DOCSLIB.ORG

Problem Set #1 Solutions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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 . (7) In order to prove that the density matrix is idempotent in an orthonormal basis, we need to first prove the following proposition. ~† ~ Proposition 1: In an orthonormal basis, CCocc occ =1. ~† ~ Proof. In order to prove that CCocc occ =1, expand the occupied MO coefficient matrices as follows: ~† ~ ~* ~ ~ * ~ ~ ~ CCCCocc occ = ∑ μ~i μ~ j = ∑ μi μ j= ∑ iμ μ j= i j =δ ij = 1 (8) μ~ μ~ μ~ ~† ~ which proves the proposition that CCocc occ =1. QED With Proposition 1, we are now in a position to directly prove that the density matrix is idempotent in an orthonormal basis. Proposition 2: The density matrix is idempotent in an orthonormal basis with P 2 = P . Proof. From the definition of the density matrix in an orthonormal basis, given by (7) above, we ~ ~ † 2 have PCC= occ occ and want to prove that P = P , i.e., the idempotency condition in an orthonormal basis. In order to do so, expand P 2 as follows 2 ~ ~ † 2 ~ ~ † ~ ~† ~ ~ † ~ ~ † ~ ~ † PCCCCCCCCCCCCP= ( occ occ )(= occ occ )( occ occ ) = occ( occ occ) occ =occ occ = (9) in which the results of Proposition 1 were used in a fairly straightforward manner (c.f. (8)). QED (b) Derive the generalization of the idempotency condition for the case of a nonorthonormal basis, with overlap matrix S. In a nonorthonormal basis, the results of Proposition 1 must be adjusted to derive the general idempotency condition for the 1-PDM. In analogy to Q1(a), we have two propositions which require proof. † Proposition 3: In a nonorthonormal basis, C.occSC occ =1 ~† ~ Proof. Using the results of Proposition 1, namely that CCocc occ =1in an orthonormal basis and ~ + 1 the fact that CSC= 2 from (5), we have: 1 1 1 1 1 1 ~ † ~ + 2 † + 2 † +2 + 2 † +2 + 2 † Cocc C occ = ( S Cocc ) ( S Cocc )= ( Cocc S )( S Cocc ) = Cocc S S Cocc = Cocc SC occ = 1 (10) which completes the proof. QED With Proposition 3 at our disposal, we are now in a position to show that PSP= P is the generalization of the idempotency condition for the case of a nonorthonormal basis. Proposition 4: The density matrix is idempotent in a nonorthonormal basis with PSP= P . Proof. From the definition of the density matrix in a nonorthonormal basis, given by (2) above, † we have PCC= occ occ and we wish to prove that PSP= P , i.e., the idempotency condition in a nonorthonormal basis. In order to do so, expand PSP as follows † † † † † PSP= ( Cocc C occ )()() S Cocc C occ = Cocc C occ SC occ C occ = Cocc C occ = P (11) which completes the proof. QED (c) Derive the eigenvalues of an idempotent matrix (you may assume an orthonormal basis). Hence write an operator form for it, and describe the function of this operator. If we write a general eigenvalue equation for the density operator, Pˆ , as Pˆ ψ= λ ψ , and acknowledge that the density operator is idempotent, i.e., PPˆ2 = ˆ , we have the following two simultaneous conditions: Pˆ ψ= λ ψ (12) and PPˆ2 ψ= ˆ ψ = λ ψ . (13) However, we could write (13) as follows: PPPPˆ2 ψ= ˆ( ˆ ψ) =ˆ()( λψ = λψPˆ )() = λλψ = λψ2 . (14) Combining (13) and (14), we note that λ ψ= λ2 ψ (15) which implies that λ= λ2 , which can only be satisfied if λ = 0 or λ = 1. Therefore, the eigenvalues of the density operator are 0 and 1—this should make sense since the matrix form of the density matrix is diagonal with 1’s in the occupied block and 0’s in the virtual block (neglecting the factor of two arising from the restricted closed-shell treatment). To find the eigenvectors of the density operator, one must consider the consequences of having an eigenspectrum comprised of {λ |λ = 0,1} . In essence, the action of the density operator is to either return an eigenvector unchanged ( λ = 1) or to annihilate an eigenvector ( λ = 0 ). This defines a projection operator, i.e., an operator of the form: Xˆ = ∑ p p (16) p which is nothing more than the analog of the unit dyadic in function space. Therefore, the density operator is a projection operator for the occupied space, O≡ Span{}ψ k with dim(ON ) = occ , with the following explicit form: Nocc ˆ P = ∑ ψk ψ k . (17) k ˆ When acting on an occupied MO such as ψ i , an eigenfunction of P that is contained within the occupied space, the density operator will return ψ i unchanged: Nocc Nocc ˆ P ψi = ∑ ψk ψ k ψ i = ∑ ψk δ ki = ψ i . (18) k k ˆ When acting on a virtual (or unoccupied) MO such as ψ a , an eigenfunction of P that is NOT contained within the occupied space, the density operator will annihilate ψ a : Nocc Nocc ˆ P ψa = ∑ ψk ψ k ψ a = ∑ ψk δ ka = 0 (19) k k since ∀k, k ≠ a . In other words, the virtual space, V≡ Span{}ψ c with dim(VN) = virt and the occupied space are orthogonal spaces, OV⊥ . Of course, one could define a projection operator corresponding to virtual space, namely, Nvirt ˆ Q = ∑ ψa ψ a (20) a from which one can show that PQˆ +ˆ = 1 ˆ . Why? (d) Suppose a matrix M (in an orthonormal basis) is approximately idempotent, with a deviation from idempotency that is D (i.e. MMD= 0 + with M 0 idempotent). Consider then a transformed matrix defined as M ′ =3MM2 − 2 3 . Derive an expression for its deviation from idempotency in terms of D (to leading order), and hence discuss the action of this transformation. 2 Since MMD=0 + , with MM0 = 0 and D the deviation from idempotency, we have: 2 2 2 M=() M0 + D = M0 + M 0 D + DM0 + D (21) which clearly illustrates that M deviates from idempotency to OD() , i.e., linear in order of D. If we now consider the transformed matrix defined as MM′ = 3 2 − 2M 3 , then the deviation from idempotency of M ′ can be derived by computing M ′2 − M ′ in terms of D. Lets first consider M ′ , in which we will need to take successive powers of M. Equation (21) provides us with an expression for M 2 , so we need to explicitly consider M 3 now: 3 2 M=( M0 + M 0 D + DM0 + D)( M0 + D ) 2 2 3 =M0 + M 0 DM 0 + DM0 + D M0 + M 0 D + M0 D + DM0 D + D (22) 2 2 ≈M0 + M 0 DM 0 + DM0 + D M0 + M 0 D + M0 D + DM0 D where, the term that was cubic in D was omitted before we proceed any further in the derivation. From (21) and (22), we have M ′ as: 2 3 2 2 2 M′ =3 M − 2 M ≈ M0 + M 0 D + DM0 +3 D − 2 M0 DM 0 −2 D M0 − 2 M0 D − 2 DM0 D . (23) From (23), we can compute the deviation from idempotency of M ′ , by considering the quantity M ′2 − M ′ directly (keeping in mind that only the leading terms in D must be considered) 2 2 2 2 2 M′ − M′ ≈3( M0 D − M0 D M0− M 0 DM 0 D− DM0 DM 0 + D M0 + DM0 D − D ) . (24) if the leading terms in D, i.e., OD( 2 ) are retained. Hence, the transformation matrix is now idempotent to first order in the deviation D from idempotency. In this sense, the transformation matrix has purified M ′ and is this procedure is referred to as a “purification of an almost idempotent density matrix” (For an extremely readable detailed review of the central role the density matrix plays in quantum chemistry, see R. McWeeny, Phys. Rev. 32, 335 (1960)). 2. Koopman’s theorem for ionization energies and electron affinities. (a) Use the Slater-Condon matrix element expressions to derive the Koopman’s expression for the energy difference between an optimized n-electron HF determinant, and the (n-1) electron determinant obtained by removing 1 electron from the ith spin-orbital.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Theoretical Studies of Charge Recombination Reactions
    Theoretical studies of charge recombination reactions Sifiso Musa Nkambule Akademisk avhandling f¨oravl¨aggandeav licentiatexamen vid Stockholms universitet, Fysikum. January, 2015. Abstract This thesis is based on theoretical studies that have been done for low-energy reactions involving small molecular systems. It is mainly focusing on mutual neutralization of oppositely charged ions and dissociative recombination of molecular ions. Both reactions involve highly excited electronic states that are coupled to each other. Employing electronic structure meth- ods, the electronic states relevant to the reactions are computed. It is necessary to go beyond the Born-Oppenheimer approximation and include non-adiabatic effects. This non-adiabatic behaviour plays a crucial role in driving the reactions. Schemes on how to go about and include these coupling elements are discussed in the thesis, which include computations of non-adiabatic and electronic couplings of the resonant states to the ionisation continuum. The nuclear dynamics are studied either semi-classically, using the Landau-Zener method or quantum mechanically, employing the time-independent and time-dependant Schr¨odinger + equations. Reactions studied here are mutual neutralization in collision of H + H−, where both the semi-classical and quantum mechanical methods are employed. Total and differential cross sections are computed for all hydrogen isotopes. We also perform quantum mechanical + studies of mutual neutralization in the collision of He + H−. For this system not only the non-adiabatic couplings among the neutral states have to be considered, but also the electronic coupling to the ionization continuum. Worth noting in the two systems is that reactants and the final products belong to different electronic states, which a coupled together, hence for the reaction to take place there must be non-adiabatic transition.
    [Show full text]
  • Quaternion Symmetry in Relativistic Molecular Calculations: the Dirac–Hartree–Fock Method T
    Quaternion symmetry in relativistic molecular calculations: The Dirac–Hartree–Fock method T. Saue, H. Jensen To cite this version: T. Saue, H. Jensen. Quaternion symmetry in relativistic molecular calculations: The Dirac–Hartree– Fock method. Journal of Chemical Physics, American Institute of Physics, 1999, 111 (14), pp.6211- 6222. 10.1063/1.479958. hal-03327446 HAL Id: hal-03327446 https://hal.archives-ouvertes.fr/hal-03327446 Submitted on 27 Aug 2021 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. JOURNAL OF CHEMICAL PHYSICS VOLUME 111, NUMBER 14 8 OCTOBER 1999 Quaternion symmetry in relativistic molecular calculations: The Dirac–Hartree–Fock method T. Sauea) and H. J. Aa Jensen Department of Chemistry, University of Southern Denmark—Main campus: Odense University, DK-5230 Odense M, Denmark ͑Received 5 April 1999; accepted 9 July 1999͒ A symmetry scheme based on the irreducible corepresentations of the full symmetry group of a molecular system is presented for use in relativistic calculations. Consideration of time-reversal symmetry leads to a reformulation of the Dirac–Hartree–Fock equations in terms of quaternion algebra. Further symmetry reductions due to molecular point group symmetry are then manifested by a descent to complex or real algebra.
    [Show full text]
  • Hartree-Fock Theory
    Hartree-Fock theory Morten Hjorth-Jensen1 National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA1 May 16-20 2016 c 2013-2016, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license Why Hartree-Fock? Hartree-Fock (HF) theory is an algorithm for finding an approximative expression for the ground state of a given Hamiltonian. The basic ingredients are I Define a single-particle basis f αg so that HF h^ α = "α α with the Hartree-Fock Hamiltonian defined as HF HF h^ = t^+u ^ext +u ^ HF I The term u^ is a single-particle potential to be determined by the HF algorithm. HF I The HF algorithm means to choose u^ in order to have HF hH^i = E = hΦ0jH^jΦ0i that is to find a local minimum with a Slater determinant Φ0 being the ansatz for the ground state. HF I The variational principle ensures that E ≥ E0, with E0 the exact ground state energy. Why Hartree-Fock? We will show that the Hartree-Fock Hamiltonian h^HF equals our definition of the operator f^ discussed in connection with the new definition of the normal-ordered Hamiltonian (see later lectures), that is we have, for a specific matrix element HF X hpjh^ jqi = hpjf^jqi = hpjt^+u ^extjqi + hpijV^jqiiAS ; i≤F meaning that HF X hpju^ jqi = hpijV^jqiiAS : i≤F The so-called Hartree-Fock potential u^HF brings an explicit medium dependence due to the summation over all single-particle states below the Fermi level F .
    [Show full text]
  • Restricted Closed Shell Hartree Fock Roothaan Matrix Method Applied to Helium Atom Using Mathematica Introduction the Hartree Fo
    European J of Physics Education Volume 5 Issue 1 2014 Acosta, Tapia, Cab Restricted Closed Shell Hartree Fock Roothaan Matrix Method Applied to Helium Atom using Mathematica César R. Acosta*1 J. Alejandro Tapia1 César Cab1 1Av. Industrias no Contaminantes por anillo periférico Norte S/N Apdo Postal 150 Cordemex Physics Engineering Department Universidad Autónoma de Yucatán *[email protected] (Received: 20.12. 2013, Accepted: 13.02.2014) Abstract Slater type orbitals were used to construct the overlap and the Hamiltonian core matrices; we also found the values of the bi-electron repulsion integrals. The Hartree Fock Roothaan approximation process starts with setting an initial guess value for the elements of the density matrix; with these matrices we constructed the initial Fock matrix. The Mathematica software was used to program the matrix diagonalization process from the overlap and Hamiltonian core matrices and to make the recursion loop of the density matrix. Finally we obtained a value for the ground state energy of helium. Keywords: STO, Hartree Fock Roothaan approximation, spin orbital, ground state energy. Introduction The Hartree Fock method (Hartree, 1957) also known as Self Consistent Field (SCF) could be made with two types of spin orbital functions, the Slater Type Orbitals (STO) and the Gaussian Type Orbital (GTO), the election is made in function of the integrals of spin orbitals implicated. We used the STO to construct the wave function associated to the helium atom electrons. The application of the quantum mechanical theory was performed using a numerical method known as the finite element method, this gives the possibility to use the computer for programming the calculations quickly and efficiently (Becker, 1988; Brenner, 2008; Thomee, 2003; Reddy, 2004).
    [Show full text]
  • 7 X 11 Long.P65
    Cambridge University Press 978-0-521-83346-2 - Computational Physics, Second Edition Jos Thijssen Index More information Index ab initio method, see electronic structure Barker algorithm, see Monte Carlo, 303 calculations, 293 Barnes–Hut algorithm, 245 Adams’ formula, 578 basin of attraction, 328 aluminium, see band structure, 126 basis sets, 37, 61, 68, 71, 130, 135, 136, 148, Amdahl’s law, 547 274, 288, 354, 359 Andersen methods, see molecular contracted, 68 dynamics(MD), 226 Gaussian type orbitals (GTO), 65–68, 73 antisymmetric wave function, 49 Gaussian product theorem, 66, 74, 76 Ashcroft empty-core pseudopotential, see band linear combination of atomic orbitals, structure calculations, 146 (LCAO), 65, 127 asymptotic freedom, see quantum minimal, 67, 173, 269, 368 chromodynamics (QCD) and quarks, molecular orbitals (MO), 65, 66 518 palarisation orbitals atomic units, 125, 137, 148, 151, 158 Slater type orbitals (STO), 67, 130, 88 augmented plane wave (APW) method, see Bethe–Salpeter method, 108 band structure calculations, 122, 168 binary hypercube, 547 binary systems, see fluid dynamics, 448 binding energy, 168 Bloch state, 128, 135, 165, 353 back-substitution, 582, 593, 594 Bloch theorem, 125 back-tracking, 494 body-centred cubic, see crystal lattices, 123 band gap, see band structure, 104 Boltzmann equation, 449, 452, 455, 460 band structure, 104, 123, 126, 130, 131, 133, Born approximation, see quantum scattering, 27 134, 142, 145, 148, 163, 169 Born–Oppenheimer approximation, 43, 45, 46, 79, 80, 134, 263, 269 aluminium, 128 boundary
    [Show full text]