Molecular bonding in strong magnetic fields

Trygve Helgaker 1, Mark Hoffmann 1,2 Kai Molecules Lange in strong 1, Alessandro magnetic fields Soncini 1,3 AMAP,, andzurich, ErikJune Tellgren 1 4 212 1 (CTCC, 1 / 35 Uni Molecular bonding in strong magnetic fields Trygve Helgaker 1, Mark Hoffmann 1,2 Kai Lange 1, Alessandro Soncini 1,3, and Erik Tellgren 1 1 CTCC, Department of Chemistry, University of Oslo, Norway 2 Department of Chemistry, University of North Dakota, Grand Forks, USA 3 School of Chemistry, University of Melbourne, Australia Accurate Methods for Accurate Properties, in celebration of the 6th birthday of Peter Taylor, University Zurich Irchel, Zurich, Switzerland, June 4 6, 212

Peter Taylor By his Pool, San Jose, April 1988 Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 2 / 35

Peter Taylor In Celest Rohlfing s Pool, Livermore, April 1988 Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 3 / 35

Peter Taylor With Andy Komornicki, NASA Ames, April 1988 Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 4 / 35

Peter Taylor In his Pool, San Diego, October 1994 Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 5 / 35

Peter Taylor Peter with his Cray, San Diego Supercomputer Center, October 1994 Jan Almlo f in his Garden, Minneapolis, October 1994 Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 6 / 35

Peter Taylor Riding the Ravenglass & Eskdale Railway July 23 Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 7 / 35

Peter Taylor Riding the Ravenglass & Eskdale Railway July 23 Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 8 / 35

Peter Taylor Slovakia September 25 Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 9 / 35

Peter Taylor Giving a Dinner Speech, Sostrup Summer School July 26 Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 1 / 35

Peter Taylor With Susan Colwell and Julia Rice, Budapest June 27 Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 11 / 35

Introduction perturbative vs. nonperturbative studies Molecular magnetism is usually studied perturbatively such an approach is highly successful and widely used in quantum chemistry molecular magnetic properties are accurately described by perturbation theory example: 2 MHz NMR spectra of vinyllithium experiment RHF 1 2 1 2 MCSCF B3LYP 1 2 1 2 We have undertaken a nonperturbative study of molecular magnetism gives 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 AMAP, Zurich, June 1 4 212 12 / 35

Introduction the electronic Hamiltonian The non-relativistic electronic Hamiltonian (a.u.) in a magnetic field B along the z axis: H = H + 1 2 BLz + Bsz + 1 8 B2 (x 2 + y 2 ) linear and quadratic B terms H is the field-free operator and L and s the orbital and spin angular momentum operators one atomic unit of B corresponds to 2.35 1 5 T = 2.35 1 9 G Coulomb regime: B 1 a.u. earth-like conditions: Coulomb interactions dominate magnetic interactions are treated perturbatively earth magnetism 1 1, NMR 1 4 ; pulsed laboratory field 1 3 a.u. Intermediate regime: B 1 a.u. white dwarves: up to about 1 a.u. the Coulomb and magnetic interactions are equally important complicated behaviour resulting from an interplay of linear and quadratic terms Landau regime: B 1 a.u. neutron stars: 1 3 1 4 a.u. magnetic interactions dominate (Landau levels) Coulomb interactions are treated perturbatively relativity becomes important for B α 2 1 4 a.u. We here consider the weak and intermediate regimes (B < 1 a.u.) For a review, see D. Lai, Rev. Mod. Phys. 73, 629 (21) Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 13 / 35

Introduction London orbitals and size extensivity The non-relativistic electronic Hamiltonian in a magnetic field: H = H + 1 2 BLz + Bsz + 1 8 B2 (x 2 + y 2 ) linear and quadratic B terms The orbital-angular momentum operator is imaginary and gauge-origin dependent L = i (r O), A O = 1 B (r O) 2 we must optimize a complex wave function and also ensure gauge-origin invariance Gauge-origin invariance is imposed by using London atomic orbitals (LAOs) ω lm (r K, B) = exp [ 1 2 ib (O K) r] χ lm (r K ) special integral code needed London orbitals are necessary to ensure size extensivity and correct dissociation H 2 dissociation FCI/un-aug-cc-pVTZ B 2.5 diamagnetic system full lines: with LAOs dashed lines: AOs with mid-bond gauge origin 1..5.5 1. B 2.5 2 4 6 8 1 B 1. B. Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 14 / 35

Introduction the London program and overview We have developed the London code for calculations in finite magnetic fields complex wave functions and London atomic orbitals Hartree Fock theory (RHF, UHF, GHF), FCI theory, and Kohn Sham theory energy, gradients, excitation energies London atomic orbitals require a generalized integral code 1 F n(z) = exp( zt 2 )t 2n dt complex argument z C++ code written by Erik Tellgren, Alessandro Soncini, and Kai Lange C 2 is a large system Overview: molecular dia- and paramagnetism helium atom in strong fields: atomic distortion and electron correlation H 2 and He 2 in strong magnetic fields: bonding, structure and orientation molecular structure in strong magnetic fields Previous work in this area: much work has been done on small atoms FCI on two-electron molecules H 2 and H (Cederbaum) Nakatsuji s free complement method no general molecular code with London orbitals Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 15 / 35

Closed-shell paramagnetic molecules molecular diamagnetism and paramagnetism.1 The Hamiltonian has paramagnetic and diamagnetic parts: a) b) x 1 3 14 12 H = H + 1 2 BLz + Bsz + 1 8 B2 (x 2.8 + y 2 ) linear and quadratic B terms.6 Most closed-shell molecules are diamagnetic their energy increases in an applied magnetic field.4 induced currents oppose the field according to Lenz s law Some closed-shell systems are paramagnetic their energy decreases in a magnetic field relaxation of the wave function lowers the energy RHF calculations of the field dependence of the energy for two closed-shell systems: a).2.1.5.5.1 c) b) x 1 3 14 d) 1 8 6 4 2.1.5.1.8.6.4.2.1.5.5.1.2 12.4.61.8 8.1.12 6.14 4.16.18 2.2.1.1.5.5.5.1.1.2.1.1.2.3.5.1.15 left: benzene: diamagnetic dependence on an out-of-plane field, χ < right: BH: c) paramagnetic dependence on a perpendicular d) field, χ > Helgaker et al. (CTCC, University.2 of Oslo) Molecules in strong magnetic.2 fields AMAP, Zurich, June 1 4 212 16 / 35

Closed-shell paramagnetic molecules diamagnetic transition at stabilizing field strength B c However, all systems become diamagnetic in sufficiently strong fields: 756.685 756.69 756.695 756.7 756.75 756.71 The transition occurs at a characteristic stabilizing critical field strength B c B c.1 for C 2 (ring conformation) above 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 Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 17 / 35

Closed-shell paramagnetic molecules paramagnetism and double minimum explained Ground and (singlet) excited states of BH along the z axis zz = 1sB 2 2σ2 BH 2p2 z, zx = 1sB 2 2σ2 BH 2pz 2px, zy = 1s2 B 2σ2 BH2pz 2py All expectation values increase quadratically in a perpendicular field in the y direction: n H + 1 2 BLy + 1 8 B2 (x 2 + z 2 ) n = En + 1 8 n x 2 + z 2 n B 2 = E n 1 2 χnb2 The zz ground state is coupled to the low-lying zx excited state by this field: zz H + 1 2 BLy + 1 8 B2 (x 2 + z 2 ) xz = 1 zz Ly xz B 2.9.9.6.6.3.3.1.9.9.6.6.3.3.1.1 A paramagnetic ground-state with a double minimum is generated by strong coupling Tellgren et al., PCCP 11, 5489 (29) Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 18 / 35

Closed-shell paramagnetic molecules induced electron rotation 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 Λ = 2E (B), Λ = r π, π = p + A Paramagnetic closed-shell molecules (here BH): E(B x ) Energy Energy!25.11!25.11!25.12!25.12!25.13!25.13 E(B x )!25.14!25.14 L x (B x ) L x (B x )!25.15!25.15!25.16!25.16.2.2.4.4 Angular momentum 1 Angular momentum 1.5.5!.5!.5.2.4 L / x / r r! x! C C nuc nuc.4.4 Nuclear shielding Nuclear shielding in Boron Boron Hydrog Hydroge.2.2!.2.2.2 there is no rotation at the field-free energy maximum: B = the onset of paramagnetic rotation Orbital energies (against the field) reduces HOMO!LUMO the energy gapfor B > Singlet excitation en the strongest paramagnetic rotation occurs at the energy.5 inflexion point the rotation comes to.1 a halt at the stabilizing field strength: B = B c.4.48 the onset of diamagnetic rotation (with the field) increases the energy for B > B.35 c.46.3!.1 LUMO diamagnetic rotation always increases.44 HOMO the energy according to Lenz s law.25 Diamagnetic closed-shell molecules:!(b x ) Helgaker et al. (CTCC, University of Oslo)!.2 Molecules in strong magnetic.42 fields AMAP, Zurich, June.21 4 212 19 / 35! gap (B x ) "(B x )

The helium atom total energy and orbital energies The total energy illustrates the complicated evolution of electronic states in the field (left) the behaviour depends on orbital and spin angular momenta eventually, all energies increase diamagnetically The orbital energies behave in an equally complicated manner (right) 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 1 Helium atom, RHF/augccpVTZ 1.6 1.8 8 2. 6 <XYZ> 2.2 2.4 Orbital energy, [Hartree] 4 2.6 2 2.8 3...2.4.6.8 1. B [a.u] 2 1 2 3 4 5 6 7 8 9 1 Field, B [au] Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 2 / 35

The helium atom natural occupation numbers and electron correlation The FCI occupation numbers approach 2 and strong fields diminishing 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 AMAP, Zurich, June 1 4 212 21 / 35

The helium atom atomic size and atomic distortion Atoms become squeezed and distorted in magnetic fields Helium 1s 2 1 S (left) is prolate in all fields Helium 1s2p 3 P (right) is oblate in weak fields and prolate in strong fields transversal size proportional to 1/ B, longitudinal size proportional to 1/ log B Atomic distortion affects chemical bonding which orientation is favored? Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 22 / 35

The H 2 molecule potential-energy curves of the 1 Σ + g (1σ2 g ) and 3 Σ + u (1σg1σ u ) states (M S = ) FCI/un-aug-cc-pVTZ curves in parallel (full) and perpendicular (dashed) orientations The singlet (blue) and triplet (red) energies increase diamagnetically in all orientations 1..5 B. 1..5 B.75 1 2 3 4 1 2 3 4.5.5 1. 1. 1..5 B 1.5 1..5 B 2.25 1 2 3 4 1 2 3 4.5.5 1. 1. The singlet triplet separation is greatest in the parallel orientation (larger overlap) the singlet state favors a parallel orientation (full line) the triplet state favors a perpendicular orientation (dashed line) and becomes bound parallel orientation studied by Schmelcher et al., PRA 61, 43411 (2); 64, 2341 (21) Hartree Fock studies by Žaucer and and Ažman (1977) and by Kubo (27) Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 23 / 35

The H2 molecule lowest singlet and triplet potential-energy surfaces E (R, Θ) I Polar plots of the singlet (left) and triplet (right) energy E (R, Θ) at B = 1 a.u. 9 9 135 45 1 18 225 2 3 4 135 45 5 315 1 18 2 3 5 225 27 4 315 27 I Bond distance Re (pm), orientation Θe ( ), diss. energy De, and rot. barrier E (kj/mol) B. 1. Re 74 66 Helgaker et al. (CTCC, University of Oslo) θe singlet De 459 594 E 83 Re 136 Molecules in strong magnetic fields triplet θe De 9 12 E 12 AMAP, Zurich, June 1 4 212 24 / 35

The H 2 molecule a new bonding mechanism: perpendicular paramagnetic bonding Consider a minimal basis of London AOs, correct to first order in the field: 1σ g/u = N g/u (1s A ± 1s B ), 1s A = N s e i 1 2 B R A r e ar 2 A In the helium limit, the bonding and antibonding MOs transform into 1s and 2p AOs: lim R 1σg = 1s, lim R 1σu = { 2p, 2p x i B 4a 2p y 2p 1, all orientations parallel orientation perpendicular orientation The magnetic field modifies the MO energy level diagram perpendicular orientation: antibonding MO is stabilized, while bonding MO is destabilized parallel orientation: both MOs are unaffected relative to the AOs Σ 1Σg 1Σ u.5 1Σg 1. 1.5 1Σ u.5 1sA 1sB.1.15 1Σ u 1Σg Molecules of zero bond order may therefore be stabilized by the magnetic field Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 25 / 35

The H 2 molecule a new bonding mechanism: perpendicular paramagnetic bonding The triplet H 2 is bound by perpendicular stabilization of antibonding MO 1σu note: Hartree Fock theory gives a qualitatively correct description EêkJmol -1-55 H 2 EêkJmol -1-1 5-5525 HF -1 7-555 FCI -1 9 1 2 3 Rêpm 1 however, there are large contributions from dynamical correlation method B R e D e UHF/un-aug-cc-pVTZ 2.25 93.9 pm 28.8 kj/mol FCI/un-aug-cc-pVTZ 2.25 92.5 pm 38.4 kj/mol UHF theory underestimates the dissociation energy but overestimates the bond length basis-set superposition error of about 8 kj/mol Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 26 / 35

The H 2 molecule Zeeman splitting of the lowest triplet state The spin Zeeman interaction contributes BM s to the energy, splitting the triplet lowest singlet (blue) and triplet (red) energy of H 2: 1,1 B 2.25 2 1 1, B 2.25 1 2, 3 4 1 1,M, B. 2 1,1 B 2.25 The ββ triplet component becomes the ground state at B.4 a.u. eventually, all triplet components will be pushed up in energy diamagnetically... Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 27 / 35

The H 2 molecule evolution of lowest three triplet states We often observe a complicated evolution of electronic states a (weakly) bound 3 Σ + u (1σg1σ u ) ground state in intermediate fields a covalently bound 3 Π u(1σ g2π u) ground state in strong fields.5 2.5 2.25 1. 2. 1.75 E (Ha) 1.5 2. 1 2 3 4 5 6 7 8 R (bohr) 1.5 1.25 1..75.5.25. Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 28 / 35

The H 2 molecule electron rotation and correlation The field induces a rotation of the electrons Λ z about the molecular axis increased rotation increases kinetic energy, raising the energy concerted rotation reduces the chance of near encounters natural occupation numbers indicate reduced importance of dynamical correlation Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 29 / 35

The helium dimer the 1 Σ + g (1σg1σ 2 u 2 ) singlet state The field-free He 2 is bound by dispersion in the ground state our FCI/un-aug-cc-pVTZ calculations give D e =.8 kj/mol at R e = 33 pm In a magnetic field, He 2 shrinks and becomes more strongly bound perpendicular paramagnetic bonding (dashed lines) as for H 2 for B = 2.5, D e = 31 kj/mol at R e = 94 pm and Θ e = 9 4. 4.5 B 2. 5. B 1.5 5.5 B 1. B.5 B. Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 3 / 35

The helium dimer the 3 Σ + u (1σ2 g 1σ u 2σg) triplet state the covalently bound triplet state becomes further stabilized in a magnetic field D e = 178 kj/mol at R e = 14 pm at B = D e = 655 kj/mol at R e = 8 pm at B = 2.25 (parallel orientation) D e = 379 kj/mol at R e = 72 pm at B = 2.25 (perpendicular orientation) 4.8 5. B. 5.2 5.4 B.5 5.6 B 1. 5.8 B 2. The molecule begins a transition to diamagnetism at B 2 eventually, all molecules become diamagnetic T = 1 2 (σ π)2 = 1 2 (σ (p + A))2 = 1 2 ( σ (p + 1 2 B r)) 2 Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 31 / 35

The helium dimer the lowest quintet state In sufficiently strong fields, the ground state is a bound quintet state at B = 2.5, it has a perpendicular minimum of D e = 1 kj/mol at R e = 118 pm 3.5 4. B. 4.5 5. 5.5 6. 6.5 B.5 B 1. B 1.5 B 2.5 In strong fields, anisotropic Gaussians are needed for a compact description for B 1 without such basis sets, calculations become speculative Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 32 / 35

Molecular structure Hartree Fock calculations on helium clusters We have studied helium clusters in strong magnetic fields (here B = 2) RHF/u-aug-cc-pVTZ level of theory all structures are planar and consist of equilateral triangles suggestive of hexagonal 2D crystal lattice ( 3 He crystallizes into an hcp structure at about 1 MPa) He 3 and He 6 bound by 3.7 and 6.8 me h per atom MOLDEN MOLDEN vibrational frequencies in the range 2 2 cm 1 (for the 4 He isotope) MOLDEN 2.53 defaults used 2.49 2.48 first point 2.44 2.42 2.61 1.951 2.293 2.6 2.84 2.86 Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 33 / 35

Molecular structure Hartree Fock calculations on larger molecules Ammonia for B.6 a.u. at the RHF/cc-pVTZ level of theory 1.89 18.7 8.6 1.888 18.6 18.5 8.55 B mol.axis B mol.axis d NH [bohr] 1.886 1.884 1.882 H N H [degrees] 18.4 18.3 18.2 18.1 18 E inv [me h ] 8.5 8.45 1.88 1.878.2.4.6.8.1.12.14.16.18.2 B [au] 17.9 17.8 17.7.2.4.6.8.1.12.14.16.18.2 B [au] 8.4 8.35.1.2.3.4.5.6 B [au] bond length (left), bond angle (middle) and inversion barrier (right) ammonia shrinks and becomes more planar (from shrinking lone pair?) in the parallel orientation, the inversion barrier is reduced by.1 cm 1 at 1 T Benzene in a field of.16 along two CC bonds (RHF/6-31G**) it becomes 6.1 pm narrower and 3.5 pm longer in the field direction agrees with perturbational estimates by Caputo and Lazzeretti, IJQC 111, 772 (211) 138.75 C 6 H 6, HF/6 31G* 138.7 138.65 d(c C) [pm] 138.6 138.55 138.5 C C other (c) C C B (d) 138.45 138.4 138.35 138.3.2.4.6.8.1.12.14.16 B [au] Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields AMAP, Zurich, June 1 4 212 34 / 35

Conclusions Summary and outlook We have developed the LONDON program for molecules in magnetic field We have studied He, H 2 and He 2 in magnetic fields atoms and molecules shrink molecules are stabilized by magnetic fields preferred orientation in the field varies from system to system We have studied molecular structure in magnetic fields bond distances are typically shortened in magnetic fields We have studied closed-shell 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 An important goal is to study the universal density functional in magnetic fields ( ) F [ρ, j] = sup u,a E[u, A] ρ(r)u(r) dr j(r) A(r) dr Support: The Norwegian Research Council and the European Research Council Helgaker et al. (CTCC, University of Oslo) Conclusions AMAP, Zurich, June 1 4 212 35 / 35