalism in the grand canonical ensemble to derive the temperature dependent expressions. Finally, we will discuss the charge transfer process from the perspective of density functional theory to show the relevance of the concepts of chemical potential and hardness in chemistry.
Development
Paul Ayers
McMaster Univ.
Peter Gill
Australian National University
Although density functional theory (DFT) has become extremely popular within the quantum chemistry community, it still suffers from some serious deficiencies and most of these are now well understood. One of the key goals of modern research, therefore, is to develop new methods that preserve the low cost of DFT calculations while offering significantly enhanced accuracy.
Many of the problems of DFT stem from the fact that most functionals are based on the uniform electron gas, a model that consists of an infinite number of electrons in an infinite volume. Unfortunately, this system does not resemble the electron density in most molecules and one route to the improvement of DFT is to replace this foundation with a new one.
Electrons-on-a-sphere is a particularly attractive model because it is defined by a single parameter (the radius R of the sphere) and varying this takes us from a weakly correlated system (small R) dominated by dynamical correlation, to a strongly correlated system (large R) dominated by static correlation.
I will review this model and show how it can be used as the starting point for a new way of understanding and improving DFT.
Miquel Huix-Rotllant
CNRS and Université Aix-Marseille
Time-dependent density functional theory (TDDFT) has become a fundamental tool in quantum chemistry to calculate excited state properties of molecular systems with up to several hundreds of atoms. Despite this success, the approximations performed at the level of the exchange-correlation functional limit its accuracy. Indeed, several important problems have been detected like incorrect description of static correlation, the wrong description of charge-transfer and/or Rydberg states, or the incorrect description of state intersections with the ground state. This course will be divided in two parts: first, I will give a simple introduction to exact TDDFT and the main approximations usually applied to solve TDDFT equations in practice, focusing on the main drawbacks that are frequently encountered in approximate TDDFT. Second, we will do a hands-on tutorial with some practical examples like analysis of excited states and simulation of absorption and emission spectra for some organic molecules.
Cyrus Umrigar
Physics Department, Cornell University
Quantum Monte Carlo Methods in Chemistry and Physics
In this lecture I convey the basic ideas behind Variational Monte Carlo (VMC) and various projector Monte Carlo (PMC) methods in a unified manner. Similarities and differences between the methods, and, choices made in designing the methods are discussed, with particular reference to the notorious Fermion sign problem. Both methods where the Monte Carlo walk is performed in a discrete space, and, methods where it is performed in a continuous space are considered.
[1] Julien Toulouse, Roland Assaraf and Cyrus J. Umrigar, Introduction to the Variational and Diffusion Monte Carlo Methods, Adv. Quant. Chem., 73, 285 (2015)
[2] C. J. Umrigar, Observations on variational and projector Monte Carlo methods, J. Chem. Phys., 143, 164105 (2015)
Materials
Carlos Cárdenas
Universidad de Chile
From molecules to solids
- Free electrons in a box.
- Born–von Karman periodic boundary condition (PBC)
- Free electrons in periodic boundary conditions: a simple model of metals. The concept of density of states will be intruded here.
- Chain Molecule. The reciprocal space will be introduced here.
- The square and cubic lattices
- Brillouin zones
- Energy Bands: insulators, semiconductors and metals.
Varinia Bernales
Gagliardi Group, Department of Chemistry, University of Minnesota Twin-cities, Minneapolis, MN 55455-0431, USA
Multiconfigurational Wave Function Theory: A perspective on the examination of transition metals in porous materials: In this lecture, we will cover the fundamentals and applications of multireference wave function theory. The basic theory and notation will be introduced and some popular multireference methods will be discussed. We will review interesting applications in which multireference methods can be applied, such as the electronic structure characterization of molecular complexes, catalysis, and force-field development. Additional comparison between density functional and multireference theory on such complicated systems will be discussed, and the advantages and limitations of these methods will be presented. In the second part of my seminar, recent research on porous materials such zeolites and metal-organic frameworks (MOFs) will be discussed. In particular, the characteristic nature of MOFs allows combinatorial flexibility and incorporation of supported metal atoms or clusters for the rational design of novel catalytic materials. The reactive intermediates have a multiconfigurational character and thus, multireference methods are able to elucidate their electronic structure.
Juan E. Peralta
Department of Physics and Science of Advanced Materials Program, Central Michigan University, Mount Pleasant, MI 48859, USA.
Magnetic Properties of Transition Metal Complexes from Density Functional Theory
The characterization of the magnetic properties of molecular systems is important to understand and make predictions of the magnetic behavior of materials where magnetism is of molecular origin.[1] In this short course we will first briefly review the physical origin and theory of magnetic properties in molecular systems, and describe current methods to evaluate these properties in complexes containing transition metal atoms using density functional theory (DFT).[1-4] We will emphasize the strengths and deficiencies of DFT for these properties, including those intrinsic of DFT and those of current approximations. Finally, we will analyze real examples using the magnetic anisotropy energy and magnetic exchange couplings in connection to observed magnetic experimental data. [5-7]
[1] O. Kahn, Molecular Magnetism, VCH: New York, NY (1993).
[2] J. Kortus, M. R. Pederson, T. Baruah, N. Bernstein, and C. S. Hellberg, Density functional studies of single molecule magnets, Polyhedron 22, 1871 (2003).
[3] C. van Wüllen, Magnetic anisotropy from density functional calculations. Comparison of different approaches: Mn12O12 acetate as a test case, J. Chem. Phys. 130, 194109 (2009).
[4] J. J. Phillips, and J. E. Peralta, The role of range-separated Hartree-Fock exchange in the calculation of magnetic exchange couplings in transition metal complexes, J. Chem. Phys. 134 034108 (2011).
[5] Jens Kortus, Tunna Baruah, Noam Bernstein, and Mark R. Pederson, Magnetic ordering, electronic structure, and magnetic anisotropy energy in the high-spin Mn10 single molecule magnet, Phys. Rev. B 66 092403 (2002).
[6] J. J. Phillips, J. E. Peralta, and G. Christou, Magnetic couplings in spin frustrated FeIII7 disklike clusters, J. Chem. Theory Comput. 9, 5585 (2013).
[7] R. P. Joshi, J. J. Phillips, and J. E. Peralta, Magnetic exchange couplings in heterodinuclear complexes based on differential local spin rotations, J. Chem. Theory Comput. 12, 1728 (2016).
Chemical Bonding
Marco García-Revilla
Univ. Guanajuato, México
Summary The study of the Chemical Bond belongs to the most important issues in Chemistry. The new methodologies in this field enable us to rationalize chemical phenomenon where the traditional models fail. The Interacting Quantum Atoms (IQA)[1,2] and the Electron Distribution Functions EDF[3] belong to such new methodologies. IQA and EDF have been shown to be successful to deal with the study of the Chemical Bond. Two studies are presented in this lecture. 1) Oxygen under extreme pressure conditions, the unexplained physicochemical behavior of O2 under pressure can be finally rationalized by the IQA and EDF methodology.[4] 2) Constructing molecular graphs from IQA bonding descriptors, the exchange-correlation energies can be used to draw molecular graphs with physical insight.[5]
[1] A. Martín Pendás, M. A. Blanco, and E. Francisco. J. Chem. Phys. 2004, 120, 4581; J. Comput. Chem. 2005, 26, 344; J. Chem. Theory Comput. 2005,1,1096; J. Comput. Chem. 2007, 28, 16;. A. Martín Pendás, M. A. Blanco, and E. Francisco. J. Chem. Phys. 2006, 125. 184112 A. Martín Pendás, M. A. Blanco,and E. Francisco. J. Comput. Chem. 2009, 30, 98; D. Tiana et al. J. Chem.Theory Comput. 2010, 6, 1064; D. Tiana et. al. Phys. Chem. Chem. Phys. 2011,13, 5068.
[2] A. Martín Pendás, E. Francisco, M. A. Blanco, and Carlo Gatti. Chem. Eur.
J. 13, 9362 (2007).
[3] E. Francisco, A. Martín Pendás, M. A. Blanco. J. Chem. Phys. 126, 094102 (2007); E. Francisco, M. A. Blanco , A. Martín Pendás. Comp. Phys. Commun. 178, 621 (2008); A. Martín Pendás, E. Francisco, M. A. Blanco, Phys. Chem. Chem. Phys. 9, 1087 (2007).
[4] M. A García-Revilla, E.Francisco, A.Martín Pendás, J.M.Recio, M.I.Hernández, J. Campos-Martínez, E. Carmona-Novillo, and R. Hernández-Lamoneda. J. Chem. Theory Comput., 9, 2179 (2013).
[5] M. A García-Revilla, E. Francisco, PL. Popelier, and A. Martín Pendás. Chemphyschem, 14,1211 (2013).
Ángel Martín Pendás
Universidad de Oviedo. Spain.
A real space perspective of how energy and electrons distribute in molecules: Interacting quantum atoms and electron distribution functions
The topological approach to chemical bonding in real space, or Quantum Chemical Topology (QCT) has now come of age. Its best known flavor, the Quantum Theory of Atoms in Molecules (QTAIM) [1] has been extremely successful, providing an orbital invariant theory of chemical bonding problems based on an observable, the electron density, amenable to experimental determination. In this course we will consider the basics of QCT as well as two development that expands its scope and predictive power: the Interacting Quantum Atoms (IQA) [2-3] approach, which provides an exact energetic decomposition within the QTAIM valid at general geometries, and the electron distribution functions (EDF) [4].
[1] R. F. W. Bader, Atoms in Molecules , Oxford University Press., Oxford (1990). [2] A. Martín Pendás, M. A. Blanco, and E. Francisco. J. Chem. Phys. 2004, 120, 4581; J. Comput. Chem. 2005, 26, 344; J. Chem. Theory Comput. 2005, 1, 1096; J. Comput. Chem. 2007, 28, 16;. A. Martín Pendás, M. A. Blanco, and E. Francisco. J. Chem. Phys. 2006, 125. 184112 A. Martín Pendás, M. A. Blanco, and E. Francisco. J. Comput. Chem. 2009, 30, 98; D. Tiana et al. J. Chem. Theory Comput. 2010, 6, 1064; D. Tiana et. al. Phys. Chem. Chem. Phys. 2011, 13, 5068. [3] A. Martín Pendás, E. Francisco, M. A. Blanco, and Carlo Gatti. Chem. Eur. J. 13, 9362 (2007). [4] E. Francisco, A. Martín Pendás, M. A. Blanco. J. Chem. Phys. 126, 094102 (2007); E. Francisco, M. A. Blanco , A. Martín Pendás. Comp. Phys. Commun. 178, 621 (2008); A. Martín Pendás, E. Francisco, M. A. Blanco, Phys. Chem. Chem. Phys. 9, 1087 (2007).
Summary
Eloy Ramos-Cordoba
University of California, Berkeley
Quantifying and characterizing the (poly)radical nature of molecules
Diradicals are molecules with two electrons occupying two near degenerate orbitals. Indeed, how close to degeneracy these orbitals are (HOMO-LUMO gap) or more generally the singlet-triplet gap is one of the characteristic features of diradical systems. Diradicals are important in chemistry since they emerge as intermediates of many chemical reactions. Pure, ideal diradicals such as a dissociated H2 singlet can be easily characterized theoretically from different indicators, depending on the nature of the wave function. However, the quantification of the diradical or diradicaloid character of short-lived singlet diradicals is not so trivial because the formally unpaired electrons do interact to some extent. More generally, a polyradical can be considered as a molecule with N electrons occupying N near degenerate orbitals. The design of organic high-spin polyradicals is of major relevance for the development of new organic and organometallic magnetic materials. As opposed to the simplest case of diradicals, the more general case of organic polyradicals adds extra complexity to the puzzle of radical character quantification, as most of the indices and quantities that have been proposed in the literature to detect and quantify the diradical character of molecular systems cannot be trivially generalized beyond diradicals. In this lecture, I will review different methods to characterize unpaired electrons in radical or polyradical systems from ab initio calculations. I will introduce the Local Spin analysis and I will show the ability of this analysis to characterize and quantify the radical nature of molecular systems.
Pedro Salvador
Universitat de Girona (Catalonia, Spain)
Effective atomic orbitals and its application to Oxidation State analysis
The concept of atom in a molecule has always craved for a proper definition, and we are lacking a single, unambiguous one. Within the LCAO approach, the atom may be identified with the subspace of the basis functions attached to it. Such approach leads to the so-called Hilbert-space analyses. A different strategy is to subdivide the physical (3D) space into atomic regions or domains, which represent (together with the nucleus) the atom. When looking for an atom in a molecule, obviously we are not merely interested in a subdivision of the 3D space into atomic volumes, but rather in assigning different physical quantities to the individual atoms (or their groups).
Probably, the most appropriate entities that serve to characterize the state of the atom within the molecule are the so-called effective atomic orbitals (“effective AOs”). In this approach, one obtains for each atom a set of orthogonal atomic hybrids and their respective occupation numbers, adding up to the net population of the atom, even if no atom-centered basis functions are used at all to expand the molecular orbitals. These atomic hybrids closely mimic the core and valence shells of the atom, as anticipated on the basis of classical notions of electron configuration of the atom/fragment within the molecule.
In this lecture we will review the basic features of the effective AOs, in the framework of fuzzy atoms and disjoints domains such as QTAIM. It is of particular conceptual relevance the fact that in the basis of such orbitals, the Hilbert-space and 3D-space analysis are numerically equivalent. We will end by seeing a direct application of these objects, for the derivation of formal oxidation states from wave function analysis.