Molecular Magnetism. Magnetic Resonance Parameters. Trygve Helgaker

Transcription

1 Molecular Magnetism Magnetic Resonance Parameters 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, 2012 Trygve Helgaker (CTCC, University of Oslo) Magnetic resonance parameters LCT, UPMC, November / 36

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 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) Magnetic resonance parameters LCT, UPMC, November / 36

3 Review: magnetic perturbations In atomic units, the molecular Hamiltonian is given by H = H 0 + 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) Magnetic resonance parameters LCT, UPMC, November / 36

4 Zeeman interactions The scalar and vector potentials of the uniform (static) fields E and B are given by: E = φ A } { = const φ(r) = E r t B = A = const A(r) = 1 2 B r Electrostatic interaction: φ(r i ) = E r i = E d e, i i d e = i Orbital paramagnetic interaction with the magnetostatic field: A p i = 1 B r 2 i p i = 1 2 B L, L = r i p i i i i r i electric dipole operator Spin paramagnetic interaction with the magnetostatic field: B s i = B S, S = s i spin ang. mom. op. i i Total paramagnetic interaction with a uniform magnetic field: orbital ang. mom. op. H z = B d m, d m = 1 (L + 2S) Zeeman interaction 2 Trygve Helgaker (CTCC, University of Oslo) Magnetic resonance parameters LCT, UPMC, November / 36

5 Hyperfine interactions The nuclear magnetic moments set up a magnetic vector potential ( 10 8 a.u.): A(r) = α 2 K M K r K rk 3, α 2 = c a.u., M K = γ K I K 10 4 a.u. This vector potential gives rise to the following paramagnetic hyperfine interaction A p = K M T K hpso K, h PSO K = α 2 r K p rk 3 = α 2 L K rk 3 paramagnetic SO (PSO) magnetic moment interacts with the electron s orbital motion about the nucleus Taking the curl of this vector potential, we obtain: B(r) = A(r) = 8πα2 3 δ(r K )M K + α 2 K K 3r K (r K M K ) r 2 K M K r 5 K the first contact term contributes only when the electron is at the nucleus the second 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(r) then gives rise to two distinct first-order triplet operators: B s = K M T K (hfc 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) Trygve Helgaker (CTCC, University of Oslo) Magnetic resonance parameters LCT, UPMC, November / 36

6 Perturbation theory 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 rk 3, A (r) = B (r), A (r) = 0 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 resonance parameters LCT, UPMC, November / 36

7 Zeeman and hyperfine interactions Trygve Helgaker (CTCC, University of Oslo) Magnetic resonance parameters LCT, UPMC, November / 36

8 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 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 ( 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 resonance parameters LCT, UPMC, November / 36

9 First-order molecular properties The first-order properties are expectation values of H (1) Permanent magnetic moment M = 0 (1) H 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 H (1) hf 0 = 8πα2 3 0 δ (r K ) s 0 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. 0 Trygve Helgaker (CTCC, University of Oslo) Magnetic resonance parameters LCT, UPMC, November / 36

10 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 resonance parameters LCT, UPMC, November / 36

11 Simulated 200 MHz NMR spectra of vinyllithium 12 C 2 H 3 6 Li experiment RHF MCSCF B3LYP Trygve Helgaker (CTCC, University of Oslo) Magnetic resonance parameters LCT, UPMC, November / 36

12 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 resonance parameters LCT, UPMC, November / 36

13 Zeeman and hyperfine interactions Trygve Helgaker (CTCC, University of Oslo) Magnetic resonance parameters LCT, UPMC, November / 36

14 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 resonance parameters LCT, UPMC, November / 36

15 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 resonance parameters LCT, UPMC, November / 36

16 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 resonance parameters LCT, UPMC, November / 36

17 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 resonance parameters LCT, UPMC, November / 36

18 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 resonance parameters LCT, UPMC, November / 36

19 NMR shieldings: normal distributions of errors LDA 0.06 BLYP 0.06 PBE KT B3LYP B B PBE CAM B3LYP O B3LYP O B O B O PBE O CAM B3LYP CCSD Normal distributions of the errors in NMR shielding constants calculated in the aug-cc-pcvqz basis relative to the CCSD(T)/aug-cc-pCV[TQ]Z benchmark data set for 26 molecules (Teale et al., JCP) Trygve Helgaker (CTCC, University of Oslo) Magnetic resonance parameters LCT, UPMC, November / 36

20 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 resonance parameters LCT, UPMC, November / 36

21 MAE (Exp.) MAE (Emp. Eq.) NMR shieldings: mean absolute errors relative to experiment MAE / ppm MAE (Exp.) MAE (Emp. Eq.) 0 Mean absolute errors (in ppm) for NMR shielding constants relative to experimental (blue) and empirical equilibrium values (red). The inclusion of vibrational corrections in the empirical equilibrium values leads to a degradation of the quality of the RHF and DFT results but to a notable improvement for the CCSD and CCSD(T) methods. (Teale et al., JCP) Trygve Helgaker (CTCC, University of Oslo) Magnetic resonance parameters LCT, UPMC, November / 36

22 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 resonance parameters LCT, UPMC, November / 36

23 Zeeman and hyperfine interactions Trygve Helgaker (CTCC, University of Oslo) Magnetic resonance parameters LCT, UPMC, November / 36

24 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 resonance parameters LCT, UPMC, November / 36

25 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% 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 resonance parameters LCT, UPMC, November / 36

26 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 O H NH3 N H CH4 C2H4 HCN C H C C N C N2 CO C2H2 C C Trygve Helgaker (CTCC, University of Oslo) Magnetic resonance parameters LCT, UPMC, November / 36

27 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 D 2p g FCI 1 P 1s2p 3 P 1s2p 1 1 g 1Σ u 2 1 u 1Σg1Σu R ionic cov ion 3 u 1Σg1Σu covalent 2 1 g 1Σ g 2 1 S 1s 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 resonance parameters LCT, UPMC, November / 36

28 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 resonance parameters LCT, UPMC, November / 36

29 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 resonance parameters LCT, UPMC, November / 36

30 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 resonance parameters LCT, UPMC, November / 36

31 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 resonance parameters LCT, UPMC, November / 36

32 Valinomycin C 54 H 90 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: the coupling constants decay in characteristic fashion, which we shall examine most of the indirect couplings beyond 500 pm are small and cannot be detected Trygve Helgaker (CTCC, University of Oslo) Magnetic resonance parameters LCT, UPMC, November / 36

33 Valinomycin C 54 H 90 N 8 O 18 One-bond spin spin couplings to CH, CO, CN, CC greater than 0.01 Hz Trygve Helgaker (CTCC, University of Oslo) Magnetic resonance parameters LCT, UPMC, November / 36

34 Valinomycin C 54 H 90 N 8 O 18 Two-bond spin spin couplings to CH, CO, CN, CC greater than 0.01 Hz Trygve Helgaker (CTCC, University of Oslo) Magnetic resonance parameters LCT, UPMC, November / 36

35 Valinomycin C 54 H 90 N 8 O 18 Three-bond spin spin couplings to CH, CO, CN, CC greater than 0.01 Hz Trygve Helgaker (CTCC, University of Oslo) Magnetic resonance parameters LCT, UPMC, November / 36

36 Valinomycin C 54 H 90 N 8 O 18 Four-bond spin spin couplings to CH, CO, CN, CC greater than 0.01 Hz Trygve Helgaker (CTCC, University of Oslo) Magnetic resonance parameters LCT, UPMC, November / 36