Molecular Magnetism Molecules in an External Magnetic Field Trygve Helgaker Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, Norway Laboratoire de Chimie Théorique, Université Pierre et Marie Curie, Paris, France November 27, 212 Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 1 / 34
Molecular Magnetism Part 1: The electronic Hamiltonian Hamiltonian mechanics and quantization electromagnetic fields scalar and vector potentials electron spin Part 2: Molecules in an external magnetic field Hamiltonian in an external magnetic field gauge transformations and London orbitals magnetizabilities diamagnetism and paramagnetism induced currents molecules and molecular bonding in strong fields Part 3: NMR parameters Zeeman and hyperfine operators nuclear shielding constants indirect nuclear spin spin coupling constants Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 2 / 34
Hamiltonian in a uniform magnetic field The nonrelativistic electronic Hamiltonian (implied summation over electrons): H = H + A (r) p + B (r) s + 1 2 A (r)2 The vector potential of the uniform (static) fields B is given by: B = A = const A O (r) = 1 2 B (r O) = 1 2 B r O note: the gauge origin O is arbitrary! The orbital paramagnetic interaction: A O (r) p = 1 2 B (r O) p = 1 2 B (r O) p = 1 2 B L O where we have introduced the angular momentum relative to the gauge origin: The diamagnetic interaction: L O = r O p 1 2 A2 (B) = 1 8 (B r O) (B r O ) = 1 8 [ B 2 r 2 O (B r O) 2] The electronic Hamiltonian in a uniform magnetic field depends on the gauge origin: H = H + 1 2 B L O + B s + 1 8 [ B 2 r 2 O (B r O) 2] a change of the origin is a gauge transformation Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 3 / 34
Gauge transformation of the Schrödinger equation What is the effect of a gauge transformation on the wave function? Consider a general gauge transformation for the electron (atomic units): A = A + f, φ = φ f t It can be shown this represents a unitary transformation of H i / t: ( H i ) ( = exp ( if ) H i ) exp (if ) t t In order that the Schrödinger equation is still satisfied ( H i ) Ψ t ( H i t ) Ψ, the new wave function must undergo a compensating unitary transformation: Ψ = exp ( if ) Ψ All observable properties such as the electron density are then unaffected: ρ = (Ψ ) Ψ = [Ψ exp( if )] [exp( if )Ψ] = Ψ Ψ = ρ Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 4 / 34
Gauge-origin transformations Different choices of gauge origin in the external vector potential are related by gauge transformations: A O (r) = 1 B (r O) 2 A G (r) = A O (r) A O (G) = A O (r) + f, f (r) = A O (G) r The exact wave function transforms accordingly and gives gauge-invariant results: Ψ exact G = exp [ if (r)] Ψ exact O = exp [ia O (G) r] Ψ exact O rapid oscillations Illustration: H 2 on the z axis in a magnetic field B =.2 a.u. in the y direction wave function with gauge origin at O = (,, ) (left) and G = (1,, ) (right).5.5.4.4.3.3 Wave function, ψ.2.1.1.2 Gauge transformed wave function, ψ".2.1.1.2.3.4 Re(ψ) Im(ψ) ψ 2.3.4 Re(ψ") Im(ψ") ψ" 2.5 1.5 1.5.5 1 1.5 Space coordinate, x (along the bond).5 1.5 1.5.5 1 1.5 Space coordinate, x (along the bond) Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 5 / 34
London atomic orbitals The exact wave function transforms in the following manner: Ψ exact G = exp [ i 1 2 B (G O) r] Ψ exact O this behaviour cannot easily be modelled by standard atomic orbitals Let us build this behaviour directly into the atomic orbitals: ω lm (r K, B, G) = exp [ i 1 2 B (G K) r] χ lm (r K ) χ lm (r K ) is a normal atomic orbital centered at K and quantum numbers lm ω lm (r K, B, G) is a field-dependent orbital at K with field B and gauge origin G Each AO now responds in a physically sound manner to an applied magnetic field indeed, all AOs are now correct to first order in B, for any gauge origin G the calculations become rigorously gauge-origin independent uniform (good) quality follows, independent of molecule size These are the London orbitals after Fritz London (1937) Questions: also known as GIAOs (gauge-origin independent AOs or gauge-origin including AOs) are London orbitals needed in atoms? why not attach the phase factor to the total wave function instead? Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 6 / 34
London orbitals Let us consider the FCI dissociation of H 2 in a magnetic field full lines: London atomic orbitals dashed lines: AOs with gauge origin between atoms dotted lines: AOs with gauge origin on one of the atoms Without London orbitals, the FCI method is not size extensive in magnetic fields Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 7 / 34
Expansion of the molecular energy in a magnetic field Assume zero nuclear magnetic moments and expand the molecular electronic energy in the external magnetic induction B: E (B) = E + B T E (1) + 1 2 BT E (2) B + The molecular magnetic moment at B is now given by M mol (B) def de (B) = db = E(1) E (2) B + = M perm + ξb +, where we have introduced the permanent magnetic moment and the magnetizability: M perm = E (1) = de db permanent magnetic moment B= describes the first-order change in the energy but vanishes for closed-shell systems ξ = E (2) = d2 E db 2 molecular magnetizability B= describes the second-order energy and the first-order induced magnetic moment First-order energies for imaginary and triplet perturbations vanish for closed-shell sysetems: c.c. ˆΩ imaginary c.c. c.c. ˆΩ triplet c.c. Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 8 / 34
The magnetizability The electronic Hamiltonian in a uniform magnetic field: H = H + 1 2 B L O + B s + 1 [ B 2 ro 2 8 (B r O) 2] The molecular magnetizability of a closed-shell system: ξ = d2 E db 2 = 2 H B 2 + 2 H B n n H B E n n E = 1 ) r O ro (r T 4 O T r O I 3 + 1 L O n n L T O 2 E }{{} n n E diamagnetic term } {{ } paramagnetic term The isotropic part of the diamagnetic term is given by: ξ dia = 1 3 Tr ξ dia = 1 x 2 6 O + yo 2 + O z2 1 = r 2 6 O Only the orbital Zeeman interaction contributions to the paramagnetic term: S singlet state system surface for 1 S systems (closed-shell atoms), the paramagnetic term vanishes altogether: 1 2 L O 1 S gauge origin at nucleus In most (but not all) systems the diamagnetic term dominates Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 9 / 34
Hartree Fock magnetizabilities and hypermagnetizabilities London orbitals are correct to first-order in the external magnetic field for this reason, basis-set convergence is usually improved, RHF magnetizabilities and hypermagnetizabilities of benzene: basis set χ xx χ yy χ zz X xxxx X yyyy X zzzz London STO-3G 8.1 8.1 23. 211 211 52 6-31G 8.2 8.2 23.1 219 219 64 cc-pvdz 8.1 8.1 22.3 236 236 12 aug-cc-pvdz 8. 8. 22.4 316 316 153 origin CM STO-3G 35.8 35.8 48.1 45 45 27 6-31G 31.6 31.6 39.4 29 29 152 cc-pvdz 15.4 15.4 26.9 9 9 241 aug-cc-pvdz 9.9 9.9 25.2 413 413 159 origin H STO-3G 35.8 176.3 116.7 45 1477 534 6-31G 31.6 144.8 88. 29 1588 5866 cc-pvdz 15.4 48. 41.6 9 2935 3355 aug-cc-pvdz 9.9 2.9 33.9 413 3321 197 The RHF model overestimates the magnitude of magnetizabilities by 5% 1%: 1 3 JT 2 HF exp. diff. H 2 O 232 218 6.4% NH 3 289 271 6.6% CH 4 315 289 9.% CO 2 374 349 7.2% PH 3 441 435 1.4% Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 1 / 34
Normal distributions of errors for magnetizabilities Normal distributions of magnetizability errors for 27 molecules in the aug-cc-pcvqz basis relative to CCSD(T)/aug-cc-pCV[TQ]Z values (Lutnæs et al., JCP 131, 14414 (29)) Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 11 / 34
Mean absolute errors for magnetizabilities Mean relative errors (MREs, %) in magnetizabilities of 27 molecules realtive to the CCSD(T)/aug-cc-pCV[TQ]Z values. The DFT results are grouped by functional type. The heights of the bars correspond to the largest MRE in each category. (Lutnæs et al., JCP 131, 14414 (29)) Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 12 / 34
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 Most closed-shell molecules are diamagnetic their energy increases in an applied magnetic field 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).6.4.2.1.5.5.1 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 perpendiculard) field, χ >.2 Trygve Helgaker (CTCC, University of Oslo).2 Molecules in an external magnetic field LCT, UPMC, November 27 212 13 / 34
.1 Closed-shell paramagnetic molecules diamagnetic transition at stabilizing field strength.5 B c.15!.3!.5 B!au".2.4.6.8.1.12 However, all systems become diamagnetic in sufficiently strong fields: c) d) W!W!au" BH W!W!au" C8H8: total energy W!W!au" a) B!au".2.5.1.15.2.25.3!.4 C12H12: total energy!.1!.2 STO!3G, Bc ".24 DZ, Bc ".22 aug!dz, Bc ".23.4.2 STO!3G, Bc ".35 6!31G, Bc ".34 cc!pvdz, Bc ".32.15.1.5 STO!3G, Bc ".18 6!31G, Bc ".18 cc!pvdz, Bc ".16!.3.1.2.3.4.5.6.5.1.15.2.25.3 B!au"!.5!.4!.2!.1 b) W!W!au" CH #!.2!.4!.6!.8!.1!.12.1.2.3.4.5.6 B!au" The transition occurs at a characteristic stabilizing critical field strength B c B c.22 perpendicular STO3!G, Bc " to.43principal axis for BH B c.32 along DZ, the principal Bc ".44 axis for antiaromatic octatetraene C 8 H 8 B c.16 along aug!dz, the principal Bc ".45 axis for antiaromatic [12]-annulene C 12 H 12 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 72 H 72 We may in principle separate such molecules by applying a field gradient W!W!au" MnO4! c).1.2.3.4.5.6.7 B!au" Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 14 / 34
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 Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 15 / 34
Closed-shell paramagnetic molecules C 2: more structure 756.685 756.69 756.695 756.7 756.75 756.71 Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 16 / 34
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 (B ) x 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!! C C x nuc nuc.4.4 Nuclear shielding Nuclear shielding in Boron Boron Hydrog Hydrogen.2.2!.2.2.2 there is no rotation at the field-free energy maximum: B = the onset of paramagnetic rotation Orbital (against energies the field) reduces the energy HOMO!LUMO for B > gap the strongest paramagnetic rotation occurs at the energy inflexion.5 point the rotation comes to a halt.1 at the stabilizing field strength: B = B c the onset of diamagnetic rotation (with the field) increases the.48energy for B > B c.46 Diamagnetic closed-shell molecules: diamagnetic rotation always!.1increases thelumo energy according.44 to Lenz s law HOMO!(B x )!.2.42 Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November.227 212 17 / 34! gap (B x ) "(B x ) Singlet excitation en.4.35.3.25
Molecules in strong magnetic fields three field regimes 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 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) Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 18 / 34
The helium atom total energy of the 1 S(1s 2 ), 3 S(1s2s), 3 P(1s2p) and 1 P(1s2p) states Electronic states evolve in a complicated manner in a magnetic field the behaviour depends on orbital and spin angular momenta eventually, all energies increase diamagnetically 1.6 1.8 2. <XYZ> 2.2 2.4 2.6 2.8 3...2.4.6.8 1. B [a.u] Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 19 / 34
The helium atom orbital energies The orbital energies behave in an equally complicated manner 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 8 6 Orbital energy, [Hartree] 4 2 2 1 2 3 4 5 6 7 8 9 1 Field, B [au] Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 2 / 34
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 Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 21 / 34
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? Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 22 / 34
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) Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 23 / 34
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 2 1 18 3 4 135 5 225 45 315 1 18 225 3 4 5 315 27 I 2 27 Bond distance Re (pm), orientation Θe ( ), diss. energy De, and rot. barrier E (kj/mol) B. 1. Re 74 66 Trygve Helgaker (CTCC, University of Oslo) θe singlet De 459 594 E 83 Re 136 triplet θe De 9 12 Molecules in an external magnetic field E 12 LCT, UPMC, November 27 212 24 / 34
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 1σg = 1s, R { lim R 1σu = 2p, 2p x i B 4a 2p y 2p 1, The magnetic field modifies the MO energy level diagram all orientations parallel orientation perpendicular orientation perpendicular orientation: antibonding MO is stabilized, while bonding MO is destabilized parallel orientation: both MOs are unaffected relative to the AOs 1Σg 1Σ u.5 1Σg 1. 1.5 Σ 1Σ u.5 1sA 1sB.1.15 1Σ u 1Σg Molecules of zero bond order may therefore be stabilized by the magnetic field Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 25 / 34
The H 2 molecule a new bonding mechanism: perpendicular paramagnetic bonding The triplet H 2 is bound by perpendicular stabilization of antibonding MO 1σ u 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 Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 26 / 34
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... Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 27 / 34
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. Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 28 / 34
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 Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 29 / 34
The helium dimer the 1 Σ + g (1σ2 g 1σ 2 u ) 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. B 2.5 4.5 B 2. 5. B 1.5 5.5 B 1. B.5 B. R Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 3 / 34
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 R Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 31 / 34
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 4 B. 5 B.5 B 1. 6 B 1.5 B 2. 7 B 2.5 In strong fields, anisotropic Gaussians are needed for a compact description for B 1 without such basis sets, calculations become speculative Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 32 / 34
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 vibrational frequencies in the range 2 2 cm 1 (for the 4 He isotope) MOLDEN MOLDEN 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 Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 33 / 34
Molecular structure Hartree Fock calculations on ammonia and benzene 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] 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) 8.4 8.35.1.2.3.4.5.6 B [au] 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] Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 212 34 / 34