GIPAW-NMR method

GIPAW-NMR method

apsi<Ari.P.Seitsonen@iki.fi>

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Outline

● PART I: Introduction to NMR● Introduction to NMR● ChemicalShielding & ElectricFieldGradients

● PART II: GIPAW method● GIPAW for CS; extensions● (GI)PAW for EFG

● PART III: Applications of GIPAW method● Examples

● PART IV: Using the GIPAW module● Implementation● Input/output of GIPAW module

PART I: Introduction to NMR

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NMR spectroscopy

● Microscopic: Gives information about the

individual atoms (~ Å)

● Sensitive to the local environment of the atoms

● … but requires modelling/theoretical input

● Solids, amorphous, liquid, gaseous samples

● Time scale: ~ ms

Nuclear Magnetic Resonance

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Principles of MR spectroscopy

B

Induced orbital currents:● structure and chemical bonds● local electronic structure

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NMR technology

Two fields applied:● Static, homogeneous field: Align magnetic moments● Oscillating field: Excite between two (or two) magnetic energy

levels

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Response to magnetic field

The interaction depends on the local environment

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Spin-polarised nuclei

Isotopic enriching

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Spin-polarised nuclei

Isotopic enriching

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NMR Hamiltonian

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NMR Hamiltonian: Chemical shift

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Chemical shift

From the NMR Hamiltonian the shielding tensor is defined as:

It can be calculated via the response:

The chemical shift is then defined by

is a reference value in a well-characterised material

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Chemical shift: Linear response

The electronic structure does not depend explicitly on the magnetic field but implicitly through the wave functions. Their contribution can be calculated in perturbation theory:

GS wfcs perturbed wfcs

induced current

induced field

DFPT to magn. field

Biot-Savart

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Converse approach to chemical shielding

● Modern Theory of the Orbital Magnetisation

● Instead of direct approach

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Macroscopic shape

● The ring currents at the surface can lead to a notiseable signal● In anisotropic samples the effect can lead to a dependence on the

shape of the sample

Example: Haldane model:

From: T Thonhauser, Davide Ceresoli, David Vanderbilt & R Resta, Phys Rev Lett 95 (2005) 137205

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NMR Hamiltonian: Electric field gradient

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EFG: Electric Field Gradient

In quadrupolar nuclei; non-zero only when no cubic symmetry

Principal axis system: Eigenvectors and -values of

Convention:

Observables:● Quadrupolar coupling constant

● Asymmetry parameter

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NMR Hamiltonian: J coupling

PART II: GIPAW method

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Chemical shielding

● An external magnetic field leads to current leads to induced field:

● Task: Calculate the induced current via perturbation theory● Condensed phase – infinite systems● Reconstruct the wave functions and density close to the nucleus

due to the pseudo potential approach

Biot-Savart's law

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Magnetic field – gauge problem

● The result depends on the atomic coordinate

● One has to cure the gauge problem

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GIPAW method

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GIPAW method

● Introduce the gauge correction into the PAW scheme:

● The observables:

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GIPAW method

● First-order, perturbing potential:

● Perturbed orbitals:

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GIPAW method

● All-electron current operator:

● In GIPAW:

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GIPAW method● Expansion:

● Expectation value of first-order current:

● After regrouping:

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GIPAW method

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GIPAW method

● Trick: Replace

● In practise finite q: Too large, not accurate; too small, numerical problems

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GIPAW method

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● Collecting the terms, using on-site approximation:

Finally:

Biot-Savart:

GIPAW method

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Macroscopic shape

● The macroscopic shape appears via the susceptibility

where

In GIPAW

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Core relaxation

Core contribution has been shown to be to a large degree a constant, independent of the environment

Thomas Gregor, Francesco Mauri & Roberto Car, J Chem Phys 111 (1999) 1815

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(GI)PAW method: EFG

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(GI)PAW method: EFG

● This is evaluated using the PAW equations for the density

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GIPAW: Extensions

● Vanderbilt ultra-soft pseudo potentials ; Jonathan R Yates, Chris

J Pickard & Francesco Mauri, Phys Rev B 76 (2007) 024401

● Metals; Knight shift ; Mayeul d'Avezac, Nicola Marzari &

Francesco Mauri, Phys Rev B 76 (2007) 165122

● J coupling ; Sian A Joyce, Jonathan R Yates, Chris J Pickard &

Francesco Mauri, J Chem Phys 127 (2007) 204107

● EPR ; C J Pickard & F Mauri, Phys Rev Lett 88 (2002) 086403 ;

Davide Ceresoli, Uwe Gerstmann, apsi & Francesco Mauri,

arXiv.org:0904.1988

● NMR ; T Thonhauser, Davide Ceresoli, Arash A Mostoli, Nicola

Marzari, R Resta & David Vanderbilt, arXiv.org:0709.4429

PART III: Applications of GIPAW

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Applications using GIPAW: Molecules

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Applications using GIPAW: Molecules

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Applications using GIPAW: SiO2

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Applications using GIPAW: SiO2

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Applications using GIPAW: SiO2

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Applications using GIPAW: MgSiO3

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Applications using GIPAW: MgSiO3

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Applications using GIPAW: MgSiO3

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Applications using GIPAW: CNT

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Applications using GIPAW: CNT

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Applications using GIPAW: Metals

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Applications using GIPAW: Metals

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Applications using GIPAW: v-B2O3

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Applications using GIPAW: v-B2O3

Fraction of borons in boroxol rings:

• Experiments:• MD simulations:(Raman, NMR, inelastic diffusion…)(empirical potentials)

f = 60­85 %f = 0­30 %

• molecular units:

OB

• boroxol rings:

first­principles MD?!

B B

B

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Applications using GIPAW: v-B2O3

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Applications using GIPAW: v-B2O3

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Applications using GIPAW: v-B2O3

Explanation for the former controversy:Explanation for the former controversy:

The simulations started from low concentration of The simulations started from low concentration of boroxyl, the quench did not allow for proper boroxyl, the quench did not allow for proper equilibrationequilibration

PART IV: Implementation of GIPAW in Q-E

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GIPAW implementations

● paratec

● CASTEP; commercial

● Quantum ESPRESSO

URL: http://www.gipaw.net/

The GIPAW module● General framework for computing magnetic resonance (MR)

spectra and more...

● Available for production in Espresso-4.1Credits: D. Ceresoli, A. P. Seitsonen, U. Gerstmann and F. Mauri

● Capabilities

Magnetic susceptibility NMR shielding tensors Electric Field Gradients (EFGs)

EPR g-tensor Hyperfine couplings

XAS (S. Fabris and Y. Yao) XANES (G. Gougoussis and M. Calandra)

● Simple input and nicely formatted output for calculated quantities

NMR

EPR

X-Ray

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GIPAW@Q-E: Limitations

● Only norm-conserving pseudo potentials currently

● The symmetry operations have to map coordinate axis to each

other, otherwise operation should be excluded!

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GIPAW pseudo potentials

● At the time of this school (July 2009), our GIPAW implementation works only with special norm-conserving PP's(eg. H.pbe-tm-gipaw.UPF)

● The PP's are generated according to existing (and well tested) NC-PP's

● They contain extra datasets, including:● core AE wfcs● valence AE and PS wfcs, 2 x angular momentum● AE and PS atomic potential

● They can be generated using the ld1.x code. A short guide is available at

http://www.impmc.upmc.fr/~software/gipaw/instructions.html

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Oxygen GIPAW pseudo potential &input title = 'O' prefix = 'O' zed = 8.0 rel = 1 config = '1s2 2s2 2p4 3s-1 3p-1 3d-1' iswitch = 3 dft = 'PBE' / &inputp pseudotype = 1 tm = .true. lloc = 2 file_pseudopw = 'O.pbe-tm-gipaw.UPF' lgipaw_reconstruction = .true. /32S 1 0 2.00 0.00 1.40 1.402P 2 1 4.00 0.00 1.40 1.403D 3 2 -1.00 -0.30 1.40 1.40 &test /42S 1 0 2.00 0.00 1.40 1.402P 2 1 4.00 0.00 1.40 1.403S 2 0 0.00 0.00 1.40 1.403P 3 1 0.00 -0.10 1.40 1.40

config.: 1s2 2s2 2p4

empty: 3s, 3p, 3d

PSEUDO: 1 proj x ang. mom.

GIPAW: 2 proj x ang. mom.3p unbound -> scattering state at -0.1 Ry

NC-PP lmax = 2, lloc = 2

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Oxygen GIPAW pseudo potential

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Oxygen GIPAW pseudo potential

NMR on benzene (C6H6)

FILE: benzene-relax.in

&control prefix = 'benzene' outdir = './scratch/'.../

&system.../

...

ATOMIC_SPECIESC 12.000 C.pbe-tm-gipaw.UPFH 2.000 H.pbe-tm-gipaw.UPF

ATOMIC_POSITIONS angstrom...

K_POINTS automatic1 1 1 0 0 0

FILE: benzene-nmr.in

&inputgipaw job = 'nmr' Prefix = 'benzene' tmp_dir = './scratch/' q_gipaw = 0.01 use_nmr_macroscopic_shape = .true./

FILE: benzene-efg.in

&inputgipaw job = 'efg' prefix = 'benzene' tmp_dir = './scratch/'/

pw.x -in benzene-scf.in > benzene-scf.outgipaw.x -in benzene-nmr.in > benzene-nmr.outgipaw.x -in benzene-efg.in > benzene-efg.out

benzene-nmr.out

Program GIPAW v.4.1 starts ... Today is 21Jul2005 at 8:52:20

Parallel version (MPI)

Number of processors in use: 4 R & G space division: proc/pool = 4

Planes per process (thick) : nr3 = 96 npp = 24 ncplane = 9216

Proc/ planes cols G planes cols G columns G Pool (dense grid) (smooth grid) (wavefct grid) 1 24 1377 82821 24 1377 82821 345 10373 2 24 1377 82819 24 1377 82819 344 10372 3 24 1377 82819 24 1377 82819 346 10374 4 24 1378 82820 24 1378 82820 346 10374 tot 96 5509 331279 96 5509 331279 1381 41493

init_gipaw_1: projectors nearly linearly dependent: ntyp = 1, l/n1/n2 = 0 2 1 0.99854824 init_gipaw_1: projectors nearly linearly dependent: ntyp = 1, l/n1/n2 = 1 2 1 0.99933412 init_gipaw_1: projectors nearly linearly dependent: ntyp = 2, l/n1/n2 = 0 2 1 0.99706935

NMR: species C, contribution to shift due to core = 200.510NMR: species H, no information on the core

benzene-nmr.out

f-sum rule: -29.9594 0.0000 0.0000 0.0000 -29.9594 0.0000 0.0000 0.0000 -29.9725

f-sum rule (symmetrized): -29.9594 0.0000 0.0000 0.0000 -29.9594 0.0000 0.0000 0.0000 -29.9725

chi_bare pGv (HH) in 10^{-6} cm^3/mol: -35.1776 0.0000 0.0000 0.0000 -35.1776 0.0000 0.0000 0.0000 -91.7273

chi_bare vGv (VV) in 10^{-6} cm^3/mol: -31.2502 0.0000 0.0000 0.0000 -31.2502 0.0000 0.0000 0.0000 -94.3768

benzene-nmr.out

NMR chemical bare shifts in ppm:

Atom 1 C pos: ( 0.131936 0.000000 0.000000) sigma: -100.1731 -181.7889 0.0000 0.0000 0.0000 -109.8400 0.0000 0.0000 0.0000 -8.8903

Atom 2 C pos: ( 0.065968 0.114260 0.000000) sigma: -100.1731 -127.8272 -31.1548 0.0000 -31.1548 -163.8017 0.0000 0.0000 0.0000 -8.8903

NMR chemical diamagnetic shifts in ppm:

Atom 1 C pos: ( 0.131936 0.000000 0.000000) sigma: 4.0081 4.0056 0.0000 0.0000 0.0000 4.0056 0.0000 0.0000 0.0000 4.0130

NMR chemical paramagnetic shifts in ppm:

Atom 1 C pos: ( 0.131936 0.000000 0.000000) sigma: -57.5707 -85.7823 0.0000 0.0000 0.0000 -59.3537 0.0000 0.0000 0.0000 -27.5761

benzene-nmr.out

Total NMR chemical shifts in ppm:

Atom 1 C pos: ( 0.131936 0.000000 0.000000) sigma: 46.7743 -63.0556 0.0000 0.0000 0.0000 35.3219 0.0000 0.0000 0.0000 168.0566

Symmetric tensor -63.0556 0.0000 0.0000 0.0000 35.3219 0.0000 0.0000 0.0000 168.0566

eigenvalue: 168.0566 eigenvector: 0.0000 0.0000 1.0000

eigenvalue: 35.3219 eigenvector: 0.0000 -1.0000 0.0000

eigenvalue: -63.0556 eigenvector: 1.0000 0.0000 0.0000

Anti-symmetric tensor 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

benzene-efg.out

Total EFG calculation:

C 1 efg -0.127585 0.000000 0.000000 C 1 efg 0.000000 -0.208497 0.000000 C 1 efg 0.000000 0.000000 0.336082

C 1 eig= 0.336082 -0.127585 -0.208497 C 1 Q= 1.00 10e-30 m^2 Cq= 0.7897 MHz eta= 0.24075

C 2 efg -0.188269 0.035036 0.000000 C 2 efg 0.035036 -0.147813 0.000000 C 2 efg 0.000000 0.000000 0.336082

C 2 eig= 0.336082 -0.127585 -0.208497 C 2 Q= 1.00 10e-30 m^2 Cq= 0.7897 MHz eta= 0.24075

benzene-efg.out

H 7 efg 0.276516 0.000000 0.000000 H 7 efg 0.000000 -0.129257 0.000000 H 7 efg 0.000000 0.000000 -0.147259

H 7 eig= 0.276516 -0.129257 -0.147259 H 7 Q= 1.00 10e-30 m^2 Cq= 0.6497 MHz eta= 0.06510

Acknowledgements

● Francesco Mauri

● Davide Ceresoli

Thibault Charpentier