Molecular electronic structure in strong magnetic fields

Trygve Helgaker 1, Mark Hoffmann 1,2 Kai Molecules Lange in strong 1, Alessandro magnetic fields Soncini 1,3, andcctcc Erik Jackson Tellgren 211 1 (CTCC, 1 / 23 Uni Molecular electronic structure 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 2th Conference on Current Trends in Computational Chemistry (CCTCC), Hilton Jackson Hotel and Conference Center, Jackson, Mississippi, USA, October 27 29, 211

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 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 CCTCC Jackson 211 2 / 23

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 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. the Coulomb and magnetic interactions are equally important complicated behaviour resulting from an interplay of linear and quadratic terms white dwarves: up to about 1 a.u. Landau regime: B 1 a.u. astrophysical conditions: magnetic interactions dominate Landau levels: harmonic-oscillator Hamiltonian with force constant B 2 /4 Coulomb interactions are treated perturbatively relativity becomes important for B α 2 1 4 a.u. neutron stars 1 3 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 CCTCC Jackson 211 3 / 23

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 ensured 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 Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson 211 4 / 23

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, Kai Lange, and Alessandro Soncini C 2 is a large system Overview: closed-shell paramagnetic molecules: transition to diamagnetism 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 conclusion Previous work in this area: much work has been done on small atoms FCI on two-electron molecules H 2 and H but without London orbitals Nakatsuji s free complement method no general molecular code with London orbitals Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson 211 5 / 23

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 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 CCTCC Jackson 211 6 / 23

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 CCTCC Jackson 211 7 / 23

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: H + 1 2 BLy + 1 8 B2 (x 2 + z 2 ) = E + 1 8 x 2 + z 2 B 2 = E 1 2 χ B 2 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 CCTCC Jackson 211 8 / 23

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 ) E(B x ) Energy!25.11!25.12!25.13!25.14!25.15 L (B ) L x (B x ) x x Angular momentum 1 1.5.5!25.16!.5!.5.2.2.4.4.2.2.4.4 no rotation at B = paramagnetic rotation against Orbital the energies field reduces the energy HOMO!LUMO for < B < gapb c maximum paramagnetic rotation at the inflection point.5 E (B) = no rotation and lowest.1 energy at B = B c.48 diamagnetic rotation with the field increases the energy for B > B c.46 Diamagnetic closed-shell molecules:!.1 LUMO.44 diamagnetic rotation always increases HOMO the energy according to Lenz s law!(b x ) L x / r! C nuc L x / r! C nuc.4.4.2.2 Nuclear shielding inte Boron Hydrogen!.2!.2.2.2 Singlet excitation en.4.35.3.25!.2.42 Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC.2 Jackson 211 9 / 23! gap (B x ) "(B x )

The helium atom total energy and orbital energies 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 (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 3 Helium atom, RHF/augccpVTZ 1 Helium atom, RHF/augccpVTZ 2 8 Energy, E [Hartree] 1 1 Orbital energy, [Hartree] 6 4 2 2 3 1 2 3 4 5 6 7 8 9 Field, B [au] 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 CCTCC Jackson 211 1 / 23

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 CCTCC Jackson 211 11 / 23

The helium atom atomic size and atomic distortion Atoms become squeezed and distorted in magnetic fields Helium 1s 2 1 S (left) becomes squeezed and prolate 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 will be favored? Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson 211 12 / 23

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 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 energy increases diamagnetically for both states in all orientations The singlet triplet separation is greatest in the parallel orientation (stronger overlap) the singlet state favors a parallel orientation (red full line) the triplet state favors a perpendicular orientation (blue dashed line) and becomes bound parallel orientation studied by Schmelcher et al., PRA 61, 43411 (2); 64, 2341 (21) Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson 211 13 / 23

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 D, and rot. barrier E (kj/mol) B. 1. Re 74 66 Helgaker et al. (CTCC, University of Oslo) θe singlet D 459 594 E 83 Re 136 Molecules in strong magnetic fields triplet θe D 9 12 E 12 CCTCC Jackson 211 14 / 23

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 (red) and triplet (blue) energy of H 2: 1. 1..5 B..5 B.75.5 1. 1.5 2. 2.5 3. 3.5.5 1. 1.5 2. 2.5 3. 3.5.5.5 1. 1. 1.5 1.5 2. 2. 1. 1..5 B 1.5.5 B 2.25.5 1. 1.5 2. 2.5 3. 3.5.5 1. 1.5 2. 2.5 3. 3.5.5.5 1. 1. 1.5 1.5 2. 2. The ββ triplet component becomes the ground state at B.25 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 CCTCC Jackson 211 15 / 23

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 CCTCC Jackson 211 16 / 23

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 CCTCC Jackson 211 17 / 23

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 =.8 kj/mol at R e = 33 pm (too short) In a magnetic field, He 2 becomes smaller and more strongly bound this effect is particularly strong in the perpendicular orientation (dashed lines) for B = 2.5, D = 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 CCTCC Jackson 211 18 / 23

The helium dimer the 3 Σ + u (1σ2 g 1σ u 2σg) triplet state He 2 in the covalently bound triplet state becomes further stabilized in a magnetic field D = 178 kj/mol at R e = 14 pm at B = D = 655 kj/mol at R e = 8 pm at B = 2.25 (parallel orientation) D = 379 kj/mol at R e = 72 pm at B = 2.25 (perpendicular orientation) The molecule begins a transition to diamagnetism at B 2 4.8 5. B. 5.2 5.4 B.5 5.6 B 1. 5.8 B 2.5 In strong magnetic fields, He 2 has a quintet ground state Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson 211 19 / 23

The helium dimer the lowest singlet, triplet and quintet states We have considered the lowest singlet, triplet and quintet states in the absence of a field, only the triplet state is covalently bound however, in sufficiently strong fields, the ground state is a bound quintet state at B = 2.5, it has a perpendicular minimum of D = 1 kj/mol at R e = 118 pm Below, we have plotted singlet (left), triplet (middle) and quintet (right) states bound states contract with shrinking size of the atoms and become more strongly bound non-covalently bound molecules energy minimum in perpendicular orientation 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 CCTCC Jackson 211 2 / 23

Molecular structure FCI calculations on the H + 3 ion We are investigating H + 3 in a magnetic field Warke and Dutta, PRA 16, 1747 (1977) equilateral triangle for B < 1; linear chain for B > 1 ground state singlet for B <.5; triplet for B >.5 the triplet state does not become bound until fields B > 1 Potential energy curves for lowest singlet (left) and triplet (right) states electronic energy as function of bond distances for equilateral triangles 2. 2.5 3 2.5 1.5 1..5 2.25 2. 1.75 1.5 2 1 2.25 2. 1.75 1.5 E (Ha). 1.25 E (Ha) 1.25.5 1. 1.5 1..75.5.25 1 2 1..75.5.25 2..5 1. 1.5 2. 2.5 R (bohr). 3.5 1. 1.5 2. 2.5 R (bohr). Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CCTCC Jackson 211 21 / 23

Molecular structure Hartree Fock calculations on larger molecules Ammonia in a field of.15 along the symmetry axis 1.15 NH 3, HF/6 31G** 18.3 NH 3, HF/6 31G** d(n H) [pm] 1.1 1.5 1 99.95 99.9 99.85 99.8 99.75 99.7 H N H angle [degrees] 18.2 18.1 18 17.9 17.8 17.7 17.6 99.65.2.4.6.8.1.12.14.16 B [au] it shrinks: bonds contract by.3 pm it becomes more planar from shrinking lone pair? 17.5.2.4.6.8.1.12.14.16 B [au] Benzene in a field of.16 along two CC bonds, 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 CCTCC Jackson 211 22 / 23

Conclusions Summary and outlook We have developed the LONDON program for molecules in magnetic field 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 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 not studied molecules in the Landau regime B 1 a.u. all molecules become linear with singly occupied orbitals of β spin 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 Postdoc position is available in my group Helgaker et al. (CTCC, University of Oslo) Conclusions CCTCC Jackson 211 23 / 23