Day 2: NMR and EPR spectroscopy (part II) Davide Ceresoli Department of Materials, Oxford University [email protected]

Outline ●





PART I: ● Basic principles of magnetic resonance spectroscopy ● Introduction to experimental NMR ● Interpretation of NMR spectra ● Solid state NMR PART II: ● Effective NMR spin hamiltonian ● The GIPAW method ● Examples ● Brief introduction to EPR spectroscopy and EPR parameters PART III: (Emine Kuçukbenli) ● GIPAW pseudopotentials ● The gipaw.x code: input file and description of the output

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

2

NMR parameters NMR parameters are: → chemical shift → nuclear quadrupole → J coupling They can be extracted by fitting the experimental spectrum or can be calculated from from first-principles, given the atomistic structure.

NMR parameters

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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Effective NMR Hamiltonian chemical shift

J coupling nuclear quadrupole (I>1/2)

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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The chemical shift From the NMR Hamiltonian the shielding tensor is defined as:

It is the second derivative of energy w.r.t. field and nuclear moment:

The chemical shift is then defined by:

ref is a reference value in a well-characterized material.

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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“Direct” and “converse” methods Direct approach (traditional): ● ● ●

linear response to external magnetic field calculate the induced current, then the induced field Mauri, Louie (1996); GIPAW: Pickard, Mauri (2001)

Converse approach: ● ● ●

no linear response, no magnetic field, no gauge-origin problem calculate the change of orbital magnetization due to nuclear magnetic moment based on the “Modern Theory of the Orbital Magnetization”

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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Outline of linear response Vector potential A(r)=(1/2)B x r is incompatible with crystal periodicity Solution: apply a long-wavelength magnetic field (q << 1)

The response to an incommensurate perturbation is obtained by Density Functional Perturbation Theory (DFPT): BiotSavart

DFPT to magn. field

GS wfcs

perturbed wfcs

induced current

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

induced field 7

EFG: electric field gradient tensor Quadrupolar nuclei (I>1/2); non-zero only when no cubic symmetry:

Principal axis system: Eigenvectors and -values of Convention: Observables: ● Quadrupolar coupling constant ●

Asymmetry parameter

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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Outline ●





PART I: ● Basic principles of magnetic resonance spectroscopy ● Introduction to experimental NMR ● Interpretation of NMR spectra ● Solid state NMR PART II: ● Effective NMR spin hamiltonian ● The GIPAW method ● Examples ● Brief introduction to EPR spectroscopy and EPR parameters PART III: (Emine Kuçukbenli) ● GIPAW pseudopotentials ● The gipaw.x code: input file and description of the output

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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GIPAW

Need to reconstruct the wavefunction near the nuclei! Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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Pseudopotential / all-electron AE wfc (blue) oscillates rapidly near the nucleus PS wfc (red) smooth, no nodes in the core region

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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PAW idea PAW (Blöchl, 1994) = projector augmented wave Idea: “reconstruct” the AE wfc from the PS wfc valence wfcs

atomic partial waves

=

+(

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

atomic projectors



) 12

Gauge Including PAW Translation in magnetic field yields a gauge phase factor:

PAW + magnetic field = GIPAW

Take-home message: (1) wfc reconstruction (2) gauge-invariance Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

13

Outline ●





PART I: ● Basic principles of magnetic resonance spectroscopy ● Introduction to experimental NMR ● Interpretation of NMR spectra ● Solid state NMR PART II: ● Effective NMR spin hamiltonian ● The GIPAW method ● Examples ● Brief introduction to EPR spectroscopy and EPR parameters PART III: (Emine Kuçukbenli) ● GIPAW pseudopotentials ● The gipaw.x code: input file and description of the output

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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Example: molecules red = GIPAW blue = Gaussian™

Pickard, Mauri PRB 63, 245101 (2001) Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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Example:

F

19

GIPAW ~ Gaussian 6Z (140 basis/atom)

Ceresoli, Marzari, Lopez, Thonhauser PRB 81, 184424 (2010) Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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Example:

Ca cement models

43

Tobermorite 1.1nm

Ca (6+1)-coord

Ca 5-coord

Tobermorite 1.1nm

Tobermorite Ca 6-coord 9nm

Jennite Ca 6-coord

Jennite

Portlandite

Ca 6-coord

Portlandite

Bowers and Kirkpatrick, JACS 92, 545 (2009) Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

max(Ca-O) = 2.8 Å

17

Outline ●





PART I: ● Basic principles of magnetic resonance spectroscopy ● Introduction to experimental NMR ● Interpretation of NMR spectra ● Solid state NMR PART II: ● Effective NMR spin hamiltonian ● The GIPAW method ● Examples ● Brief introduction to EPR spectroscopy and EPR parameters PART III: (Emine Kuçukbenli) ● GIPAW pseudopotentials ● The gipaw.x code: input file and description of the output

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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EPR/ESR spectroscopy Electron Paramagnetic Resonance / Electron Spin Resonance





Tipical fields Resonance

~0.5 T ~14 GHz



Paramagnetic defects in solids, radicals in proteins Sensitive to local geometry and electronic structure (charge, hybridization) Non destructive, small samples, dilute spins

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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Interpretation of EPR spectra L

Btot = Bext + Bind

SpinOrbit, SOO

S

+ Bnucl + Bother-spin

S

I

S

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

Hyperfine

S Zero Field Splitting

20

Effective Spin Hamiltonian

g-tensor

hyperfine

ZFS

Selection rules: |ΔMS| = 1

|ΔMI| = 0

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

21

Theory of EPR parameters The effective spin Hamiltonian allows to extract EPR parameters from the experiments, but in order to calculate EPR parameters, we need a “physical” Hamiltonian. The simplest non-relativistic Hamiltonian that accounts for spin is the Pauli Hamiltonian: Mass velocity and Darwin

Zeeman and Zeeman-KE

Spin orbit ge = 2.0023192778

Spin other-orbit

g' = 2(ge- 1)

c = 1/α = 137.03599

S = (ħ/2)σ (σ = Pauli matrixes) Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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... plus other terms

From: Pekka Manninen PhD thesis, University of Oulo, Finland (2004). http://herkules.oulu.fi/isbn9514274318/ Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

23

Zero Field Splitting In case: S > 1/2



Two electron integrals



Usually small (?)



Not implemented in Espresso

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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g-tensor: the easy part From the effective spin Hamiltonian, the g-tensor is defined as:

The Zeeman and Zeeman-Kinetic Energy term yield:

Kinetic energy of occupied orbitals Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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g-tensor: SO and SOO The SO and SOO don't depend explicitly on the magnetic field, but implicitly through the wavefunctions. Their contribution can be calculated in perturbation theory: BiotSavart

DFPT to magn. field

GS wfcs

perturbed wfcs

induced current

induced field

Finally:

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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Converse approach to the g-tensor

2.002319...

gSO



“Converse” method: ∂Morb/∂S ≈ 1/(2S) [Morb(S=) - Morb(S=)]



Modern Theory of the Orbital Magnetization



GS calc. including Spin-Orbit; no magnetic field, no linear response



Implemented in an experimental version of Espresso (ask me)

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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EPR of diatomic radicals L=0 L=1 LR = linear response This work = converse method

2S

C

L

F

Important when g very different from 2 (i.e. transition metals impurities)!

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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Hyperfine coupling

Isotropic (Fermi-contact)

Dipolar (traceless)

Spin density: Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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Spin densities of isolated atoms Atom Li C N O Na Si Mn Mn+ Mn2+ Mn3+ Mn4+ ● ● ●

Configuration 2s(1,0) 2s2 2p(2,0) 2s2 2p(3,0) 2s2 2p(3,1) 3s(1,0) 3s2 3p(2,0) 4s2 3d(5,0) 4s1 3d(5,0) 4s0 3d(5,0) 4s0 3d(4,0) 4s0 3d(3,0)

1s 0.003

2s 0.219

3s

4s

Total 0.222

-0.199 -0.440 -0.417

0.178 0.429 0.437

0.019 -0.094 -0.008

-0.030 0.056 -2.203

0.810 -0.193 0.913

1.384

0.799 -0.232 0.086

0.040 -0.018 -0.018 -0.017

-2.215 -2.294 -2.103 -1.793

0.833 0.888 0.942 0.904

7.194 0.000 0.000 0.000

5.852 -1.424 -1.179 -0.907

-0.021 -0.011 0.019

Calculated with ld1.x, extrapolated at the nucleus Spin-polarized LDA values in elec./bohr3

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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Mn orbitals and VXC

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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Core relaxation ●

Project valence density around atoms and add GIPAW reconstruc.



Calculate VXC from projected spherical density



Core spin density from Incomplete Perturbation Theory

Not yet available in Espresso-4.2. Experimental! Ask me. Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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Hyperfine couplings of 2 Molecule Atom C CH● H OH● O H CH3● C CN● H2CN●

nd

row radicals

no core relax core relax experiment 204 90 47 -50 -50 -58 -142 -61 -45 -56 -56 -73

H C N

210 -69 636 -7

122 -69 581 -18

107 -70 588 -13

C N H

-73 70 230

-67 34 230

-81 26 234

Fermi contact (Aiso), values in MHz OH●

CN●

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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Hyperfine couplings of 2

nd

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

row radicals

34

“Converse” EPR: DFT+U on MnOx (x=1..3) ●

Using experimental geometry (if unavailable, CCSD(T) geometry)



Comparison only to available experimental data





MnO (S=5/2) Aiso(Mn)

Aiso(O)

MnO3 (S=1/2) Aiso(Mn)

Aiso(O)

PBE

578

-8

PBE

2058

197

PBE+Uscf

461

5

PBE+Uscf

899

364

Expt. (MHz)

480

-8

Expt. (MHz)

1613

81

☺ MnO2 (S=3/2) Aiso(Mn)

Adip(Mn)

PBE

836

-129

PBE+Uscf

665

-97

Expt. (MHz)

603

-126

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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Bibliography and references

Books ●

P. T. Callaghan, Principles of Nuclear Magnetic Resonance Microscopy , Clarendon Press



T. N. Mitchell and B. Costisella, NMR – From Spectra to Structures, Springer



G. S. Rule and T. K. Hitchens, Fundamentals of Protein NMR Spectroscopy, Springer



N. Jacobsen, NMR spectroscopy explained, Wiley



M. Duer, Solid state NMR spectroscopy, Blackwell



J. A. Weil and J. R. Bolton, Electron Paramagnetic Resonance, Wiley



M. Knaupp, M. Bühl and V. G. Malkin, Calculation of NMR and EPR Parameters, VileyVCH

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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Hamiltonians The simplest Hamiltonian describing all the physics of NMR and EPR is the Dirac-Breit Hamiltonian. The DB Hamiltonian is fully relativistic (four-component) and difficult to solve. In order to make it numerically tractable, it must be reduced to a two-component non-relativistic Hamiltonian by some transformations and approximations. Physicists apply the Foldy-Wouthuysen transformation to obtain the Pauli Hamiltonian. Chemists prefer the Douglas-Kroll-Hess transformation. Another popular approximation is the ZORA (zeroth-order regular approximation). These transformation lead to different expressions for the Hamiltonian terms, that are numerically very close. ●

Pekka Manninen PhD thesis, University http://herkules.oulu.fi/isbn9514274318/

of

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

Oulo,

Finland

(2004).

38

Theory of EPR parameters ●

G. Schreckenbach and T. Ziegler, J. Phys. Chem. A 101, 3388 (1997)



C. J. Pickard and F. Mauri, Phys. Rev. Lett. 88, 086403 (2002)



S. Patchkovskii, R. T. Strong, C. J. Pickard and S. Un, J. Chem. Phys. 122, 214101 (2005)

GIPAW and core-relaxation ●

C. J. Pickard and F. Mauri, Phys. Rev. B 65, 245101 (2001)



C. J. Pickard and F. Mauri, Phys. Rev. Lett. 91, 106401 (2003)



M. d'Avezac, N. Marzari and F. Mauri, Phys. Rev. B 76, 165122 (2007)



J. R. Yates, C. J. Pickard and F. Mauri, Phys. Rev. B 76, 024401 (2007)



M. S. Barhamy, M. H. F. Sluiter and Y. Kawazoe, Phys. Rev. B 76, 035124 (2007)

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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Converse approach ●





D. Ceresoli, U. Gerstmann, A. P. Seitsonen and F. Mauri, First-principles theory of the orbital magnetization, PRB 81, 060409 (2010) T. Thonhauser, D. Ceresoli, A. A. Mostofi, N. Marzari, R. Resta and D. Vanderbilt, A converse approach to the calculation of NMR shielding tensors , JCP 131, 101101 (2009) D. Ceresoli, N. Marzari, M. G. Lopez and T. Thonhauser, Ab-initio converse NMR for pseudopotentials, PRB 81, 184424 (2010).

Modern Theory of the Orbital Magnetization ●





T. Thonhauser, D. Ceresoli, D. Vanderbilt and R. Resta, Orbital magnetization in periodic insulators, Phys. Rev. Lett. 95, 137205 (2005) D. Ceresoli, T. Thonhauser, D. Vanderbilt and R. Resta, Orbital magnetization in crystalline solids: multi-band insulators, Chern insulators, and metals , Phys. Rev. B 74, 024408 (2006) I. Souza and D. Vanderbilt, Dichroic f-sum rule and the orbital magnetization of crystals, Phys. Rev. B 77, 054438 (2008)

Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

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Online resources: blogs, codes, lectures ●

NMR Wiki: http://nmrwiki.org



Software: http://edunmrsoft.blogsome.com







NMR periodic table: http://www.bruker-nmr.de/guide/eNMR/chem/NMRnuclei.html Solid state NMR literature blog: http://ssnmr.blogspot.com Other blogs: http://nmr-software.blogspot.com http://u-of-o-nmr-facility.blogspot.com http://scienceblogs.com/scientificactivist/2006/11/nmr_blogs.php

… and of course: www.quantum-espresso.org www.gipaw.net http://qe-forge.org/projects/qe-gipaw Computational spectroscopy using Quantum Espresso and related codes, SISSA, July 2010

41

Day 2: NMR and EPR spectroscopy (part II)

... spectroscopy using Quantum Espresso and related codes, SISSA, July 2010. Hyperfine couplings of 2nd row radicals. Molecule Atom no core relax core relax.

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