Magnetism in a relativistic perspective

Transcription

1 Magnetism in a relativistic perspective Trond Saue Laboratoire de Chimie et Physique Quantiques Université de Toulouse 3 (Paul Sabatier) 118 route de Narbonne, F Toulouse, France trond.saue@irsamc.ups-tlse.fr

2 The DIRAC code P rogram for A tomic and M olecular D I R A C irect terative elativistic ll-electron alculations Web site: New release DIRAC10 in september!!! Wave functions: [HF, MP2, RASCI, MCSCF, CCSD(T), FSCCSD] + DFT [LDA, GGAs, hybrids] HF/DFT: Electric and magnetic properties: expectation values, linear and quadratic response functions, single excitation energies

3 The DIRAC village

4 Outline Particles and fields: Particles: the Dirac Hamiltonian Fields: Maxwell s equations The non-relativistic limit of electrodynamics Para- and diamagnetic contributions Multipolar gauge Homogeneous magnetic fields: origin dependence in the static and time-dependent case

5 Particles and fields Complete Hamiltonian: H = H particles + H interaction + H fields

6 Free particle Non-relativistic free particle: H NR particle = p2 2m Relativistic free particle: H R particle = m 2 c 4 + c 2 p 2 = mc p2 (mc) 2 = mc2 + p2 2m p4 8m 3 c Non-relativistic limit (shift of zero of energy): lim c HR particle = lim c ( H R particle mc 2) = p2 2m

7 The Dirac equation Sehr viel unglücklicher bin ich über die Frage nach der relativistischen Formulierung und über die Inkonsequenz der Dirac-Theorie... Also ich find die gengewärtige Lage ganz absurd und hab mich deshalb, quasi aus Verzweiflung, auf ein anderes gebiet, das der Ferromagnetismus begeben. W. Heisenberg (1928) Lorentz covariant form (4-vector notation): (iγ µ µ mc) ψ = 0; µ = (, i c t γ µ = (βα, iβ) ) Convential form: [ β mc 2 c (α p) i ] ψ = 0; ψ = t ψ Lα ψ Lβ ψ Sα ψ Sβ Dirac matrices: β = β mc 2 ; β = [ ] I2 0 ; α = 0 I 2 [ 0 σ σ 0 ]

8 Fields Maxwell s microscopic equations (SI units): B = 0 E + B t = 0 E = ρ/ε 0 B 1 c 2 E t = 1 ε 0 c 2j (the electric and magnetic constants are related through ε 0 µ 0 = 1/c 2 ) Boundary conditions: E and B go to zero at infinite distance from the sources

9 Fields General solutions (with retarded time t r = t r 12 c ) : E(r 1 ) = 1 4πε 0 B(r 1 ) = 1 4πε 0 c² { ρ (r 2, t r ) r 12 r 3 12 { r 12 j (r 2, t r ) r ρ (r 2, t r ) r 12 r 2 12 j } (r 2, t r ) c 2 dτ 2 r 12 + r 12 j } (r 2, t r ) dτ 2 cr²12 to which the homogeneous soutions (electromagnetic waves) may be added. It is, however, more conventient to work with electromagnetic potentials E = φ A t ; B = A

10 Fields Maxwell s microscopic equations are obtained from the Lagrangian density with L = j α A α c 2 ε 0 F αβ F αβ /4 4-current: j α = ( (j, icρ) 4-potential: A α = A, i c φ) Electromagnetic field tensor: F αβ = α A β β A α by invoking the Euler-Lagrange equations for continuous systems L A α β ( L ) ( β A α ) = 0 The final result is: β F αβ = j α /c 2 ε 0 ; (Lorentz gauge: µ A µ = 0 2 A α = j α /c 2 ε 0 )

11 Particles and fields Complete Hamiltonian H = H particles + H interaction + H fields Fields specified: (iγ µ µ mc) ψ = 0 Dirac equation Non-relativistic limit ( p 2 2m i ) ψ = 0 t Schrödinger equation Particles (sources) specified: Non-relativistic limit β F αβ = j α /c 2 ε 0??? Maxwell s equations

12 The non-relativistic limit of electrodynamics T. Saue, Adv. Quantum Chem. 48 (2005) 383 B = 0 B = 0 E + B = 0 c E = 0 t E = 4πρ/ε 0 E = 4πρ/ε 0 B 1 E c 2 = 4π t ε 0 c 2j B = 0 In the strict non-relativistic limit there are no magnetic fields and no effects of retardation! The Coulomb gauge bears its name because it singles out the instantaneous Coulomb interaction, which constitutes the proper non-relativistic limit of electrodynamics and which is the most important interaction in chemistry. All retardation effects as well as magnetic interactions are to be considered corrections of a perturbation series of the total interaction (in 1/c 2 ).

13 The principle of minimal electromagnetic substitution (M. Gell-Mann, Nuovo Cimento Suppl. 4 (1956) 848) Interaction Hamiltonian (K. Schwarzschild, Gött. Nach. Math.-Phys. Kl. (1903) 126) H interaction = j α A α dτ = (ρφ j A) dτ The Hamiltonian of a particle interacting with external fields is obtained from the freeparticle Hamiltonian through the substitutions: p µ p µ qa µ Electron: q = e p p + ea E E + eφ The relativistic coupling of particles to electromagnetic fields is employed in both nonrelativistic and relativistic frameworks. However, in the non-relativistic framework there is no mechanism for the generation of magnetic fields.

14 Magnetic interaction Non-relativistic magnetic operator: ĥ NR A = e e (p A + A p) + (σ B) + e2 A 2 } 2m {{ 2m }} 2m {{} paramagnetic diamagnetic Relativistic magnetic operator: ĥ R A = ec (α A) Velocity operator: dr dt = i [ r, H NR] = π m ; π = p + ea i [ r, H R] = cα

15 The diamagnetic contribution: one-electron system M. M. Sternheim, Phys. Rev. 128 (1962) 676 2nd order perturbation theory [ĥr A = ec (α A)] : E (2) = (+) (+) 0 ĥr A n n ĥr A 0 E 0 E n 0 ĥr A n n ĥr A 0 + ( ) + ( ) 0 ĥr A n n ĥr A 0 E 0 E n 0 ĥr A n n ĥr A 0 E 0 E n 2mc 2 0 ĥr A n n ĥr A 0 e 2 A E (+) 0 E n 2m 0 }{{}}{{} paramagnetic diamagnetic The approximation becomes exact in the non-relativistic limit.

16 The diamagnetic contribution: many-electron systems G. Aucar, T. Saue, H. J. Aa. Jensen and L. Visscher, J. Chem. Phys. 110 (1999) 9677 More complicated: the N-particle basis (Slater determinants) should be constructed from positive-energy orbitals only, but the orbitals allowed to relax with an update of the potential. This implies that the exact solution can not be obtained from full CI, rather from a MCSCF approach. Orbital relaxation: φ i = p φ p U pi ; U = U 1 Exponential parametrization: U = exp [ κ] ; κ = κ (amplitudes of orbital rotations) First-order induced current density: j B = e i { φ B i cαφ i + φ i cαφb i }

17 First-order induced current density in pyridine Radovan Bast, Jonas Jusélius and Trond Saue, Chemical Physics 356 (2009) 187 First-order orbitals: φ B i (1) = d φ i = φ a κ B ai db B=0 z = 0 bohr z = 1 bohr

18 First-order induced current density in pyridine Radovan Bast, Jonas Jusélius and Trond Saue, Chemical Physics 356 (2009) 187 First-order orbitals: φ B i (1) = d φ i = φ a κ B ai db B=0 κ ++ rotations with positive-energy virtuals κ + rotations with negative-energy virtuals κ ++ ; z = 0 bohr κ + ; z = 0 bohr

19 Multipolar gauge F. Bloch, in W. Heisenberg und die Physik unserer Zeit, Braunschweig 1961 L. D. Barron and C. G. Gray, J. Phys. A 6 (1973) 59 W. E. Brittin, W. Rodman Smythe and W. Wyss, Am. J. Phys. 50 (1982) 693 The energy of the interaction between particles and fields is expressed in terms of external potentials, not fields: E int = [ρ(r)φ(r) j(r) A(r)] dτ Is is possible to express the interaction energy directly in terms of the fields? The answer is: Yes, using multipolar gauge. Consider a Taylor expansion of the scalar potential (implicit summation) φ(r) = φ [0] + r i φ [1] i r ir j φ [2] ij +... = n=0 1 n! r j 1 r j2... r jn φ [n] j 1 j 2...j n ; X [n] j 1 j 2...j n = n X r j1 r j2... jn 0

20 Multipolar gauge 2 Using the relation E = φ A t we obtain φ(r) = φ [0] r i E [0] i 1 2 r ir j E [1] i;j... A [0] i r i t 1 2 r ir j A [0] i;j t... which can be written more compactly as φ(r) = φ (r) χ t where appears χ = n=0 1 ( ) (n + 1)! r j 1 r j2... r jn r A [n] j 1 j 2...j n = 1 0 r A (ur) du

21 Multipolar gauge 3 We carry out the gauge-transformation φ (r) = φ(r) + χ t which gives φ (r) = φ [0] n=0 1 ( ) (n + 1)! r j 1 r j2... r jn r E [n] j 1 j 2...j n Note that we may set φ [0] to zero.

22 Multipolar gauge 4 The vector potential can likewise be Taylor-expanded A i (r) = A i (0) + r j A [1] i;j +... = n=0 1 n! r j 1 r j2... r jn A [n] i;j 1 j 2...j n We next carry out the gauge transformation A = A χ, noting that i χ = A [0] i r ja [1] i;j r ja [1] j;i r jr k A [2] i;jk r jr k A {2] j;ki +... Carrying out the gauge-transformation gives A i = 1 2 r j ( ) A [1] i;j A[1] j;i + 1 ( ) 3 r jr k A [1] i;jk A[1] j;ik +...

23 Multipolar gauge 5 Consider a specific contribution A x = 1 2 = 1 2 { ( ) x A [1] x;x A [1] x;x { } yb z [1] zb y [1] + y ( ) A [1] x;y A [1] y;x +... = z ( r B [0]) x +... ( )} A [1] x;z A [1] z;x +... The gauge-transformed vector potential can thus be written more compactly as A = A χ = n=0 n + 1 ( ) (n + 2)! r j 1 r j2... r jn r B [n] j 1 j 2...j n Notice that r A = 0

24 Multipolar gauge 6 In multipolar gauge the potentials are accordingly expressed in terms of the fields as φ(r) = φ [0] A(r) = n=0 n=0 1 ( ) (n + 1)! r j 1 r j2... r jn r E [n] j 1 j 2...j n n + 1 ( ) (n + 2)! r j 1 r j2... r jn r B [n] j 1 j 2...j n In the case of a homogeneous electric field E we get φ(r) = r E In the case of a homogeneous magnetic field B we get A(r) = 1 2 B r

25 Multipoles Inserting potentials in multipolar gauge into the interaction energy gives E int = [ρ(r)φ(r) j(r) A(r)] dτ = Q [0] φ [0] (electric monopole) n=1 1 n! Q[n] j 1...j n 1 E [n 1] j 1...j n 1 n=1 1 n! m[n] j 1...j n 1 B [n 1] j 1...j n 1 (electric multipoles) (magnetic multipoles) Coulomb gauge is generally not satisfied A = n=0 (n + 2) (n + 1) ( ) r j1 r j2... r jn r ( B) [n] j (n + 3)! 1 j 2...j n Gauge freedom is not lost! It is now given by the choice of expansion point.

26 Multipole expansions The interaction of a molecule with nonuniform fields can be handled through multipole expansions by truncation to given order based on the variation of the field over the molecular volume. Consider the interaction of a molecule with light: E = A T t ; B = A T Notice that the multipole expansion arises in this case from an expansion of the purely transversal vector potential about some expansion point chosen within the molecular volume: A [0] Q [1] E [0] (electric dipole approximation) A [1] 1 2 etc... ij ( ) Q [2] ij E[1] i;j m [1] B [0]

27 Magnetic multipoles m [n] j 1 j 2...j n 1 ;j n = n n + 1 r j1 r j2... r jn 1 (r j(r)) jn dτ (symmetric in all but one index) Magnetic dipole: m [1] i = 1 2 (r j(r)) i dτ Magnetic quadrupole: m [2] ij = 2 3 r i (r j(r)) j dτ A point magnetic dipole is generated from a current loop, letting the coil area go to zero.

28 Static homogeneous magnetic field Vector potential: A (r) = 1 2 (B r G) ; r G = r G Non-relativistic formulation: H NR B = e 2m B (r G p) + e 2m Relativistic formulation: 1 (σ B) + 8m BT [I 3 rg 2 r G r T ] G B H R B = ec (α A) = m [1] B; m [1] = 1 2 e (r G cα) In a linear molecule, the parallel component of paramagnetic shielding and magnetizability is zero so that the perpendicular components can be determined from experimental measurements of the spin-rotation constant and the rotational g factor, respectively. However, in relativistic systems there is no longer proportionality between the electronic magnetic dipole moment and angular momentum and so the parallel component of shielding and magnetizability is generally non-zero and not accessible by such measurements.

29 Zeeman effect A colleague who met me strolling rather aimlessly in the beautiful streets of Copenhagen [1922] said to me in a friendly manner, You look very unhappy ; whereupon I answered tercely, How can one look happy when he is thinking about the anomalous Zeeman effect? WOLFGANG PAULI (1945) Non-relativistic theory: [ e 2m B l, (n l) ] = ie 2m l (B n) = im[1]orb (B n) [ e 2m B l, l2] = 0 Relativistic theory: [ ] 1 ecb (r α), (n j) 2 [ ] 1 ecb (r α), j2 2 = i 2 ec (r α) (B n) = im[1] (B n) = 0

30 Static magnetic dipole: origin dependence By change of origin (r G r P ) of a magnetic dipole we obtain m [1] A = 1 2 = 1 2 {r P j(r)} dτ = 1 {(r G + R GP ) j(r)} dτ 2 {r G j(r)} dτ + R GP 1 j(r)dτ 2 Stationary exact case: 1 ψ i 2 (r P ecα) ψ i B = 1 ψ i 2 (r G ecα) ψ i B+ 1 2 (B R GP ) ψ i ecα ψ i However ψ i cα ψ i = i ψ i [ r, H R ] ψi = i (Ei E i ) ψ i r ψ i = 0

31 Dynamic magnetic dipole: origin dependence Consider now a time-dependent homogeneous magnetic field described by φ (r, t) = 0; A (r, t) = 1 2 (B (t) r G) ; r G = r G The corresponding interaction Hamiltonian reads: H int = eφ + ec (α A) = B (t) 1 2 e (r G cα) and has the form of a scalar product of the magnetic field and the magnetic dipole moment with respect to the arbitrary gauge origin. However, this form is somewhat deceptive since we know that the magnetic field is accompanied by a time-dependent inhomogeneous electric field E(r, t) = φ A t = 1 2 (Ḃ rg ) ; Ḃ = B t

32 Dynamic magnetic dipole: origin dependence 2 We consider an expansion of the vector potential about some chosen expansion point P A (r, t) = A (P, t) + r P ;m [ m A i (r, t)] r=p = A [0] (t) + r P ;m A [1] m (t) = 1 2 (B (t) R P G) (B (t) r P ) Note that we get zero magnetic field from the zeroth-order vector potential, and the full magnetic field from the first-order term The corresponding electric field is E (r, t) = E [0] (t) + r P ;m E [1] m (t) = 1 2 ( ) B t (t) R P G 1 2 ( ) B t (t) r P and is therefore expressed as a Taylor [ expansion ] of the electric field about expansion point P. The field gradient is E [1] i;m = m Ȧ i = 1 2 ɛ mijḃj r=p

33 Dynamic magnetic dipole: origin dependence 3 The switch from gauge origin G to P in the vector potential is achieved by subtracting the gradient of the gauge function χ (r, t) = r P A [0] (t) = r P 1 2 (B (t) R P G). However, since the gauge function is time-dependent there will be a modification of the scalar potential as well! φ = + χ t = r P Ȧ[0] = r P 1 2 (Ḃ (t) RP G ) The total interaction Hamiltonian then reads H int = er 1 2 ) (Ḃ (t) RP G e (r P cα) B (t) = Q [1] E [0] m [1] B [0] What about the interaction with the electric field gradient?

34 Dynamic magnetic dipole: origin dependence 4 To flush out things correctly we go to multipolar gauge. The gauge function reads χ (r, t) = r P A [0] r P ;mr P A [1] m and leads to the transformed potentials φ (r, t) = χ t = r P Ȧ[0] r P ;mr P Ȧ[1] m ; A (r, t) = A (r, t) χ = 1 2 (r P B (t)) The interaction Hamiltonian now reads H int = eφ + ec (α A) = er P Ȧ[0] 1 2 er P ;mr P Ȧ[1] m e (r P cα) B (t) = Q [1] E [0] Q [1] m E [1] m m [1] B [0]

35 Confused? You won t be... First version: φ (r, t) = 0; A i (r, t) = 1 2 (B (t) r G) i Second version (change of gauge origin): χ (r, t) = r P A [0] (t) φ (r, t) = r P Ȧ[0] ; A i (r, t) = r P ;m A [1] i;m = 1 2 (B (t) r P ) i Third version (multipolar gauge): χ (r, t) = r P A [0] r P ;mr P A [1] m φ (r, t) = r P Ȧ[0] r P ;mr P Ȧ[1] m ; A i (r, t) = 1 2 r P ;m ( ) A [1] i;m A[1] m;i The important message is that a change of origin of an electric or magnetic multipole constitute a gauge transformation. In the time-dependent case electric and magnetic fields couple and origin independence can only be assured for the total interaction.

36 Conclusion Magnetism is a relativistic effect The conventional non-relativistic framework combines a non-relativistic description of particles with a relativistic description of their coupling to electromagnetic fields The relativistic perspective is thus interesting not only interesting because it introduces relativistic corrections to such a description, but also because it is a more consistent framework. The separation of para- and diamagnetic contributions is possible in the framework of relativistic perturbation theory. Due to the coupling of magnetic and electric fields in the time-dependent case, the origin invariance of dynamic magnetic multipoles can not be investigated separately from that of electric ones. Acknowledgement: Radovan Bast