DOCSLIB.ORG

Understanding and Interpreting Molecular Electron Density Distributions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Understanding and Interpreting Molecular Electron Density Distributions Research: Science and Education Understanding and Interpreting Molecular Electron Density Distributions C. F. Matta and R. J. Gillespie* Department of Chemistry, McMaster University, Hamilton, ON L8S 4M1, Canada; *[email protected] Advances in the power and speed of computers have made concerned in this paper. A more formal discussion of electron ab initio and density functional theory (DFT) calculations density is presented in Appendix 2. an almost routine procedure on a PC (1). From these calcu- The electron density is key to the bonding and geometry lations the equilibrium geometry, the energy, and the wave of a molecule because the forces holding the nuclei together function of a molecule can be determined. From the wave in a molecule are the attractive forces between the electrons and function one can obtain all the properties of the molecule, the nuclei. These attractive forces are opposed by the repulsions including the distribution of electronic charge, or electron between the electrons and the repulsions between the nuclei. density. However, the electron density is often not calculated In the equilibrium geometry of a molecule these electrostatic or discussed, perhaps because it is not widely realized that forces just balance. The fundamentally important Hellman– very useful information on bonding and geometry can be Feynman theorem (4–7) states that the force on a nucleus in a obtained from it. It seems particularly important to discuss molecule is the sum of the Coulombic forces exerted by the other electron densities in introductory chemistry courses because nuclei and by the electron density distribution ρ. This means students can grasp the concept of electron density much more that the energy of interaction of the electrons with the nuclei readily than the abstract mathematical concept of an orbital. It can be found by a consideration of the classical electrostatic is also not as widely understood as it should be that orbitals forces between the nuclei and the electronic charge cloud. There are not physical observables but only mathematical constructs are no mysterious quantum mechanical forces, and no other that cannot be determined by experiment (2). In contrast, the force, such as the gravitational force, is of any importance in electron density distribution in a molecule or crystal can be holding the atoms in a molecule together. The atoms are held observed by electron diffraction and X-ray crystallography together by the electrostatic force exerted by the electronic charge (3); and it can also, and often more readily, be obtained from on the nuclei. But it is quantum mechanics, and particularly ab initio and density functional theory calculations. the Pauli principle, that determines the distribution of elec- This article gives a simple introduction to the electron tronic charge, as we shall see. densities of molecules and how they can be analyzed to ob- tain information on bonding and geometry. More detailed The Representation of the Electron Density discussions can be found in the books by Bader (4), Popelier (5), and Gillespie and Popelier (6 ). Computational details The electron density (ρ) varies in three dimensions (i.e., to reproduce the results presented in this paper are presented it is a function of the three spatial coordinates [x,y,z]), so a in Appendix 1. full description of how ρ varies with position requires a fourth dimension. A common solution to this problem is to show how ρ The Electron Density varies in one or more particular planes of the molecule. Figure 1a shows a relief map of the electron density, ρ, of the σ Quantum mechanics allows the determination of the SCl2 molecule in the v(xz) plane. The most striking features probability of finding an electron in an infinitesimal volume of this figure are that ρ is very large in an almost spherical surrounding any particular point in space (x,y,z); that is, the region around each nucleus while assuming relatively very probability density at this point. Since we can assign a prob- small values, and at first sight featureless topology, between ability density to any point in space, the probability density these nuclear regions. The high electron density in the nearly defines a scalar field, which is known as the probability density spherical region around each nucleus arises from the tightly distribution. When the probability density distribution is held core electrons; the relatively very small and more diffuse multiplied by the total number of electrons in the molecule, density between these regions arises from the more weakly N, it becomes what is known as the electron density distribution held bonding electrons. In fact, it was necessary to truncate or simply the electron density and is given the symbol ρ(x,y,z). the very high maxima in Figure 1a (at ρ = 2.00 au)1 to make It represents the probability of finding any one of the N it possible to show the features of the electron density distri- electrons in an infinitesimal volume of space surrounding bution between the nuclei. In particular, there is a ridge of the point (x,y,z), and therefore it yields the total number of increased electron density between the sulfur atom and each electrons when integrated over all space. The electron density of the chlorine atoms. The electron density has values of can be conveniently thought of as a cloud or gas of negative 3.123 × 103 and 2.589 × 103 au at the S and Cl nuclei, re- × ᎑1 charge that varies in density throughout the molecule. Such a spectively, but a value of only 1.662 10 au at the minimum charge cloud, or an approximate representation of it, is often of the ridge between the peak around the sulfur nucleus and used in introductory texts to represent the electron density each of the chlorine nuclei. This ridge of increased electron ψ2 of an atomic orbital. It is also often used incorrectly to density between the S atom and each of the Cl atoms, small as depict the orbital ψ itself. In a multielectron atom or molecule it is, is the density in the bonding region that is responsible for only the total electron density can be experimentally observed pulling the nuclei together. Along a line at the top of this ridge or calculated, and it is this total density with which we are the electron density is locally greater than in any direction JChemEd.chem.wisc.edu • Vol. 79 No. 9 September 2002 • Journal of Chemical Education 1141 Research: Science and Education a Figure 1. Relief maps of the electron density of (a) SCl2 and (b) H2O in the plane of the nuclei (density and distances from the origin of the coordinate system in au). Isodensity contour lines are shown in the order 0.001, 0.002, 0.004, 0.008 (four outermost contours); 0.02, 0.04, 0.08 (next three); 0.2, 0.4, 0.8 (next three). The density is truncated at 2.00 au (innermost contour). These contours are b shown in blue, violet, magenta, and green, respectively, on the Figure 2. Contour maps of the electron density of (a) SCl2 and (b) figure in the table of contents (p 1028). H2O. The density increases from the outermost 0.001 au isodensity contour in steps of 2 × 10n, 4 × 10n, and 8 × 10n au with n start- ing at ᎑3 and increasing in steps of unity. The lines connecting the away from the line. This line coincides with the bond between nuclei are the bond paths, and the lines delimiting each atom are the S and Cl atoms as it is normally drawn and is called a the intersection of the respective interatomic surface with the plane bond path (4–6, 10). The point of minimum electron density of the drawing. The same values for the contours apply to subse- along the bond path is called the bond critical point. quent contour plots in this paper. Figure 1b shows a relief map of the electron density of the water molecule. Its features are similar to those of the density map for SCl2, but the electron density around the hydrogen of the electron population. Thus this outer contour shows atoms is much smaller than around the oxygen atom, as we the shape of the molecule in the chosen plane. In a condensed would expect. The electron density at the maximum at the phase the effective size of a molecule is a little smaller. We oxygen nucleus has a value of 2.947 × 102 au, whereas that see more clearly here that the bond paths (the lines along at the hydrogen nucleus is only 4.341 × 10᎑1 au, which is only the top of the density ridges between the nuclei) coincide slightly greater than the value of 3.963 × 10᎑1 au at the with the bonds as they are normally drawn. minimum at the bond critical point. The very small electron Figure 3 shows the electron density contour maps for density surrounding the hydrogen nucleus is due to less than the period 2 fluorides LiF, BF3, CF4, OF2, and for the iso- one electron because the more electronegative oxygen atom lated B atom. In LiF each atom is almost spherical, consis- attracts electron density away from the hydrogen atom so that tent with the usual model of this molecule as consisting of it has a positive charge. the ions Li+ and F᎑. The volume of the lithium atom is much Another common way to represent the electron density smaller than that of the F atom, again consistent with the ionic distribution is as a contour map, analogous to a topographic model. We will see later that we can also obtain the atomic contour map representing the relief of a part of the earth’s charges from the electron density and that the charges on the surface.
Load more

Recommended publications
  • Intuitive Chemical Topology Concepts. / In
    1 Intuitive Chemical Topology Concepts Eugene V. Babaev Moscow State University, Moscow 119899, RUSSIA 1. Introduction During last two decades, chemistry underwent a strong influence from nonroutine mathematical methods. Chemical applications of graph theory [1–10], topology [11–18], and related fields of fundamental mathematics [21–27] are growing rapidly. This trend may indicate the birth of a novel interdisciplinary field, mathematical chemistry, imperceptibly flourishing on the border of chemistry and pure mathematics. The term “chemical topology” (the title of this volume) has become frequent in papers, books, and scientific conferences. Similarly, the term molecular topology (differing in its sense from molecular geometry) has become frequently used, although there may exist diverse viewpoints on the meaning of this concept. The key difference between topology and geometry is that in geometry the concepts of distances and angles are important, whereas in topology they are not. Instead, the essential topological properties are connectedness and continuity. Chemical structural formulas are good examples of topological models of molecules: the pattern of connectivity between atoms is essential, whereas the interatomic distances and angles are less important and may be neglected. A colloquial definition of topology is “rubber” geometry. Indeed, one may consider a topological object made of an ideal elastic material, so that any deformation (like stretching or distortion) retains points of this elastic object “close enough” one to another. In contrast, cutting separates previously close points, and any joining of parts makes points “closer” one to another. Such a continuous transformation of an object without cutting and joining any parts is called homeomorphism. (The term should not be confused with homomorphism.) The study of properties that are invariant to homeomorphic transformations is the fundamental goal of topology.
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • From Chemical Topology to Molecular Machines Nobel Lecture, December 8, 2016 by Jean-Pierre Sauvage University of Strasbourg, Strasbourg, France
    From Chemical Topology to Molecular Machines Nobel Lecture, December 8, 2016 by Jean-Pierre Sauvage University of Strasbourg, Strasbourg, France. o a large extent, the eld of “molecular machines” started aer several groups T were able to prepare reasonably easily interlocking ring compounds (named catenanes for compounds consisting of interlocking rings and rotaxanes for rings threaded by molecular laments or axes). Important families of molecular machines not belonging to the interlocking world were also designed, prepared and studied but, for most of them, their elaboration was more recent than that of catenanes or rotaxanes. Since the creation of interlocking ring molecules is so important in relation to the molecular machinery area, we will start with this aspect of our work. e second part will naturally be devoted to the dynamic properties of such systems and to the compounds for which motions can be directed in a controlled manner from the outside, i.e., molecular machines. We will restrict our discussion to a very limited number of examples which we con- sider particularly representative of the eld. CHEMICAL TOPOLOGY Generally speaking, chemical topology refers to molecules whose graph (i.e., their representation based on atoms and bonds) is non-planar [1–2]. A planar graph cannot be represented in a plane or on a sheet of paper without crossing points. In topology, the object can be distorted as much as one likes but its topo- logical properties are not modied as long as no cleavage occurs [3]. In other 111 112 The Nobel Prizes words, a circle and an ellipse are topologically identical.
    [Show full text]
  • 10 Mar 2015 Density Functional Theory for Field Theorists I
    RUNHETC-2015-07 SCIPP 15/11 Density Functional Theory for Field Theorists I Tom Banks NHETC and Department of Physics Rutgers University, Piscataway, NJ 08854-8019 and Department of Physics and SCIPP University of California, Santa Cruz, CA 95064 E-mail: [email protected] Abstract I summarize Density Functional Theory (DFT) in a language familiar to quantum field theorists, and introduce several apparently novel ideas for constructing systematic approximations for the density functional. I also note that, at least within the large K approximation (K is the number of electron spin components), it is easier to compute the quantum effective action of the Coulomb photon field, which is related to the density functional by algebraic manipulations in momentum space. arXiv:1503.02925v1 [cond-mat.mtrl-sci] 10 Mar 2015 0 Contents 1 Density Functional Theory for Quantum Field Theorists 1 2 Expansions in the Number of Spin Components 5 2.1 The Large K Expansion ............................. 5 2.2 SmallKExpansion ................................ 8 2.3 TheKSEquations ................................ 9 3 Expansions of the Functional Determinant 10 4 1/K expansion of the HEG 11 5 Conclusions 12 6 Acknowledgments 13 1 Density Functional Theory for Quantum Field The- orists Much of atomic, molecular and (quantum) condensed matter physics, reduces, in the non- relativistic limit, to the problem of solving the Schr¨odinger equation for point-like nuclei and electrons, interacting via the Coulomb potential. The electrons are fermions, and are best treated by introducing an electron field ik x ψi(x)= ai(k)e · , (1.1) Xk where i is a spin label. If we introduce dimensionless space-time coordinates, by using the Bohr radius to measure length, and the Rydberg to measure energy, the only parameters in me the problem are the nuclear charges Za and the mass ratios ma .
    [Show full text]
  • Density Functional Theory
    NEA/NSC/R(2015)5 Chapter 12. Density functional theory M. Freyss CEA, DEN, DEC, Centre de Cadarache, France Abstract This chapter gives an introduction to first-principles electronic structure calculations based on the density functional theory (DFT). Electronic structure calculations have a crucial importance in the multi-scale modelling scheme of materials: not only do they enable one to accurately determine physical and chemical properties of materials, they also provide data for the adjustment of parameters (or potentials) in higher-scale methods such as classical molecular dynamics, kinetic Monte Carlo, cluster dynamics, etc. Most of the properties of a solid depend on the behaviour of its electrons, and in order to model or predict them it is necessary to have an accurate method to compute the electronic structure. DFT is based on quantum theory and does not make use of any adjustable or empirical parameter: the only input data are the atomic number of the constituent atoms and some initial structural information. The complicated many-body problem of interacting electrons is replaced by an equivalent single electron problem, in which each electron is moving in an effective potential. DFT has been successfully applied to the determination of structural or dynamical properties (lattice structure, charge density, magnetisation, phonon spectra, etc.) of a wide variety of solids. Its efficiency was acknowledged by the attribution of the Nobel Prize in Chemistry in 1998 to one of its authors, Walter Kohn. A particular attention is given in this chapter to the ability of DFT to model the physical properties of nuclear materials such as actinide compounds.
    [Show full text]
  • Electronic Structure and Physical Properties of 13C Carbon Composite
    Electronic structure and physical properties of 13C carbon composite ABSTRACT: This review is devoted to the application of graphite and graphite composites in science and technology. Structure and electrical properties, as so technological aspects of producing of high-strength artificial graphite and dynamics of its destruction are considered. These type of graphite are traditionally used in the nuclear industry. Author was focused on the properties of graphite composites based on carbon isotope 13C. Generally, the review relies on the original results and concentrates on actual problems of application and testing of graphite materials in modern nuclear physics and science and its technology applications. Translated by author from chapters 5 of the Russian monograph by Zhmurikov E.I., Bubnenkov I.A., Pokrovsky A.S. et al. Graphite in Science and Nuclear Technique// eprint arXiv:1307.1869, 07/2013 (BC 2013arXiv1307.1869Z Author: Evgenij I. Zhmurikov Address: 68600 Pietarsaari, Finland E-mail: [email protected] KEYWORDS: Structure and properties of carbon; Isotope 13C; Radioactive ion beams 2 1. Introduction In order to explore ever-more exotic regions of the nuclear chart, towards the limits of stability of nuclei, European nuclear physicists have built several large-scale facilities in various countries of the European Union. Today they are collaborating in planning of a new radioactive ion beam (RIB) facility which will permit them to investigate hitherto unreachable parts of the nuclear chart. This European ISOL (isotope-separation-on-line) facility is called EURISOL [1]. At the present moment experiments with RIB of the first generation have yielded important results. But the first generation RIB facilities are often limited by the low intensity of the beams.
    [Show full text]
  • Topological Analysis of Electron Density
    Jyväskylä Summer School on Charge Density August 2007 Topological Analysis of Electron Density Louis J Farrugia Jyväskylä Summer School on Charge Density August 2007 What is topological analysis ? A method of obtaining chemically significant information from th e electron density ρ (rho ) ρ is a quantum -mechanical observable , and may also be obtained from experiment “It seems to me that experimental study of the scattered radiati on, in particular from light atoms, should get more attention, since in this way it should be possible to determine the arrangement of the electrons in the atoms.” P. Debye, Ann. Phys. (1915) 48 , 809. Jyväskylä Summer School on Charge Density August 2007 Why analyse the charge density ? The traditional way of approaching the theoretical basis of chem istry is though the wavefunction and the molecular orbitals obtained through (approximate) solutions to the Schr ödinger wave equation HHΨΨ == EE ΨΨ The Hohenberg -Kohn theorem confirmed that the density, ρ(r), is the fundamental property that characterises the ground state of a sy stem - once ρ(r) is known, the energy of the system is uniquely defined, and from there a diverse range of molecular properties can, in princ iple, be deduced. Thus a knowledge of ρ(r) opens the door to understanding of all the key challenges of chemistry. ρ is a quantum -mechanical observable. 9 P. Hohenberg, W. Kohn Phys. Rev. 1964 , 136 , B864. Jyväskylä Summer School on Charge Density August 2007 Can we ever observe orbitals ? Simply not ever possible - see E. Scerri (2000) J. Chem. Ed. 77 , 1492 . However ?? - see J.
    [Show full text]
  • Theoretical Basis of Quantum-Mechanical Modeling of Functional Nanostructures
    S S symmetry Review Theoretical Basis of Quantum-Mechanical Modeling of Functional Nanostructures Aleksey Fedotov 1,2 , Alexander Vakhrushev 1,2,*, Olesya Severyukhina 1,2, Anatolie Sidorenko 2 , Yuri Savva 2, Nikolay Klenov 3,4 and Igor Soloviev 3 1 Department Simulation and Synthesis Technology Structures, Institute of Mechanics, Udmurt Federal Research Center of the Ural Branch of the Russian Academy of Sciences, 426067 Izhevsk, Russia; [email protected] (A.F.); [email protected] (O.S.) 2 Laboratory of Functional Nanostructures, Orel State University named after I.S. Turgenev, 302026 Orel, Russia; [email protected] (A.S.); su_fi[email protected] (Y.S.) 3 Laboratory of Nanostructure Physics, Department of Microelectronics, Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, 119991 Moscow, Russia; [email protected] (N.K.); [email protected] (I.S.) 4 Science and Research Department, Moscow Technical University of Communication and Informatics, 111024 Moscow, Russia * Correspondence: [email protected] Abstract: The paper presents an analytical review of theoretical methods for modeling functional nanostructures. The main evolutionary changes in the approaches of quantum-mechanical modeling are described. The foundations of the first-principal theory are considered, including the stationery and time-dependent Schrödinger equations, wave functions, the form of writing energy operators, and the principles of solving equations. The idea and specifics of describing the motion and inter- action of nuclei and electrons in the framework of the theory of the electron density functional are Citation: Fedotov, A.; Vakhrushev, A.; Severyukhina, O.; Sidorenko, A.; presented. Common approximations and approaches in the methods of quantum mechanics are Savva, Y.; Klenov, N.; Soloviev, I.
    [Show full text]