Tutorial Pickard

8/3/2019 Tutorial Pickard

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GIPAW

A first rinci les theor of solid state NMR

Chris J PickardSchool of Physics and Astronomy

University of St Andrews


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2

n = 2d sin

Experiment

Theory

Structure

Theinverseproblem


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Magnetic resonance

NMR EPR

shielding, EFGs, g-tensor, hyperfine


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What kind of theory?

The trained spectroscopist

(Semi-)empirical computational methods

Interpolative

First principles or ab initio

Extrapolative and predictive


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Magnetic resonance is

quantum mechanical

Bint(

r)

Bext

J(r)

B = 0J

Born-Oppenheimer approximation: only treat electrons quantum mechanically


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Quantum mechanics

is difficult

|000+ b |001+ c |011+ d |111+

e |110+ f |100+ g |101+ h |010

|0+ |1|0

|1One particle, two states

Three particles, two states

(1 |0+ 1 |1)(2 |0+ 2 |1)(3 |0+ 3 |1)

Number of coefficients goes as 2N

In a material, the number of states and particles is very large


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Density functional theoryHohenberg-Kohn

E[] = F[] + Vext(r)(r)d3r

Kohn-Sham

(r) =

n

n

(r)n(r)Ts[] = 12

n

n

2nd3r

F[] = Ts[] +

(r)(r)

|r r|d3rd

3r

+ Exc[]

Changes the problem to the easy independent

particle option


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Density functional theory

But we still dont know whatExc[] is

The local density approximation

Exc[] = xc()(r)d3r

This is first principles

Exc[] = xc(,)(r)d3r

Generalised gradient approximations

These may or may not be first principles


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Extended Systems

Glass Ice Polymer

Simple crystal Zeolite Solution


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Cluster approximations

The only option for traditional methods

But you have to worry about:

making the cluster small

terminating the cluster

long range electrostatics


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Periodic boundary conditionsAl Bandstructure

1st Brillouin zone

Use the translational

symmetry of the crystal

nk(

r

) =eikr

u

nk(r

)

Blochs theorem


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Supercells


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Plane waves

ei(k+G)

r

Basis functions

Unlike atom centred atomic orbitals or gaussian

functions, there is no origin to a plane wave

There is no gauge origin problem

1

2 |k+G

|

2

Ecut

The quality of the basis is controlled by a single number

A Fourier series


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Pseudopotentials

Na

The core electrons are frozen and the valence

orbitals smoothed within the core radius

All-electron potential

Pseudopotential

rc

All-electron wavefunction

Pseudowavefunction


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A pseudopotential theory

The core electrons contribution to the

shielding must not be chemically sensitive

We must fix up the wavefunction near the nucleus

| = T | where T = 1 +

n

(|n |n)n|

Projector augmented waves

These break gauge invariance


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GIPAW

Gauge Including Projector Augmented Waves

PAW with gauge invariancerestored

Position operator rdescribed aslong wavelength limit of sin(qr)

Scalar relativistic effects includedusing ZORA

Response calculated in terms ofoccupied orbitals only (not sumover all states)

Implemented in: Paratec, (NMR-)CASTEP


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Doing calculations

Converge

Basis set

Brillouin zone integration

Supercell size

ChooseComputer

Code

DFT functionalPseudopotentials

100 atoms on a PC

200 atoms on a cluster

1000 atoms on a supercomputer?

My preferences

CASTEP, the GGA-PBE functional and ultrasoft pseudopotentials


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Structure

Where do you get your structure from?

The phase space of possible structures is large

Many applications start with X-ray structures

CAUTION: Check forces on atoms

Protons are typically misplaced

NMR chemical shifts are verysensitive to structure


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Dynamics

Calculations are typically performed at 0K

Experiments are typically performed at 300K

CAUTION: the atoms are really moving

This might be tackled via lattice dynamicscalculations or molecular dynamics simulations

An area of active research


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Density functional theory

Structures are typically good to 1-2%

Chemical shifts might be good to 2% of the shift range

An Inverse Lottery

When things gowrong they can go

very wrong

It is normally good


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Applications

Minerals

Amino acids

Protein fragmentsPharmaceutical

polymorphs

Boron carbidesGlasses

Bone


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Structure determination

Poster 342: Lyndon Emsley et al

Rose: from database

-L-Aspartyl-L-alanine

Orange: protons relaxed usingCASTEP

C

H


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Structure determination

PSD

PSD + CASTEP

Poster 342: Lyndon Emsley et al

H

C


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Conclusion

Good theory makesexperiments better

What do theorists get out of this?

NMR gives us a wealth of high quality data

which challenges us to improve our theories

We have quite a good theory- but it could be better


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Bibliography

Solid State PhysicsAshcroft andMermin Brooks Cole (1976)

Electronic Structure: basic theory and practical methods Martin Cambridge University Press (2004)

Iterative minimization techniques for ab initio total-energy

calculations: molecular dynamics and conjugate gradients

Payne et al Rev. Mod. Phys. 64, 1045 - 1097 (1992)

Calculation of NMR andEPR parameters Kaupp et al Wiley-VCH (2004)

All-electron magnetic response with pseudopotentials: NMRchemical shifts

Pickard and Mauri Phys. Rev. B. 63, 245101 (2001)

First principles methods using CASTEP Clark et alZeitschrift fur Kristallographie, 220,

567-570 (2005)

http://www.atypon-link.com/OLD/doi/abs/10.1524/zkri.220.5.567.65075

Direct link to Clarket al
http://www.atypon-link.com/OLD/doi/abs/10.1524/zkri.220.5.567.65075http://www.atypon-link.com/OLD/doi/abs/10.1524/zkri.220.5.567.65075http://www.atypon-link.com/OLD/doi/abs/10.1524/zkri.220.5.567.65075http://www.atypon-link.com/OLD/doi/abs/10.1524/zkri.220.5.567.65075http://www.atypon-link.com/OLD/doi/abs/10.1524/zkri.220.5.567.65075http://www.atypon-link.com/OLD/doi/abs/10.1524/zkri.220.5.567.65075http://www.atypon-link.com/OLD/doi/abs/10.1524/zkri.220.5.567.65075http://www.atypon-link.com/OLD/doi/abs/10.1524/zkri.220.5.567.65075http://www.atypon-link.com/OLD/doi/abs/10.1524/zkri.220.5.567.65075http://www.atypon-link.com/OLD/doi/abs/10.1524/zkri.220.5.567.65075http://www.atypon-link.com/OLD/doi/abs/10.1524/zkri.220.5.567.65075http://www.atypon-link.com/OLD/doi/abs/10.1524/zkri.220.5.567.65075


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Acknowledgments

Theorists Experimentalists Castep

Francesco Mauri

Jonthan Yates

Sian Joyce

Anne-Christine Uldry

Robin Harris

Sharon AshbrookSteven Brown

Lyndon Emsley

Melinda Duer

Ray Dupree

Ian Farnan

The CDG

Stewart Clark

Matt Probert

Phil HasnipMatt Segall

Keith Refson

Mike Payne

Accelrys

Victor MilmanGeorge Fitzgerald

http://www.castep.org/
http://www.castep.org/http://www.castep.org/http://www.castep.org/

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