DOCSLIB.ORG

Hartree-Fock Method

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Hartree-Fock Method KH Computational Physics- 2009 Hartree-Fock Method Hartree-Fock It is probably the simplest method to treat the many-particle system. The dynamic many particle problem is replaced by an effective one-electron problem: electron is moving in an effective static potential field. It can be viewed as a variational method where the full many-body wave function is replaced by a single Slater determinant. The elements of the determinant are one electron orbitals with orbital and spin part. It can also be viewed as the lowest order term in perturbative expansion (linear in U) with respect to interaction between electrons. These two views will be substantiated below by equations. But first we will use decoupling method to derive the Hartree-Fock equations. Kristjan Haule, 2009 –1– KH Computational Physics- 2009 Hartree-Fock Method In the second quantization, the many electron problem takes the form 1 2 1 H = drΨ†(r)[ +V (r)]Ψ(r)+ drdr′Ψ†(r)Ψ†(r′)v (r r′)Ψ(r′)Ψ(r) −2∇ ext 2 c − Z Z (1) where Ψ(r) is the field operator of electron, Vext(r) is the potential of nucleous and v (r r′) is the Coulomb interaction 1/ r r′ . We assumed Born-Oppenheimer c − | − | approximation for nuclei motion (freezing them since their kinetic energy is of the order of Mnuclei/me). The Hartree-Fock approximation replaces the two-body interaction term by an effective one body term Ψ†(r)Ψ†(r′)Ψ(r′)Ψ(r) Ψ†(r)Ψ(r) Ψ†(r′)Ψ(r′)+ Ψ†(r′)Ψ(r′) Ψ†(r)Ψ(r) (2) → h i h i Ψ†(r)Ψ(r′) Ψ†(r′)Ψ(r) Ψ†(r′)Ψ(r) Ψ†(r)Ψ(r′) (3) −h i − h i With the generalized density ρ(r, r′)= Ψ†(r)Ψ(r′) the effective Hartree-Fock h i Hamiltonian takes the form eff 1 2 H = drdr′Ψ†(r) + V (r)+ dr′′ρ(r′′, r′′)v (r′′ r) δ(r r′) −2∇ ext c − − Z Z ρ(r′, r)vc(r′ r) Ψ(r′) (4) Kristjan Haule, 2009 − − } –2– KH Computational Physics- 2009 Hartree-Fock Method H is a one-particle Hamiltonian • eff The effective potential is non-local (very different from the LDA approximation) • Higher order terms in perturbative expansion are completely neglected (so-called • correlation part in Density Functional Theory) If we write the field operator Ψ in one electron basis Ψ(r)= ψα(r)cα (5) α X where cα creates an electron in one electron state ψα(r) we get eff eff H = Hαβ cα† cβ (6) Xαβ where eff H = H0 + (U U )n (7) αβ αβ aαbβ − aαβb ab Xab with n = c† c and matrix elements ab h a bi 1 H0 = ψ 2 + V ψ (8) αβ h α|− 2∇ ext| βi Kristjan Haule, 2009 –3– KH Computational Physics- 2009 Hartree-Fock Method U = ψ (r)ψ (r′) v (r r′) ψ (r)ψ (r′) (9) αβγδ h α β | c − | γ δ i If the one electron orbitals ψα are othogonal, we are done. First, we need to calculate matrix elements of H0 and U . Then we can proceed to eff eff construct Hαβ according to Eq. (7) and finally we can diagonalize the matrix Hαβ . In Eq. (7) the sum over a and b runs over all occupied states only (since nab = δabna is 0 for unoccupied and 1 for occupied eigenstates). Weather the orbital is occupied or not, we eff know only after we calculate the one-electron levels, i.e., diagonalize Hαβ by ψ Heff ε ψ =0. Hence, the system of equations needs to be solved h α| − | βi self-consistently by iteration. Kristjan Haule, 2009 –4– KH Computational Physics- 2009 Hartree-Fock Method Interpretation in terms of electron self-energy In many-body problems, one usually defines the so-called self-energy. It is the quantity that needs to be added to non-interacting Hamiltonian to get the interacting effective Hamiltonian eff 0 H = drdr′Ψ†(r) H (r)δ(r r′)+Σ(r, r′) Ψ(r′) (10) − Z From Eq. (4) we can see that Σ(r, r′)= δ(r r′) dr′′ρ(r′′, r′′)v (r′′ r) ρ(r′, r)v (r′ r) (11) − c − − c − Z This term is just the lowest order term in perturbation expansion of self-energy in powers of Coulomb repulsion and its diagrammatic representation in terms of Feyman diagrams is Kristjan Haule, 2009 –5– KH Computational Physics- 2009 Hartree-Fock Method Derivation for beginners We mentioned in the previous chapter (second quantization) that Hartree Fock approximation corresponds to the approximation where one finds single ”properly symmetrized” Slater determinant that minimizes the total energy for the interacting Hamiltonian E = Φn1n2, nN H Φn1n2, nN (12) h ··· | | ··· i We can vary one-electron basis functions with constraint that they are normalized. In previous chapter we called one-electron basis functions uk(x). Here, we called them ψ(x) therefore we will stick to later definition. We remember from previous lecture 0 1 E = ψ∗ (x)H (x)ψ (x)dx + dxdx′ψ∗ (x)ψ∗ (x′)v (x x′)ψ (x)ψ (x′) α α 2 α β C − α β α X Xαβ Z 1 dxdx′ψ∗ (x)ψ∗ (x′)v (x x′)ψ (x)ψ (x′) −2 α β C − β α Xαβ Z Kristjan Haule, 2009 –6– KH Computational Physics- 2009 Hartree-Fock Method We would like to minimize E ǫ dxψ∗ (x)ψ (x)= min (15) − α α α α X Z We look for the best ψβ, so we will vary the functional with respect to ψβ∗ δE 0 0= = H (x)ψ (x)+ dx′ψ∗(x′)v (x x′)ψ (x′) ψ (x) (16) δψ β γ C − γ β β∗ " γ # X Z dx′ψ∗(x′)v (x x′)ψ (x′)ψ (x) ǫ ψ (x) (17) − γ C − β γ − β β γ X Z We can multiply the equation by ψα(x) and integrate over x to get 0 ψ∗ (x)H (x)ψ (x)dx + dxdx′ψ∗ (x)ψ∗(x′)v (x x′)ψ (x′)ψ (x) (18) α β α γ C − γ β γ Z X Z dxdx′ψ∗ (x)ψ∗(x′)v (x x′)ψ (x′)ψ (x) ǫ ψ∗ (x)ψ (x)dx (19) − α γ C − β γ − β α β γ X Z Z Kristjan Haule, 2009 –7– KH Computational Physics- 2009 Hartree-Fock Method or H0 + (U U )= ǫ δ (20) αβ γαγβ − γαβγ α αβ γ occupied ∈ X This is equivalent to Eq.(7). One needs to be carefull and sum only over occupied states γ. This is because in this derivation we worked with fixed number of particles N and all states of the Slater determinant must be occupied. This is hardly a surprise since the minimization gives ground state only, thus, this is zero temperature analog of the above more general perturbation formalism. Kristjan Haule, 2009 –8– KH Computational Physics- 2009 Hartree-Fock Method Nonorthogonal base In actual calculation, the one electron orbitals are choosen very carefully because we would like to get (at least) the ground state with only few orbitals (The computation time increases as N 3, where N is the number of orbitals - solving eigenvalue problem). Those optimally chosen orbitals are almost never orthogonal! One way around would be to orthogonalize basis functions first. This is a time consuming task and also leads to basis functions which are less localized. It is more convenient to generalize the equations to non-orthogonal basis functions than to orthogonalize basis functions. Let us call the non-orthogonal basis functions Φp(r). Further, let us write the transformation from non-orthogonal base to eigenbase-base as ψα(r)= ApαΦp(r) (21) p X We want to solve ψ Heff ε ψ =0 (22) h α| − | βi Kristjan Haule, 2009 –9– KH Computational Physics- 2009 Hartree-Fock Method and using Eq. (21) we get eff A∗ (H εO )A =0 (23) pα pq − pq qβ where O is the overlap matrix O = Φ Φ . This is the generalized eigenvalue pq pq h p| qi problem Heff A = εOA, where H is Hermitian and O is positive definite matrix. The eigenvectors A are not orthogonal, but rather satisfy the relation A†OA =1 which can be deduced from the relation ψ ψ = δ . h α| βi αβ We also need the density matrix in the non-orthogonal base. It follows from ρ(r, r′)= Ψ†(r)Ψ(r′) = ψ∗ (r)n ψ (r′)= Φ∗(r)A∗ n A Φ (r′), h i α αβ β p pα αβ qβ q αβ pq,αβ X X (24) and since n = δ f(ε µ) we conclude ρ = A∗ A f(ε µ). αβ αβ α − qp pα qα α − The total energy of the system is not α εα! The right total energy in the eigen-base is 1 P 1 E = H0 n + (U U )n n = Tr[(Heff + H0)n] (25) αα α 2 aαaα − aααa a α 2 α αa X X Kristjan Haule, 2009 –10– KH Computational Physics- 2009 Hartree-Fock Method In the non-orthogonal basis, the total energy takes the form 1 1 0 1 1 0 E = ε n + (A†H A) n = ε n + H ρ (26) 2 α α 2 αα α 2 α α 2 pq qp α α α pq X X X X Now we sketch algorithm to solve the general Hartree-Fock problem 1 Pick a set of one electron orbitals φp(r) and add them the spin part of the wave function to get Φ (r)= φ (r) s .
Load more

Recommended publications
  • 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.
    [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]
  • Chapter 18 the Single Slater Determinant Wavefunction (Properly Spin and Symmetry Adapted) Is the Starting Point of the Most Common Mean Field Potential
    Chapter 18 The single Slater determinant wavefunction (properly spin and symmetry adapted) is the starting point of the most common mean field potential. It is also the origin of the molecular orbital concept. I. Optimization of the Energy for a Multiconfiguration Wavefunction A. The Energy Expression The most straightforward way to introduce the concept of optimal molecular orbitals is to consider a trial wavefunction of the form which was introduced earlier in Chapter 9.II. The expectation value of the Hamiltonian for a wavefunction of the multiconfigurational form Y = S I CIF I , where F I is a space- and spin-adapted CSF which consists of determinental wavefunctions |f I1f I2f I3...fIN| , can be written as: E =S I,J = 1, M CICJ < F I | H | F J > . The spin- and space-symmetry of the F I determine the symmetry of the state Y whose energy is to be optimized. In this form, it is clear that E is a quadratic function of the CI amplitudes CJ ; it is a quartic functional of the spin-orbitals because the Slater-Condon rules express each < F I | H | F J > CI matrix element in terms of one- and two-electron integrals < f i | f | f j > and < f if j | g | f kf l > over these spin-orbitals. B. Application of the Variational Method The variational method can be used to optimize the above expectation value expression for the electronic energy (i.e., to make the functional stationary) as a function of the CI coefficients CJ and the LCAO-MO coefficients {Cn,i} that characterize the spin- orbitals.
    [Show full text]
  • Arxiv:1902.07690V2 [Cond-Mat.Str-El] 12 Apr 2019
    Symmetry Projected Jastrow Mean-field Wavefunction in Variational Monte Carlo Ankit Mahajan1, ∗ and Sandeep Sharma1, y 1Department of Chemistry and Biochemistry, University of Colorado Boulder, Boulder, CO 80302, USA We extend our low-scaling variational Monte Carlo (VMC) algorithm to optimize the symmetry projected Jastrow mean field (SJMF) wavefunctions. These wavefunctions consist of a symmetry- projected product of a Jastrow and a general broken-symmetry mean field reference. Examples include Jastrow antisymmetrized geminal power (JAGP), Jastrow-Pfaffians, and resonating valence bond (RVB) states among others, all of which can be treated with our algorithm. We will demon- strate using benchmark systems including the nitrogen molecule, a chain of hydrogen atoms, and 2-D Hubbard model that a significant amount of correlation can be obtained by optimizing the energy of the SJMF wavefunction. This can be achieved at a relatively small cost when multiple symmetries including spin, particle number, and complex conjugation are simultaneously broken and projected. We also show that reduced density matrices can be calculated using the optimized wavefunctions, which allows us to calculate other observables such as correlation functions and will enable us to embed the VMC algorithm in a complete active space self-consistent field (CASSCF) calculation. 1. INTRODUCTION Recently, we have demonstrated that this cost discrep- ancy can be removed by introducing three algorithmic 16 Variational Monte Carlo (VMC) is one of the most ver- improvements . The most significant of these allowed us satile methods available for obtaining the wavefunction to efficiently screen the two-electron integrals which are and energy of a system.1{7 Compared to deterministic obtained by projecting the ab-initio Hamiltonian onto variational methods, VMC allows much greater flexibil- a finite orbital basis.
    [Show full text]
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • Iia. Systems of Many Electrons
    EE5 in 2008 Lecture notes Univ. of Iceland Hannes J´onsson IIa. Systems of many electrons Exchange of particle labels Consider a system of n identical particles. By identical particles we mean that all intrinsic properties of the particles are the same, such as mass, spin, charge, etc. Figure II.1 A pairwise permutation of the labels on identical particles. In classical mechanics we can in principle follow the trajectory of each individual particle. They are, therefore, distinguishable even though they are identical. In quantum mechanics the particles are not distinguishable if they are close enough or if they interact strongly enough. The quantum mechanical wave function must reflect this fact. When the particles are indistinguishable, the act of labeling the particles is an arbi- trary operation without physical significance. Therefore, all observables must be unaffected by interchange of particle labels. The operators are said to be symmetric under interchange of labels. The Hamiltonian, for example, is symmetric, since the intrinsic properties of all the particles are the same. Let ψ(1, 2,...,n) be a solution to the Schr¨odinger equation, (Here n represents all the coordinates, both spatial and spin, of the particle labeled with n). Let Pij be an operator that permutes (or ‘exchanges’) the labels i and j: Pijψ(1, 2,...,i,...,j,...,n) = ψ(1, 2,...,j,...,i,...,n). This means that the function Pij ψ depends on the coordinates of particle j in the same way that ψ depends on the coordinates of particle i. Since H is symmetric under interchange of labels: H(Pijψ) = Pij Hψ, i.e., [Pij,H]=0.
    [Show full text]
  • 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.
    [Show full text]
  • Slater Decomposition of Laughlin States
    Slater Decomposition of Laughlin States∗ Gerald V. Dunne Department of Physics University of Connecticut 2152 Hillside Road Storrs, CT 06269 USA [email protected] 14 May, 1993 Abstract The second-quantized form of the Laughlin states for the fractional quantum Hall effect is discussed by decomposing the Laughlin wavefunctions into the N-particle Slater basis. A general formula is given for the expansion coefficients in terms of the characters of the symmetric group, and the expansion coefficients are shown to possess numerous interesting symmetries. For expectation values of the density operator it is possible to identify individual dominant Slater states of the correct uniform bulk density and filling fraction in the physically relevant N limit. →∞ arXiv:cond-mat/9306022v1 8 Jun 1993 1 Introduction The variational trial wavefunctions introduced by Laughlin [1] form the basis for the theo- retical understanding of the quantum Hall effect [2, 3]. The Laughlin wavefunctions describe especially stable strongly correlated states, known as incompressible quantum fluids, of a two dimensional electron gas in a strong magnetic field. They also form the foundation of the hierarchy structure of the fractional quantum Hall effect [4, 5, 6]. In the extreme low temperature and strong magnetic field limit, the single particle elec- tron states are restricted to the lowest Landau level. In the absence of interactions, this Landau level has a high degeneracy determined by the magnetic flux through the sample [7]. ∗UCONN-93-4 1 Much (but not all) of the physics of the quantum Hall effect may be understood in terms of the restriction of the dynamics to fixed Landau levels [8, 3, 9].
    [Show full text]
  • PART 1 Identical Particles, Fermions and Bosons. Pauli Exclusion
    O. P. Sushkov School of Physics, The University of New South Wales, Sydney, NSW 2052, Australia (Dated: March 1, 2016) PART 1 Identical particles, fermions and bosons. Pauli exclusion principle. Slater determinant. Variational method. He atom. Multielectron atoms, effective potential. Exchange interaction. 2 Identical particles and quantum statistics. Consider two identical particles. 2 electrons 2 protons 2 12C nuclei 2 protons .... The wave function of the pair reads ψ = ψ(~r1, ~s1,~t1...; ~r2, ~s2,~t2...) where r1,s1, t1, ... are variables of the 1st particle. r2,s2, t2, ... are variables of the 2nd particle. r - spatial coordinate s - spin t - isospin . - other internal quantum numbers, if any. Omit isospin and other internal quantum numbers for now. The particles are identical hence the quantum state is not changed under the permutation : ψ (r2,s2; r1,s1) = Rψ(r1,s1; r2,s2) , where R is a coefficient. Double permutation returns the wave function back, R2 = 1, hence R = 1. ± The spin-statistics theorem claims: * Particles with integer spin have R =1, they are called bosons (Bose - Einstein statistics). * Particles with half-integer spins have R = 1, they are called fermions (Fermi − - Dirac statistics). 3 The spin-statistics theorem can be proven in relativistic quantum mechanics. Technically the theorem is based on the fact that due to the structure of the Lorentz transformation the wave equation for a particle with spin 1/2 is of the first order in time derivative (see discussion of the Dirac equation later in the course). At the same time the wave equation for a particle with integer spin is of the second order in time derivative.
    [Show full text]
  • Spin Balanced Unrestricted Kohn-Sham Formalism Artëm Masunov Theoretical Division,T-12, Los Alamos National Laboratory, Mail Stop B268, Los Alamos, NM 87545
    Where Density Functional Theory Goes Wrong and How to Fix it: Spin Balanced Unrestricted Kohn-Sham Formalism Artëm Masunov Theoretical Division,T-12, Los Alamos National Laboratory, Mail Stop B268, Los Alamos, NM 87545. Submitted October 22, 2003; [email protected] The functional form of this XC potential remains unknown to ABSTRACT: Kohn-Sham (KS) formalism of Density the present day. However, numerous approximations had been Functional Theory is modified to include the systems with suggested,4 the remarkable accuracy of KS calculations results strong non-dynamic electron correlation. Unlike in extended from this long quest for better XC functional. One may classify KS and broken symmetry unrestricted KS formalisms, these approximations into local (depending on electron density, or cases of both singlet-triplet and aufbau instabilities are spin density), semilocal (including gradient corrections), and covered, while preserving a pure spin-state. The nonlocal (orbital dependent functionals). In this classification HF straightforward implementation is suggested, which method is just one of non-local XC functionals, treating exchange consists of placing spin constraints on complex unrestricted exactly and completely neglecting electron correlation. Hartree-Fock wave function. Alternative approximate First and the most obvious reason for RKS performance approach consists of using the perfect pairing inconsistency for the “difficult” systems, mentioned above, is implementation with the natural orbitals of unrestricted KS imperfection of the approximate XC functionals. Second, and less method and square roots of their occupation numbers as known reason is the fact that KS approach is no longer valid if configuration weights without optimization, followed by a electron density is not v-representable,5 i.e., there is no ground posteriori exchange-correlation correction.
    [Show full text]
  • Arxiv:1411.4673V1 [Hep-Ph] 17 Nov 2014 E Unn Coupling Running QED Scru S Considerable and Challenge
    Attempts at a determination of the fine-structure constant from first principles: A brief historical overview U. D. Jentschura1, 2 and I. N´andori2 1Department of Physics, Missouri University of Science and Technology, Rolla, Missouri 65409-0640, USA 2MTA–DE Particle Physics Research Group, P.O.Box 51, H–4001 Debrecen, Hungary It has been a notably elusive task to find a remotely sensical ansatz for a calculation of Sommer- feld’s electrodynamic fine-structure constant αQED ≈ 1/137.036 based on first principles. However, this has not prevented a number of researchers to invest considerable effort into the problem, despite the formidable challenges, and a number of attempts have been recorded in the literature. Here, we review a possible approach based on the quantum electrodynamic (QED) β function, and on algebraic identities relating αQED to invariant properties of “internal” symmetry groups, as well as attempts to relate the strength of the electromagnetic interaction to the natural cutoff scale for other gauge theories. Conjectures based on both classical as well as quantum-field theoretical considerations are discussed. We point out apparent strengths and weaknesses of the most promi- nent attempts that were recorded in the literature. This includes possible connections to scaling properties of the Einstein–Maxwell Lagrangian which describes gravitational and electromagnetic interactions on curved space-times. Alternative approaches inspired by string theory are also dis- cussed. A conceivable variation of the fine-structure constant with time would suggest a connection of αQED to global structures of the Universe, which in turn are largely determined by gravitational interactions. PACS: 12.20.Ds (Quantum electrodynamics — specific calculations) ; 11.25.Tq (Gauge field theories) ; 11.15.Bt (General properties of perturbation theory) ; 04.60.Cf (Gravitational aspects of string theory) ; 06.20.Jr (Determination of fundamental constants) .
    [Show full text]