Introduction to Electronic Structure Methods
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Correlation.Pdf
INTRODUCTION TO THE ELECTRONIC CORRELATION PROBLEM PAUL E.S. WORMER Institute of Theoretical Chemistry, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands Contents 1 Introduction 2 2 Rayleigh-SchrÄodinger perturbation theory 4 3 M¿ller-Plesset perturbation theory 7 4 Diagrammatic perturbation theory 9 5 Unlinked clusters 14 6 Convergence of MP perturbation theory 17 7 Second quantization 18 8 Coupled cluster Ansatz 21 9 Coupled cluster equations 26 9.1 Exact CC equations . 26 9.2 The CCD equations . 30 9.3 CC theory versus MP theory . 32 9.4 CCSD(T) . 33 A Hartree-Fock, Slater-Condon, Brillouin 34 B Exponential structure of the wavefunction 37 C Bibliography 40 1 1 Introduction These are notes for a six hour lecture series on the electronic correlation problem, given by the author at a Dutch national winterschool in 1999. The main purpose of this course was to give some theoretical background on the M¿ller-Plesset and coupled cluster methods. Both computational methods are available in many quantum chemical \black box" programs. The audi- ence consisted of graduate students, mostly with an undergraduate chemistry education and doing research in theoretical chemistry. A basic knowledge of quantum mechanics and quantum chemistry is pre- supposed. In particular a knowledge of Slater determinants, Slater-Condon rules and Hartree-Fock theory is a prerequisite of understanding the following notes. In Appendix A this theory is reviewed briefly. Because of time limitations hardly any proofs are given, the theory is sketchily outlined. No attempt is made to integrate out the spin, the theory is formulated in terms of spin-orbitals only. -
More Insight in Multiple Bonding with Valence Bond Theory K
More insight in multiple bonding with valence bond theory K. Hendrickx, B. Braida, P. Bultinck, P.C. Hiberty To cite this version: K. Hendrickx, B. Braida, P. Bultinck, P.C. Hiberty. More insight in multiple bonding with va- lence bond theory. Computational and Theoretical Chemistry, Elsevier, 2015, 1053, pp.180–188. 10.1016/j.comptc.2014.09.007. hal-01627698 HAL Id: hal-01627698 https://hal.archives-ouvertes.fr/hal-01627698 Submitted on 21 Nov 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. More insight in multiple bonding with valence bond theory ⇑ ⇑ K. Hendrickx a,b,c, B. Braida a,b, , P. Bultinck c, P.C. Hiberty d, a Sorbonne Universités, UPMC Univ Paris 06, UMR 7616, LCT, F-75005 Paris, France b CNRS, UMR 7616, LCT, F-75005 Paris, France c Department of Inorganic and Physical Chemistry, Ghent University, Krijgslaan 281 (S3), 9000 Gent, Belgium d Laboratoire de Chimie Physique, UMR CNRS 8000, Groupe de Chimie Théorique, Université de Paris-Sud, 91405 Orsay Cédex, France abstract An original procedure is proposed, based on valence bond theory, to calculate accurate dissociation ener- gies for multiply bonded molecules, while always dealing with extremely compact wave functions involving three valence bond structures at most. -
Theoretical Methods That Help Understanding the Structure and Reactivity of Gas Phase Ions
International Journal of Mass Spectrometry 240 (2005) 37–99 Review Theoretical methods that help understanding the structure and reactivity of gas phase ions J.M. Merceroa, J.M. Matxaina, X. Lopeza, D.M. Yorkb, A. Largoc, L.A. Erikssond,e, J.M. Ugaldea,∗ a Kimika Fakultatea, Euskal Herriko Unibertsitatea, P.K. 1072, 20080 Donostia, Euskadi, Spain b Department of Chemistry, University of Minnesota, 207 Pleasant St. SE, Minneapolis, MN 55455-0431, USA c Departamento de Qu´ımica-F´ısica, Universidad de Valladolid, Prado de la Magdalena, 47005 Valladolid, Spain d Department of Cell and Molecular Biology, Box 596, Uppsala University, 751 24 Uppsala, Sweden e Department of Natural Sciences, Orebro¨ University, 701 82 Orebro,¨ Sweden Received 27 May 2004; accepted 14 September 2004 Available online 25 November 2004 Abstract The methods of the quantum electronic structure theory are reviewed and their implementation for the gas phase chemistry emphasized. Ab initio molecular orbital theory, density functional theory, quantum Monte Carlo theory and the methods to calculate the rate of complex chemical reactions in the gas phase are considered. Relativistic effects, other than the spin–orbit coupling effects, have not been considered. Rather than write down the main equations without further comments on how they were obtained, we provide the reader with essentials of the background on which the theory has been developed and the equations derived. We committed ourselves to place equations in their own proper perspective, so that the reader can appreciate more profoundly the subtleties of the theory underlying the equations themselves. Finally, a number of examples that illustrate the application of the theory are presented and discussed. -
Site-Selective Electronic Correlation in Α-Plutonium Metal
ARTICLE Received 18 Jun 2013 | Accepted 19 Sep 2013 | Published 18 Oct 2013 DOI: 10.1038/ncomms3644 Site-selective electronic correlation in a-plutonium metal Jian-Xin Zhu1, R.C. Albers1, K. Haule2, G. Kotliar2 & J.M. Wills1 An understanding of the phase diagram of elemental plutonium (Pu) must include both, the effects of the strong directional bonding and the high density of states of the Pu 5f electrons, as well as how that bonding weakens under the influence of strong electronic correlations. Here we present electronic-structure calculations of the full 16-atom per unit cell a-phase structure within the framework of density functional theory together with dynamical mean- field theory. Our calculations demonstrate that Pu atoms sitting on different sites within the a-Pu crystal structure have a strongly varying site dependence of the localization– delocalization correlation effects of their 5f electrons and a corresponding effect on the bonding and electronic properties of this complicated metal. In short, a-Pu has the capacity to simultaneously have multiple degrees of electron localization/delocalization of Pu 5f electrons within a pure single-element material. 1 Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA. 2 Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA. Correspondence and requests for materials should be addressed to J.-X.Z. (email: [email protected]). NATURE COMMUNICATIONS | 4:2644 | DOI: 10.1038/ncomms3644 | www.nature.com/naturecommunications 1 & 2013 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms3644 ure plutonium has long been considered to be one of the strongly localized. -
An Introduction to Hartree-Fock Molecular Orbital Theory
An Introduction to Hartree-Fock Molecular Orbital Theory C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology June 2000 1 Introduction Hartree-Fock theory is fundamental to much of electronic structure theory. It is the basis of molecular orbital (MO) theory, which posits that each electron's motion can be described by a single-particle function (orbital) which does not depend explicitly on the instantaneous motions of the other electrons. Many of you have probably learned about (and maybe even solved prob- lems with) HucÄ kel MO theory, which takes Hartree-Fock MO theory as an implicit foundation and throws away most of the terms to make it tractable for simple calculations. The ubiquity of orbital concepts in chemistry is a testimony to the predictive power and intuitive appeal of Hartree-Fock MO theory. However, it is important to remember that these orbitals are mathematical constructs which only approximate reality. Only for the hydrogen atom (or other one-electron systems, like He+) are orbitals exact eigenfunctions of the full electronic Hamiltonian. As long as we are content to consider molecules near their equilibrium geometry, Hartree-Fock theory often provides a good starting point for more elaborate theoretical methods which are better approximations to the elec- tronic SchrÄodinger equation (e.g., many-body perturbation theory, single-reference con¯guration interaction). So...how do we calculate molecular orbitals using Hartree-Fock theory? That is the subject of these notes; we will explain Hartree-Fock theory at an introductory level. 2 What Problem Are We Solving? It is always important to remember the context of a theory. -
The V State of Ethylene: Valence Bond Theory Takes up the Challenge Wei Wu, Huaiyu Zhang, Benoît Braïda, Sason Shaik, Philippe C
The V state of ethylene: valence bond theory takes up the challenge Wei Wu, Huaiyu Zhang, Benoît Braïda, Sason Shaik, Philippe C. Hiberty To cite this version: Wei Wu, Huaiyu Zhang, Benoît Braïda, Sason Shaik, Philippe C. Hiberty. The V state of ethylene: valence bond theory takes up the challenge. Theoretical Chemistry Accounts: Theory, Computation, and Modeling, Springer Verlag, 2014, 133 (3), 10.1007/s00214-013-1441-x. hal-01627701 HAL Id: hal-01627701 https://hal.archives-ouvertes.fr/hal-01627701 Submitted on 21 Nov 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The V state of ethylene: valence bond theory takes up the challenge Wei Wu • Huaiyu Zhang • Benoıˆt Braı¨da • Sason Shaik • Philippe C. Hiberty Abstract The ground state and first singlet excited state with the same compact VB wave function. Furthermore, of ethylene, so-called N and V states, respectively, are the measure of the spatial extension of the V state wave 2 studied by means of modern valence bond methods. It is function, 19.14 a0, is in the range of accepted values found that extremely compact wave functions, made of obtained by large-scale state-of-the-art molecular orbital- three VB structures for the N state and four structures for based methods. -
Theory and Technique
Lawrence Berkeley National Laboratory Lawrence Berkeley National Laboratory Title COMPUTATIONAL METHODS FOR MOLEUCLAR STRUCTURE DETERMINATION: THEORY AND TECHNIQUE Permalink https://escholarship.org/uc/item/3b7795px Author Lester, W.A. Publication Date 1980-03-14 Peer reviewed eScholarship.org Powered by the California Digital Library University of California CONTENTS Foreword v List of Invited Speakers vii Workshop Participants viii LECTURES 1 Introduction to Computational Quantum Chemistry Ernest R. Davidson 1-1 2/3 Introduction to SCF Theory Ernest R. Davidson 2/3-1 4 Semi-Empirical SCF Theory Michael C. Zerner 4-1 5 An Introduction to Some Semi-Empirical and Approximate Molecular Orbital Methods Miehael C. Zerner 5-1 6/7 Ab Initio Hartree Fock John A. Pople 6/7-1 8 SCF Properties Ernest R. Davidson 8-1 9/10 Generalized Valence Bond William A, Goddard, III 9/10-1 11 Open Shell KF and MCSCF Theory Ernest R. Davidson 11-1 12 Some Semi-Empirical Approaches to Electron Correlation Miahael C. Zerner 12-1 13 Configuration Interaction Method Ernest R. Davidson 13-1 14 Geometry Optimization of Large Systems Michael C. Zerner 14-1 15 MCSCF Calculations: Results Boaen Liu 15-1 lfi/18 Empirical Potentials, Semi-Empirical Potentials, and Molecular Mechanics Norman L. Allinger 16/18-1 iv 17 CI Calculations: Results Bowen Liu 17-1 19 Computational Quantum Chemistry: Future Outlook Ernest R. Davidson 19-1 V FOREWORD The National Resource for Computation in Chemistry (NRCC) was established as a Division of Lawrence Berkeley Laboratory (LBL) in October 1977. The functions of the NRCC may be broadly categorized as follows: (1) to make information on existing and developing computa tional methodologies available to all segments of the chemistry community, (2) to make state-of-the-art computational facilities (both hardware and software] accessible to the chemistry community, and (3) to foster research and development of new computational methods for application to chemical problems. -
An Experimental Chemist's Guide to Ab Initio Quantum Chemistry
J. Phys. Chem. 1991, 95, 1017-1029 1017 FEATURE ARTICLE An Experimental Chemist’s Guide to ab Initio Quantum Chemistry Jack Simons Chemistry Department, University of Utah, Salt Lake City, Utah 84112 (Received: October 5, 1990) This article is not intended to provide a cutting edge, state-of-the-art review of ab initio quantum chemistry. Nor does it offer a shopping list of estimates for the accuracies of its various approaches. Unfortunately, quantum chemistry is not mature or reliable enough to make such an evaluation generally possible. Rather, this article introduces the essential concepts of quantum chemistry and the computationalfeatures that differ among commonly used methods. It is intended as a guide for those who are not conversant with the jargon of ab initio quantum chemistry but who are interested in making use of these tools. In sections I-IV, readers are provided overviews of (i) the objectives and terminology of the field, (ii) the reasons underlying the often disappointing accuracy of present methods, (iii) and the meaning of orbitals, configurations, and electron correlation. The content of sections V and VI is intended to serve as reference material in which the computational tools of ab initio quantum chemistry are overviewed. In these sections, the Hartree-Fock (HF), configuration interaction (CI), multiconfigurational self-consistent field (MCSCF), Maller-Plesset perturbation theory (MPPT), coupled-cluster (CC), and density functional methods such as X, are introduced. The strengths and weaknesses of these methods as well as the computational steps involved in their implementation are briefly discussed. 1. What Does ab Initio Quantum Chemistry Try To Do? quantitative predictions to be made. -
The Avogadro Constant to Be Equal to Exactly 6.02214X×10 23 When It Is Expressed in the Unit Mol −1
[august, 2011] redefinition of the mole and the new system of units zoltan mester It is as easy to count atomies as to resolve the propositions of a lover.. As You Like It William Shakespeare 1564-1616 Argentina, Austria-Hungary, Belgium, Brazil, Denmark, France, German Empire, Italy, Peru, Portugal, Russia, Spain, Sweden and Norway, Switzerland, Ottoman Empire, United States and Venezuela 1875 -May 20 1875, BIPM, CGPM and the CIPM was established, and a three- dimensional mechanical unit system was setup with the base units metre, kilogram, and second. -1901 Giorgi showed that it is possible to combine the mechanical units of this metre–kilogram–second system with the practical electric units to form a single coherent four-dimensional system -In 1921 Consultative Committee for Electricity (CCE, now CCEM) -by the 7th CGPM in 1927. The CCE to proposed, in 1939, the adoption of a four-dimensional system based on the metre, kilogram, second, and ampere, the MKSA system, a proposal approved by the ClPM in 1946. -In 1954, the 10th CGPM, the introduction of the ampere, the kelvin and the candela as base units -in 1960, 11th CGPM gave the name International System of Units, with the abbreviation SI. -in 1970, the 14 th CGMP introduced mole as a unit of amount of substance to the SI 1960 Dalton publishes first set of atomic weights and symbols in 1805 . John Dalton(1766-1844) Dalton publishes first set of atomic weights and symbols in 1805 . John Dalton(1766-1844) Much improved atomic weight estimates, oxygen = 100 . Jöns Jacob Berzelius (1779–1848) Further improved atomic weight estimates . -
Diagonalization, Eigenvalues Problem, Secular Equation
Diagonalization, eigenvalues problem, secular equation Diagonalization procedure of a matrix A with dimensionality n × n (e.g. the Hessian matrix) 2 3 a11 a12 : : : a1n 6 7 6 a21 a22 : : : a2n 7 6 7 A = 6 . 7 6 . 7 4 5 an1 an2 : : : ann consists in finding a matrix C such, that the matrix D=C−1AC is diagonal: 2 3 d1 0 ::: 0 6 7 6 0 d2 ::: 0 7 −1 6 7 C AC = D = 6 . 7 6 . 7 4 5 0 0 : : : dn Equation C−1AC=D can also be written as AC = CD Denoting the elements of matrix C by cij and using te fact that D is diagonal (i.e., its elements dij are of the form djδij, where δij denotes the Kronecker delta, δii=1 i δij=0 for i6=j) we obtain X X X (AC)ij = aik ckj (CD)ij = cik dkj = cik dj δkj = djcij k k k Diagonalization, eigenvalues problem, secular equation Employing the equation (AC)ij = (CD)ij we obtain X aik ckj = djcij k In matrix notation this equation takes the form ACj = dj Cj where Cj is the jth column of matrix C. This is equation for the eigenvalues (dj) and eigenvectors (Cj) of matrix A. Solving this equation, that is the solving the so called eigenproblem for matrix A, is equivalent to diagonalization of matrix A. This is because the matrix C is built from the (column) eigenvectors C1;C2;:::;Cn: C = [C1;C2;:::;Cn] The equation for eigenvectors can also be written as( A − dj E) Cj = 0, where E is the unit matrix with elements δij. -
Page 1 of 29 Dalton Transactions
Dalton Transactions Accepted Manuscript This is an Accepted Manuscript, which has been through the Royal Society of Chemistry peer review process and has been accepted for publication. Accepted Manuscripts are published online shortly after acceptance, before technical editing, formatting and proof reading. Using this free service, authors can make their results available to the community, in citable form, before we publish the edited article. We will replace this Accepted Manuscript with the edited and formatted Advance Article as soon as it is available. You can find more information about Accepted Manuscripts in the Information for Authors. Please note that technical editing may introduce minor changes to the text and/or graphics, which may alter content. The journal’s standard Terms & Conditions and the Ethical guidelines still apply. In no event shall the Royal Society of Chemistry be held responsible for any errors or omissions in this Accepted Manuscript or any consequences arising from the use of any information it contains. www.rsc.org/dalton Page 1 of 29 Dalton Transactions Core-level photoemission spectra of Mo 0.3 Cu 0.7 Sr 2ErCu 2Oy, a superconducting perovskite derivative. Unconventional structure/property relations a, b b b a Sourav Marik , Christine Labrugere , O.Toulemonde , Emilio Morán , M. A. Alario- Manuscript Franco a,* aDpto. Química Inorgánica, Facultad de CC.Químicas, Universidad Complutense de Madrid, 28040- Madrid (Spain) bCNRS, Université de Bordeaux, ICMCB, 87 avenue du Dr. A. Schweitzer, Pessac, F-33608, France -
Guide for the Use of the International System of Units (SI)
Guide for the Use of the International System of Units (SI) m kg s cd SI mol K A NIST Special Publication 811 2008 Edition Ambler Thompson and Barry N. Taylor NIST Special Publication 811 2008 Edition Guide for the Use of the International System of Units (SI) Ambler Thompson Technology Services and Barry N. Taylor Physics Laboratory National Institute of Standards and Technology Gaithersburg, MD 20899 (Supersedes NIST Special Publication 811, 1995 Edition, April 1995) March 2008 U.S. Department of Commerce Carlos M. Gutierrez, Secretary National Institute of Standards and Technology James M. Turner, Acting Director National Institute of Standards and Technology Special Publication 811, 2008 Edition (Supersedes NIST Special Publication 811, April 1995 Edition) Natl. Inst. Stand. Technol. Spec. Publ. 811, 2008 Ed., 85 pages (March 2008; 2nd printing November 2008) CODEN: NSPUE3 Note on 2nd printing: This 2nd printing dated November 2008 of NIST SP811 corrects a number of minor typographical errors present in the 1st printing dated March 2008. Guide for the Use of the International System of Units (SI) Preface The International System of Units, universally abbreviated SI (from the French Le Système International d’Unités), is the modern metric system of measurement. Long the dominant measurement system used in science, the SI is becoming the dominant measurement system used in international commerce. The Omnibus Trade and Competitiveness Act of August 1988 [Public Law (PL) 100-418] changed the name of the National Bureau of Standards (NBS) to the National Institute of Standards and Technology (NIST) and gave to NIST the added task of helping U.S.