Molecular Magnetic Properties

Molecular Magnetic Properties Trygve Helgaker Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway The 12th Sostrup Summer School Quantum Chemistry and Molecular Properties July 1 13 212 Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 1 / 44

Magnetic perturbations In atomic units, the molecular Hamiltonian is given by H = H + A i (r i ) p i + B i (r i ) s i i i i }{{}}{{} orbital paramagnetic spin paramagnetic There are two kinds of magnetic perturbation operators: φ i (r i ) + 1 A 2 i 2 (r i ) i }{{} diamagnetic the paramagnetic operator is linear and may lower or raise the energy the diamagnetic operator is quadratic and always raises the energy There are two kinds of paramagnetic operators: the orbital paramagnetic operator couples the field to the electron s orbital motion the spin paramagnetic operator couples the field to the electron s spin In the study of magnetic properties, we are interested in two types of perturbations: uniform external magnetic field B, with vector potential A ext(r) = 1 2 B r leads to Zeeman interactions nuclear magnetic moments M K, with vector potential A nuc(r) = α 2 K M K r K r 3 K leads to hyperfine interactions where α 1/137 is the fine-structure constant Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 2 / 44

Molecules in an external magnetic field Part 1: Molecules in an external magnetic field Hamiltonian in an external magnetic field gauge transformations and London orbitals diamagnetism and paramagnetism induced currents Part 2: Magnetic resonance parameters Zeeman and hyperfine operators magnetizabilities nuclear shielding constants indirect nuclear spin spin coupling constants Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 3 / 44

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) Molecular Magnetic Properties 12th Sostrup Summer School (212) 4 / 44

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) Molecular Magnetic Properties 12th Sostrup Summer School (212) 5 / 44

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) Molecular Magnetic Properties 12th Sostrup Summer School (212) 6 / 44

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) Molecular Magnetic Properties 12th Sostrup Summer School (212) 7 / 44

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) Molecular Magnetic Properties 12th Sostrup Summer School (212) 8 / 44

Diamagnetism and paramagnetism The Hamiltonian has paramagnetic and diamagnetic parts: H = H + 1 2 BLz + Bsz + 1 8 B2 (x 2 + y 2 ) linear and quadratic B terms 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) a).1.8.6.2.1.5.5.1 c) b) x 1 3 14 b) d) 14 12 1 8 6 4 2 x 1 3.1.5.1.8.6.4.2.1.5.5.1.2 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, ξ < c) d) right: BH: paramagnetic dependence on a perpendicular field, ξ >.2.2 Trygve Helgaker (CTCC, University of Oslo).4 Molecular Magnetic Properties 12th Sostrup Summer School (212) 9 / 44

Transition from para- to diamagnetism However, all closed-shell systems become diamagnetic in sufficiently strong fields: 756.75 756.71 756.685 756.69 756.695 756.7 One atomic unit field strength is 2.2 1 5 T highest fields created in laboratories is 1 3 a.u. 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 B c should be observable for C 72 H 72 Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 1 / 44

Closed-shell paramagnetic molecules 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) Molecular Magnetic Properties 12th Sostrup Summer School (212) 11 / 44

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): Energy Angular momentum Nuclear shielding Energy 1 Angular momentum Nuclear shielding in!25.11 1 Boron!25.11.4 Boron Hydrog.4 Hydrogen!25.12!25.12.5.2!25.13.5.2!25.13!25.14!25.14!25.15!25.15!.2!25.16!25.16!.5!.5.2.2.4.4.2.4.2.2 there is no rotation at the field-free energy maximum: B = Orbital energies HOMO!LUMO gap the onset of paramagnetic rotation (against the field) reduces the energy for B > Singlet excitation en the strongest paramagnetic rotation occurs at the energy.5 inflexion point.1.4 the rotation comes to a halt at the stabilizing field.48 strength: B = B c the onset of diamagnetic rotation (with the field) increases the energy for B > B.35 c.46.3!.1 LUMO.44 diamagnetic rotation always increases HOMO the energy according to Lenz s law.25!.2.42.2 E(B x ) E(B x ) Diamagnetic closed-shell molecules:!(b x ) Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties.4 12th Sostrup Summer School (212) 12 / 44 L x (B x ) L (B ) x x! gap (B x ) L x / / r r!! C C x nuc nuc "(B x )

Expansion of the molecular energy in a magnetic field 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 systems: c.c. ˆΩ imaginary c.c. c.c. ˆΩ triplet c.c. Such molecules therefore do not have a permanent magnetic moment. Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 13 / 44

The magnetizability The electronic Hamiltonian in a uniform magnetic field: H = H + 1 2 B L O + B s + 1 [ 8 B 2 ro 2 (B r O) 2] The molecular magnetizability of a closed-shell system: ξ = d2 E db 2 = 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 The isotropic part of the diamagnetic term is given by: ξ dia = 1 3 Tr ξ dia = 1 6 xo 2 + y O 2 + O z2 = 1 6 } {{ } paramagnetic term r 2 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: 1 3 JT 2 RHF exp. diff. H 2O 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) Molecular Magnetic Properties 12th Sostrup Summer School (212) 14 / 44

The H2 molecule I I I Earth magnetism is 1 1 a.u., whereas NMR uses 1 3 a.u. In stellar atmospheres, much stronger fields exist 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 5 225 315 1 18 2 225 27 I 45 3 4 5 315 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 Molecular Magnetic Properties triplet θe De 9 12 E 12 12th Sostrup Summer School (212) 15 / 44

Part 2: Magnetic resonance parameters Part 1: Molecules in an external magnetic field Hamiltonian in an external magnetic field gauge transformations and London orbitals diamagnetism and paramagnetism induced currents Part 2: Magnetic resonance parameters Zeeman and hyperfine operators magnetizabilities nuclear shielding constants indirect nuclear spin spin coupling constants Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 16 / 44

Magnetic perturbations In atomic units, the molecular Hamiltonian is given by H = H + A i (r i ) p i + B i (r i ) s i i i i }{{}}{{} orbital paramagnetic spin paramagnetic There are two kinds of magnetic perturbation operators: φ i (r i ) + 1 A 2 i 2 (r i ) i }{{} diamagnetic the paramagnetic operator is linear and may lower or raise the energy the diamagnetic operator is quadratic and always raises the energy There are two kinds of paramagnetic operators: the orbital paramagnetic operator couples the field to the electron s orbital motion the spin paramagnetic operator couples the field to the electron s spin In the study of magnetic properties, we are interested in two types of perturbations: uniform external magnetic field B, with vector potential A ext(r) = 1 2 B r leads to Zeeman interactions nuclear magnetic moments M K, with vector potential A nuc(r) = α 2 K M K r K r 3 K leads to hyperfine interactions where α 1/137 is the fine-structure constant Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 17 / 44

Perturbation theory The nonrelativistic electronic Hamiltonian: H = H + H (1) + H (2) = H + A (r) p + B (r) s + 1 2 A (r)2, B (r) = A (r) Second-order Rayleigh Schrödinger perturbation theory: E (1) = A p + B s E (2) = 1 2 A 2 n A p + B s n n A p + B s E n E Zeeman interactions with uniform external field: A (r) = 1 2 B r O H (1) Z = 1 2 B L O + B s Hyperfine interactions with nuclear magnetic moments (1 8 a.u.) A K (r) = α 2 M K r K r 3 K H (1) hf = α 2 M K L K r 3 K }{{} paramagnetic spin orbit (PSO) + 8πα2 3 δ (r K ) M K s + α 2 3(s r K )(r K M K ) (M K s)rk 2 r }{{} K 5 }{{} Fermi contact (FC) spin dipole (SD) Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 18 / 44

Zeeman and hyperfine interactions Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 19 / 44

Taylor expansion of the energy Expand the energy in the presence of an external magnetic field B and nuclear magnetic moments M K around zero field and zero moments: E (B, M) = E + perm. magnetic moments {}}{ B T E (1) + + 1 2 BT E (2) B + 1 2 }{{} K magnetizability hyperfine coupling {}}{ K B T E (11) K M K } {{ } shieldings + 1 M T K E(1) K + 1 M T K 2 E(2) KL M L + KL }{{} spin spin couplings First-order terms vanish for closed-shell systems because of symmetry they shall be considered only briefly here Seond-order terms are important for many molecular properties magnetizabilities nuclear shieldings constants of NMR nuclear spin spin coupling constants of NMR electronic g tensors of EPR (not dealt with here) Higher-order terms are negligible since the perturbations are tiny: 1) the magnetic induction B is weak ( 1 4 a.u.) 2) the nuclear magnetic moments M K couple weakly (µ µ N 1 8 a.u.) Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 2 / 44

First-order molecular properties The first-order properties are expectation values of H (1) Permanent magnetic moment M = (1) H Z = 1 2 L O + s permanent magnetic moment dominates the magnetism of molecules the molecule reorients itself and enters the field such molecules are therefore paramagnetic Hyperfine coupling constants A K = H (1) hf = 8πα2 3 δ (r K ) s M K + measure spin density at the nucleus important in electron paramagnetic resonance (EPR) recall: there are three hyperfine mechanisms: FC, SD and PSO H (1) hf = 8πα2 δ (r 3 K ) M K s + α 2 3(s r K )(r K M K ) (M K s)rk 2 rk 5 + α 2 M K L K rk 3 Note: there are no first-order Zeeman or hyperfine couplings for closed-shell molecules c.c. ˆΩ imaginary c.c. c.c. ˆΩ triplet c.c. Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 21 / 44

High-resolution NMR spin Hamiltonian Consider a molecule in an external magnetic field B along the z axis and with nuclear spins I K related to the nuclear magnetic moments M K as: M K = γ K I K 1 4 a.u. where γ K is the magnetogyric ratio of the nucleus. Assuming free molecular rotation, the nuclear magnetic energy levels can be reproduced by the following high-resolution NMR spin Hamiltonian: where we have introduced H NMR = γ K (1 σ K )BI K z + γ K γ L 2 K KL I K I L K K>L }{{}}{{} nuclear Zeeman interaction nuclear spin spin interaction the nuclear shielding constants σ K the (reduced) indirect nuclear spin spin coupling constants K KL This is an effective nuclear spin Hamiltonian: it reproduces NMR spectra without considering the electrons explicitly the spin parameters σ K and K KL are adjusted to fit the observed spectra we shall consider their evaluation from molecular electronic-structure theory Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 22 / 44

Simulated 2 MHz NMR spectra of vinyllithium 12 C 2 H 3 6 Li experiment RHF 1 2 1 2 MCSCF B3LYP 1 2 1 2 Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 23 / 44

Nuclear shielding constants Expansion of closed-shell energy in an external field B and nuclear magnetic moments M K : Here E (11) K E (B, M) = E + 1 2 BT E (2) B + 1 2 K B T E (11) K M K + 1 2 KL M T K E(2) KL M L + describes the coupling between the applied field and the nuclear moments: in the absence of electrons (i.e., in vacuum), this coupling is identical to I 3 : HZ nuc = B M K the purely nuclear Zeeman interaction K in the presence of electrons (i.e., in a molecule), the coupling is modified slightly: E (11) K = I 3 + σ K the nuclear shielding tensor Shielding constants arise from a hyperfine interaction between the electrons and the nuclei they are of the order of α 2 5 1 5 and are measured in ppm The nuclear Zeeman interaction does not enter the electronic problem compare with the nuclear nuclear Coulomb repulsion Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 24 / 44

Zeeman and hyperfine interactions Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 25 / 44

Ramsey s expression for the nuclear shielding tensors Ramsey s expression for nuclear shielding tensors of a closed-shell system: σ K = d2 E el = 2 H dbdm K B M 2 H B n n H M K K E n n E = α2 ro T r K I 3 r O rk T 2 rk 3 α L 2 O n n r 3 K LT K E n n E }{{}}{{} diamagnetic term paramagnetic term The (usually) dominant diamagnetic term arises from differentiation of the operator: A (B) A (M K ) = 1 2 α2 r 3 K (B r O) (M K r K ) As for the magnetizability, there is no spin contribution for singlet states: S singlet state For 1 S systems (closed-shell atoms), the paramagnetic term vanishes completely and the shielding is given by (assuming gauge origin at the nucleus): σ Lamb = 1 1 3 α2 S r 1 K 1 S Lamb formula Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 26 / 44

Benchmark calculations of BH shieldings (ppm) σ( 11 B) σ( 11 B) σ( 1 H) σ( 1 H) HF 261.3 69.1 24.21 14.15 MP2 22.7 629.9 24.12 14.24 CCSD 166.6 549.4 24.74 13.53 CCSD(T) 171.5 555.2 24.62 13.69 CCSDT 171.8 557.3 24.59 13.72 CCSDTQ 17.1 554.7 24.6 13.7 CISD 182.4 572.9 24.49 13.87 CISDT 191.7 587. 24.35 14.6 CISDTQ 17.2 554.9 24.6 13.7 FCI 17.1 554.7 24.6 13.7 TZP+ basis, R BH = 123.24 pm, all electrons correlated J. Gauss and K. Ruud, Int. J. Quantum Chem. S29 (1995) 437 M. Kállay and J. Gauss, J. Chem. Phys. 12 (24) 6841 Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 27 / 44

Calculated and experimental shielding constants (ppm) HF CAS MP2 CCSD CCSD(T) exp. HF F 413.6 419.6 424.2 418.1 418.6 41 ± 6 (3K) H 28.4 28.5 28.9 29.1 29.2 28.5 ±.2 (3K) H 2O O 328.1 335.3 346.1 336.9 337.9 323.6 ± 6 (3K) H 3.7 3.2 3.7 3.9 3.9 3.5 ±.2 NH 3 N 262.3 269.6 276.5 269.7 27.7 264.5 H 31.7 31. 31.4 31.6 31.6 31.2 ± 1. CH 4 C 194.8 2.4 21. 198.7 198.9 198.7 H 31.7 31.2 31.4 31.5 31.6 3.61 F 2 F 167.9 136.6 17. 171.1 186.5 192.8 N 2 N 112.4 53. 41.6 63.9 58.1 61.6 ±.2 (3K) CO C 25.5 8.2 1.6.8 5.6 3. ±.9 (eq) O 87.7 38.9 46.5 56. 52.9 56.8 ± 6 (eq) mean 11 6 5.1 mean abs 14 7 6 3 For references and details, see Chem. Rev. 99 (1999) 293. for exp. CO and H 2 O values, see Wasylishen and Bryce, JCP 117 (22) 161 Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 28 / 44

Kohn Sham shielding constants (ppm) HF LDA BLYP B3LYP KT2 CCSD(T) exp. HF F 413.6 416.2 41. 48.1 411.4 418.6 41 ± 6 H 28.4 28.5 28.9 29.1 29.2 28.5 ±.2 H 2O O 328.1 334.8 318.2 325. 329.5 337.9 323.6 ± 6 H 3.7 3.2 3.7 3.9 3.9 3.5 ±.2 NH 3 N 262.3 266.3 254.6 259.2 264.6 27.7 264.5 H 31.7 31. 31.4 31.6 31.6 31.2 ± 1. CH 4 C 194.8 193.1 184.2 188.1 195.1 198.9 198.7 H 31.7 31.2 31.4 31.5 31.6 3.61 F 2 F 167.9 284.2 336.7 28.3 211. 186.5 192.8 N 2 N 112.4 91.4 89.8 86.4 59.7 58.1 61.6 ±.2 CO C 25.5 2.3 19.3 17.5 7.4 5.6 3. ±.9 (eq) O 87.7 87.5 85.4 78.1 57.1 52.9 56.8 ± 6 (eq) mean 11 17 26 12 7 mean abs 14 17 26 12 7 Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 29 / 44

Direct and indirect nuclear spin spin couplings The last term in the expansion of the molecular electronic energy in B and M K E (B, M) = E + 1 2 BT E (2) B + 1 2 K BT E (11) K M K + 1 2 KL MT K E(2) KL M L + describes the coupling of the nuclear magnetic moments in the presence of electrons There are two distinct contributions to the coupling: the direct and indirect contributions E (2) KL = D KL + K KL The direct coupling occurs by a classical dipole mechanism: D KL = α 2 R 5 ( KL R 2 KL I 3 3R KL R T ) KL 1 12 a.u. it is anisotropic and vanishes in isotropic media such as gases and liquids The indirect coupling arises from hyperfine interactions with the surrounding electrons: it is exceedingly small: K KL 1 16 a.u. 1 Hz it does not vanish in isotropic media it gives the fine structure of high-resolution NMR spectra Experimentalists usually work in terms of the (nonreduced) spin spin couplings J KL = h γ K 2π γ L 2π K KL isotope dependent Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 3 / 44

Zeeman and hyperfine interactions Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 31 / 44

Ramsey s expression for indirect nuclear spin spin coupling tensors The indirect nuclear spin spin coupling tensors of a closed-shell system are given by: K KL = d2 E el = 2 H dm K dm L M K M 2 H M n n H K M L L E n n E Carrying out the differentiation of the Hamiltonian, we obtain Ramsey s expression: K KL = α 4 rk T r LI 3 r K rl T rk 3 r L 3 2α r 3 4 K L K n n r 3 L LT L E n n E }{{}}{{} diamagnetic spin orbit (DSO) paramagnetic spin orbit (PSO) 8π 3 δ(r K )s + 3r K rk T r K 2 I 3 s n n 8π 3 δ(r L)s T + 3r LrL T r L 2 I 3 s T r 5 K r 5 L 2α 4 E n n E }{{} Fermi contact (FC) and spin dipole (SD) the isotropic FC/FC term often dominates short-range coupling constants the FC/SD and SD/FC terms often dominate the anisotropic part of K KL the orbital contributions (especially DSO) are usually but not invariably small for large internuclear separations, the DSO and PSO contributions cancel Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 32 / 44

Relative importance of the contributions to spin spin coupling constants The isotropic indirect spin spin coupling constants can be uniquely decomposed as: J KL = J DSO KL + J PSO KL + J FC KL + JSD KL The spin spin coupling constants are often dominated by the FC term Since the FC term is relatively easy to calculate, it is tempting to ignore the other terms. However, none of the contributions can be a priori neglected (N 2 and CO)! 2 PSO 1 FC PSO FC FC FC FC FC SD FC FC PSO FC SD FC -1 SD H2 HF H2O OH NH3 NH CH4 C2H4 HCN CH CC NC N2 CO C2H2 CC Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 33 / 44

Restricted Hartree Fock theory and triplet instabilities The correct description of triplet excitations is important for spin spin coupling constants In restricted Hartree Fock (RHF) theory, triplet excitations are often poorly described upon H 2 dissociation, RHF does not describe the singlet ground state correctly but the lowest triplet state dissociates correctly, leading to triplet instabilities more generally, the lowest RHF triplet excitations are underestimated 1 D2p 2 1 1 g FCI 1 P1s2p 3 P1s2p 1 1 g 1Σ u 2 1 u 1Σg1Σu 2 4 6 R ionic covion 3 u 1Σg1Σu covalent 2 1 g 1Σ g 2 1 S1s 2 g FCI Near such instabilities, the RHF description of spin interactions becomes unphysical C 2H 4/Hz 1 J CC 1 J CH 2 J CH 2 J HH 3 J cis 3 J trans exp. 68 156 2 2 12 19 RHF 127 755 572 344 36 4 CAS 76 156 6 2 12 18 B3LYP 75 165 1 3 14 21 Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 34 / 44

Calculation of indirect nuclear spin spin coupling constants The calculation of spin spin coupling constants is a challenging task Spin spin coupling constants depend on many coupling mechanisms: 3 singlet response equations and 7 triplet equations for each nucleus for shieldings, only 3 equations are required, for molecules of all sizes Spin spin coupling constants require a proper description of static correlation the Hartree Fock model fails abysmally MCSCF theory treats static correlation propertly but is expensive Spin spin couplings are sensitive to the basis set the FC contribution requires an accurate electron density at the nuclei steep s functions must be included in the basis Spin spin couplings are sensitive to the molecular geometry equilibrium structures must be chosen carefully large vibrational corrections (often 5% 1%) For heavy elements, a relativistic treatment may be necessary. However, there is no need for London orbitals since no external magnetic field is involved. Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 35 / 44

Reduced spin spin coupling constants by wave-function theory RHF CAS RAS SOPPA CCSD CC3 exp vib HF 1 K HF 59.2 48. 48.1 46.8 46.1 46.1 47.6 3.4 CO 1 K CO 13.4 28.1 39.3 45.4 38.3 37.3 38.3 1.7 N 2 1 K NN 175. 5.7 9.1 23.9 2.4 2.4 19.3 1.1 H 2O 1 K OH 63.7 51.5 47.1 49.5 48.4 48.2 52.8 3.3 2 K HH 1.9.8.6.7.6.6.7.1 NH 3 1 K NH 61.4 48.7 5.2 51. 48.1 5.8.3 2 K HH 1.9.8.9.9 1..9.1 C 2H 4 1 K CC 1672. 99.6 9.5 92.5 92.3 87.8 1.2 1 K CH 249.7 51.5 5.2 52. 5.7 5. 1.7 2 K CH 189.3 1.9.5 1. 1..4.4 2 K HH 28.7.2.1.1..2. 3 K cis 3. 1. 1. 1. 1..9.1 3 K tns 33.3 1.5 1.5 1.5 1.5 1.4.2 abs. 18.3 3.3 1.6 1.8 1.2 1.6 at R e % 579 6 14 24 23 6 SOPPA: second-order polarization-propagator approximation Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 36 / 44

Reduced spin spin coupling constants by density-functional theory LDA BLYP B3LYP PBE B97-3 RAS exp vib HF 1 K HF 35. 34.5 38.9 32.6 4.5 48.1 47.6 3.4 CO 1 K CO 65.4 55.7 47.4 62. 43.4 39.3 38.3 1.7 N 2 1 K NN 32.9 46.6 2.4 43.2 12.5 9.1 19.3 1.1 H 2O 1 K OH 4.3 44.6 47.2 41.2 46.3 47.1 52.8 3.3 2 K HH.3.9.7.5.6.6.7.1 NH 3 1 K NH 41. 49.6 52.3 47. 5.1 5.2 5.8.3 2 K HH.4.7.9.7.8.9.9.1 C 2H 4 1 K CC 66.6 9.3 96.2 83.4 92.9 9.5 87.8 1.2 1 K CH 42.5 55.3 55. 5. 51.4 5.2 5. 1.7 2 K CH.4..5.2.3.5.4.4 2 K HH.4.4.3.3.3.1.2. 3 K cis.8 1.1 1.1 1. 1. 1..9.1 3 K tns 1.2 1.7 1.7 1.6 1.5 1.5 1.4.2 abs. 11.2 5.9 3.1 6.4 2.6 1.6 at R e % 72 48 14 33 14 14 Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 37 / 44

Comparison of density-functional and wave-function theory Normal distributions of errors for indirect nuclear spin spin coupling constants for the same molecules as on the previous slides HF LDA BLYP B3LYP 3 3 3 3 3 3 3 3 CAS RAS SOPPA CCSD 3 3 3 3 3 3 3 3 Some observations: LDA underestimates only slightly, but has a large standard deviation BLYP reduces the LDA errors by a factor of two B3LYP errors are similar to those of CASSCF The CCSD method is slightly better than the SOPPA method Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 38 / 44

The Karplus curve Vicinal (three-bond) spin spin coupling constants depend critically on the dihedral angle: 3 J HH in ethane as a function of the dihedral angle: 14 12 1 8 6 4 2 empirical DFT 25 5 75 1 125 15 175 Good agreement with the (empirically constructed) Karplus curve Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 39 / 44

Valinomycin C 54 H 9 N 8 O 18 DFT can be applied to large molecular systems such as valinomycin (168 atoms) there are a total of 7587 spin spin couplings to the carbon atoms in valinomycin below, we have plotted the magnitude of the reduced LDA/6-31G coupling constants on a logarithmic scale, as a function of the internuclear distance: 1 19 1 16 1 13 5 1 15 the coupling constants decay in characteristic fashion, which we shall examine most of the indirect couplings beyond 5 pm are small and cannot be detected Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 4 / 44

Valinomycin C 54 H 9 N 8 O 18 One-bond spin spin couplings to CH, CO, CN, CC greater than.1 Hz 1 3 1 3 1.3.1.3 1 2 3 4 5 6 Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 41 / 44

Valinomycin C 54 H 9 N 8 O 18 Two-bond spin spin couplings to CH, CO, CN, CC greater than.1 Hz 1 3 1 3 1.3.1.3 1 2 3 4 5 6 Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 42 / 44

Valinomycin C 54 H 9 N 8 O 18 Three-bond spin spin couplings to CH, CO, CN, CC greater than.1 Hz 1 3 1 3 1.3.1.3 1 2 3 4 5 6 Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 43 / 44

Valinomycin C 54 H 9 N 8 O 18 Four-bond spin spin couplings to CH, CO, CN, CC greater than.1 Hz 1 3 1 3 1.3.1.3 1 2 3 4 5 6 Trygve Helgaker (CTCC, University of Oslo) Molecular Magnetic Properties 12th Sostrup Summer School (212) 44 / 44