Density Functional Theory
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Atomic Orbitals
Visualization of Atomic Orbitals Source: Robert R. Gotwals, Jr., Department of Chemistry, North Carolina School of Science and Mathematics, Durham, NC, 27705. Email: [email protected]. Copyright ©2007, all rights reserved, permission to use for non-commercial educational purposes is granted. Introductory Readings Where are the electrons in an atom? There are a number of Models - a description of models that look to describe the location and behavior of some physical object or electrons in atoms. It is important to know this information, as behavior in nature. Models it helps chemists to make statements and predictions about are often mathematical, describing the behavior of how atoms will react with other atoms to form molecules. The some event. model one chooses will oftentimes influence the types of statements one can make about the behavior – that is, the reactivity – of an atom or group of atoms. Bohr model - states that no One model, the Bohr model, describes electrons as small two electrons in an atom can billiard balls, rotating around the positively charged nucleus, have the same set of four much in the way that planets quantum numbers. orbit around the Sun. The placement and motion of electrons are found in fixed and quantifiable levels called orbits, Orbits- where electrons are or, in older terminology, shells. believed to be located in the There are specific numbers of Bohr model. Orbits have a electrons that can be found in fixed amount of energy, and are identified with the each orbit – 2 for the first orbit, principle quantum number 8 for the second, and so forth. -
Chemical Bonding & Chemical Structure
Chemistry 201 – 2009 Chapter 1, Page 1 Chapter 1 – Chemical Bonding & Chemical Structure ings from inside your textbook because I normally ex- Getting Started pect you to read the entire chapter. 4. Finally, there will often be a Supplement that con- If you’ve downloaded this guide, it means you’re getting tains comments on material that I have found espe- serious about studying. So do you already have an idea cially tricky. Material that I expect you to memorize about how you’re going to study? will also be placed here. Maybe you thought you would read all of chapter 1 and then try the homework? That sounds good. Or maybe you Checklist thought you’d read a little bit, then do some problems from the book, and just keep switching back and forth? That When you have finished studying Chapter 1, you should be sounds really good. Or … maybe you thought you would able to:1 go through the chapter and make a list of all of the impor- tant technical terms in bold? That might be good too. 1. State the number of valence electrons on the following atoms: H, Li, Na, K, Mg, B, Al, C, Si, N, P, O, S, F, So what point am I trying to make here? Simply this – you Cl, Br, I should do whatever you think will work. Try something. Do something. Anything you do will help. 2. Draw and interpret Lewis structures Are some things better to do than others? Of course! But a. Use bond lengths to predict bond orders, and vice figuring out which study methods work well and which versa ones don’t will take time. -
Density Functional Theory
Density Functional Theory Fundamentals Video V.i Density Functional Theory: New Developments Donald G. Truhlar Department of Chemistry, University of Minnesota Support: AFOSR, NSF, EMSL Why is electronic structure theory important? Most of the information we want to know about chemistry is in the electron density and electronic energy. dipole moment, Born-Oppenheimer charge distribution, approximation 1927 ... potential energy surface molecular geometry barrier heights bond energies spectra How do we calculate the electronic structure? Example: electronic structure of benzene (42 electrons) Erwin Schrödinger 1925 — wave function theory All the information is contained in the wave function, an antisymmetric function of 126 coordinates and 42 electronic spin components. Theoretical Musings! ● Ψ is complicated. ● Difficult to interpret. ● Can we simplify things? 1/2 ● Ψ has strange units: (prob. density) , ● Can we not use a physical observable? ● What particular physical observable is useful? ● Physical observable that allows us to construct the Hamiltonian a priori. How do we calculate the electronic structure? Example: electronic structure of benzene (42 electrons) Erwin Schrödinger 1925 — wave function theory All the information is contained in the wave function, an antisymmetric function of 126 coordinates and 42 electronic spin components. Pierre Hohenberg and Walter Kohn 1964 — density functional theory All the information is contained in the density, a simple function of 3 coordinates. Electronic structure (continued) Erwin Schrödinger -
Electron Configurations, Orbital Notation and Quantum Numbers
5 Electron Configurations, Orbital Notation and Quantum Numbers Electron Configurations, Orbital Notation and Quantum Numbers Understanding Electron Arrangement and Oxidation States Chemical properties depend on the number and arrangement of electrons in an atom. Usually, only the valence or outermost electrons are involved in chemical reactions. The electron cloud is compartmentalized. We model this compartmentalization through the use of electron configurations and orbital notations. The compartmentalization is as follows, energy levels have sublevels which have orbitals within them. We can use an apartment building as an analogy. The atom is the building, the floors of the apartment building are the energy levels, the apartments on a given floor are the orbitals and electrons reside inside the orbitals. There are two governing rules to consider when assigning electron configurations and orbital notations. Along with these rules, you must remember electrons are lazy and they hate each other, they will fill the lowest energy states first AND electrons repel each other since like charges repel. Rule 1: The Pauli Exclusion Principle In 1925, Wolfgang Pauli stated: No two electrons in an atom can have the same set of four quantum numbers. This means no atomic orbital can contain more than TWO electrons and the electrons must be of opposite spin if they are to form a pair within an orbital. Rule 2: Hunds Rule The most stable arrangement of electrons is one with the maximum number of unpaired electrons. It minimizes electron-electron repulsions and stabilizes the atom. Here is an analogy. In large families with several children, it is a luxury for each child to have their own room. -
Electron Density Functions for Simple Molecules (Chemical Bonding/Excited States) RALPH G
Proc. Natl. Acad. Sci. USA Vol. 77, No. 4, pp. 1725-1727, April 1980 Chemistry Electron density functions for simple molecules (chemical bonding/excited states) RALPH G. PEARSON AND WILLIAM E. PALKE Department of Chemistry, University of California, Santa Barbara, California 93106 Contributed by Ralph G. Pearson, January 8, 1980 ABSTRACT Trial electron density functions have some If each component of the charge density is centered on a conceptual and computational advantages over wave functions. single point, the calculation of the potential energy is made The properties of some simple density functions for H+2 and H2 are examined. It appears that for a diatomic molecule a good much easier. However, the kinetic energy is often difficult to density function would be given by p = N(A2 + B2), in which calculate for densities containing several one-center terms. For A and B are short sums of s, p, d, etc. orbitals centered on each a wave function that is composed of one orbital, the kinetic nucleus. Some examples are also given for electron densities that energy can be written in terms of the electron density as are appropriate for excited states. T = 1/8 (VP)2dr [5] Primarily because of the work of Hohenberg, Kohn, and Sham p (1, 2), there has been great interest in studying quantum me- and in most cases this integral was evaluated numerically in chanical problems by using the electron density function rather cylindrical coordinates. The X integral was trivial in every case, than the wave function as a means of approach. Examples of and the z and r integrations were carried out via two-dimen- the use of the electron density include studies of chemical sional Gaussian bonding in molecules (3, 4), solid state properties (5), inter- quadrature. -
Chemistry 1000 Lecture 8: Multielectron Atoms
Chemistry 1000 Lecture 8: Multielectron atoms Marc R. Roussel September 14, 2018 Marc R. Roussel Multielectron atoms September 14, 2018 1 / 23 Spin Spin Spin (with associated quantum number s) is a type of angular momentum attached to a particle. Every particle of the same kind (e.g. every electron) has the same value of s. Two types of particles: Fermions: s is a half integer 1 Examples: electrons, protons, neutrons (s = 2 ) Bosons: s is an integer Example: photons (s = 1) Marc R. Roussel Multielectron atoms September 14, 2018 2 / 23 Spin Spin magnetic quantum number All types of angular momentum obey similar rules. There is a spin magnetic quantum number ms which gives the z component of the spin angular momentum vector: Sz = ms ~ The value of ms can be −s; −s + 1;:::; s. 1 1 1 For electrons, s = 2 so ms can only take the values − 2 or 2 . Marc R. Roussel Multielectron atoms September 14, 2018 3 / 23 Spin Stern-Gerlach experiment How do we know that electrons (e.g.) have spin? Source: Theresa Knott, Wikimedia Commons, http://en.wikipedia.org/wiki/File:Stern-Gerlach_experiment.PNG Marc R. Roussel Multielectron atoms September 14, 2018 4 / 23 Spin Pauli exclusion principle No two fermions can share a set of quantum numbers. Marc R. Roussel Multielectron atoms September 14, 2018 5 / 23 Multielectron atoms Multielectron atoms Electrons occupy orbitals similar (in shape and angular momentum) to those of hydrogen. Same orbital names used, e.g. 1s, 2px , etc. The number of orbitals of each type is still set by the number of possible values of m`, so e.g. -
“Band Structure” for Electronic Structure Description
www.fhi-berlin.mpg.de Fundamentals of heterogeneous catalysis The very basics of “band structure” for electronic structure description R. Schlögl www.fhi-berlin.mpg.de 1 Disclaimer www.fhi-berlin.mpg.de • This lecture is a qualitative introduction into the concept of electronic structure descriptions different from “chemical bonds”. • No mathematics, unfortunately not rigorous. • For real understanding read texts on solid state physics. • Hellwege, Marfunin, Ibach+Lüth • The intention is to give an impression about concepts of electronic structure theory of solids. 2 Electronic structure www.fhi-berlin.mpg.de • For chemists well described by Lewis formalism. • “chemical bonds”. • Theorists derive chemical bonding from interaction of atoms with all their electrons. • Quantum chemistry with fundamentally different approaches: – DFT (density functional theory), see introduction in this series – Wavefunction-based (many variants). • All solve the Schrödinger equation. • Arrive at description of energetics and geometry of atomic arrangements; “molecules” or unit cells. 3 Catalysis and energy structure www.fhi-berlin.mpg.de • The surface of a solid is traditionally not described by electronic structure theory: • No periodicity with the bulk: cluster or slab models. • Bulk structure infinite periodically and defect-free. • Surface electronic structure theory independent field of research with similar basics. • Never assume to explain reactivity with bulk electronic structure: accuracy, model assumptions, termination issue. • But: electronic -
Lecture 3. the Origin of the Atomic Orbital: Where the Electrons Are
LECTURE 3. THE ORIGIN OF THE ATOMIC ORBITAL: WHERE THE ELECTRONS ARE In one sentence I will tell you the most important idea in this lecture: Wave equation solutions generate atomic orbitals that define the electron distribution around an atom. To start we need to simplify the math by switching to spherical polar coordinates (everything = spherical) rather than Cartesian coordinates (everything = at right angles). So the wave equations generated will now be of the form ψ(r, θ, Φ) = R(r)Y(θ, Φ) where R(r) describes how far out on a radial trajectory from the nucleus you are. r is a little way out and is small Now that we are extended radially on ψ, we need to ask where we are on the sphere carved out by R(r). Think of blowing up a balloon and asking what is going on at the surface of each new R(r). To cover the entire surface at projection R(r), we need two angles θ, Φ that get us around the sphere in a manner similar to knowing the latitude & longitude on the earth. These two angles yield Y(θ, Φ) which is the angular wave function A first solution: generating the 1s orbit So what answers did Schrodinger get for ψ(r, θ, Φ)? It depended on the four quantum numbers that bounded the system n, l, ml and ms. So when n = 1 and = the solution he calculated was: 1/2 -Zr/a0 1/2 R(r) = 2(Z/a0) *e and Y(θ, Φ) = (1/4π) . -
Simple Molecular Orbital Theory Chapter 5
Simple Molecular Orbital Theory Chapter 5 Wednesday, October 7, 2015 Using Symmetry: Molecular Orbitals One approach to understanding the electronic structure of molecules is called Molecular Orbital Theory. • MO theory assumes that the valence electrons of the atoms within a molecule become the valence electrons of the entire molecule. • Molecular orbitals are constructed by taking linear combinations of the valence orbitals of atoms within the molecule. For example, consider H2: 1s + 1s + • Symmetry will allow us to treat more complex molecules by helping us to determine which AOs combine to make MOs LCAO MO Theory MO Math for Diatomic Molecules 1 2 A ------ B Each MO may be written as an LCAO: cc11 2 2 Since the probability density is given by the square of the wavefunction: probability of finding the ditto atom B overlap term, important electron close to atom A between the atoms MO Math for Diatomic Molecules 1 1 S The individual AOs are normalized: 100% probability of finding electron somewhere for each free atom MO Math for Homonuclear Diatomic Molecules For two identical AOs on identical atoms, the electrons are equally shared, so: 22 cc11 2 2 cc12 In other words: cc12 So we have two MOs from the two AOs: c ,1() 1 2 c ,1() 1 2 2 2 After normalization (setting d 1 and _ d 1 ): 1 1 () () [2(1 S )]1/2 12 [2(1 S )]1/2 12 where S is the overlap integral: 01 S LCAO MO Energy Diagram for H2 H2 molecule: two 1s atomic orbitals combine to make one bonding and one antibonding molecular orbital. -
Electron Charge Density: a Clue from Quantum Chemistry for Quantum Foundations
Electron Charge Density: A Clue from Quantum Chemistry for Quantum Foundations Charles T. Sebens California Institute of Technology arXiv v.2 June 24, 2021 Forthcoming in Foundations of Physics Abstract Within quantum chemistry, the electron clouds that surround nuclei in atoms and molecules are sometimes treated as clouds of probability and sometimes as clouds of charge. These two roles, tracing back to Schr¨odingerand Born, are in tension with one another but are not incompatible. Schr¨odinger'sidea that the nucleus of an atom is surrounded by a spread-out electron charge density is supported by a variety of evidence from quantum chemistry, including two methods that are used to determine atomic and molecular structure: the Hartree-Fock method and density functional theory. Taking this evidence as a clue to the foundations of quantum physics, Schr¨odinger'selectron charge density can be incorporated into many different interpretations of quantum mechanics (and extensions of such interpretations to quantum field theory). Contents 1 Introduction2 2 Probability Density and Charge Density3 3 Charge Density in Quantum Chemistry9 3.1 The Hartree-Fock Method . 10 arXiv:2105.11988v2 [quant-ph] 24 Jun 2021 3.2 Density Functional Theory . 20 3.3 Further Evidence . 25 4 Charge Density in Quantum Foundations 26 4.1 GRW Theory . 26 4.2 The Many-Worlds Interpretation . 29 4.3 Bohmian Mechanics and Other Particle Interpretations . 31 4.4 Quantum Field Theory . 33 5 Conclusion 35 1 1 Introduction Despite the massive progress that has been made in physics, the composition of the atom remains unsettled. J. J. Thomson [1] famously advocated a \plum pudding" model where electrons are seen as tiny negative charges inside a sphere of uniformly distributed positive charge (like the raisins|once called \plums"|suspended in a plum pudding). -
Nature of Noncovalent Carbon-Bonding Interactions Derived from Experimental Charge-Density Analysis
Nature of Noncovalent Carbon-Bonding Interactions Derived from Experimental Charge-Density Analysis Eduardo C. Escudero-Adán, Antonio Bauzffl, Antonio Frontera, and Pablo Ballester E. C. Escudero-Adán,+ Prof. P. Ballester X-Ray Diffraction Unit Institute of Chemical Research of Catalonia (ICIQ) Avgda. Països Catalans 16, 43007 Tarragona (Spain) E-mail : [email protected] A. Bauzffl,+ Prof. A. Frontera Department of Chemistry Universitat de les Illes Balears Crta. de Valldemossa km 7.5, 07122 Palma de Mallorca (Spain) E-mail : [email protected] Prof. P. Ballester Catalan Institution for Research and Advanced Studies (ICREA) Passeig Lluïs Companys 23, 08010 Barcelona (Spain) In an effort to better understand the nature of noncovalent carbon-bonding interactions, we undertook accurate high-res- olution X-ray diffraction analysis of single crystals of 1,1,2,2-tet- racyanocyclopropane. We selected this compound to study the fundamental characteristics of carbon-bonding interactions, because it provides accessible s holes. The study required ex- tremely accurate experimental diffraction data, because the in- teraction of interest is weak. The electron-density distribution around the carbon nuclei, as shown by the experimental maps of the electrophilic bowl defined by a (CN)2C-C(CN)2 unit, was assigned as the origin of the interaction. This fact was also evidenced by plotting the D21(r) distribution. Taken together, the obtained results clearly indicate that noncovalent carbon bonding can be explained as an interaction between confront- ed oppositely polarized regions. The interaction is, thus elec- trophilic–nucleophilic (electrostatic) in nature and unambigu- ously considered as attractive. Attractive intermolecular electrostatic interactions encompass electron-rich and electron-poor regions of two molecules that complement each other.[1] Electron-rich entities are typically anions or lone-pair electrons and the most well-known elec- tron-poor entity is the hydrogen atom. -
Electronic Structure Methods
Electronic Structure Methods • One-electron models – e.g., Huckel theory • Semiempirical methods – e.g., AM1, PM3, MNDO • Single-reference based ab initio methods •Hartree-Fock • Perturbation theory – MP2, MP3, MP4 • Coupled cluster theory – e.g., CCSD(T) • Multi-configurational based ab initio methods • MCSCF and CASSCF (also GVB) • MR-MP2 and CASPT2 • MR-CI •Ab Initio Methods for IPs, EAs, excitation energies •CI singles (CIS) •TDHF •EOM and Greens function methods •CC2 • Density functional theory (DFT)- Combine functionals for exchange and for correlation • Local density approximation (LDA) •Perdew-Wang 91 (PW91) • Becke-Perdew (BP) •BeckeLYP(BLYP) • Becke3LYP (B3LYP) • Time dependent DFT (TDDFT) (for excited states) • Hybrid methods • QM/MM • Solvation models Why so many methods to solve Hψ = Eψ? Electronic Hamiltonian, BO approximation 1 2 Z A 1 Z AZ B H = − ∑∇i − ∑∑∑+ + (in a.u.) 2 riA iB〈〈j rij A RAB 1/rij is what makes it tough (nonseparable)!! Hartree-Fock method: • Wavefunction antisymmetrized product of orbitals Ψ =|φ11 (τφτ ) ⋅⋅⋅NN ( ) | ← Slater determinant accounts for indistinguishability of electrons In general, τ refers to both spatial and spin coordinates For the 2-electron case 1 Ψ = ϕ (τ )ϕ (τ ) = []ϕ (τ )ϕ (τ ) −ϕ (τ )ϕ (τ ) 1 1 2 2 2 1 1 2 2 2 1 1 2 Energy minimized – variational principle 〈Ψ | H | Ψ〉 Vary orbitals to minimize E, E = 〈Ψ | Ψ〉 keeping orbitals orthogonal hi ϕi = εi ϕi → orbitals and orbital energies 2 φ j (r2 ) 1 2 Z J = dr h = − ∇ − A + (J − K ) i ∑ ∫ 2 i i ∑ i i j ≠ i r12 2 riA ϕϕ()rr () Kϕϕ()rdrr= ji22 () ii121∑∫ j ji≠ r12 Characteristics • Each e- “sees” average charge distribution due to other e-.