First Principles Calculationsof NMR Chemical Shifts

Methods and Applications

Daniel Sebastiani

Approche théorique et expérimentale des phénomènes magnétiques et des

spectroscopies associées

Max Planck Institute for Polymer Research · Mainz · Germany

1

Outline Part I

Introduction and principles of electronic structure calculations

I. Introduction to NMR chemical shielding tensors

Phenomenological approach

II. Overview electronic structure methods

HF, post-HF, DFT. Basis set types

III. External fields: perturbation theory

2

Outline Part II

Magnetic fields in electronic structure calculations

I. Perturbation Theory for magnetic fields

in particular: magnetic density functional perturbation theory

II. Gauge invariance

Dia- and paramagnetic currents

Single gauge origin, GIAO, IGLO, CSGT

III. Condensed phases: position operator problem

3

Outline Part III

Applications

I. Current densities

II. Chemical shifts of hydrogen bonded systems:

• Water cluster• Liquid water under standard and supercritical conditions• Proton conducting materials: imidazole derivatives• Chromophore: yellow dye

4

Nature of the chemical shielding

• External magnetic field Bext

• Electronic reaction: induced current j(r)

=⇒ inhomogeneous magnetic field Bind(r)

• Nuclear spin µµµ Up/Downenergy level splitting

Β=0Β=Β0 h̄ω

∆E = 2µµµ ·B = h̄ω

Bext

Bind

jind

5

Chemical shifts – chemical bonding

• NMR shielding tensor σ:definition through induced field

Btot(R) = Bext + Bind(R)

σ(R) = − ∂Bind(R)∂Bext

� 1

• Strong effect of chemical bondingHydrogen atoms: H-bonds

=⇒ NMR spectroscopy:unique characterization

of local microscopic structure (liquid water)

6

Chemical shielding tensor

σ(R) = −

∂Bindx (R)

∂Bextx

∂Bindx (R)∂Bexty

∂Bindx (R)∂Bextz

∂Bindy (R)

∂Bextx

∂Bindy (R)

∂Bexty

∂Bindy (R)

∂Bextz

∂Bindz (R)∂Bextx

∂Bindz (R)∂Bexty

∂Bindz (R)∂Bextz

• Tensor is not symmetric

=⇒ symmetrization =⇒ diagonalization =⇒ Eigenvalues

• Isotropic shielding: Tr σ(R)

• Isotropic chemical shift: δ(R) = TrσTMS − Trσ(R)

7

First principles calculations: Electronic structure

Methods

• Hartree-Fock

• Møller-PlessetPerturbation Theory

• Highly correlated methodsCI, coupled cluster, . . .

• Density functional theory

Basis sets

• Slater-type functions:Y ml exp−r/a0

• Gaussian-type functions:Y ml exp−(αr)2

• Plane waves:exp ig · r

8

Kohn-Sham density functional theory (DFT)

Central quantity: electronic density, total energy functional

No empirical parameters

EKS[{ϕi}] = −12

∑i

∫d3r 〈ϕi|∇2|ϕi〉

+12

∫d3r d3r′

ρ(r)ρ(r′)|r− r′|

+∑at

qat

∫d3r

ρ(r)|r−Rat|

+ Exc[ρ]

ρ(r) =∑

i

|ϕi(r)|2

9

DFT: Variational principle

• Variational principle: selfconsistent Kohn-Sham equations

〈ϕi|ϕj〉 = δijδ

δ ϕi(r)(EKS[ϕ]− Λkj〈ϕj|ϕk〉) = 0

Ĥ[ρ] |ϕi〉 = εi|ϕi〉

Iterative total energy minimization

• DFT: Invariant of orbital rotation

ψi = Uij ϕj

E [ϕ] = E [ψ]

10

Perturbation theory

External perturbation changes the state of the system

Expansions in powers of the perturbation (λ):

Ĥ 7→ Ĥ(0) + λĤ(1) + λ2Ĥ(2) + . . .ϕ 7→ ϕ(0) + λϕ(1) + . . .E 7→ E(0) + λE(1) + λ2E(2) + . . .

11

Perturbation theory in DFT

Perturbation expansion

E[ϕ] = E(0)[ϕ] + λ E

λ[ϕ] + . . .

ϕ = ϕ(0) + λ ϕλ + . . .

ρλ(r) = 2 <[ϕ

(0)i (r) ϕ

λi (r)

]Ĥ = Ĥ(0) + λ Ĥλ + ĤC

[ρλ]+ . . .

E[ϕ] = E(0)[ϕ] + λ E

λ[ϕ(0)]

+12λ2 E(2)[ϕ] . . .

12

Perturbation theory in DFT

• unperturbed wavefunctions ϕ(0) known:

min{ϕ}

E [ϕ] ⇐⇒ min{ϕ(1)}

E(2)[ϕ(0), ϕ(1)

]

E(2) = ϕ(1) δ2E(0)

δϕ δϕϕ(1) +

δEλ

δϕϕ(1)

• orthogonality 〈ϕ(0)j |ϕ(1)k 〉 = 0 ∀ j, k

13

Perturbation theory in DFT

Iterative calculation(Ĥ(0) δij − ε(0)ij

)ϕλj + ĤC[ρλ] ϕ

(0)i = −Ĥ

λ ϕ(0)i

Formal solution

ϕλi = Ĝij Ĥλ ϕ(0)j

14

Magnetic field perturbation

• Magnetic field perturbation: vector potential A

A = −12

(r−Rg)×B

Ĥλ = − em

p̂ · Â

= ih̄e

2mB · (r̂−Rg)× ∇̂

• Cyclic variable: gauge origin Rg

• Perturbation Hamiltonian purely imaginary =⇒ ρλ = 0

15

Magnetic field perturbation

Resulting electronic current density:

ĵr′ =e

2m

[π̂|r′〉〈r′|+ |r′〉〈r′|π̂

]ĵr′ =

e

2m

[(p̂− eÂ)|r′〉〈r′|+ |r′〉〈r′|(p̂− eÂ)

]j(r′) =

∑k

〈ϕ(0)k | ĵ(2)r′ |ϕ

(0)k 〉+ 2 〈ϕ

(0)k | ĵ

(1)r′ |ϕ

(1)k 〉

= jdia(r′) + jpara(r′)

Dia- and paramagnetic contributions:

zero and first order wavefunctions

16

The Gauge origin problem

• Gauge origin Rg theoretically not relevant

• In practice: very important: jdia(r′) ∝ R2g

• GIAO: Gauge Including Atomic Orbitals

• IGLO: Individual Gauges for Localized Orbitals

• CSGT: Continuous Set of Gauge Transformations: Rg = r′

• IGAIM: Individual Gauges for Atoms In Molecules

17

Magnetic field under periodic boundary conditions

• Basis set: plane waves(approach from condensed matter physics)

• Single unit cell (window)taken as a representative for the full crystal

• All quantities defined in reciprocal space (periodic operators)

• Position operator r̂ not periodic

• non-periodic perturbation Hamiltonian Ĥλ

18

PBC: Individual r̂-operators for localized orbitals

• Localized Wannier orbitals ϕi via unitary rotation:

ϕi = Uij ψj

orbital centers of charge di

• Idea:

Individual

position

operators

a(x)

^a

r̂ (x)b

b(x)

(x)

ϕ

r (x)

ϕ

L0 2Ld db a

19

Magnetic fields in electronic structure

• Variational principle 7→ electronic response orbitals

• Perturbation Hamiltonian Ĥλ: Â = −12 (r̂−Rg)×B

• Response orbitals 7→ electronic ring currents

• Ring currents 7→ NMR chemical shielding

• Reference to standard 7→ NMR chemical shift

20

Electronic current density

jk(r′) = 〈ϕ(0)k | ĵr′(|ϕ(α)k 〉 − |ϕ

(β)k 〉+ |ϕ

(∆)k 〉

)ĵr′ =

e

2m

[p̂|r′〉〈r′|+ |r′〉〈r′|p̂

]

modulus of current |j|

B-field along Oz

21

Current and induced magnetic field in graphite

Electronic current density |j| Induced magnetic field BzIdentification of atom-centered and aromatic current densities

Nucleus independent chemical shift maps

22

Isolated molecules

• Isolated organic molecules, 1H and 13C chemical shifts

• Comparison with Gaussian 98 calculation,(converged basis set DFT/BLYP)

23 24 25 26 27 28 29 30 31 32

σH[ppm] - exp

23

24

25

26

27

28

29

30

31

32

σH[p

pm] -

cal

c

Gaussian (DFT)this workMPL method

C6H6

C2H4

C2H2

C2H6

H2O

CH4

40 60 80 100 120 140 160 180 200

σC [ppm] - exp

40

60

80

100

120

140

160

180

200

σC [p

pm]

- c

alc

Gaussian (DFT)this workMPL method

C6H6

C2H6

C2H2

C2H4

CH4

23

Example system: Water cluster

• Water cluster: water moleculesurrounded by 6 neighbors

• Strong hydrogen bonding,nonsymmetric geometry

24

Example system: Water cluster

• Hydrogen bonding effectsstrongly affect the proton

chemical shieldings

• Large range ofindividual shieldings

25

Extended system: liquid water

• Most important solvent on earth

• Complex, dynamic hydrogenbonding

• Configuration: single snapshotfrom molecular dynamics

• Complex hydrogen bonding,strong electrostatic effects

• NMR experiment: average overentire phase space

32 water molecules atρ=1g/cm3, under periodicboundary conditions

26

Supercritical water: hydrogen bond network

8/2002

CPCHFT 110 (8) 643 – 724 (2002) · ISSN 1439-4235 · Vol. 3 · No. 8 · August 16, 2002 D55711

Concept: Conductance Calculations for Real Nanosystems(F. Grossmann)

Highlight: Terahertz Biosensing Technology(X.-C. Zhang)

Conference Report: Femtochemistry V(M. Chergui)

2001 Physics

NOBEL LECTURE

in this issue

• ab-initio MD:3×9ps, 32 moleculesP.L. Silvestrelli et al.,

Chem.Phys.Lett. 277, 478 (1997)

M. Boero et al.,

Phys.Rev.Lett. 85, 3245 (2000)

• NMR sampling:3×30 configurations3×2000 proton shifts

• Experimental data:N. Matubayashi et al.,

Phys.Rev.Lett. 78, 2573 (1997)

27

Supercritical water: chemical shift distributions

-2-101234567891011121314δH [ppm]

0

5

10

15

20

25

30

35

40

45

-2-101234567891011121314δH [ppm]

05

101520253035404550556065

-2-101234567891011121314δH [ppm]

0

10

20

30

40

50

60

70

80

ρ=1 g/cm3, T=303K ρ=0.73 g/cm3, T=653K ρ=0.32 g/cm3, T=647K

• Standard conditions: broad Gaussian distribution,continuous presence of hydrogen bonding

• Supercritical states: narrow distribution,hydrogen bonding “tails”

28

Supercritical water: gas – liquid shift

• Qualitatively reducedhydrogen bond network in

supercritical water

• Excellent agreement withexperiment

• Slight overestimation ofH-bond strength at T◦−

BLYP overbinding ?

Insufficient relaxation ?

0 0.2 0.4 0.6 0.8 1ρ [g / cm3]

0

1

2

3

4

5

6

δH

[pp

m]

calculated δliq (this work)calculated δliq (MPL)experimental δliq

=⇒ confirmation of simulation

29

Ice Ih: gas – solid shift

• Ice Ih: hexagonal lattice withstructural disorder

• 16 molecules unit cell,full relaxation

• Experimental/computedHNMR shifts [ppm]:

Exp Exp MPL this work

7.4 9.7 8.0 6.6

30

Crystalline imidazole

18 14 10[ppm]

6 2 0 −2

(a)

(b)

(c)

experimental

calculated

(crystal)

calculated

(molecule)

• Molecular hydrogen-bonded crystal

• Very good reproductionof experimental spectrum

• HNMR: π-electron – proton interactions, mobile imidazole

31

Crystalline Imidazole-PEO

• Imidazole – [Ethyleneoxide]2 – Imidazole• Strongly hydrogen bonded dimers,

complex packing structure

• Anisotropic proton conductivity (fuel cell membranes)

32

Crystalline Imidazole-PEO: NMR spectra

top: experimentalmiddle: calculated (crystal)

bottom: calculated (molecule)

• Particular hydrogen bonding:two types of high-field resonances,

intra-pair / inter-pair

• Partly amorphous regions (10ppm):mobile Imidazole-PEO molecules

• Packing effect at 0ppm

• Quantitative reproductionof experimental spectrum

33

Chromophore crystal: yellow dye

• Material for photographic films

• Unusual CH· · ·O bondunusual packing effects

• 244 atoms / unit cell

34

Chromophore NMR spectrum

top: experimentalbottom: calculated

• Full resolution of experimental spectrum,unique assignment of resonances

• Strong packing effectsfrom aromatic ring currents:

CH3 · · · Ar, ArH · · · Ar

• H-bonding too weak (9ppm):insufficient geometry optimization,

temperature effects

• Starting point for polycrystalline phase

35

Conclusion

• NMR chemical shifts from ab-initio calculations

• Gas-phase, liquid, amorphous and crystalline systems

• Assignment of experimental shift peaks to specific atoms

• Verification of conformational possibilities by their NMR patternStrong dependency on geometric parameters (bonds, angles, . . . )

• Quantification of hydrogen bonding

36