Molecules in strong magnetic fields

Molecules in strong magnetic fields Trygve Helgaker, Kai Lange, Alessandro Soncini, and Erik Tellgren Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway 14th European Symposium on Gas-Phase Electron Diffraction M. V. Lomonosov Moscow State University Moscow, June 24 28 2011 Trygve Helgaker, Kai Lange, Alessandro Soncini, and Erik Tellgren Molecules (CTCC, in strong University magnetic of Oslo) fields 14th European Symposium on GED 1 / 22

Outline 1 Molecular magnetism 2 The LONDON program 3 Molecular diamagnetism and paramagnetism 4 The helium atom 5 The H 2 and He 2 molecules 6 Excitations in strong fields 7 Conclusions Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 2 / 22

Molecules in weak magnetic fields Molecular magnetism is usually studied perturbatively E(B) = E(0) α mαbα 1 2 χ αβb αβ αb β + such an approach is highly successful and widely used in quantum chemistry molecular magnetic properties are accurately described by perturbation theory Example: 200 MHz NMR spectra of vinyllithium experiment RHF 0 100 200 0 100 200 MCSCF B3LYP 0 100 200 0 100 200 Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 3 / 22

Molecules in strong magnetic fields However, the field dependence of the energy can be complicated the energy of C 20 (ring conformation) plotted against B (atomic units) 756.705 756.710 756.685 756.690 756.695 756.700 such behaviour cannot be described or understood perturbatively We have undertaken a nonperturbative study of molecular magnetism gives new insight into molecular electronic structure describes atoms and molecules observed in astrophysics (stellar atmospheres) provides a framework for studying the current dependence of the universal density functional enables evaluation of many properties by finite-difference techniques Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 4 / 22

The electronic Hamiltonian The non-relativistic electronic Hamiltonian (atomic units) in a magnetic field B is given by H = T + V Coulomb + 1 2 BLz + Bsz + 1 8 i (x2 i + yi 2 )B2 T is the kinetic-energy operator and V Coulomb = V en + V ee + V nn the Coulomb operator L and s are the orbital and spin angular momentum operators one atomic unit of B corresponds to 2.2 10 5 T = 2.2 10 9 G Coulomb regime: B 0 a.u. earth-like conditions: Coulomb interactions dominate magnetic interactions are treated perturbatively earth magnetism 10 10, loudspeaker 10 5, NMR 10 4 ; pulsed laboratory field 10 3 a.u. Landau regime: B 1 a.u. astrophysical conditions: magnetic interactions dominate Coulomb interactions are treated perturbatively harmonic-oscillator Hamiltonian with force constant B 2 /4: Landau levels the electrons are confined in the transverse directions by the magnetic field neutron stars 10 3 10 4 a.u.; magnetars 10 5 a.u. Intermediate regime: B 1 a.u. the Coulomb and magnetic interactions are equally important complicated behaviour resulting from interplay of linear and quadratic terms white dwarves: up to about 1 a.u. We here consider the weak and intermediate regimes (B 15 a.u.) Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 5 / 22

The LONDON program The electronic Hamiltonian in a magnetic field H = T + V Coulomb + 1 2 B L + B s + 1 8 contains the orbital-angular momentum operator i (B r i ) (B r i ) L = i i (r i O) i (1) this operator is imaginary and depends on an arbitrary gauge origin O our results should be invariant to the choice of O Standard quantum-chemistry program cannot treat molecules in finite magnetic fields we must optimize a complex wave function we must use London atomic orbitals to ensure gauge-origin invariance ω lm (r K, B) = exp [ 1 2 ib (O K) r] χ lm (r K ) We have developed the LONDON code for this purpose complex wave functions with London atomic orbitals Hartree Fock (RHF, UHF, GHF) and Kohn Sham theories FCI and MCSCF theories first-order properties including molecular gradients for geometry optimizations linear response theory The code has been written by Erik Tellgren, Kai Lange, and Alessandro Soncini mostly C++, some Fortran 77 modular but not highly optimized yet C 20 is a large system Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 6 / 22

Diamagnetism and paramagnetism When a magnetic field is applied to a molecule, one of two things can happen: the energy is lowered: molecular paramagnetism the energy is raised: molecular diamagnetism Open-shell molecules are paramagnetic the molecule reorients the permanent magnetic moment and lowers the energy Closed-shell molecules may be diamagnetic or paramagnetic the induced magnetic moment may oppose or enhance the external field 0.02 most molecules are diamagnetic, opposing the external field by Lenz law in some systems, orbital unquenching results in paramagnetism RHF calculations of the field-dependence for three closed-shell systems: a) b) 14 x 10 3 a) 0.1 0.08 0.06 0.04 c) 0 0.1 0.05 0 0.05 0.1 0 b) d) 14 12 10 8 6 4 2 0 0.1 x 10 3 0.1 12 0.002 0.004 0.02 0.08 10 0.006 0.01 0.06 8 0.008 0.01 0 6 0.012 0.01 0.04 0.02 4 2 0.014 0.016 0.018 0.02 0.03 0 0.1 0.05 0 0.05 0.1 0 0.1 0.05 0 0.05 0.1 0.02 0.1 0.05 0 0.05 0.1 0 0.0 ac) benzene (aug-cc-pvdz): typical d) diamagnetic quadratic dependence on an out-of-plane field 0 b cyclobutadiene (aug-cc-pvdz): diamagnetic non-quadratic dependence on an out-of-plane field 0.002 0.02 c BH (aug-cc-pvtz): paramagnetic dependence on a perpendicular field 0.004 0.01 0.006 Helgaker et al. 0.008 (CTCC, University of Oslo) Molecules 0 in strong magnetic fields 14th European Symposium on GED 7 / 22

0.015!0.003 0.01!0.004 Stabilizing field strength for paramagnetic molecules 0.005 a) 0.02 0.04 0.06 0.08 0.1 0.12 However, plots over a larger range reveal that all closed-shell systems become diamagnetic in sufficiently strong fields: c) d) W!W0!au" BH W!W0!au" C8H8: total energy W!W0!au" C12H12: total energy B!au" 0.002 0.05 0.1 0.15 0.2 0.25 0.3 B!au"!0.005!0.01!0.02 STO!3G, Bc " 0.24 DZ, Bc " 0.22 aug!dz, Bc " 0.23 0.004 0.002 STO!3G, Bc " 0.035 6!31G, Bc " 0.034 cc!pvdz, Bc " 0.032 0.0015 0.001 0.0005 STO!3G, Bc " 0.018 6!31G, Bc " 0.018 cc!pvdz, Bc " 0.016!0.03 0.01 0.02 0.03 0.04 0.05 0.06 0.005 0.01 0.015 0.02 0.025 0.03 B!au"!0.0005!0.04!0.002!0.001 W!W0!au" CH b) # B!au" 0.1 0.2 0.3 0.4 0.5 0.6!0.02 STO3!G, Bc " 0.43!0.04 DZ, Bc " 0.44!0.06 aug!dz, Bc " 0.45!0.08!0.1!0.12 W!W0!au" MnO4! c) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 B!au" The transition occurs at a characteristic stabilizing critical field strength B c B c 0.22 perpendicular to principal axis for BH B c 0.032 along the principal axis for antiaromatic octatetraene C 8H 8 B c 0.016 along the principal axis for antiaromatic [12]-annulene C 12H 12 The stabilizing field strength decreases with increasing molecular size strongest laboratory fields attainable: 10 3 a.u. B c is inversely proportional to the area of the molecule normal to the field we estimate that B c should be observable for C 72H 72 We may in principle separate such molecules by applying a field gradient!0.1 STO!3G, Bc " 0.45 Helgaker et al. (CTCC, UniversityWachters, of Oslo) Bc " 0.50 Molecules in strong magnetic fields 14th European Symposium on GED 8 / 22

Analytical model for the diamagnetic transition The diamagnetic transition arises from an interplay between linear and quadratic terms it can be understood from a simple two-level model Molecular orbitals relevant for BH: Ground and excited states: 1s B, 2σ BH, 2p x, 2p y, 2p z (molecule along z axis) 0 = 1sB 2 2σ2 BH 2p2 z, 1 x = 1sB 2 2σ2 BH 2pz 2px, 1y = 1s2 B 2σ2 BH2pz 2py Let us apply a perpendicular magnetic field in the y direction: H B = 1 2 BLy + 1 8 B2 (x 2 + z 2 ) This leads to the following Hamiltonian matrix where 1 = 1 x : ( ) ( 0 H 0 0 H 1 E0 1 H(B) = = 2 χ 0B 2 ) iµb 1 H 0 1 H 1 iµb E 1 1 2 χ 1B 2 where we have introduced Interpretation: χ 0 = 1 4 0 x2 + z 2 0 < 0, χ 1 = 1 4 1 x2 + z 2 1 < 0, µ = 1 i 0 Ly 1 2 the diabatic states 0 and 1 x are uncoupled at B = 0 and diamagnetic in a magnetic field, they become coupled by L y the resulting adiabatic ground state may be either diamagnitic or paramagnetic Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 9 / 22

Energy levels of the two-level model The excited state assumed E = 0.02 above the ground-state at B = 0 magnetizabilities of the diabatic states: χ 0 = 28 (red) and χ 1 = 16 (yellow) Plots for different values of the coupling-matrix element µ: uncoupled (0), diamagnetic (0.2), nonmagnetic (0.374), and paramagnetic (0.6) 0.09 0.09 0.06 0.06 0.03 0.03 0.1 0.09 0.09 0.06 0.06 0.03 0.03 0.1 0.1 the magnetizability of the adiabatic ground state: χ GS = χ 0 + µ 2 / E = 7, 5, 0, 11 Paramagnetism results from avoided crossings at finite magnetic fields Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 10 / 22

Induced rotation of the electrons The magnetic field induces a rotation of the electrons about the field direction: the amount of rotation is the expectation value of the kinetic angular-momentum operator 0 Λ 0 = 2E (B) = 2M, Λ = r π, π = p + A positive rotation increases the energy; negative rotation reduces the energy Diamagnetic closed-shell molecules: 0 Λ 0 aligns with the field according to Lenz law, increasing the energy 1 2 0 Λ 0 B Paramagnetic closed-shell molecules: 0 Λ 0 first aligns against the field, decreasing the energy 0 Λ 0 goes through a minimum (maximum rotation) at the inflection point E (B) = 0 0 Λ 0 then increases again until rotation vanishes at B c 0 Λ 0 finally aligns with the field, making the system diamagnetic E(B x ) Energy!25.11!25.12!25.13!25.14!25.15!25.16 0 0.2 0.4 L x (B x ) 1 0.5 0 Angular momentum!0.5 0 0.2 0.4 L x / r! C nuc 0.4 0.2 0!0.2 Nuclear shielding integral Boron Hydrogen 0 0.2 0.4 there is no net rotation at the stationary points B = 0 and B = B c Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 11 / 22

C 20 : more structure 756.685 756.690 756.695 756.700 756.705 756.710 Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 12 / 22

10 8 6 4 2 0 2 Helium atom, RHF/augccpVTZ 0 1 2 3 4 5 6 7 8 9 10 Field, B [au] The helium atom The helium energy behaves in simple manner in magnetic fields (left) an initial quadratic diamagnetic behaviour is followed by a linear increase in B this is expected from the Landau levels (harmonic potential increases as B 2 ) The orbital energies behave in a more complicated manner (middle) the initial behaviour is determined by the angular momentum beyond B 1, all energies increase with increasing field HOMO LUMO gap increases, suggesting a decreasing importance of electron correlation The atom becomes squeezed in the magnetic field (right) this is particularly true for the transverse directions the cyclotron radius is proportional to B 1/2 special basis sets are need for strong fields (Landau orbitals) 3 Helium atom, RHF/augccpVTZ 0.8 Helium atom, RHF/augccpVTZ 2 0.7 Energy, E [Hartree] 1 0 1 Orbital energy, [Hartree] Quadrupole moment [bohr 2 ] 0.6 0.5 0.4 0.3 Qxx Qyy Qzz 2 0.2 3 0 1 2 3 4 5 6 7 8 9 Field, B [au] 0.1 0 1 2 3 4 5 6 7 8 9 Field, B [au] Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 13 / 22

The helium natural occupation numbers The natural occupation numbers are eigenvalues of the one-electron density matrix The FCI occupation numbers approach 2 and 0 strong fields decreasing importance of dynamical correlation in magnetic fields the two electrons rotate in the same direction about the field direction Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 14 / 22

The H 2 molecule Magnetic fields strongly affect chemical bonding and potetial energy surfaces Potential energy curves of H 2 in a parallel field B = 0, 5, 10 a.u. FCI/aug-cc-pVDZ calculations with energy set to zero at full dissociation H 2/aug-cc-pVDZ, lowest singlet, field parallel to bond 0.5 H 2/aug-cc-pVDZ, lowest triplet, field parallel to bond 0.0 0.5 0.0 1.0 0.5 E (Ha) 1.5 E (Ha) 1.0 2.0 2.5 B = 0.0 a.u. B = 5.0 a.u. B = 10.0 a.u. 1 2 3 4 5 6 7 8 R (bohr) 1.5 2.0 B = 0.0 a.u. B = 5.0 a.u. B = 10.0 a.u. 1 2 3 4 5 6 7 8 R (bohr) H 2 becomes strongly bound in strong fields the basis set is too small to give quantitative results The low-spin triplet component (ββ) becomes the ground state at about 0.25 a.u. the Zeeman interaction contributes BM s to the energy Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 15 / 22

Electron rotation and correlation in H 2 The energy of H 2 in the singlet state increases in a parallel field (left) the field induces a rotation of the electrons 0 Λ z 0 about the molecular axis (right) The occupation numbers indicate reduced importance of dynamical correlation the two electrons rotate in the same direction, reducing the chance of near encounters 2.00 H 2/cc-pVTZ, optimized geometry, field parallel to bond Cummulative occupation number 1.98 1.96 1.94 1.92 1.90 0 1 2 3 4 B (a.u.) Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 16 / 22

The covalently bound He 2 molecule We have also performed FCI/aug-cc-pVDZ calculations on the helium dimer energy set to zero for full dissociation in all cases We have considered the lowest singlet, triplet and quintet states in the absence of a field, only the triplet state is covalently bound in the presence of a field, all states become strongly bound He 2/aug-cc-pVDZ, lowest singlet, field parallel to bond He 2/aug-cc-pVDZ, lowest triplet, field parallel to bond He 2/aug-cc-pVDZ, lowest quintet, field parallel to bond 0.0 0.3 0.0 0.2 0.2 0.1 0.0 E (Ha) 0.5 E (Ha) 0.4 E (Ha) 0.1 1.0 B = 0.0 a.u. B = 5.0 a.u. B = 10.0 a.u. B = 15.0 a.u. 1 2 3 4 5 6 7 8 R (bohr) 0.6 0.8 B = 0.0 a.u. B = 5.0 a.u. B = 10.0 a.u. B = 15.0 a.u. 1 2 3 4 5 6 7 8 R (bohr) 0.2 0.3 0.4 0.5 B = 0.0 a.u. B = 5.0 a.u. B = 10.0 a.u. B = 15.0 a.u. 1 2 3 4 5 6 7 8 R (bohr) In magnetic fields, spin multiplets are split within each multiplet, the low-spin component has lowest energy Increasing magnetic fields favor low-spin components of high-spin states in weak fields, the ground state is a singlet in strong fields, the ground state is the low-spin quintet Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 17 / 22

Distortion of the H 2 O molecule H 2 O, HF/6 31G** H 2 O, HF/6 31G** 94.3 94.25 oop in plane, C2 axis in plane, C2 axis 107 106.8 oop in plane, C2 axis in plane, C2 axis 94.2 106.6 d(o H) [pm] 94.15 94.1 94.05 H O H angle [degrees] 106.4 106.2 106 105.8 94 105.6 93.95 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 B [au] 105.4 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 B [au] The molecule shrinks OH bonds contract by 0.25 pm in a field of 0.16 a.u. The molecule becomes more linear except for fields along the symmetry axis Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 18 / 22

Distortion of the NH 3 molecule Magnetic field applied along the symmetry axis 100.15 NH 3, HF/6 31G** 108.3 NH 3, HF/6 31G** d(n H) [pm] 100.1 100.05 100 99.95 99.9 99.85 99.8 99.75 99.7 H N H angle [degrees] 108.2 108.1 108 107.9 107.8 107.7 107.6 99.65 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 B [au] 107.5 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 B [au] As expected the molecule shrinks bonds contract by 0.3 pm in a field of 0.15 a.u. Less intuitively, the molecule becomes more planar could this be caused by the shrinking lone pair? Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 19 / 22

Distortion of the C 6 H 6 molecule At B = 0.16, the molecule is 6.1 pm narrower and 3.5 pm longer in the field direction in agreement with perturbational estimates by Caputo and Lazzeretti, IJQC 111, 772 (2011) 107.6 107.5 C 6 H 6, HF/6 31G* C H B C H other 107.4 107.3 d(c H) [pm] 107.2 107.1 107 106.9 106.8 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 B [au] 138.75 138.7 138.65 C 6 H 6, HF/6 31G* 120.5 120 C 6 H 6, HF/6 31G* C C C (a) C C H (b) d(c C) [pm] 138.6 138.55 138.5 138.45 138.4 138.35 C C other (c) C C B (d) bond angle [degrees] 119.5 119 118.5 118 117.5 138.3 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 B [au] 117 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 B [au]

Linear response theory We have implemented linear response theory (RPA) in finite magnetic fields lowest CH 3 radical states in magnetic fields (middle) transitions to the green state are electric-dipole allowed oscillator strength for transition to green CH 3 state length (red) and velocity (blue) gauges En [Ha] 39.25 39.30 39.35 39.40 39.45 39.50 39.55 39.60 39.65 0.00 0.15 0.30 0.45 B [a.u.] f E1 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.00 0.15 0.30 0.45 B [a.u.] Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields 14th European Symposium on GED 21 / 22

Conclusions We have developed the LONDON program complex wave functions London atomic orbitals HF and FCI theories molecular gradients and linear response theory The LONDON program may be used for finite-difference alternative to analytical derivatives studies of molecules in strong magnetic fields studies of exchange correlation functional in magnetic fields We have studied the behaviour of paramagnetic molecules in strong fields all paramagnetic molecules attain a global minimum at a characteristic field B c B c decreases with system size and should be observable for C 72H 72 We have studied H 2 and He 2 using FCI theory molecules become more strongly bound in magnetic fields the helium molecule becomes covalently bound We have studied molecular structure in magnetic fields bond distances are typically shortened in magnetic fields We have studied excitation energies in magnetic fields excitation energies typically increase in magnetic fields Helgaker et al. (CTCC, University of Oslo) Conclusions 14th European Symposium on GED 22 / 22