Lecture 11: calculation of magnetic parameters, part II

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1 TO DO IS TO BE SOCRATES TO BE IS TO DO SARTRE OO BE DO BE DO SINATRA Lecture 11: calculation of magnetic parameters, part II classification of magnetic perturbations, nuclear quadrupole interaction, J-coupling, g-tensor Dr Ilya Kuprov, University of Southampton, 2012 (for all lecture notes and video records see

2 Magnetic perturbation operators There is a huge number of these. They are neatly summarized in Jensen s book:

3 Magnetic perturbation operators From the vector potential relations introduced in the previous lecture after a lot of effort we can obtain (i,j indices runs over electrons and n,k indices run over nuclei): SZ ˆ SZ ˆ ˆ OZ 1 e B, ˆ N, ˆ ˆ H ˆ i g SiB Hn gn SnB Hi A pi rig pi B, 2 T T ˆ DM T T DS N ˆ, ˆ g r n ig rin rinr ig Hi A rig rig ri grig B Hn, i B S 2 3 n 2 8 2c r in 2 ˆ DSO gg n k T T H nk, i N ˆ rin rik rinr ik ˆ FC 8 N ˆ ˆ, ˆ gegnb S 2 n 3 3 Sk H ni, 2 rin SiSn 2c rinrik 3c ˆ T T ˆ PSO gn N ˆ rin pi SD N ˆ 3 ˆ,, ˆ gegnb rinrin rinr in Hni S 2 n H 3 ni, S 2 i S 5 n c rin c rin T 2 2 T 2 2 ˆ SD g Hij e B ˆ rij rij 3rijr ij ˆ FC 4 e B ˆ ˆ, ˆ g S 2 i S 5 j Hij 2 rij SiSj 2c r ij 3c ˆ ˆ T T ˆ 2 SO ge B ˆ rij pi rij pj DS e B, ˆ g rig rij rij r ig ˆ Hij S 2 i H 3 ij B 2 3 S 2c rij 2c r i ij

4 Properties: nuclear quadrupolar interaction Nuclei with spin higher than ½ have non-spherical charge density distribution. The resulting quadrupole moment Q r 3r r r d r nk interacts with the electric field gradient at the nucleus point: n k 2 3 nk 1 E U Q V V 6 2 n nk nk nk nk rk r nucleus nrk nucleus The direction of the quadrupole moment of the nucleus depends on the direction of its spin, meaning that an essentially electrostatic interaction becomes also a magnetic interaction with the following Hamiltonian: ˆ 1 ˆ ˆ eq 3 ˆ ˆ ˆ ˆ ˆ H QnkVnk Qnk SnSk SkSn nks 6 nk S S where eq is the quantity known as the nuclear quadrupole moment. It is a fundamental property of the nucleus. Any method for computing NQI must therefore correctly predict the derivatives of the electrostatic potential.

5 Properties: nuclear quadrupolar interaction W.C. Bailey ( Electron density distribution must be captured very precisely, so a large and flexible basis set is necessary at least triple-zeta, with multiple sets of diffuse and polarization functions as well as core polarization functions (e.g. in aug-cc-pcvtz basis).

6 Properties: nuclear quadrupolar interaction All factors affecting the electrostatic potential (in particular, crystal lattice and solvent effects) must be accounted for. Explicit first solvation shell is in most cases required for PCM calculations. M. Pavanello, B. Mennucci, J. Tomasi (

7 Properties: J-coupling NMR spectroscopists often forget that J-coupling is a tensor (that its anisotropy is often inconsequential is a different matter). It is defined as the second derivative of the total energy with respect to the nuclear dipole moments: E 2 A B AB E J, Jnk A B n k The contributions to perturbation theory integrals (summed over electrons) are: ˆ S diamagnetic spin-orbit PSO PSO 0 A m m B 0 0 Hˆ AB 0 ˆ m E0 Em A J B SD FC SD FC T ˆ ˆ ˆ ˆ 0 HA HA m m HB HB 0 DSO S S AB S S S m T paramagnetic spin-orbit spindipole Hˆ Hˆ E S 0 E m T S T Fermi contact The Fermi contact term dominates in CHNO molecules (up to 80% of the total).

8 Properties: J-coupling Polarized core basis sets (aug-cc-pcvnz) with large cardinal numbers are essential the basis set convergence for J-coupling tends to be slow. Non-FC terms are often prominent for long-range J-couplings.

9 Properties: J-coupling Highly correlated response methods (EOM-CCSD, SOPPA/CCSD) are often required.

10 Properties: g-tensor The g-tensor is defined in a similar manner to chemical shielding as a second derivative of the energy with respect to the applied magnetic field and the electron magnetic moment: 2 e E E gb, gnk, g g e e1 g B n k In addition to the perturbation terms listed above, two extra relativistic terms are often essential as perturbations the electron-nuclear spin-orbit term and the relativistic correction to the electron Zeeman operator: Hˆ g Z ˆ r pˆ, ˆ g S H S B p ˆ 2 ˆ SO e B n in i ZR e B ni, 2 i 3 i 2 i i 2mc rin 2mc These terms become important for the inner electrons of heavy elements, for which the distance to the nucleus is small, momentum is large and the nuclear charge is much greater than 1. For this reason, the SO term is often too large to be treated as a perturbation and needs to be included into the primary Hamiltonian.

11 Properties: g-tensor The contributions to perturbation theory integrals are: 1 2 m g Hˆ Hˆ DS RZ 0 0 Hˆ Lˆ Lˆ Hˆ SO SO 0 m m g 0 0 g m m 0 E 0 E m For heavy elements this breaks down and a relativistic description is necessary. The four-component relativistic treatment shows no systematic improvement over the two-component and the scalar approximations.

12 Properties: g-tensor

13 Properties: g-tensor The perturbation theory approach is not applicable to systems with g-shifts in excess of 20,000 ppm. For heavy atoms, ZORA is preferred to the Pauli approximation. High level treatment of electron correlation is essential (orbital energies occur in the perturbation theory denominators).

14 Properties: zero-field splitting ZFS is a quadratic spin coupling with the following spin Hamiltonian: ˆ ˆ ˆ H D S S S 1/3E S S SZ S ˆ 2 ˆ 2 ˆ 2 Z X Y The elements of the ZFS tensor are defined via the derivatives of the total energy with respect to the components of the electron magnetic moment: Z kl E Hˆ Hˆ Hˆ k m m l k l k l m E0 Em The primary contribution is from the inter-electron point dipole interaction: ˆ r r 3r r H S S 2 2 T T SD ge B ˆ ij ij ij ij ˆ ij 2 i 5 j 2c r ij This contribution is a ground state property and is therefore quite cheap. It vanishes in closed-shell systems and systems with only one unpaired electron.

15 Properties: zero-field splitting Another (often smaller) contribution comes from the spin-orbit coupling: ˆ ˆ 2 ˆ ˆ g ˆ r p g ˆ r p r p H S S SO n N in i e B ij i ij j nij, 2 n 3 2 i 3 c rin 2c rij All the usual health warnings about not treating large spin-orbit couplings with perturbation theory apply. It is also usually advantageous to take perturbation theory denominators from a higher level treatment.

16 Properties: zero-field splitting

17 Properties: zero-field splitting A specific (though not completely understood at the moment) observation for ZFS is: do not use spin-unrestricted DFT.

18 Properties: exchange interaction The inter-electron spin coupling Hamiltonian has three parts: ˆ ˆ ˆ ˆ ˆ ˆ ˆ H L DS 2JL S dl S X The first term is the usual dipole-dipole interaction. The second term ˆ ˆ Hˆ 2J LS 2J Lˆ Sˆ Lˆ Sˆ Lˆ Sˆ X X X Y Y Z Z is known as symmetric exchange coupling, it comes from the non-classical spindependent part of the Coulomb interaction. For a two-electron system: 1 1 2J T T S S r12 r12 ELS EHS * 1 * J x x x x dvdv r12 J Sˆ 2 2 HS Sˆ LS Because J is related to singlet-triplet (or, more generally, HS-LS) energy gap, it may be computed quite simply using the Yamaguchi equation above. All the usual warnings about charge transfer excitations apply.

19 Properties: exchange interaction The antisymmetric component arises from the spin-orbit coupling: ˆ ˆ 2 ˆ ˆ g ˆ r p g ˆ r p r p H S S SO n N in i e B ij i ij j nij, 2 n 3 2 i 3 c rin 2c rij The exchange interaction is often difficult to disentangle from the dipolar coupling and zero-field splitting. For this reason, it is often consigned to the role of a fudge factor, which is unfortunately far too often necessary to obtain a decent data fit.

20 Properties: magnetic circular dichroism MCD measures differential absorption of left circularly polarized (LCP) and right circularly polarized (RCP) photons from the electromagnetic field E 2 2 Ni N j i mlcp j i mrcp j f E i j which is induced in a sample by a strong magnetic field applied parallel to the direction of light propagation. To first order in B this can be rewritten as: f E C E E kt 0 B A1 B0 f E where the three coefficients depend on second-order perturbation theory integrals involving spin operators and electric dipole moments (see Frank Neese s recent work). MCD is a very specialized area little practical guidance has so far been published.