Non perturbative properties of molecules in strong magnetic fields Alessandro Soncini INPAC Institute for NanoscalePhysics and Chemistry, University of Leuven, elgium Lb ti Nti ld Ch M éti I t Laboratoire National des Champs Magnétiques Intenses Toulouse, 14 September 010
Origin dependence of magnetic propertiesp 1) In principle magnetic properties calculated with variational wavefunctions are origin independent. ) The use of finite i basis sets results in magnetic properties that depend on where the origin is chosen. STO 3G origin on He nucleus STO 3G origin at 1 a 0 from He
aug cc pvdz origin at 1a from He aug cc pvtz origin at 1a from He 0 0 aug cc pvqz origin at 1a 0 from He STO 3G/CTOCD PZ shifted origin
London orbitals 1) Gaussian basis functions augmented by a field dependent plane wave (basis functions become complex): r, H k k k k r C x y z e e l m n i HCr / k C C C C )Inprinciple, thischoice leads to complex molecular integrals 3) In response theory new complex integrals are avoided d by tki taking derivatives with respect to H, then letting H 0. 4) If H can be arbitrarily strong, complex integrals must be computed. No quantum chemistry program appears to be available for this task!
The program LONDON We developed our own ab initio code in Oslo (implementation by E. Tellgren and A. Soncini) : Molecular integrals over the field dependent hybrid basis within a generalised McMurchie Davidson scheme see: E. Tellgren, A. Soncini and T. Helgaker, JCP 19, 154114 (008) E. Tellgren, T. Helgaker and A. Soncini, PCCP 11, 5489 (009) Hartree Fock wavefunction/energy in strong fields Non linear magnetic properties from numerical differentiation Excitation frequencies (RPA, diagonalization of electronic Hessian) Induced current density in strong fields More recent advancements: DFT code for energy in strong fields Time dependent d Hartree Fock k(dynamic density matrix iterative i formalism) in strong magnetic fields (magneto optical response)
What happens to diamagnetism and paramagnetism of closed shell molecules in strong fields? The energy of a closed shell hllmolecule l in a magnetic field z : 1 1 4 W ( z ) W(0) zz z X zzzz z! 4! Diamagnetic molecules (repelled by a magnet): tr, W( z ) increases with increasing z, Most closed shell molecules are diamagnetic! Paramagnetic molecules (attracted by a magnet): tr > 0, W( z ) decreases with increasing z. Known examples are: MnO 4, H, CH we ll see some more.
Ab initio energy of closed shell molecules in strong fields: diamagnetic i systems The benzene molecule in a perpendicular magnetic field (ringcurrent enhanced diamagnetism): W au enzene HFcc-pVDZ London W au enzene HFcc-pVDZ London -30.575-30.575-30.6-30.6-30.65-30.65-30.65-30.65-30.675-30.675 0.0 0.04 0.06 0.08 0.1 0.1 au 0.0 0.04 0.06 0.08 0.1 0.1 au -30.75-30.75 1 W W 4 W W X 0 Z 0 1 1 z 4 X zzzz (aug cc pcvqz) = 97.9 X zzzz (6 31G/London) = 64.3, X zzzz (6 31G) = 5866 zzzz
Ab initio energy of closed shell molecules in strong fields: paramagnetic systems H, is a closed shell paramagnetic system ( ) W-W 0 au H 0.05 0.1 0.15 0. 0.5 0.3 au M z -0.01-0.0 STO-3G, c = 0.4 DZ, c = 0. aug-dz, c = 03 0.3 0. 0.1-0.03 003 0.05 0.1 0.15 0. 0.5 0.3 au -0.04-0.1 The paramagnetic system turns diamagnetic!
Analytical MO model for the diamagnetic transition Closed shell paramagnetic systems: rotationally allowed HOMO LUMO transition ii H y L y H 5 p y 4 p x H 3 L y 4 0 H 3 H i On the HOMO/LUMO basis the interaction Hamiltonian reads: H H i L z H 3 y 3 0 x 1 L 4 y 4 0 y H
1 1 W g W a a u W a u u g H L H L 4 0.05 0.04 0.03 0.0 0.01 0-0.01 0.03 0.0 001 0.01 0-0.01 m = 0.0,, m ÅÅÅÅÅÅÅÅÅ < c0 0 m=0., D 0.05 0.04 0.03 0.0 0.01-0.1-0.05 0 0.05 0.1 au m = 0.374, a u W a 0-0.01 m ÅÅÅÅÅÅÅÅÅ = D º c 0 m 0.6, 0.03 W a u u 0.0 001 0.01 0-0.01 m ÅÅÅÅÅÅÅÅÅ < c 0 D -0.1-0.05 0 0.05 0.1 au m ÅÅÅÅÅÅÅÅÅ 0 D > c 0-0.1-0.05 0 0.05 0.1 au -0.1-0.05 0 0.05 0.1 au
Polynomial fit of degree 4 (brown dashed line), degree 8 (red dashed), d) degree 10 (green solid line). Strongly non linear regime! W-WW 0 au 0.1 0. 0.3 0.4 au -0.01-0.0-0.03-0.04
Paramagnetic (antiaromatic) polycyclic dianions Acepleyadylene dianion C 16 H 10 Corannulene dianion C 0 H 10 E. Tellgren, T. Helgaker and A. Soncini, PCCP 11, 5489 (009)
Antiaromatic systems: [4n] annulenes D 8h - OPEN-SHELL D 4h - CLOSED-SHELL 4h rotational pair rotationally allowed HOMO LUMO transition rotational pair LUMO HOMO rotational pair HOMO Lz LUMO 0
Paramagnetic Diamagnetic transition: electron correlation effects with ihdft c (HF) PSE(HF) HF/cc pvdz c (DFT) > c (HF) PSE(DFT) > PSE (HF) 3LYP/cc pvdz c (LYP) PSE (LYP) LYP/cc pvdz
Non perturbative magneto optics: The Cotton Mouton effect revisited E n 7 E n OPTICALLY ISOTROPIC MEDIUM n = n 7 n m C
Microscopic origin of the Cotton Mouton effect: Molecular differential polarisability N n n n V Π differential polarisability : Π d u, e e e e exp d exp u, k T k T u 1 1, A. D. uckingham and J. A. Pople, Proc. Phys. Soc. LXIX, 1133 (1956)
uckingham Pople theory: quadratic dependence on N n n V n Π Π Π 0 Π 5 1,, 3 3 k T 1 Assumption: u() and () are quadratic in A. D. uckingham and J. A. Pople, Proc. Phys. Soc. LXIX, 1133 (1956)
What if u() and () are not quadratic in? Π d u, e e e e exp d exp u, k T k T u, Can be non perturbative (paramagnetic closed shell)??? (now we can compute it with LONDON) Π can now be integrated t numerically with LONDON O
He atom DATAPOINTS: TDHF/aug cc pvqz/london Lebedev grid 14 pts = 63 nm quadratic fit
COT molecule (closed shell shell paramagnetic) DATAPOINTS: TDHF/cc pvdz/london Lebedev grid 14 pts = 63 nm T = 300 K Degree 4 polynomial fit
Conclusions New ab initio/dft code LONDON for the accurate non perturbative calculation of molecular energies and response properties (TDHF) in strong magnetic fields. Strong magnetic field can induce a paramagnetic to diamagnetic to transition (at c ) in closed shell paramagnetic molecules. Effect of electron correlation, estimated via DFT, is to increase critical transition field c and the paramagnetic stabilisation energy (PSE). Non quadratic behaviour of the optical anisotropy induced by a strong magnetic field (Cotton Mouton effect) has been detected via the new code. More work needed to investigate the origin of higher order contributions.
II. Field induced i d dmagnetoelastic ti instabilities in antiferromagnetic wheels
Molecular wheel CsFe 8 Space group P 4/n Tetragonal site symmetry (C 4v ) for each CsFe 8 Each CsFe 8 is surrounded by 8 nn n.n. wheels R.W. Saalfrank et al., Angew. Chem. 36, 48 (1997) 3
Lowest spin levels in CsFe 8 H=-J S S D S gs i i1 i,z E (cm 1 ) S = 40 30 0-1 1 0 From experiment (T>1.5K): magnetic susceptibility high field torque O. Waldmann, et al., Inorg. Chem. 40, 966 (001) S = 1 S = 0 10-4 6 8 10 0 (T) Inelastic neutron scattering O. Waldmann, et al., Phys. Rev. Lett. 95, 0570 (005); Phys. Rev. 95, 05449 (006) 0 J 0.6K, D=-0. 56K 4
Magnetic torque of CsFe 8 (T < 1K) 3.3 Z 3.3 Z 93.6 Z O. Waldmann, et al., Phys. Rev. Lett. 96, 0706 (006) 5
Torque anomalies at LC in CsFe 8 (T < 1K) Singlet triplet level crossing Z 6
Suggestion: magnetoelastic instability Active vibrational mode on each CsFe 8 VˆV vib Spin Jahn Teller effect: 1 (, ) ( ) / b Q S KQ ( ) Q/ S 1( b, ) Q 0 Q 0 O. Waldmann, et al., Phys. Rev. Lett. 96, 0706 (006) 7
Hypothesis: cooperative spin Jahn Teller effect in CF CsFe 8 crystal tl Interaction between active vibrational modes of nearest neighbour CsFe 8 The Hamiltonian close to S/S+1 LC: ˆ ˆ n VQ Q H= H ĤH n n nm 1 KQ n M Qn n m 0 S iwqn iwqn 0 S 1 n 8 - the CsFe site 0 - the field at LC 8
Ĥˆ Lowest spin vibronic levels on each CsFe 8 Mean field Hamiltonian i for ˆ ˆ (0) ˆ (1) Qn Qn Qn, HHdyn H singlet/triplet LC: (0) 1 1 0 iwq dyn 0 K Q zv Q zv Q Q M Q iwq 0 1 0 0 1 (1) ĤH 0 Diagonalization of H ˆ : ˆ H (0) dyn (0) dyn V ( Q) 0 m Q 0 V ( Q) 1 1 V( Q) K QQ KQ ( ) 1 Q V Q W K 0 K / M 9
Results from mean field model Q - the order parameter Structural phase transition: Q 0 SS, 1 Q: T K magnetoelastic instability = structural ordered phase 06 0.6 0.5 Relation between critical parameters: 04 0.4 max zv kt E K z V max max c c 0 JT where 8, JT / z E W K 0.3 0. 0.1-0.75-0.5-0.5 0.5 0.5 0.75 0 W 1cm V 10cm K 100cm T 1 1 1 Good agreement with experiment bu t no cusp-like feat ure! 30
What if S = 0 and S = 1 are permanently mixed? ˆ n ˆ n 0 1 0 i H H Re W0 + ImW0 1 0 i 0 DO T 0, T max DO 0, 0 max Solutions for any T and, NO PHASE TRANSITION! 0 Kk T solve for Q : K z V Q W Im W W V Q tanh 0 W K W V Q K W K with 4 Im 4 Re 0 0 0 calculate free energy F Q, T, ENDDO ENDDO calculate and store the valueof the magnetictorqueas : F F 31
Numerical simulations of the torque experiment with permanent spin mixing F ( ) ( ) a ) b) T 1 T > T 1 c) d ) F ( ) ( ) e) f) 3 4 F ( ) ( ) 3 3 g) h) Q 3
Hˆ Permanent spin mixing interaction: cusp reproduced! n Hˆ Re W +Im n 0 0 1 1 0 0 i i 0 W0 W V K 1cm 10cm 100cm 1 1 1 Cusp-like feature appears for imaginarypermanent ent coupling No phase transition : SS, 1, Q 0 A. Soncini and L.F. Chibotaru, Phys. Rev. Lett. 99, 07704 (007) 33
Field dependent vibronic constants Field-independent vibronic constants: Field-dependent vibronic constants: W( ) W( ) W 0 1 0 In order to recover the experimental slope of d /d, the vibronic constants should be field-dependent 34
Permanent spin mixing mechanism: DM exchange H d,, 1 z i i sxksyk syks xk Due to C C : Antisymmetric Exchange H d s s s s 8 4 DM,, 1,, 1 i W 0 DM sin d 5 N 8, s, S 0 S 1 Ns 1 S 1 N S 3 3 d 001 0.01 J F. Cinti et al., Eur. Phys. J 30, 461 (00) -1 For / W 1.07d 0. cm 0 i 1, i d ii, 1 z dz d d d ii, 1 i1, i z z Dzyaloshinsky-Moriya mechanism: Is relevant for permanent spin mixing Is not relevant for vibronic coupling: DM - vibronic constants are too weak W / W Q 0.1 - no field dependence of vibronic constants is expected 0 35
Vibronic spin mixing mechanism: ZFS interaction Strong field dependence due to denominators: H ˆ ZFS H ˆ ZFS/ Q 0, M, 1 1 0.1 - estimation S MS M s W1 WM i 1-0 S 0 0 E0M 0 0.0606 - simulation S E M W S The M S mixing path: Zero-field splitting mechanism: Is relevant for both permanent and vibronic spin mixing Provides field-dependent coupling constants A. Soncini and L.F. Chibotaru PRL 99,07704(007) 36
Conclusions We proposed a microscopic i theory of spin mixing ii instabilities i in AFM wheels indicating three basic ingredients: 1) spin mixing ii vibroniccoupling i with ihfield dependent d d vibronicconstants i ) intermolecular elastic interaction 3) vibration independent spin mixing interactions The vibronic coupling mainly originates from second order ZFS mechanisms. The permanent coupling is due to Dzyaloshinsky Moriya i exchange, explaining the cusp feature in the experimental phase diagram. The observed effects cannot be explained in terms of a single molecule l l theory, cooperative effects must be accounted for. 37
Thanks to. For funding: CTCC Centre for Theoretical and Computational Chemistry(Oslo) INPAC Institute for Nanoscale Physics and Chemistry (Leuven) Dr. E. Tellgren and Prof. T. Helgaker (CTCC University of Oslo) Prof. L. Chibotaru (Katholieke Universiteit Leuven) You for your attention!