A very efficient two-density approach to atomistic simulations and a proof of principle for small atoms and molecules

Transcription

1 A very efficient two-density approach to atomistic simulations and a proof of principle for small atoms and molecules Werner A Hofer and Thomas Pope School of Natural and Environmental Sciences Newcastle University Web:

2 Overview A brief history Comparison with the standard model Many-body framework Proof of principle Summary 2

3 Two crucial questions Question 1: Why do electrons change their wavelength when they change their velocity? (09/2009) Can only be answered if electrons are described by two and not one density Mass or charge density r Spin density S r + S = n(dft)

4 Two crucial questions Question 1: Why do electrons change their wavelength when they change their velocity? (09/2009) Can only be answered if electrons are described by two and not one density Mass or charge density r Spin density S r + S = n(dft)

5 Two crucial questions Question 1: Why do electrons change their wavelength when they change their velocity? (09/2009) Can only be answered if electrons are described by two and not one density Mass or charge density r Spin density S r + S = n(dft)

6 Two crucial questions Question two: Are triangles real? Answer: Triangles are not real, they are abstract mathematical objects in mathematical space The problem is that the theory is too strong, too compelling. I feel we are missing a basic point. The next generation, as soon as they will have found that point, will knock on their heads and say: How could they have missed that? Isidor Rabi 1985

7 Two different spaces Mathematical Space Real three-dimensional Space Objects: Triangles, matrices, wavefunctions, spins, operators, spinors, (most) particles, strings Objects: Densities, masses, momenta, energies, magnetic moments, electromagnetic fields, gravitational fields Key difference: Objects in mathematical space are not physical objects Objects in real three-dimensional space are physical objects

8 Theoretical models Mathematical Space String theory Real three-dimensional Space Classical Mechanics Classical Electrodynamics Statistical Thermodynamics

9 Theoretical models Mathematical Space String theory Real three-dimensional Space Classical Mechanics Classical Electrodynamics Statistical Thermodynamics Quantum Mechanics Nuclear Physics Density Functional Theory High Energy Physics

10 Theoretical models Mathematical Space String theory Real three-dimensional Space Classical Mechanics Classical Electrodynamics Statistical Thermodynamics Quantum Mechanics Nuclear Physics Density Functional Theory High Energy Physics Mathematics creates Reality

11 Theoretical models Mathematical Space String theory Real three-dimensional Space Classical Mechanics Classical Electrodynamics Statistical Thermodynamics Density Functional Theory 2.0 Quantum Mechanics Nuclear Physics High Energy Physics arxiv: Mathematics creates Reality

12 Theoretical models Mathematical Space String theory Real three-dimensional Space Classical Mechanics Classical Electrodynamics Statistical Thermodynamics Density Functional Theory 2.0 Quantum Mechanics Nuclear Physics High Energy Physics Mathematics creates Reality

13 Spin in the standard model Stern-Gerlach experiments: Silver atoms, with no orbital magnetic moment, are deflected by magnetic fields in Stern-Gerlach experiments 1 This is due to the spin of the outer electron 1. Walter Gerlach and Otto Stern (1922) 13

14 Standard Model What is spin? Spin is a vector: Spin is affected by a magnetic field, therefore it has the properties of a magnetic moment, therefore it is a vector. Spin is isotropic: The results of the Stern-Gerlach experiment are the same if the magnetic field is rotated, therefore spin is isotropic, therefore it is not a vector. Problems: How can spin be isotropic before the measurement? How does spin become a vector in the measurement? 14

15 Standard Model Pauli Matrices Spin is modelled with the help of Pauli Matrices: With corresponding eigenvectors for the eigenvalues +/-1: Which act on spinors: 15

16 Standard Model Stern-Gerlach experiments explained When the spin of this particle is measured with respect to a given axis, a=x, y, z, the probability that a spin of ±1/2 will be measured is, Spin can have one of two values because the Pauli matrices span the space of observables of the 2-dimensional Hilbert space. The wavefunction collapses on the eigenvector, which leads to the two trajectories in the Stern-Gerlach experiments. 16

17 Standard Model Consecutive measurements If a new measurement is performed on axis b, the probability of measuring the same spin value is, and the probability of measure the opposite spin value is, This is due to the non-commutativity of the Pauli matrices 17

18 Standard Model Problem with this explanation: No physical explanation what a collapse of the wavefunction actually means, therefore a host of theoretical speculation in terms of mathematical models: 1. Ghirardi-Rimini-Weber model: wavefunction amplitudes change with distance 2. Penrose interpretation: link between quantum effects and spacetime curvature 3. Copenhagen Interpretation: there is no causality in physics 4. De Broglie-Bohm approach: wavefunctions create potentials and change instantaneously, and over arbitrary distances, if experimental conditions change 5. Everett: every measurement leads to a different universe 6. Dowe: backwards causation 7. many more attempts to solve the measurement problem Missing: a physical model which determines the cause of a particular trajectory in a Stern-Gerlach experiment 18

19 Standard Model The Einstein-Podolsky-Rosen problem: In the EPR thought experiment 1, a source emits an electron pair. Two measurements are performed, which depend on one another. Because of the non-commutativity of the Pauli matrices: measurements performed on the same axis are correlated measurements performed on different axes are uncorrelated How does the second electron know what axis was chosen measuring the first? 1. Albert Einstein, Boris Podolsky and Nathan Rosen, Physical Review 47, 777 (1935) 19

20 Extended electrons Geometric algebra In three space dimensions, in geometric algebra the three directions are described by three perpendicular unit vectors, e 1, e 2, e 3. Multiplying vectors is anti-commutative and gives a so-called bivector, which is a twodimensional area: Multiplying all 3 vectors together gives a pseudoscalar, which is equal to the imaginary unit: i=e 1 e 2 e 3 Multiplying a vector with a pseudoscalar gives the bivector composed of the two other unit vectors: The algebra of unit vectors in geometric algebra is the same as the algebra of Pauli matrices. 20

21 Extended electrons Spin densities and wavefunctions The electron has field components, e E and e H that are perpendicular to the vector of motion, e v, and one another. Because e H e E =-e E e H, we define the spin unit vector, which is parallel or antiparallel to the vector of motion, The electron is described by an effective wavevector in three dimensional space containing mass and spin densities and a spin bivector: 21

22 Extended electrons Spin vectors The spin vector is defined as: Spin is either parallel or antiparallel to the direction of motion. The spin vector is contained in the bivector term of the wavefunction. Spin is isotropic in relation to rotations in the bivector plane. A statistical manifold of equal number spin-up and spin-down electrons is isotropic. 22

23 Extended electrons Stern-Gerlach experiments In a magnetic field, the direction of the spin vector changes: Modified Landau Lifschitz equation 23

24 Extended electrons Stern-Gerlach experiments In a magnetic field, the direction of the spin vector changes: The induced spin vector, S', is: Stern-Gerlach experiments measure the spin induced by magnetic fields 24

25 Extended electrons Einstein-Podolsky-Rosen experiments The measurements performed today on photons contain rotations of electromagnetic fields in the plane perpendicular to the direction of motion, which are described by a rotor: To account for the two rotations, we take the product of the rotors for each photon: The correlation probability is: 25

26 Advantages Spin has vector properties and is isotropic. Stern-Gerlach experiments are well explained in terms of cause and effect. Anti-commutativity of spin measurements is well understood. No spooky action at a distance in EPR experiments. No need for many worlds or retrocausality. 26

27 Advantages Spin has vector properties and is isotropic. Stern-Gerlach experiments are well explained in terms of cause and effect. Anti-commutativity of spin measurements is well understood. No spooky action at a distance in EPR experiments. No need for many worlds or retrocausality. 27

28 Many-body problem Scaling of DFT simulations Kohn-Sham Formulation Good accuracy Not orbital free, so poor scaling Order-N DFT Linear scaling Limited scope due to approximations Orbital free DFT Linear scaling Poor accuracy Extended electron model Linear scaling No orbitals Potentially as accurate as KS DFT

29 Many-body problem Governing equations The effective wavefunction is a combination of scalar (mass density) and bivector (spin density) terms Hamiltonian similar to standard Hamiltonian with addition of a bivector potential

30 Many-body problem Governing equations The bivector potential and the spin bivector in the wavefunction give rise to two types of objects: bivectors and scalars The equations only have a solution if the spinvector e s is aligned to the bivector e b Note that the bivector potential couples mass and spin densities: both are solutions of the same equation if v b vanishes (single electron case).

31 Many-body problem Functional equations The approach is described by two functional equations: The first equation describes the total energy functional, the second the bivector functional. One can explicitly calculate the bivector potential v b from these equations:

32 Total energy functional The total energy functional is derived from the Kohn-Sham-like equations by solving the variation: This leads to the total energy functional:

33 Bivector energy functional Similarly, the bivector energy functional is derived from the Kohn-Sham-like equations by solving the variation: This leads to the bivector energy functional:

34 Self-consistent solution A self-consistent method for solving the many-body problem within the framework of many electrons relies on minimising the contributions from both, mass and spin densities, with both functionals: There is no bivector term in the equation for the total energy. Instead, the bivector functional affects the densities, which in turn are used to calculate the total energy.

35 Method Implementation Overlap integral Hartree-Fock method: integrals Kinetic integral Nuclear Integral Coulomb/Exchange Integral

36 Method Implementation Hartree-Fock method: solver Construct the Fock matrix from nuclear, kinetic, Coulomb, and exchange integrals Diagonalise the overlap matrix perform the same unitary operation on the Fock matrix Calculate the eigenvectors and transform into the original basis

37 Method Implementation Extended electrons: solver Construct the diagonal Fock matrix from nuclear, kinetic, Coulomb, and exchange integrals Construct the off-diagonal Fock matrix from the bivector term, which is given in terms of the kinetic integral Repeat the Hartree-Fock method for the full Fock matrix

38 Hydrogen molecule: bond length

39 Small atoms: energies

40 Summary We have formulated a many-body approach which encodes exchange effects in mass and spin densities. Correlation effects still to be included. In general, the formulation is much simpler than present formulations it contains only four independent variables. The implementation is free of auxiliary assumptions and able to reproduce reliable results for small systems. Due to the theoretical simplifications the method is truly linear scaling, and much more efficient than the current standard. The interaction energy between mass density and spin density has the same value as the exchange energy in the Hartree-Fock theory.

41 Thank you!