Restricted Closed Shell Hartree Fock Roothaan Matrix Method Applied to Helium Atom Using Mathematica Introduction the Hartree Fo
Total Page：16
File Type：pdf, Size：1020Kb
Load more
Recommended publications
-
![Arxiv:1809.04476V2 [Physics.Chem-Ph] 18 Oct 2018 (Sub)States](https://docslib.b-cdn.net/cover/8692/arxiv-1809-04476v2-physics-chem-ph-18-oct-2018-sub-states-198692.webp)
Arxiv:1809.04476V2 [Physics.Chem-Ph] 18 Oct 2018 (Sub)States
Quantum System Partitioning at the Single-Particle Level Quantum System Partitioning at the Single-Particle Level Adrian H. M¨uhlbach and Markus Reihera) ETH Z¨urich, Laboratorium f¨ur Physikalische Chemie, Vladimir-Prelog-Weg 2, CH-8093 Z¨urich,Switzerland (Dated: 17 October 2018) We discuss the partitioning of a quantum system by subsystem separation through unitary block- diagonalization (SSUB) applied to a Fock operator. For a one-particle Hilbert space, this separation can be formulated in a very general way. Therefore, it can be applied to very different partitionings ranging from those driven by features in the molecular structure (such as a solute surrounded by solvent molecules or an active site in an enzyme) to those that aim at an orbital separation (such as core-valence separation). Our framework embraces recent developments of Manby and Miller as well as older ones of Huzinaga and Cantu. Projector-based embedding is simplified and accelerated by SSUB. Moreover, it directly relates to decoupling approaches for relativistic four-component many-electron theory. For a Fock operator based on the Dirac one-electron Hamiltonian, one would like to separate the so-called positronic (negative-energy) states from the electronic bound and continuum states. The exact two-component (X2C) approach developed for this purpose becomes a special case of the general SSUB framework and may therefore be viewed as a system- environment decoupling approach. Moreover, for SSUB there exists no restriction with respect to the number of subsystems that are generated | in the limit, decoupling of all single-particle states is recovered, which represents exact diagonalization of the problem. -
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. -
A New Scalable Parallel Algorithm for Fock Matrix Construction
A New Scalable Parallel Algorithm for Fock Matrix Construction Xing Liu Aftab Patel Edmond Chow School of Computational Science and Engineering College of Computing, Georgia Institute of Technology Atlanta, Georgia, 30332, USA [email protected], [email protected], [email protected] Abstract—Hartree-Fock (HF) or self-consistent field (SCF) ber of independent tasks and using a centralized dynamic calculations are widely used in quantum chemistry, and are scheduler for scheduling them on the available processes. the starting point for accurate electronic correlation methods. Communication costs are reduced by defining tasks that Existing algorithms and software, however, may fail to scale for large numbers of cores of a distributed machine, particularly in increase data reuse, which reduces communication volume, the simulation of moderately-sized molecules. In existing codes, and consequently communication cost. This approach suffers HF calculations are divided into tasks. Fine-grained tasks are from a few problems. First, because scheduling is completely better for load balance, but coarse-grained tasks require less dynamic, the mapping of tasks to processes is not known communication. In this paper, we present a new parallelization in advance, and data reuse suffers. Second, the centralized of HF calculations that addresses this trade-off: we use fine- grained tasks to balance the computation among large numbers dynamic scheduler itself could become a bottleneck when of cores, but we also use a scheme to assign tasks to processes to scaling up to a large system [14]. Third, coarse tasks reduce communication. We specifically focus on the distributed which reduce communication volume may lead to poor load construction of the Fock matrix arising in the HF algorithm, balance when the ratio of tasks to processes is close to one. -
Domain Decomposition and Electronic Structure Computations: a Promising Approach
Domain Decomposition and Electronic Structure Computations: A Promising Approach G. Bencteux1,4, M. Barrault1, E. Canc`es2,4, W. W. Hager3, and C. Le Bris2,4 1 EDF R&D, 1 avenue du G´en´eral de Gaulle, 92141 Clamart Cedex, France {guy.bencteux,maxime.barrault}@edf.fr 2 CERMICS, Ecole´ Nationale des Ponts et Chauss´ees, 6 & 8, avenue Blaise Pascal, Cit´eDescartes, 77455 Marne-La-Vall´ee Cedex 2, France, {cances,lebris}@cermics.enpc.fr 3 Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, USA, [email protected] 4 INRIA Rocquencourt, MICMAC project, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, FRANCE Summary. We describe a domain decomposition approach applied to the spe- cific context of electronic structure calculations. The approach has been introduced in [BCH06]. We survey here the computational context, and explain the peculiari- ties of the approach as compared to problems of seemingly the same type in other engineering sciences. Improvements of the original approach presented in [BCH06], including algorithmic refinements and effective parallel implementation, are included here. Test cases supporting the interest of the method are also reported. It is our pleasure and an honor to dedicate this contribution to Olivier Pironneau, on the occasion of his sixtieth birthday. With admiration, respect and friendship. 1 Introduction and Motivation 1.1 General Context Numerical simulation is nowadays an ubiquituous tool in materials science, chemistry and biology. Design of new materials, irradiation induced damage, drug design, protein folding are instances of applications of numerical sim- ulation. For convenience we now briefly present the context of the specific computational problem under consideration in the present article. -
Matrix Algebra for Quantum Chemistry
Matrix Algebra for Quantum Chemistry EMANUEL H. RUBENSSON Doctoral Thesis in Theoretical Chemistry Stockholm, Sweden 2008 Matrix Algebra for Quantum Chemistry Doctoral Thesis c Emanuel Härold Rubensson, 2008 TRITA-BIO-Report 2008:23 ISBN 978-91-7415-160-2 ISSN 1654-2312 Printed by Universitetsservice US AB, Stockholm, Sweden 2008 Typeset in LATEX by the author. Abstract This thesis concerns methods of reduced complexity for electronic structure calculations. When quantum chemistry methods are applied to large systems, it is important to optimally use computer resources and only store data and perform operations that contribute to the overall accuracy. At the same time, precarious approximations could jeopardize the reliability of the whole calcu- lation. In this thesis, the selfconsistent eld method is seen as a sequence of rotations of the occupied subspace. Errors coming from computational ap- proximations are characterized as erroneous rotations of this subspace. This viewpoint is optimal in the sense that the occupied subspace uniquely denes the electron density. Errors should be measured by their impact on the over- all accuracy instead of by their constituent parts. With this point of view, a mathematical framework for control of errors in HartreeFock/KohnSham calculations is proposed. A unifying framework is of particular importance when computational approximations are introduced to eciently handle large systems. An important operation in HartreeFock/KohnSham calculations is the calculation of the density matrix for a given Fock/KohnSham matrix. In this thesis, density matrix purication is used to compute the density matrix with time and memory usage increasing only linearly with system size. The forward error of purication is analyzed and schemes to control the forward error are proposed. -
Problem Set #1 Solutions
PROBLEM SET #2 SOLUTIONS by Robert A. DiStasio Jr. Q1. 1-particle density matrices and idempotency. (a) A matrix M is said to be idempotent if M 2 = M . Show from the basic definition that the HF density matrix is idempotent when expressed in an orthonormal basis. An element of the HF density matrix is given as (neglecting the factor of two for the restricted closed-shell HF density matrix): * PCCμν = ∑ μi ν i . (1) i In matrix form, (1) is simply equivalent to † PCC= occ occ (2) where Cocc is the occupied block of the MO coefficient matrix. To reformulate the nonorthogonal Roothaan-Hall equations FC= SCε (3) into the preferred eigenvalue problem ~~ ~ FC= Cε (4) the AO basis must be orthogonalized. There are several ways to orthogonalize the AO basis, but I will only discuss symmetric orthogonalization, as this is arguably the most often used procedure in computational quantum chemistry. In the symmetric orthogonalization procedure, the MO coefficient matrix in (3) is recast as ~ + 1 − 1 ~ CSCCSC=2 ⇔ = 2 (5) which can be substituted into (3) yielding the eigenvalue form of the Roothaan-Hall equations in (4) after the following algebraic manipulation: FC = SCε − 1 ~ − 1 ~ FSCSSC( 2 )(= 2 )ε − 1 − 1 ~ −1 − 1 ~ SFSCSSSC2 ( 2 )(= 2 2 )ε . (6) −1 − 1 ~ −1 − 1 ~ ()S2 FS2 C= () S2 SS2 Cε ~~ ~ FC = Cε ~ ~ −1 − 1 −1 − 1 In (6), the dressed Fock matrix, F , was clearly defined as F≡ S2 FS 2 and S2 SS2 = S 0 =1 was used. In an orthonormal basis, therefore, the density matrix of (2) takes on the following form: ~ ~† PCC= occ occ . -
Relativistic Cholesky-Decomposed Density Matrix
Relativistic Cholesky-decomposed density matrix MP2 Benjamin Helmich-Paris,1, 2, a) Michal Repisky,3 and Lucas Visscher2 1)Max-Planck-Institut f¨urKohlenforschung, Kaiser-Wilhelm-Platz 1, D-45470 M¨ulheiman der Ruhr 2)Section of Theoretical Chemistry, Vrije Universiteit Amsterdam, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands 3)Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, UiT The Arctic University of Norway, N-9037 Tromø Norway (Dated: 15 October 2018) In the present article, we introduce the relativistic Cholesky-decomposed density (CDD) matrix second-order Møller–Plesset perturbation theory (MP2) energies. The working equations are formulated in terms of the usual intermediates of MP2 when employing the resolution-of-the-identity approximation (RI) for two-electron inte- grals. Those intermediates are obtained by substituting the occupied and vir- tual quaternion pseudo-density matrices of our previously proposed two-component atomic orbital-based MP2 (J. Chem. Phys. 145, 014107 (2016)) by the corresponding pivoted quaternion Cholesky factors. While working within the Kramers-restricted formalism, we obtain a formal spin-orbit overhead of 16 and 28 for the Coulomb and exchange contribution to the 2C MP2 correlation energy, respectively, compared to a non-relativistic (NR) spin-free CDD-MP2 implementation. This compact quaternion formulation could also be easily explored in any other algorithm to compute the 2C MP2 energy. The quaternion Cholesky factors become sparse for large molecules and, with a block-wise screening, block sparse-matrix multiplication algorithm, we observed an effective quadratic scaling of the total wall time for heavy-element con- taining linear molecules with increasing system size. -
Lecture 16 Hartree Fock Implementation
Quantum Chemistry Lecture 16 Implementation of Hartree-Fock Theory NC State University The linear combination of atomic orbitals (LCAO) Calculations of the energy and properties of molecules requires hydrogen-like wave functions on each of the nuclei. The Hartree-Fock method begins with assumption That molecular orbitals can be formed as a linear combination of atomic orbitals. The basis functions fm are hydrogen-like atomic orbitals that have been optimized by a variational procedure. The HF procedure is a variational procedure to minimize the coefficients Cmi. Note that we use the index m for atomic orbitals and i or j for molecular orbitals. Common types of atomic orbitals Slater-type orbitals (STOs) The STOs are like hydrogen atom wave functions. The problem with STOs arises in multicenter integrals. The Coulomb and exchange integrals involve electrons on different nuclei and so the distance r has a different origin. Gaussian-type orbitals (GTOs) Gaussian orbitals can be used to mimic the shape of exponentials, i.e. the form of the solutions for the hydrogen atom. Multicenter Gaussian integrals can be solved analytically. STOs vs GTOs GTOs are mathematically easy to work with STOs vs GTOs GTOs are mathematically easy to work with, but the shape of a Gaussian is not that similar to that of an exponential. STOs vs GTOs Therefore, linear combinations of Gaussians are used to imitate the shape of an exponential. Shown is a representation of the 3-Gaussian model of a STO. Double-zeta basis sets Since the remaining atoms have a different exponential dependence than hydrogen it is often convenient to include more parameters. -
Hartree-Fock Approach
The Wave equation The Hamiltonian The Hamiltonian Atomic units The hydrogen atom The chemical connection Hartree-Fock theory The Born-Oppenheimer approximation The independent electron approximation The Hartree wavefunction The Pauli principle The Hartree-Fock wavefunction The L#AO approximation An example The HF theory The HF energy The variational principle The self-consistent $eld method Electron spin &estricted and unrestricted HF theory &HF versus UHF Pros and cons !er ormance( HF equilibrium bond lengths Hartree-Fock calculations systematically underestimate equilibrium bond lengths !er ormance( HF atomization energy Hartree-Fock calculations systematically underestimate atomization energies. !er ormance( HF reaction enthalpies Hartree-Fock method fails when reaction is far from isodesmic. Lack of electron correlations!! Electron correlation In the Hartree-Fock model, the repulsion energy between two electrons is calculated between an electron and the a!erage electron density for the other electron. What is unphysical about this is that it doesn't take into account the fact that the electron will push away the other electrons as it mo!es around. This tendency for the electrons to stay apart diminishes the repulsion energy. If one is on one side of the molecule, the other electron is likely to be on the other side. Their positions are correlated, an effect not included in a Hartree-Fock calculation. While the absolute energies calculated by the Hartree-Fock method are too high, relative energies may still be useful. Basis sets Minimal basis sets Minimal basis sets Split valence basis sets Example( Car)on 6-3/G basis set !olarized basis sets 1iffuse functions Mix and match %2ective core potentials #ounting basis functions Accuracy Basis set superposition error Calculations of interaction energies are susceptible to basis set superposition error (BSSE) if they use finite basis sets. -
Fock Matrix Construction for Large Systems
Fock Matrix Construction for Large Systems Elias Rudberg Theoretical Chemistry Royal Institute of Technology Stockholm 2006 Fock Matrix Construction for Large Systems Licentiate thesis c Elias Rudberg, 2006 ISBN 91-7178-535-3 ISBN 978-91-7178-535-0 Printed by Universitetsservice US AB, Stockholm, Sweden, 2006 Typeset in LATEX by the author Abstract This licentiate thesis deals with quantum chemistry methods for large systems. In particu- lar, the thesis focuses on the efficient construction of the Coulomb and exchange matrices which are important parts of the Fock matrix in Hartree–Fock calculations. The meth- ods described are also applicable in Kohn–Sham Density Functional Theory calculations, where the Coulomb and exchange matrices are parts of the Kohn–Sham matrix. Screening techniques for reducing the computational complexity of both Coulomb and exchange com- putations are discussed, as well as the fast multipole method, used for efficient computation of the Coulomb matrix. The thesis also discusses how sparsity in the matrices occurring in Hartree–Fock and Kohn– Sham Density Functional Theory calculations can be used to achieve more efficient storage of matrices as well as more efficient operations on them. As an example of a possible type of application, the thesis includes a theoretical study of Heisenberg exchange constants, using unrestricted Kohn–Sham Density Functional Theory calculations. iii Preface The work presented in this thesis has been carried out at the Department of Theoretical Chemistry, Royal Institute of Technology, Stockholm, Sweden. List of papers included in the thesis Paper 1 Efficient implementation of the fast multipole method, Elias Rudberg and Pawe l Sa lek,J. -
![Arxiv:1912.01335V4 [Physics.Atom-Ph] 14 Feb 2020](https://docslib.b-cdn.net/cover/0707/arxiv-1912-01335v4-physics-atom-ph-14-feb-2020-1230707.webp)
Arxiv:1912.01335V4 [Physics.Atom-Ph] 14 Feb 2020
Fast apparent oscillations of fundamental constants Dionysios Antypas Helmholtz Institute Mainz, Johannes Gutenberg University, 55128 Mainz, Germany Dmitry Budker Helmholtz Institute Mainz, Johannes Gutenberg University, 55099 Mainz, Germany and Department of Physics, University of California at Berkeley, Berkeley, California 94720-7300, USA Victor V. Flambaum School of Physics, University of New South Wales, Sydney 2052, Australia and Helmholtz Institute Mainz, Johannes Gutenberg University, 55099 Mainz, Germany Mikhail G. Kozlov Petersburg Nuclear Physics Institute of NRC “Kurchatov Institute”, Gatchina 188300, Russia and St. Petersburg Electrotechnical University “LETI”, Prof. Popov Str. 5, 197376 St. Petersburg Gilad Perez Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot, Israel 7610001 Jun Ye JILA, National Institute of Standards and Technology, and Department of Physics, University of Colorado, Boulder, Colorado 80309, USA (Dated: May 2019) Precision spectroscopy of atoms and molecules allows one to search for and to put stringent limits on the variation of fundamental constants. These experiments are typically interpreted in terms of variations of the fine structure constant α and the electron to proton mass ratio µ = me/mp. Atomic spectroscopy is usually less sensitive to other fundamental constants, unless the hyperfine structure of atomic levels is studied. However, the number of possible dimensionless constants increases when we allow for fast variations of the constants, where “fast” is determined by the time scale of the response of the studied species or experimental apparatus used. In this case, the relevant dimensionless quantity is, for example, the ratio me/hmei and hmei is the time average. In this sense, one may say that the experimental signal depends on the variation of dimensionful constants (me in this example). -
The Hartree-Fock Method
An Iterative Technique for Solving the N-electron Hamiltonian: The Hartree-Fock method Tony Hyun Kim Abstract The problem of electron motion in an arbitrary field of nuclei is an important quantum mechanical problem finding applications in many diverse fields. From the variational principle we derive a procedure, called the Hartree-Fock (HF) approximation, to obtain the many-particle wavefunction describing such a sys- tem. Here, the central physical concept is that of electron indistinguishability: while the antisymmetry requirement greatly complexifies our task, it also of- fers a symmetry that we can exploit. After obtaining the HF equations, we then formulate the procedure in a way suited for practical implementation on a computer by introducing a set of spatial basis functions. An example imple- mentation is provided, allowing for calculations on the simplest heteronuclear structure: the helium hydride ion. We conclude with a discussion of how to derive useful chemical information from the HF solution. 1 Introduction In this paper I will introduce the theory and the techniques to solve for the approxi- mate ground state of the electronic Hamiltonian: N N M N N 1 X 2 X X ZA X X 1 H = − ∇i − + (1) 2 rAi rij i=1 i=1 A=1 i=1 j>i which describes the motion of N electrons (indexed by i and j) in the field of M nuclei (indexed by A). The coordinate system, as well as the fully-expanded Hamiltonian, is explicitly illustrated for N = M = 2 in Figure 1. Although we perform the analysis 2 Kim Figure 1: The coordinate system corresponding to Eq.