Molecular Magnetic Properties

Transcription

1 Molecular Magnetic Properties Trygve Helgaker Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, Norway European Summer School in Quantum Chemistry (ESQC) 2015 Torre Normanna, Sicily, Italy September 6 19, 2015 Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 1 / 74

2 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 diamagnetism and paramagnetism molecules in strong fields DFT in magnetic fields Part 3: Molecular Magnetic Properties magnetizabilities Zeeman and hyperfine operators nuclear shielding constants indirect nuclear spin spin coupling constants Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 2 / 74

3 Hamiltonian mechanics In classical Hamiltonian mechanics, a system of particles is described in terms their positions q i and conjugate momenta p i. For each such system, there exists a scalar Hamiltonian function H(q i, p i ) such that the classical equations of motion are given by: q i = H p i, ṗ i = H q i (Hamilton s equations of motion) Example: a single particle of mass m in a conservative force field F (q) the Hamiltonian function is constructed from the corresponding scalar potential: H(q, p) = p2 2m + V (q), V (q) F (q) = q Note: Hamilton s equations are equivalent to Newton s equations: q = H(q,p) = p } p m ṗ = H(q,p) V (q) = m q = F (q) (Newton s equations of motion) = q q Newton s equations are second-order differential equations Hamilton s equations are first-order differential equations the Hamiltonian function is not unique! Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 3 / 74

4 Quantization of a particle in conservative force field The Hamiltonian formulation is more general than the Newtonian formulation: it is invariant to coordinate transformations it provides a uniform description of matter and field it constitutes the springboard to quantum mechanics The Hamiltonian function (the total energy) of a particle in a conservative force field: H(q, p) = p2 2m + V (q) Standard rule for quantization (in Cartesian coordinates): carry out the substitutions p i, H i t multiply the resulting expression by the wave function Ψ(q) from the right: i Ψ(q) ] = [ 2 t 2m 2 + V (q) Ψ(q) This approach is sufficient for a treatment of electrons in an electrostatic field it is insufficient for nonconservative systems it is therefore inappropriate for systems in a general electromagnetic field Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 4 / 74

5 The Lorentz force and Maxwell s equations In the presence of an electric field E and a magnetic field (magnetic induction) B, a classical particle of charge z experiences the Lorentz force: F = z (E + v B) since this force depends on the velocity v of the particle, it is not conservative The electric and magnetic fields E and B satisfy Maxwell s equations ( ): Note: E = ρ/ε 0 B ε 0 µ 0 E/ t = µ 0 J B = 0 E + B/ t = 0 Coulomb s law Ampère s law with Maxwell s correction Faraday s law of induction when the charge and current densities ρ(r, t) and J(r, t) are known, Maxwell s equations can be solved for E(r, t) and B(r, t) on the other hand, since the charges (particles) are driven by the Lorentz force, ρ(r, t) and J(r, t) are functions of E(r, t) and B(r, t) We shall consider the motion of particles in a given (fixed) electromagnetic field. Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 5 / 74

6 Scalar and vector potentials The second, homogeneous pair of Maxwell s equations involves only E and B: 1 Eq. (1) is satisfied by introducing the vector potential A: B = 0 (1) E + B t = 0 (2) B = 0 = B = A vector potential (3) 2 inserting Eq. (3) in Eq. (2) and introducing a scalar potential φ, we obtain ( E + A ) = 0 = E + A t t = φ scalar potential The second pair of Maxwell s equations is thus automatically satisfied by writing E = φ A t B = A The potentials (φ, A) contain four rather than six components as in (E, B). They are obtained by solving the first, inhomogeneous pair of Maxwell s equations, which contains ρ and J. Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 6 / 74

7 Gauge transformations The scalar and vector potentials φ and A are not unique. Consider the following transformation of the potentials: φ = φ f } t A f = f (q, t) gauge function of position and time = A + f This gauge transformation of the potentials does not affect the physical fields: E = φ A t = φ + f t A t f t B = A = (A + f ) = B + f = B = E We are free to choose f (q, t) to make the potentials satisfy additional conditions Typically, we require the vector potential to be divergenceless: A = 0 = (A + f ) = 0 = 2 f = A Coulomb gauge We shall always assume that the vector potential satisfies the Coulomb gauge: A = B, A = 0 Coulomb gauge Note: A is still not uniquely determined, the following transformation being allowed: A = A + f, 2 f = 0 Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 7 / 74

8 The Hamiltonian in an electromagnetic field We must construct a Hamiltonian function such that Hamilton s equations are equivalent to Newton s equation with the Lorentz force: q i = H p i & ṗ i = H q i ma = z (E + v B) To this end, we introduce scalar and vector potentials φ and A such that E = φ A t, B = A In terms of these potentials, the classical Hamiltonian function becomes H = π2 + zφ, π = p za kinetic momentum 2m Quantization is then accomplished in the usual manner, by the substitutions p i, H i t The time-dependent Schrödinger equation for a particle in an electromagnetic field: i Ψ t = 1 ( i za) ( i za) Ψ + zφ Ψ 2m Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 8 / 74

9 Electron spin The nonrelativistic Hamiltonian for an electron in an electromagnetic field is then given by: H = π2 2m eφ, π = i + ea However, this description ignores a fundamental property of the electron: spin. Spin was introduced by Pauli in 1927, to fit experimental observations: (σ π)2 π2 H = eφ = 2m 2m + e 2m B σ eφ where σ contains three operators, represented by the two-by-two Pauli spin matrices ( 0 1 σ x = 1 0 ) ( 0 i, σ y = i 0 ) ( 1 0, σ z = 0 1 ) The Schrödinger equation now becomes a two-component equation: ( ) π 2 e eφ + 2m 2m Bz e (Ψα ) ( ) (Bx iby ) 2m Ψα e 2m (Bx + iby ) π 2 = E e eφ 2m 2m Bz Ψ β Ψ β Note: the two components are only coupled in the presence of an external magnetic field Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 9 / 74

10 Spin and relativity The introduction of spin by Pauli in 1927 may appear somewhat ad hoc By contrast, spin arises naturally from Dirac s relativistic treatment in 1928 is spin a relativistic effect? However, reduction of Dirac s equation to nonrelativistic form yields the Hamiltonian H = (σ π)2 2m π2 eφ = 2m + e 2m B σ eφ π2 2m eφ in this sense, spin is not a relativistic property of the electron but we note that, in the nonrelativistic limit, all magnetic fields disappear... We interpret σ by associating an intrinsic angular momentum (spin) with the electron: s = σ/2 Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 10 / 74

11 Molecular Hamiltonian The nonrelativistic Hamiltonian for an electron in an electromagnetic field is therefore H = π2 2m + e B s eφ, π = p + ea, p = i m expanding π 2 and assuming the Coulomb gauge A = 0, we obtain π 2 Ψ = (p + ea) (p + ea) Ψ = p 2 Ψ + ep AΨ + ea pψ + e 2 A 2 Ψ = p 2 Ψ + e(p A)Ψ + 2eA pψ + e 2 A 2 Ψ = ( p 2 + 2eA p + e 2 A 2) Ψ in molecules, the dominant electromagnetic contribution is from the nuclear charges: φ = 1 Z K e 4πɛ 0 K + φ r ext K Summing over all electrons and adding pairwise Coulomb interactions, we obtain H = 1 2m p2 i e2 Z K + e2 r 1 ij 4πɛ i 0 r Ki ik 4πɛ 0 i>j + e A i p i + e B i s i e φ i m m i i i + e2 A 2 i 2m i zero-order Hamiltonian first-order Hamiltonian second-order Hamiltonian Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 11 / 74

12 Magnetic perturbations In atomic units, the molecular Hamiltonian is given by H = H 0 + A(r i ) p i + B(r i ) s i i i i }{{}}{{} orbital paramagnetic spin paramagnetic There are two kinds of magnetic perturbation operators: φ(r i ) + 1 A 2 (r i ) 2 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) Magnetic Properties ESQC15 12 / 74

13 Review We have studied the nonrelativistic electronic Hamiltonian: H = H 0 + H (1) + H (2) = H 0 + A (r) p + B (r) s A (r)2 where the vector potential in the Coulomb gauge is not unique but satisfies the relations A (r) = B (r), A (r) = 0 We shall consider uniform external field and the nuclear magnetic moments: A (r) = 1 2 B r O, A K (r) = α 2 M K r K rk 3, First- and second-order Rayleigh Schrödinger perturbation theory gives: E (1) = 0 A p + B s 0 E (2) = 1 0 A n 0 A p + B s n n A p + B s 0 E n E 0 We shall first consider external magnetic fields and next nuclear magnetic fields Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 13 / 74

14 Molecular Magnetic Properties 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 diamagnetism and paramagnetism molecules in strong fields DFT in magnetic fields Part 3: Molecular Magnetic Properties magnetizabilities Zeeman and hyperfine operators nuclear shielding constants indirect nuclear spin spin coupling constants Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 14 / 74

15 Review We have studied the nonrelativistic electronic Hamiltonian: H = H 0 + H (1) + H (2) = H 0 + A (r) p + B (r) s A (r)2 where the vector potential in the Coulomb gauge is not unique but satisfies the relations A (r) = B (r), A (r) = 0 We shall consider uniform external field and the nuclear magnetic moments: A (r) = 1 2 B r O, A K (r) = α 2 M K r K rk 3, First- and second-order Rayleigh Schrödinger perturbation theory gives: E (1) = 0 A p + B s 0 E (2) = 1 0 A n 0 A p + B s n n A p + B s 0 E n E 0 We shall first consider external magnetic fields and next nuclear magnetic fields Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 15 / 74

16 Hamiltonian in a uniform magnetic field The nonrelativistic electronic Hamiltonian (implied summation over electrons): H = H 0 + A (r) p + B (r) s 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 (r) = 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 B L O + B s [ B 2 r 2 O (B r O) 2] a change of the origin is a gauge transformation Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 16 / 74

17 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) Magnetic Properties ESQC15 17 / 74

18 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 = 0.2 a.u. in the y direction wave function with gauge origin at O = (0, 0, 0) (left) and G = (100, 0, 0) (right) Wave function, ψ Gauge transformed wave function, ψ" Re(ψ) Im(ψ) ψ Re(ψ") Im(ψ") ψ" Space coordinate, x (along the bond) Space coordinate, x (along the bond) Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 18 / 74

19 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 modeled 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) Magnetic Properties ESQC15 19 / 74

20 London orbitals Let us consider the FCI dissociation of H 2 in a magnetic field full lines: with London atomic orbitals dashed lines: without London atomic orbitals B B B 0.0 Without London orbitals, the FCI method is not size extensive in magnetic fields Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 20 / 74

21 Molecular diamagnetism and paramagnetism The Hamiltonian has paramagnetic and diamagnetic parts: H = H BLz + Bsz B2 (x 2 + y 2 ) para- and diamagnetic parts Second-order perturbation theory gives the magnetizability expression: ξ zz = d2 E db 2 = x 2 + y L z n n L z 0 a) dia- andb) paramagnetic parts 2 x 10 E n n E Most closed-shell molecules are diamagnetic with 0.08 ξ zz < 0 (below left) their energy increases in an applied magnetic field the induced currents oppose the field Some closed-shell systems are paramagnetic with ξ zz > 0 (below right) their energy decreases in a magnetic field relaxation of the wave function lowers the energy, dominating the diamagnetic term RHF calculations of the field dependence of the energy for two closed-shell systems: a) c) b) 14 0 x 10 3 d) Trygve Helgaker (CTCC, University of c) Oslo) Magnetic Properties d) ESQC15 21 / 74

22 Closed-shell paramagnetism explained The Hamiltonian has a linear and quadratic dependence on the magnetic field: H = H BLz + Bsz B2 (x 2 + y 2 ) linear and quadratic B terms Consider the effect of B on the ground state and on the first excited state: the quadratic term raises the energy of both states, creating diabatic energy curves depending on the states, diabatic curve crossings may occur at a given field strength the linear term couples the two states, creating adiabatic states and avoiding crossings A sufficiently strong coupling gives a paramagnetic ground-state with a double minimum Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 22 / 74

23 C 20 in a perpendicular magnetic field All systems become diamagnetic in sufficiently strong fields: The transition occurs at a characteristic stabilizing critical field strength B c B c 0.01 for C 20 (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 72 H 72 We may in principle separate such molecules by applying a field gradient Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 23 / 74

24 Induced electron rotation The magnetic field induces a rotation of the electrons about the field direction: the derivative of the energy with the field is half the kinetic angular-momentum operator E (B) = r π 0, 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! Angular momentum 1 Angular momentum !0.5! L x / / r r!! C C x nuc nuc Nuclear shielding Nuclear shielding in Boron Boron Hydrog Hydrogen ! there is no rotation at the field-free energy maximum: B = 0 the onset of paramagnetic rotation Orbital (against energies the field) reduces the energy HOMO!LUMO for B > 0gap the strongest paramagnetic rotation occurs at the energy inflection 0.5 point the rotation comes to a halt 0.1 at the stabilizing field strength: B = B c the onset of diamagnetic rotation in the opposite direction increases 0.48 the energy for B > B c Diamagnetic closed-shell molecules: induced diamagnetic rotation!0.1 always increases LUMOthe energy HOMO!(B x )! Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties 0.4 ESQC15 24 / 74! gap (B x ) 0.44 "(B x ) Singlet excitation en

25 Molecules in strong magnetic fields The non-relativistic electronic Hamiltonian (a.u.) in a magnetic field B along the z axis: H = H BLz + Bsz B2 (x 2 + y 2 ) linear and quadratic B terms one atomic unit of B corresponds to T = G Coulomb regime: B 1 a.u. earth-like conditions: Coulomb interactions dominate magnetic interactions are treated perturbatively earth magnetism 10 10, NMR 10 4 ; pulsed laboratory field 10 3 a.u. familiar chemistry Intermediate regime: B 1 a.u. white dwarfs: 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 exotic chemistry Landau regime: B 1 a.u. neutron stars: a.u. magnetic interactions dominate (Landau levels) Coulomb interactions are treated perturbatively relativity becomes important for B α a.u. alien chemistry Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 25 / 74

26 The H2 molecule in a strong magnetic field I Singlet H2 1σg2 aligns itself with the field and becomes more strongly bound I Triplet H2 1σg 1σu i becomes bound in a perpendicular field orientation B I Re 74 pm 66 pm singlet De 459 kj/mol 594 kj/mol Erot 0 83 kj/mol Re 136 pm triplet De 0 12 kj/mol Erot 0 12 kj/mol Polar plots of E (R, Θ) in the singlet (left) and triplet (right) states I red for high energy; blue for low energy Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 26 / 74

27 Perpendicular paramagnetic bonding The magnetic field modifies the MO energy level diagram, creating a zero-bond-order bond perpendicular orientation: antibonding MO is stabilized, while bonding MO is (less) destabilized parallel orientation: both MOs are unaffected relative to the AOs EêkJmol -1 EêkJmol Σ H 2 1Σ u He HF s A HF 1s B FCI Rêpm Σ g FCI Rêpm In the helium limit, the bonding and antibonding MOs transform into 1s and 2p AOs: lim 1σg = 1s, R 0 { lim R 0 1σ u = 2p 0, 2p 1, all orientations parallel orientation perpendicular orientation (stabilized) Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 27 / 74

28 Calculated H 2 dissociation curves in a magnetic field FCI/cc-pVTZ energy curves of the lowest electronic states of H 2 H 2 dissociation, B = B 0 = 0 MG -0.4 H 2 dissociation, B = B 0 = 235 MG E [hartree] E [hartree] R and 12.4-R [bohr] R and 12.4-R [bohr] red curves are energies in perpendicular orientation plotted against bond distance R; blue curves are energies in parallel orientation plotted against 12.4 R crosses indicate triplet states Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 28 / 74

29 DFT in a magnetic field: current-density-functional theory (CDFT) In a magnetic field, the Hohenberg Kohn variation principle takes the form ( E(v, A) = inf F (ρ, κ) + (v + 1 ρ,κ 2 A2 ρ) + (A κ) ) Vignale Rasolt functional F (ρ, κ) depends on the density and paramagnetic current density: ρ(r) = Ψ (r)ψ(r) κ(r) = ReΨ (r)pψ(r) the paramagnetic current density κ is not observable but uniquely determined by Ψ The observable physical current density is related to the paramagnetic current density by j = κ + ρ A j is observable but not uniquely determined by Ψ The Hohenberg Kohn theorem: there (probably) is no Hohenberg Kohn theorem in a magnetic field note: Ψ is not uniquely determined by the Hamiltonian Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 29 / 74

30 Kohn Sham CDFT and representability A Kohn Sham decomposition of the CDFT functional yields: F (ρ, κ) = T s(ρ, κ) + J(ρ) + E xc(ρ, ν), ν(r) = κ(r) ρ(r) vorticity the exchange correlation functional depends on the density and the gauge-invariant vorticity Vignale and Rasolt, PRL 59, 2360 (1987) Warning: closed-shell two-electron systems are not representable by a single determinant density ρ (top), physical current j 2 (middle), and vorticity ν (bottom) of H 2 Hartree Fock Full CI When are (ρ, κ) representable by a Slater determinant? always when the system contains more than four electrons by ensembles for fewer electrons Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 30 / 74

31 VRG exchange correlation functional The Vignale Rasolt Geldart (VRG) local vorticity exchange correlation functional E VRG (ν) = g(ρ(r)) ν(r) 2 dr, ν(r) = κ(r) ρ(r) based on the response of a uniform electron gas to a uniform magnetic field the function g(ρ) is fitted to RPA reference values Dissociation of triplet H 2 in a strong magnetic field: E(R) E( ) [me h ] FCI HF B3LYP BLYP BLYP VRG(TP) BLYP VRG(LHC) LDA LDA VRG(TP) LDA VRG(LHC) R [bohr] evaluation of the vorticity is problematic for small densities Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 31 / 74

32 CDFT meta-gga functionals The meta-gga functionals depend on the kinetic-energy density: N occ τ σ(r) = ϕ iσ (r) 2 τ σ(r) = τ σ(r) κσ(r) 2 2ρ i=1 σ(r) Dissociation of Ne 2 in a perpendicular field: cb TPSS-based forms give a remarkable accuracy mggas seem to deliver a reliable description of strong-field effects Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 32 / 74

33 Molecular Magnetic Properties 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 diamagnetism and paramagnetism molecules in strong fields DFT in magnetic fields Part 3: Molecular Magnetic Properties Derivatives and perturbation theory Zeeman and hyperfine operators magnetizabilities nuclear shielding constants indirect nuclear spin spin coupling constants Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 33 / 74

34 Derivatives and perturbation theory I Consider a normalized CI wave function: c = n=0 cn n, ct c = 1, m n = δ mn The basis functions n are the normalized CI eigenstates of the unperturbed problem: m H n = δ mne n, E 0 E 1 E 2 We here assume that the ground-state energy function depends on two external parameters: c H(x, y) c = cm m H(x, y) n cn, mn n c 2 n = 1 We construct a variational CI Lagrangian: L(x, y, c, µ) = ( ) c m m H(x, y) n c n µ cn 2 1 mn n The stationary conditions are given by L(x, y, c, µ) = 0 c n = 2 n H(x, y) c 2µc n = 0 = H(x, y)c = E 0 (x, y)c L(x, y, c, µ) = 0 µ = cn 2 1 = 0 n = c T c = 1 the first condition is the CI eigenvalue problem with ground-state energy E 0(x, y) = µ the second condition is the CI normalization condition Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 34 / 74

35 Derivatives and perturbation theory II Using the CI Lagrangian, we calculate CI energy derivative in the usual way: de dx = L x, d 2 E dxdy = 2 L x y + n 2 L x c n c n x, 2 L c n c n m c n x = 2 L x c m By inverting the electronic Hessian, we obtain the more compact expression: d 2 E dxdy = 2 L x y 2 [ L 2 ] 1 L 2 L x c mn m c m c n c n y We next evaluate the various partial derivatives at x = y = 0 where c = 0 : L x = 0 H x 0, 2 L x y = 0 2 H x y 0, 2 L c m c n = 2 m H E 0 n = 2(E n E 0 )δ mn 2 L = 2 n H x c x 0 n Inserted above, we recover Rayleigh Schrödinger perturbation theory to second order: de dx = 0 H d 2 E x 0, dxdy = 0 2 H x y H x n n H y 0 E n n E 0 Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 35 / 74

36 Review magnetic perturbations In atomic units, the molecular Hamiltonian is given by H = H 0 + A(r) p + B(r) s + 1 }{{}}{{} 2 A2 (r) }{{} orbital paramagnetic spin paramagnetic diamagnetic There are two kinds of magnetic perturbation operators: paramagnetic (may lower or raise energy) and diamagnetic (always raises energy) There are two kinds of paramagnetic operators: the orbital paramagnetic and spin paramagnetic In the study of magnetic properties, we are interested in two types of perturbations: A O (r) = 1 2 Bext r O A K (r) = α 2 M K r K r 3 K external field leads to Zeeman interactions nuclear magnetic moment leads to hyperfine interactions where α 1/137 is the fine-structure constant First- and second-order Rayleigh Schrödinger perturbation theory gives: E (1) = 0 A p + B s 0 E (2) = A 2 0 n 0 A p + B s n n A p + B s 0 E n E 0 Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 36 / 74

37 Orbital paramagnetic interactions: A p There are two sources of vector potential: the external field and the magnetic moments: A O = 1 2 Bext r O, Typical magnitudes of the perturbations: A K = α 2 M K r K r 3 K the external field is typically about 10 4 a.u. (NMR experiments) the nuclear vector potential is exceedingly small (about 10 8 a.u.) since: α 2 = c a.u., M K = γ K I K 10 4 a.u. We obtain the following orbital paramagnetic (Zeeman and hyperfine) operators A O p = 1 2 Bext r O p = 1 2 Bext r O p = 1 2 Bext L O orbital Zeeman A K p = α 2 M K r K p r 3 K = α 2 M K r K p r 3 K = α 2 M K L K r 3 K paramagnetic spin orbit (PSO) interactions depend on angular momenta L O and L K relative to O and R K, respectively we often write the PSO interaction in the manner: A K p = M K h PSO K, h PSO K = α 2 L K rk 3 These are imaginary singlet operators they have zero expectation values of closed-shell states they generate complex wave functions Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 37 / 74

38 Spin paramagnetic interactions: B s The spin interaction with the external uniform field B ext is trivial: B ext s spin Zeeman interaction should be compared with the orbital Zeeman interaction 1 2 Bext L O note the different prefactors in the two Zeeman terms! Taking the curl of the nuclear dipolar vector potential A K, we obtain the nuclear field: B K = 8πα2 3 δ(r K )M K + α 2 3r K (r K M K ) rk 2 M K rk 5 the first term is a contact interaction and contributes only at the nucleus the second term is a classical dipole field and contributes at a distance This magnetic field B K thus gives rise to two spin hyperfine interactions: B K s = M K (h FC K + hsd K ), h FC K h SD = 8πα2 3 δ(r K ) s Fermi contact (FC) K = α2 3r K rk T r K 2 I 3 r K 5 s spin dipole (SD) these are the Fermi-contact and spin dipole interactions the FC operator contributes when the electron passes through the nucleus the SD operator is a classical dipolar interaction, decaying as r 3 K These are real triplet operators, which change the spin of the wave function they have zero expectation values of closed-shell states they couple closed-shell states to triplet states Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 38 / 74

39 Diamagnetic operators: 1 2 A2 From A O and A K, we obtain three diamagnetic operators: A O A O = 1 4 (Bext r O) (B ext r O ) A O A K = α2 2 (B ext r O ) (M K r K ) r 3 K A K A L = α 4 (M K r K ) (M L r L ) r 3 K r 3 L typical magnitude 10 8 typical magnitude typical magnitude These are all real singlet operators their expectation values contribute to second-order magnetic properties they are all exceedingly small but nonetheless all observable Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 39 / 74

40 Perturbation theory with Zeeman and hyperfine operators The nonrelativistic electronic Hamiltonian: H = H 0 + H (1) + H (2) = H 0 + A (r) p + B (r) s A (r)2 Second-order Rayleigh Schrödinger perturbation theory: E (1) = 0 A p + B s 0 E (2) = 1 0 A n 0 A p + B s n n A p + B s 0 E n E 0 Vector potentials of the uniform external field and the nuclear magnetic moments: A(r) = 1 2 B r O, A K (r) = α 2 M K r K r 3 K henceforth, we use the notation B for B ext Orbital and spin Zeeman interactions with the external magnetic field: H (1) Z = 1 2 B L O + B s Orbital and spin hyperfine interactions with the nuclear magnetic moments: H (1) hf = α 2 M K L K } rk 3 {{ } PSO + 8πα2 3 δ (r K ) M K s }{{} FC + α 2 3(s r K )(r K M K ) (M K s)rk 2 } rk 5 {{ } SD Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 40 / 74

41 Zeeman and hyperfine interactions Zeeman SO SS, SO, OO SO PSO FC+SD hyperfine FC+SD PSO Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 41 / 74

42 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 0 + perm. magnetic moments {}}{ B T E (10) BT E (20) B }{{} K magnetizability hyperfine coupling {}}{ K B T E (11) K M K } {{ } shieldings + 1 M T K E(01) K + 1 M T K 2 E(02) 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 Second-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 ( 10 4 a.u.) 2) the nuclear magnetic moments M K couple weakly (µ 0 µ N 10 8 a.u.) Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 42 / 74

43 First-order molecular properties The first-order properties are expectation values of H (1) Permanent magnetic moment M = 0 H (1) Z 0 = L O + s 0 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 = 0 (1) H hf 0 = 8πα δ (rk ) s 0 MK + measure spin density at the nucleus important in electron paramagnetic resonance (EPR) recall: there are three hyperfine mechanisms: FC, SD and PSO Note: there are no first-order Zeeman or hyperfine couplings for closed-shell molecules all expectation values vanish for imaginary operators and triplet operators: c.c. ˆΩ imaginary c.c. c.c. ˆΩ triplet c.c. 0 Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 43 / 74

44 Molecular magnetizabilities Expand the molecular electronic energy in the external magnetic field: E (B) = E 0 B T M 1 2 BT ξb + The magnetizability describes the second-order energy: 0.1 ξ = d2 E db 2 = 0 2 H B H B n n H 0.08 B E n n E 0 = 1 0 ro T 4 r OI 3 r O ro T L O n n L T O E }{{} n n E 0 } 0 {{} diamagnetic term paramagnetic term The magnetizability describes the curvature at zero magnetic field: a) a) c) b) x b) d) x left: diamagnetic dependence on the field (ξ < 0); right: paramagnetic dependence on the field (ξ > 0) c) 0 Trygve Helgaker (CTCC, University of Oslo) d) Magnetic Properties ESQC15 44 / 74

45 Basis-set convergence of Hartree Fock magnetizabilities London orbitals are correct to first-order in the external magnetic field For this reason, basis-set convergence is usually improved RHF magnetizabilities of benzene: basis set χ xx χ yy χ zz London STO-3G G cc-pvdz aug-cc-pvdz origin CM STO-3G G cc-pvdz aug-cc-pvdz origin H STO-3G G cc-pvdz aug-cc-pvdz Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 45 / 74

46 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, (2009)) Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 46 / 74

47 Mean absolute errors for magnetizabilities Mean relative errors (MREs, %) in magnetizabilities of 27 molecules relative 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, (2009)) Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 47 / 74

48 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 10 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) Magnetic Properties ESQC15 48 / 74

49 Simulated 200 MHz NMR spectra of vinyllithium 12 C 2 H 3 6 Li Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 49 / 74

50 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 BT E (20) B K B T E (11) K M K KL M T K E(02) 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 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) Magnetic Properties ESQC15 50 / 74

51 Zeeman and hyperfine interactions Zeeman SO SS, SO, OO SO PSO FC+SD hyperfine FC+SD PSO Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 51 / 74

52 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 = 0 2 H dbdm K B M H B n n H M 0 K K E n n E 0 = α2 ro T 0 r K I 3 r O rk T 2 rk 3 0 α 0 L 2 O n n r 3 K LT K 0 E n n E 0 }{{}}{{} 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 0 0 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 = α2 S r 1 K 1 S Lamb formula Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 52 / 74

53 Benchmark calculations of BH shieldings (ppm) σ( 11 B) σ( 11 B) σ( 1 H) σ( 1 H) HF MP CCSD CCSD(T) CCSDT CCSDTQ CISD CISDT CISDTQ FCI TZP+ basis, R BH = pm, all electrons correlated J. Gauss and K. Ruud, Int. J. Quantum Chem. S29 (1995) 437 M. Kállay and J. Gauss, J. Chem. Phys. 120 (2004) 6841 Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 53 / 74

54 Coupled-cluster convergence of shielding constants in CO (ppm) CCSD CCSD(T) CCSDT CCSDTQ CCSDTQ5 FCI σ( 13 C) σ( 13 C) σ( 17 O) σ( 17 O) All calculations in the cc-pvdz basis and with a frozen core. Kállay and Gauss, J. Chem. Phys. 120 (2004) Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 54 / 74

55 Calculated and experimental shielding constants (ppm) HF CAS MP2 CCSD CCSD(T) exp. HF F ± 6 (300K) H ± 0.2 (300K) H 2O O ± 6 (300K) H ± 0.02 NH 3 N H ± 1.0 CH 4 C H F 2 F N 2 N ± 0.2 (300K) CO C ± 0.9 (eq) O ± 6 (eq) For references and details, see Chem. Rev. 99 (1999) 293. for exp. CO and H 2 O values, see Wasylishen and Bryce, JCP 117 (2002) Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 55 / 74

56 Kohn Sham shielding constants (ppm) HF LDA BLYP B3LYP KT2 CCSD(T) exp. HF F ± 6 H 2O O ± 6 NH 3 N CH 4 C F 2 F N 2 N ± 0.2 CO C ± 0.9 (eq) O ± 6 (eq) Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 56 / 74

57 MAE (Exp.) MAE (Emp. Eq.) NMR: mean absolute errors relative to experiment Mean absolute errors relative to experimental (blue) and empirical equilibrium values (red) MAE / ppm MAE (Exp.) MAE (Emp. Eq.) 0 Kohn Sham calculations give shielding constants of uneven quality errors increase when vibrational corrections are applied diamagnetic error is negligible (except for LDA and KT2) paramagnetic error is an order of magnitude larger: 10 ppm lack of current dependence contributes about 40% poor density dependence contributes about 60% Teale et al. JCP 138, (2013), Reimann et al. PCCP 17, (2015) Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 57 / 74

58 Mean absolute NMR shielding errors relative to empirical equilibrium values RHF LDA MAE / ppm CC GGA Hybrid OEP-Hybrid H C N O F All 10 0 Mean absolute errors (in ppm) for NMR shielding constants relative to empirical equilibrium values for H (white), C (grey), N (blue), O (red), and F (yellow). The total mean absolute errors over all nuclear types are shown by the purple bars. The DFT methodologies are arranged in the categories LDA, GGA, hybrid and OEP-hybrid. (Teal et al., JCP) Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 58 / 74

59 CDFT-mGGA shielding constants Box-whisker plot of errors in (C)DFT NMR shielding constants relative to extrapolated CCSD(T) values. All calculations use the aug-cc-pcvtz basis set. HF LDA PBE PBEVRGTP KT3 KT3VRGTP B972 cb98 ctpss ctpssh Error ppm Observations functionals are numerically stable to evaluate and physically motivated only modest improvements underlying flaws of functionals may dominate difficult to un-pick current effects included in a non-negotiable gauge correction Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 59 / 74

60 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 BT E (20) B K BT E (11) K M K KL MT K E(02) 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 (02) 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 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 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) Magnetic Properties ESQC15 60 / 74

61 Zeeman and hyperfine interactions Zeeman SO SS, SO, OO SO PSO FC+SD hyperfine FC+SD PSO Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 61 / 74

62 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 = 0 2 H dm K dm L M K M H M n n H K M 0 L L E n n E 0 Carrying out the differentiation of the Hamiltonian, we obtain Ramsey s expression: K KL = α 0 4 rk T r LI 3 r K rl T rk 3 r L 3 0 2α 0 r 3 4 K L K n n r 3 L LT L 0 E n n E 0 }{{}}{{} diamagnetic spin orbit (DSO) paramagnetic spin orbit (PSO) 0 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 0 r 5 K r 5 L 2α 4 E n n E 0 }{{} 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) Magnetic Properties ESQC15 62 / 74

63 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)! 200 PSO FC PSO FC FC FC FC FC SD FC FC PSO FC SD FC -100 SD H2 HF H2O OH NH3 NH CH4 C2H4 HCN CH CC NC N2 CO C2H2 CC Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 63 / 74

64 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 properly 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% 10%) 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) Magnetic Properties ESQC15 64 / 74

65 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 1 D2p g FCI 1 P1s2p 3 P1s2p 1 1 g 1Σ u 2 1 u 1Σg1Σu 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 RHF CAS B3LYP Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 65 / 74

66 Reduced spin spin coupling constants by wave-function theory RHF CAS RAS SOPPA CCSD CC3 exp vib HF 1 K HF CO 1 K CO N 2 1 K NN H 2O 1 K OH K HH NH 3 1 K NH K HH C 2H 4 1 K CC K CH K CH K HH K cis K tns abs at R e % SOPPA: second-order polarization-propagator approximation Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 66 / 74

67 Reduced spin spin coupling constants by density-functional theory LDA BLYP B3LYP PBE B97-3 RAS exp vib HF 1 K HF CO 1 K CO N 2 1 K NN H 2O 1 K OH K HH NH 3 1 K NH K HH C 2H 4 1 K CC K CH K CH K HH K cis K tns abs at R e % Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 67 / 74

68 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 CAS RAS SOPPA CCSD 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) Magnetic Properties ESQC15 68 / 74

69 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: empirical DFT Good agreement with the (empirically constructed) Karplus curve Trygve Helgaker (CTCC, University of Oslo) Magnetic Properties ESQC15 69 / 74