Molecular Magnetic Properties. The 11th Sostrup Summer School. Quantum Chemistry and Molecular Properties July 4 16, 2010

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1 1 Molecular Magnetic Properties The 11th Sostrup Summer School Quantum Chemistry and Molecular Properties July 4 16, 2010 Trygve Helgaker Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Oslo, Norway

2 2 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 φ i (r i )+ 1 A 2 i(r i ) 2 i }{{} diamagnetic In the study of magnetic properties, we are interested in the following two types of perturbations: uniform external magnetic field B, with vector potential A ext (r) = 1 B r leads to Zeeman interactions 2 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 in both cases, we may set φ = 0 In the following, we shall consider each of these perturbations in turn.

3 3 Zeeman interactions The scalar and vector potentials of the uniform (static) fields E and B are given by: E = φ A t = const B = A = const φ(r) = E r A(r) = 1 2 B r Interaction with the electrostatic field: φ(r i ) = E r i = E d e, d e = i i i r i electric dipole operator 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 orbital ang. mom. op. i i i 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: H z = B d m, d m = 1 (L+2S) Zeeman interaction 2

4 4 Hyperfine interactions The nuclear moments set up a magnetic vector potential ( 10 8 a.u.): A(r) = α 2 K M K r K r 3 K, α 2 = c a.u., M K = γ K hi K 10 4 a.u. This vector potential gives rise to the following paramagnetic hyperfine interaction A p = K M T K hpso K, hpso K = α2r K p r 3 K = α 2L K r 3 K 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 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(r) then gives rise to two distinct first-order triplet operators: B s = K M T K (hfc K +hsd K ), r 5 K h FC K = 8πα2 3 δ(r K )s Fermi contact (FC) h SD K = α2 3r Kr T K r2 K I 3 r K 5 s spin dipole (SD)

5 5 Review The nonrelativistic electronic Hamiltonian: H = H 0 +H (1) +H (2) = H 0 +A(r) p+b(r) s+ 1 2 A(r)2 Rayleigh Schrödinger perturbation theory to second order: E (1) = 0 A p+b s 0 E (2) = A 2 0 A p+b s n n A p+b s 0 0 n 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 2M K r K r 3 K, A(r) = B(r), A(r) = 0 Orbital and spin Zeeman interactions with the external magnetic field: H (1) Zeeman = 1 2 B L O +B s Orbital and spin hyperfine interactions with the nuclear magnetic moments: H (1) hyperfine = α2m K L K r 3 K }{{} PSO + 8πα2 δ(r K )M K s 3 }{{} FC +α 23(s r K)(r K M K ) (M K s)r 2 K r 5 K }{{} SD

6 The many interactions: Coulomb, Zeeman and hyperfine 6

7 7 Gauge transformation of the Schrödinger equation 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 H i ) Ψ, t t the new wave function must be related to the old one by a compensating unitary transformation: Ψ = exp( if)ψ No observable properties such as the electron density are then affected: ρ = (Ψ ) Ψ = [Ψexp( if)] [exp( if)ψ] = Ψ Ψ = ρ

8 8 Gauge-origin transformations Different choices of gauge origin in the external vector potential are related by a gauge transformation: A O (r) = 1 2 B (r O) A K (r) = A O (r) A O (K) = A O (r)+ f, f (r) = A O (K) r the exact wave function transforms accordingly and gives gauge-invariant results: Ψ exact K = exp[ia O (K) r]ψ exact O approximate wave functions are in general not able to carry out this transformation: Ψ approx K exp[ia O (K) r]ψ approx O different gauge origins may therefore give different results! We might contemplate attaching an explicit phase factor to the wave function: Ψ approx K def = exp[ia O (K) r]ψ approx O for any K, this approach produces the same result as with the gauge origin at O however, no natural, best gauge origin can usually be identified (except for atoms) in any case, we might as well carry out the calculation with the origin at O! applied to individual AOs, however, this approach makes much more sense!

9 9 Natural gauge origin for AOs Assume AOs positioned at K with the following properties: H 0 χ lm = E 0 χ lm, L K z χ lm = m l χ lm, L K = i(r K) We first choose the gauge origin to be at K: A K (r) = 1 2 B (r K) the AOs χ lm are then correct to first order in B: [ H K (B)χ lm = H BLK z +O ( B 2)] χ lm = Next, we put the gauge origin at O K: A O (r) = 1 2 B (r O) [ E m lb +O ( B 2)] χ lm the AOs χ lm are now correct only to zero order in B: [ H O (B)χ lm = H BLO z +O( B 2)] [ χ lm E m lb +O ( B 2)] χ lm Standard AOs are biased towards K!

10 10 London orbitals A traditional AO gives best description with the gauge origin at its position K. Attach to each AO a phase factor that represents the gauge-origin transformation to its position K from the global origin O: ω lm = exp[ia K (O) r]χ lm = exp [ i 1 2 B (O K) r] χ lm Each AO now behaves as if the global gauge origin were at its position! In particular, all AOs are now correct to first order in B, for any global origin O. The calculations become gauge-origin independent uniform (good) quality follows These are the London orbitals (1937), also known as GIAOs (gauge-origin independent AOs or gauge-origin including AOs).

11 11 Review The nonrelativistic electronic Hamiltonian: H = H 0 +H (1) +H (2) = H 0 +A(r) p+b(r) s+ 1 2 A(r)2 Rayleigh Schrödinger perturbation theory to second order: E (1) = 0 A p+b s 0 E (2) = A 2 0 A p+b s n n A p+b s 0 0 n 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 2M K r K r 3 K, A(r) = B(r), A(r) = 0 Orbital and spin Zeeman interactions with the external magnetic field: H (1) Zeeman = 1 2 B L O +B s Orbital and spin hyperfine interactions with the nuclear magnetic moments: H (1) hyperfine = α2m K L K r 3 K }{{} PSO + 8πα2 δ(r K )M K s 3 }{{} FC +α 23(s r K)(r K M K ) (M K s)r 2 K r 5 K }{{} SD

12 12 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 BT E (20) B }{{} perm. magnetic moments {}}{ B T E (10) + magnetizability K hyperfine coupling {}}{ K B T E (11) K M K } {{ } shieldings + 1 M T K E(01) K KL M T K E(02) KL M L } {{ } spin spin couplings The first-order terms vanish for closed-shell systems because of symmetry: c.c. ˆΩ c.c. imaginary c.c. ˆΩ c.c. triplet 0 Higher-order terms are negligible since the perturbations are tiny: 1) the magnetic induction B is weak ( 10 5 a.u.) 2) the nuclear magnetic moments M K couple weakly (µ 0 µ N 10 8 a.u.) We shall therefore consider only the second-order terms: the magnetizability, the shieldings, and the spin spin couplings +

13 13 The magnetizability Assume zero nuclear magnetic moments and expand the molecular electronic energy in the external magnetic induction B: E(B) = E 0 +B T E (10) BT E (20) B+ The molecular magnetic moment at B is now given by M mol (B) def = de(b) db = E(10) E (20) B+ = M perm +ξb+, where we have introduced the permanent magnetic moment and the magnetizability: M perm = E (10) = de permanent magnetic moment db B=0 describes the first-order change in the energy but vanishes for closed-shell systems ξ = E (20) = d2 E db 2 molecular magnetizability B=0 describes the second-order energy and the first-order induced magnetic moment The magnetizability is responsible for molecular diamagnetism, important for molecules without a permanent magnetic moment.

14 14 Paramagnetic interactions for the magnetizability For magnetizability we need the Zeeman interactions 1 B L and B S 2 the triplet operator B S does not contribute for closed shells

15 15 The calculation of magnetizabilities The molecular magnetizability of a closed-shell system: 0 ξ = d2 E db 2 = 0 2 H B n = 1 0 r O r T O (r 4 T 0 O r O )I }{{} n diamagnetic term H B n n H B 0 E n E 0 0 L O n n L T O 0 E n E 0 } {{ } paramagnetic term The (often) dominant diamagnetic term arises from differentiation of the operator: 1 [ B 2 r 2 O (B r O)(B r O ) ] 2 A2 (B) = 1 8 (B r O) (B r O ) = 1 8 the isotropic part of the diamagnetic contribution is given by: ξ dia = 1 3 Trξ dia = 1 0 x 2 6 O +yo 2 O +z2 1 0 = 0 r O Only the orbital Zeeman interaction contributions to the paramagnetic term: 1 2 L O S 0 0 singlet state system surface for 1 S systems (closed-shell atoms), the paramagnetic term vanishes altogether: 1 S 0 gauge origin at nucleus

16 16 Hartree Fock magnetizabilities Basis-set requirements for magnetizabilities are modest if London orbitals are used: basis cc-pvdz cc-pvtz cc-pvqz HF basis-set error 2.8% 1.0% 0.4% The HF model overestimates the magnitude of magnetizabilities by 5% 10%: JT 2 HF exp. diff. H 2 O % NH % CH % CO % PH % H 2 S % C 3 H % CSO % CS % compare with polarizabilities, which require large basis sets and are underestimated

17 17 Nonperturbative calculations of molecules in magnetic fields It is possible to carry out nonperturbative studies of molecules in magnetic fields Below we have plotted the energy of C 20 as a function of the magnetic field strength C 20 is paramagnetic in weak fields but diamagnetic in strong fields this behaviour cannot be reproduced by expansions, which diverge for strong fields

18 18 Nonperturbative calculations of molecules in magnetic fields Strong magnetic fields affect molecules in many ways Below we have plotted the potential energy curve of F 2 in different perpendicular fields at short internuclear separations, the electronic system is always diamagnetic at large separations, it is paramagnetic for weak and diamagnetic for strong fields the molecule is stretched by the applied field

19 19 High-resolution NMR spin Hamiltonian Consider a molecule in the presence of an external 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 hi 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: H NMR = K γ K h(1 σ K )BI K z + K>Lγ K γ L h 2 K KL I K I L where we have introduced } {{ } 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

20 20 Simulated 200 MHz NMR spectra of vinyllithium experiment RHF MCSCF B3LYP

21 21 Nuclear shielding constants Recall the energy expansion for a closed-shell molecule in the presence of an external field B and nuclear magnetic moments M K : E(B,M) = E BT E (20) B+ 1 2 In this expansion, E (11) K magnetic moments: 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 in the absence of electrons (i.e., in vacuum), this coupling is identical to I 3 : HZeeman 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 Since the nuclear shielding constants arise from a hyperfine interaction between the electrons and the nuclei, it is proportional to α and is measured in ppm. The nuclear Zeeman interaction, which does not enter the electronic problem, has here been introduced in a purely ad hoc fashion. Its status is otherwise similar to that of the Coulomb nuclear nuclear repulsion operator.

22 22 Paramagnetic interactions for shielding constants For shieldings, we must combine Zeeman and hyperfine interactions for closed shells, only the singlet operators 1 2 B L and M K r 3 K L K contribute

23 23 The calculation of nuclear shielding tensors Nuclear shielding tensors of a closed-shell system: 0 2 H B M 0 2 K n r T O 0 r KI 3 r O r T K 2 rk 3 0 α 2 n }{{} diamagnetic term σ K = d2 E el dbdm K = = α2 0 H B n n E n E 0 0 L O n n r 3 H M K 0 K LT K 0 E n E 0 } {{ } 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 the shielding is given by (assuming gauge origin at the nucleus): σ Lamb = 1 3 α2 1 S r 1 1 S Lamb formula K

24 24 Benchmark calculations of BH shieldings 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

25 25 Calculated and experimental shielding constants HF CAS MP2 CCSD CCSD(T) exp. HF F ± 6 (300K) H ± 0.2 (300K) H 2 O 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 experimental CO and H 2 O, see Wasylishen and Bryce, JCP 117 (2002) 10061

26 26 DFT shielding constants HF LDA BLYP B3LYP KT2 CCSD(T) exp. HF F ± 6 H 2 O O ±6 NH 3 N CH 4 C F 2 F N 2 N ±0.2 CO C ± 0.9 (eq) O ± 6 (eq)

27 27 Coupled-cluster convergence of shielding constants in CO 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) 6841.

28 28 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+ 1 2 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

29 29 Paramagnetic interactions for spin spin coupling constants For spin spin coupling constants, only hyperfine interactions contribute both singlet (PSO) and triplet (FC and SD) operators are now needed

30 30 K KL = The calculation of indirect nuclear spin spin coupling tensors The indirect nuclear spin spin coupling tensor of a closed-shell system is given by: 0 H n n M H 0 K M L d 2 E el dm K dm L = 0 2 H M K M 0 2 L n Carrying out the differentiation, we obtain: K KL = α 0 4 r T K r LI 3 r K r T L rk 3 r3 0 2α 4 L n }{{} diamagnetic spin orbit (DSO) 0 r 3 K L K E n E 0 n n r 3 L LT L 0 E n E 0 } {{ } paramagnetic spin orbit (PSO) 2α 4 0 8π 3 δ(r K)s+ 3r Kr T K r2 K I 3 r K 5 s n n 8π 3 δ(r L)s T + 3r Lr T L r2 L I 3 r L 5 s T 0 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

31 31 Calculations of indirect nuclear spin spin coupling constants The calculation of spin spin coupling constants is a challenging task: triplet as well as singlet perturbations are involved electron correlation important the Hartree Fock model fails abysmally the dominant FC contribution requires an accurate description of the electron density at the nuclei (large decontracted s sets) We must solve a large number of response equations: 3 singlet equations and 7 triplet equations for each nucleus for shieldings, only 3 equations are required, for molecules of all sizes Spin spin couplings are very sensitive to the molecular geometry: equilibrium structures must be chosen carefully large vibrational corrections (often 5% 10%) However, unlike in shielding calculations, there is no need for London orbitals since no external magnetic field is involved. For heavy elements, a relativistic treatment may be necessary.

32 32 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 +JPSO KL +JFC 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 C H C2H4 C C HCN N C N2 CO C2H2 C C

33 33 RHF and the triplet instability problem RHF does not in general work for spin spin calculations: the RHF wave function often becomes triplet unstable at or close to such instabilities, the RHF description of spin interactions becomes unphysical the spin spin coupling constants of C 2 H 4 : Hz 1 J CC 1 J CH 2 J CH 2 J HH 3 J cis 3 J trans exp RHF CAS B3LYP

34 34 Reduced spin spin coupling constants (10 19 kg m 2 s 2 Å 2 ) RHF CAS RAS SOPPA CCSD CC3 exp vib HF 1 J HF CO 1 J CO N 2 1 J NN H 2 O 1 J OH J HH NH 3 1 J NH J HH C 2 H 4 1 J CC J CH J CH J HH J cis J tns abs at R e %

35 35 Reduced spin spin coupling constants (10 19 kg m 2 s 2 Å 2 ) LDA BLYP B3LYP PBE B97-3 RAS exp vib HF 1 J HF CO 1 J CO N 2 1 J NN H 2 O 1 J OH J HH NH 3 1 J NH J HH C 2 H 4 1 J CC J CH J CH J HH J cis J tns abs at R e %

36 36 Comparison of density-functional and wave-function theory normal distributions of errors for these molecules and some other systems for which vibrational corrections have been made: HF LDA BLYP B3LYP CAS RAS SOPPA CCSD some observations: HF has a very broad distribution and overestimates strongly LDA underestimates only slightly, but has a large standard deviation BLYP reduces the LDA errors by a factor of two B3LYP improves upon GGA (but not as dramatically as for other properties) B3LYP errors are similar to those of CASSCF and about twice those of the dynamically correlated methods RASSCF, SOPPA, and CCSD the most accurate method appears to be CCSD the situation is much less satisfactory than for geometries and atomization energies

37 37 The Karplus curve Vicinal couplings depend critically on the dihedral angle: 3 J HH in ethane as a function of the dihedral angle: 14 empirical 12 DFT The agreement with the empirical Karplus curve is good.

38 38 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

39 39 Valinomycin LDA/6-31G spin spin couplings to CH, CO, CN, CC greater than 0.01 Hz

40 40 Valinomycin LDA/6-31G spin spin couplings to CH, CO, CN, CC greater than 0.01 Hz

41 41 Valinomycin LDA/6-31G spin spin couplings to CH, CO, CN, CC greater than 0.01 Hz

42 42 Valinomycin LDA/6-31G spin spin couplings to CH, CO, CN, CC greater than 0.01 Hz

43 43 Valinomycin LDA/6-31G spin spin couplings to CH, CO, CN, CC greater than 0.01 Hz

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