Lecture 10: calculation ofmagnetic parameters, part I

vector potential, gauge invariance problem,magnetic susceptibility, chemical shielding, hyperfine coupling

History I believe furnishes no example of a priest-ridden people

maintaining a free civil government. This marks the lowest grade of

ignorance, of which their political as well as religious leaders will

always avail themselves for their own purpose.

Thomas Jefferson

Dr Ilya Kuprov, University of Southampton, 2012

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

Magnetic field in quantum mechanics

The canonical quantization procedure (skipped here, see textbooks) then yields thefollowing Hamiltonian for each particle in the system:

32

k k

k k

r rB rA rr r

The scalar potential is the usual collection of Coulomb terms. The vector potentialhas contributions from the external (assumed uniform) magnetic field as well asfrom the field of each magnetic dipole in the system:

The electric and the magnetic components of the electromagnetic field may bedescribed in terms of the scalar and the vector potential (see Maxwell’s equations):

,, , , ,

A r tE r t r t B r t A r t

t

21 ˆˆ ˆ ˆ , ,

2H T U p qA r t q r t

m

Note the divergence in the vector potential of the uniform field at infinity.

Magnetic field in quantum mechanicsFor a given magnetic induction, the vector potential is only defined up to a gradi-ent of an arbitrary “gauge function”, because the curl of a gradient is zero:

( ) 0 f B A A f

This is indeed the case in a complete basis set, but in practical calculations,particularly in atomic basis sets, the quality of description varies from point topoint and the origin invariance is broken as a result. For example:

All physical observables only depend on magnetic induction and must therefore beindependent of the gauge function. In particular, the observables must beinvariant under the change of the coordinate system origin, because

0 03 3

0

( ) ( ) ( ), ...2 2 | | | |

k k k k

k k

B r r r r r r rB rr r r r r

xˆ[ , ]nm n k k m n k k mk

p x i x xx x

Numerical accuracy of this commutator clearly depends on how good the resolu-tion of the identity is and therefore on the local quality of the basis set.

We can require without loss of generality (“Coulomb gauge”) that

The Hamiltonian has zeroth-, first- and second-order terms in vector potential:

2

2 21 ˆ ˆˆ ˆ2 2 2

q qH p p A A p Am m m

ˆ0 0A p A i A

With this in place, the Hamiltonian simplifies to

22 21 ˆˆ ˆ

2 2 2q qH p A p A

m m m

Within the non-relativistic quantum theory, the interaction of spin with themagnetic vector potential must be added artificially:

2

2 21 ˆˆˆ ˆ2 2 2

q qH p A p A S Am m m

where is the magnetogyric ratio of the particle and S is the spin operator.

Magnetic field in quantum mechanics

Gauge invariance problem

After this transformation the wavefunction acquires a coordinate-dependent phase(long derivation, but may be verified by direct inspection):

2 10 0 0

1 exp2 2g g

if r B r r r r B r r r r

This transformation has no observable physical consequences, but it allows eachbasis function to have its own gauge origin. This improves numerical accuracy.

The most general form for a gauge transformation is:

1 2 1

g g gA r A r A r f r

In the simplest case where the gauge origin is translated in space:

1 2 1

expg g gr r if r r

but observables and probabilities are invariant because phase terms cancel:

1 1 2 2 1 2

2 2ˆ ˆ , g g g g g gr O r r O r r r

Gauge invariance problemThe common solution is to give each atomic or molecular orbital its own gaugeorigin. This does not formally affect the observables, but improves numericalaccuracy in atomic basis sets. Plane wave basis sets do not have this problem.

K.K. Baldridge, J.S. Siegel (http://dx.doi.org/10.1021/jp990523x)

Properties: magnetic susceptibilityThe magnetic susceptibility tensor may be defined in terms of energy derivatives:

2

T0 0

1 1 EB E B E B BB B

χ χ χ

It is therefore a second-order response property with respect to the applied magneticfield. After perturbation theory derivatives are taken, the following terms survive:

where i index runs over electrons and g indexdenotes the gauge origin. Only first derivatives withrespect to the magnetic field need in practice becomputed. Nuclear contributions are ignored. R

ingcurrentdensityinhexabenzocoronene.

OZ OZ0 0DM

0 00

OZ

DM T T

ˆ ˆˆ2 2

1 ˆˆ2

1ˆ8

m m

m m

i ig i

i ig ig ig ig

H HH

E E

H r p

H r r r r

χ

J. Juselius, D. Sundholm, J. Gauss (http://dx.doi.org/10.1063/1.1773136)

Properties: magnetic susceptibility

Magnetic susceptibility is not hard tocompute – DFT methods work well.

Y. Fujiwara, Y. Tanimoto (http://dx.doi.org/10.1088/1742-6596/156/1/012022)

Properties: chemical shieldingChemical shielding tensor may be expressed as the second derivative of the totalenergy with respect to the external magnetic field and the nuclear dipole moment:

2

T, EE BB

1 δ 1 δ

After differentiating the second-order perturbation theory expression for the totalenergy, the following terms survive (their names are historical and have little to dowith the proper meaning of dia- and paramagnetism):

T20 0

0 0T0

ˆ ˆˆk n k

nk kn

H H BHE EB

δ

T2 22

3 3T

ˆ ˆ ˆ, ,

2 2ig in ig inig i in i

i i in in inn

r r r rr p r pH H Hr rB B

1

“paramagnetic term”

“diamagnetic term”

where i index runs over electrons, n index over nuclei, k index over energy levels is the fine structure constant and g index denotes the gauge origin.

Properties: chemical shieldingThe accuracy of a shielding calculation method depends on the nucleus and thetype of the molecule – always check the literature beforehand to see what works.

An inconvenient truth is that the “accu-racy” of shielding calculations is oftenthe result of error compensation, mea-ning that a higher level method does notnecessarily yield a better result – thebest method for aromatic 19F CSA still is,notoriously, Hartree-Fock.

M.E. Harding, M. Lenhart, A.A. Auer, J. Gauss (http://dx.doi.org/10.1063/1.2943145)

Properties: chemical shielding

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J.F. Stanton, J. Gauss, H.-U. Siehl (http://dx.doi.org/10.1016/0009-2614(96)01077-9)

Properties: chemical shielding

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F.A.A. Mulder, M. Filatov (http://dx.doi.org/10.1039/B811366C)

Properties: hyperfine couplingsHFC is a bilinear coupling between a nucleus and an electron in an open-shellsystem. The ground state Fermi contact component usually dominates:

The other important component comes from the electron-nuclear dipole couplingintegrated over the unpaired electron density distribution:

HFCs are a ground state property and aretherefore very cheap to compute.The Fermi contact component is isotropic anddipole-dipole component is traceless, but thehyperfine coupling often has non-zero rhombi-city as a result of combining axialities withdifferent orientations.

Stereo plot of hyperfine tensors in 3-fluorotyrosyl radical.

FC

8 ˆˆˆ0 0 ( )3 e n n n n

nH L S

30DD e 5 3

ˆ ˆˆ ˆ3( )( ) ( )ˆ0 04

en en n nn e e e

n en en

L r r S L SH r r d rr r

Properties: hyperfine couplings

Solvation and conformationalaveraging effects are essentialfor the correct prediction ofexperimental HFCs.A good example is the CH2group in aromatic amino acidradicals: the proton HFC canbe anywhere between –2.0 and+20 Gauss, depending on con-formation.

In anharmonic systems (famously, nitroxides) fullvibrational averaging may be necessary.

Because the wavefunction cusp at the nucleuspoint must be accurately sampled, Fermi con-tact coupling requires specialized basis sets(EPR-II and EPR-III in Gaussian) and is thehardest part to compute – large HFCs tend to beunderestimated in single-point calculations.The dipolar component of HFC (i.e. the hyperfineanisotropy) is easier and generally comes outright in double zeta and larger basis sets.Unrestricted calculation is required in all cases.

Hyperfine couplings: basis set and solvent effects

Contractions of the EPR-II basis set have been optimized to reproduce expe-rimental data specifically with the B3LYP exchange-correlation functional. Compa-rable accuracy with other functionals is not guaranteed.

N. Rega, M. Cossi, V. Barone (http://dx.doi.org/10.1063/1.472906)