first principles calculations of nmr chemical shifts

First Principles Calculationsof NMR Chemical Shifts

Methods and Applications

Daniel Sebastiani

Approche theorique et experimentale des phenomenes magnetiques et des

spectroscopies associees

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/Down

energy level splitting

Β=0Β=Β0 hω

∆E = 2µµµ ·B = hω

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 bonding

Hydrogen atoms: H-bonds

=⇒ NMR spectroscopy:

unique characterization

of local microscopic structure (liquid water)

6

Chemical shielding tensor

σ(R) = −

∂Bind

x (R)∂Bext

x

∂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-Plesset

Perturbation Theory

• Highly correlated methods

CI, 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〉 = δij

δ

δ ϕi(r)(EKS[ϕ]− Λkj〈ϕj|ϕk〉) = 0

H[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):

H 7→ H(0) + λH(1) + λ2H(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 Eλ

[ϕ] + . . .

ϕ = ϕ(0) + λ ϕλ + . . .

ρλ(r) = 2 <[ϕ

(0)i (r) ϕλ

i (r)]

H = H(0) + λ Hλ + HC[ρλ]

+ . . .

E[ϕ] = E(0)[ϕ] + λ Eλ

[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(H(0) δij − ε

(0)ij

)ϕλ

j + HC[ρλ]

ϕ(0)i = −Hλ ϕ

(0)i

Formal solution

ϕλi = Gij Hλ ϕ

(0)j

14

Magnetic field perturbation

• Magnetic field perturbation: vector potential A

A = −12

(r−Rg)×B

Hλ = − e

mp · A

= ihe

2mB · (r−Rg)× ∇

• Cyclic variable: gauge origin Rg

• Perturbation Hamiltonian purely imaginary =⇒ ρλ = 0

15

Magnetic field perturbation

Resulting electronic current density:

jr′ =e

2m

[π|r′〉〈r′|+ |r′〉〈r′|π

]jr′ =

e

2m

[(p− eA)|r′〉〈r′|+ |r′〉〈r′|(p− eA)

]j(r′) =

∑k

〈ϕ(0)k | j(2)r′ |ϕ(0)

k 〉+ 2 〈ϕ(0)k | j(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 r not periodic

• non-periodic perturbation Hamiltonian Hλ

18

PBC: Individual r-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

r (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 Hλ: A = −12 (r−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 | jr′

(|ϕ(α)

k 〉 − |ϕ(β)k 〉+ |ϕ(∆)

k 〉)

jr′ =e

2m

[p|r′〉〈r′|+ |r′〉〈r′|p

]

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 molecule

surrounded by 6 neighbors

• Strong hydrogen bonding,

nonsymmetric geometry

24

Example system: Water cluster

• Hydrogen bonding effects

strongly affect the proton

chemical shieldings

• Large range of

individual shieldings

25

Extended system: liquid water

• Most important solvent on earth

• Complex, dynamic hydrogen

bonding

• Configuration: single snapshot

from molecular dynamics

• Complex hydrogen bonding,

strong electrostatic effects

• NMR experiment: average over

entire 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 molecules

P.L. Silvestrelli et al.,

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

M. Boero et al.,

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

• NMR sampling:

3×30 configurations

3×2000 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 reduced

hydrogen bond network in

supercritical water

• Excellent agreement with

experiment

• Slight overestimation of

H-bond strength at T◦−

BLYP overbinding ?

Insufficient relaxation ?

0 0.2 0.4 0.6 0.8 1ρ [g / cm

3]

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 with

structural disorder

• 16 molecules unit cell,

full relaxation

• Experimental/computed

HNMR 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 reproduction

of 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 reproduction

of experimental spectrum

33

Chromophore crystal: yellow dye

• Material for photographic films

• Unusual CH· · ·O bond

unusual packing effects

• 244 atoms / unit cell

34

Chromophore NMR spectrum

top: experimentalbottom: calculated

• Full resolution of experimental spectrum,

unique assignment of resonances

• Strong packing effects

from 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 pattern

Strong dependency on geometric parameters (bonds, angles, . . . )

• Quantification of hydrogen bonding

36