Lecture 11: calculation ofmagnetic parameters, part II

classification of magnetic perturbations,nuclear quadrupole interaction, J-coupling, g-tensor

“TO DO IS TO BE” – SOCRATES“TO BE IS TO DO” – SARTRE“OO BE DO BE DO” – SINATRA

Dr Ilya Kuprov, University of Southampton, 2012

(for all lecture notes and video records see http://spindynamics.org)

Magnetic perturbation operatorsThere is a huge number of these. They are neatly summarized in Jensen’s book:

Magnetic perturbation operatorsFrom the vector potential relations introduced in the previous lecture after a lot ofeffort we can obtain (i,j indices runs over electrons and n,k indices run over nuclei):

e B N

T T2 T T N

S

2

SZ SZ OZ

DM3

DS,

D O,

1ˆ ˆ ˆ ˆ, , ˆ ˆ ,2

1 1 ˆ, 2 8 2

ˆ

ˆ ˆ

ˆ

i n n i ig i

ig in in igni

i

g ig

n

ig ig nin

n k

i n i

i n i

k

H H H

H Hr

H

g S B g S B A p r p B

r r rgA r r r r B B Sc r

g g

2 T TN N4 3 3 2T T

N,

N, 2 3 2 5

FC,

PSO SD

SD

8ˆ ˆ ˆ ˆ, 2 3

ˆ 3ˆ ˆ ˆ,

ˆ

ˆ ˆ

ˆ

in ik in ik e n Bn k in i n

in ik

n in i e n B in in in inn i n

in in

n i

n i n i

ij

r r r r g gS S r S Sc r r c

g r p g g r r r rr

H

H HS S Sc r

H

c

BFCSO

T T2 2 2 2e B e

2 5 2

T Te B e B

2 3 2 3DS

3 4ˆ ˆ ˆ ˆ, 2 3

ˆ ˆ2ˆ , 2

ˆ

ˆ 2

ˆ

ij ij ij iji j ij i j

ij

ij i ij

j

j ig ij ij i

i

i

ijg

iij

ijj

r

H

r r rg gS S r S Sc r c

r p r p r r r rg gS Bc r c r

H

H

ˆiS

Properties: nuclear quadrupolar interaction

interacts with the electric field gradient at the nucleus point:

21 3ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ

6 2 1 2nk nk nk n k k n nknkeQH Q V Q S S S S S

S S

The direction of the quadrupole moment of the nucleus depends on the direction ofits spin, meaning that an essentially electrostatic interaction becomes also amagnetic interaction with the following Hamiltonian:

Nuclei with spin higher than ½ have non-spherical charge density distribution.The resulting quadrupole moment

2 33nk kn nkQ r r r r d r

2

nucleus nucleus

1 6

nnk nk nk

nk k n k

EU Q V Vr r r

where eQ is the quantity known as the “nuclear quadrupole moment”. It is afundamental property of the nucleus. Any method for computing NQI musttherefore correctly predict the derivatives of the electrostatic potential.

Properties: nuclear quadrupolar interaction

Electron density distribution must be captured veryprecisely, so a large and flexible basis set is necessa-ry – at least triple-zeta, with multiple sets of diffuseand polarization functions as well as core polariza-tion functions (e.g. in aug-cc-pCVTZ basis).

W.C. Bailey (http://dx.doi.org/10.1016/S0301-0104(99)00342-0)

Properties: nuclear quadrupolar interactionAll factors affecting the electrostatic potential (in particular, crystal lattice andsolvent effects) must be accounted for.

Explicit first solvation shell is in most ca-ses required for PCM calculations.

M. Pavanello, B. Mennucci, J. Tomasi (http://dx.doi.org/10.1007/s00214-006-0117-1)

Properties: J-couplingNMR spectroscopists often forget that J-coupling is a tensor (that its anisotropy isoften inconsequential is a different matter). It is defined as the second derivative ofthe total energy with respect to the nuclear dipole moments:

2A B AB

A B, nkn k

EE J

J

The contributions to perturbation theory integrals (summed over electrons) are:

“Fermi contact”

S S

S S

S S

T T

TPSO PSO0 A B 0DSO

0 AB 00AB

A B TSD FC SD FC0 A A B B 0

0

ˆ ˆˆ

ˆ ˆˆ ˆ ˆ ˆ

m m

m m

m m

m m

H HH

E ES S

H H H H

E E

J

“spin-dipole”

“diamagnetic spin-orbit”

“paramagnetic spin-orbit”

The Fermi contact term dominates in CHNO molecules (up to 80% of the total).

Properties: J-coupling

Polarized core basis sets(aug-cc-pCVnZ) with largecardinal numbers areessential – the basis setconvergence for J-couplingtends to be slow.Non-FC terms are oftenprominent for long-rangeJ-couplings.

Properties: J-coupling

Highly correlated response methods (EOM-CCSD, SOPPA/CCSD) are often required.

Properties: g-tensorThe g-tensor is defined in a similar manner to chemical shielding – as a secondderivative of the energy with respect to the applied magnetic field and the electronmagnetic moment:

2e

ee, , nkn k

EE B g gB

g g 1 g

In addition to the perturbation terms listed above, two extra relativistic terms areoften essential as perturbations – the electron-nuclear spin-orbit term and therelativistic correction to the electron Zeeman operator:

These terms become important for the inner electrons of heavy elements, for whichthe distance to the nucleus is small, momentum is large and the nuclear charge ismuch greater than 1. For this reason, the SO term is often too large to be treatedas a perturbation and needs to be included into the primary Hamiltonian.

2e B e BSO ZR, 2 3 2ˆˆ ˆ ˆ, 2ˆ 2ˆn i in in ii i iing Z r p gS S B pmc r

H Hmc

Properties: g-tensor

For heavy elements this breaks down and a relativistic description is necessary.

DS RZ0 0

SO SO0 0 0 0

0

ˆ ˆ

ˆ ˆ ˆ ˆ12

m m g g m m

m m

H H

H L L HE E

g

The contributions to perturbation theory integrals are:

The four-component relativistic treatment shows no systematic improvement overthe two-component and the scalar approximations.

Properties: g-tensor

Properties: g-tensorThe perturbation theory approach is not appli-cable to systems with g-shifts in excess of20,000 ppm.

For heavy atoms, ZORA is preferred to thePauli approximation.

High level treatment of electron correlation isessential (orbital energies occur in the pertur-bation theory denominators).

Properties: zero-field splitting

T T2 2SD e B

2 5

3ˆ ˆˆ2

ij ij ij ijij i j

ij

r r r rgH S Sc r

The primary contribution is from the inter-electron point dipole interaction:

ZFS is a quadratic spin coupling with the following spin Hamiltonian:

2 2 2Z X Y ˆ ˆˆ ˆ ˆˆ 1 / 3H D S S S E S S S S Z

This contribution is a ground state property and is therefore quite cheap. Itvanishes in closed-shell systems and systems with only one unpaired electron.

The elements of the ZFS tensor are defined via the derivatives of the total energywith respect to the components of the electron magnetic moment:

2 20 0

0 00

ˆ ˆˆk m m l

klmk l k l m

H HE HZE E

Properties: zero-field splittingAnother (often smaller) contribution comes from the spin-orbit coupling:

SO N e B, 2 3 2 3

ˆ ˆˆ 2ˆ ˆˆ2

ij i ij jn in in ij n i

in ij

r p r pg r p gH S Sc r c r

All the usual health warnings about not treating large spin-orbit couplings withperturbation theory apply. It is also usually advantageous to take perturbationtheory denominators from a higher level treatment.

Properties: zero-field splitting

Properties: zero-field splitting

A specific (though not completelyunderstood at the moment) ob-servation for ZFS is: do not usespin-unrestricted DFT.

Properties: exchange interaction

The first term is the usual dipole-dipole interaction. The second term

X X X Y Y Z Zˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆ2 2H J L S J L S L S L S

is known as “symmetric exchange coupling”, it comes from the non-classical spin-dependent part of the Coulomb interaction. For a two-electron system:

T T S S12 12 LS HS

2 2* *

HS LS1 1 2 1 2 2 1 2 1 212

1 1 2 ˆ ˆ1

Jr r E EJ

S SJ x x x x dV dVr

Because J is related to singlet-triplet (or, more generally, HS-LS) energy gap, itmay be computed quite simply using the Yamaguchi equation above. All the usualwarnings about charge transfer excitations apply.

The inter-electron spin coupling Hamiltonian has three parts:

Xˆ ˆ ˆˆ ˆ ˆˆ 2H L S JL S dL S D

Properties: exchange interactionThe antisymmetric component arises from the spin-orbit coupling:

SO N e B, 2 3 2 3

ˆ ˆˆ 2ˆ ˆˆ2

ij i ij jn in in ij n i

in ij

r p r pg r p gH S Sc r c r

The exchange interaction is often difficult todisentangle from the dipolar coupling andzero-field splitting. For this reason, it is oftenconsigned to the role of a fudge factor, whichis unfortunately far too often necessary toobtain a decent data fit.

Properties: magnetic circular dichroismMCD measures differential absorption of left circularly polarized (LCP) and rightcircularly polarized (RCP) photons from the electromagnetic field

2 2LCP RCPi j i j i ji jN N m m f E

E

which is induced in a sample by a strong magnetic field applied parallel to thedirection of light propagation. To first order in B this can be rewritten as:

01 0

f E CB A B f EE E kT

where the three coefficients depend onsecond-order perturbation theory integralsinvolving spin operators and electric dipolemoments (see Frank Neese’s recent work).

MCD is a very specialized area – little prac-tical guidance has so far been published.