First Principles Calculations of NMR Chemical Shifts Daniel Sebastiani

First Principles Calculations of NMR Chemical Shifts

Methods and Applications

Daniel Sebastiani

Approche théorique et expérimentaledes phénomènesmagnétiques 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 Bext • Electronic reaction: induced current j(r) Bind =⇒ inhomogeneous magnetic ﬁeld Bind(r) jind

• Nuclear spin µµ Up/Down energy level splitting

Β=0 Β=Β 0 ¯hω

∆E = 2µµ · B =hω ¯

5 Chemical shifts – chemical bonding

• NMR shielding tensor σ: definition through induced field Btot(R) = Bext + Bind(R) ∂Bind(R) σ(R) = − 1 ∂Bext • Strong effect of chemical bonding Hydrogen atoms: H-bonds

=⇒ NMR spectroscopy: unique characterization of local microscopic structure (liquid water)

6 Chemical shielding tensor

 ind ind ind  ∂Bx (R) ∂Bx (R) ∂Bx (R) ext ext ext  ∂Bx ∂By ∂Bz   ind ind ind  σ(R) = −  ∂By (R) ∂By (R) ∂By (R)   ∂Bext ∂Bext ∂Bext   x y z  ind ind ind  ∂Bz (R) ∂Bz (R) ∂Bz (R)  ext ext ext ∂Bx ∂By ∂Bz

• 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 Basis sets • Hartree-Fock • Slater-type functions: m Yl exp −r/a0 • Møller-Plesset Perturbation Theory • Gaussian-type functions: m 2 Yl exp −(αr) • Highly correlated methods CI, coupled cluster, . . . • Plane waves: exp ig · r • Density functional theory

8 Kohn-Sham density functional theory (DFT)

Central quantity: electronic density, total energy functional No empirical parameters

1 X Z E [{ϕ }] = − d3r hϕ |∇2|ϕ i KS i 2 i i i 1 Z ρ(r)ρ(r0) + d3r d3r0 2 |r − r0| X Z ρ(r) + q d3r + E [ρ] at |r − R | xc at at X 2 ρ(r) = |ϕi(r)| i

9 DFT: Variational principle

• Variational principle: selfconsistent Kohn-Sham equations

hϕi|ϕji = δij δ (EKS[ϕ] − Λkjhϕj|ϕki) = 0 δ ϕi(r) ˆ H[ρ] |ϕii = εi|ϕii

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):

Hˆ 7→ Hˆ(0) + λHˆ(1) + λ2Hˆ(2) + ... ϕ 7→ ϕ(0) + λϕ(1) + ... E 7→ E(0) + λE(1) + λ2E(2) + ...

11 Perturbation theory in DFT

Perturbation expansion

(0) λ E[ϕ] = E[ϕ] + λ E[ϕ] + ... ϕ = ϕ(0) + λ ϕλ + ... λ h (0) λ i ρ (r) = 2 < ϕi (r) ϕi (r) ˆ ˆ(0) ˆλ ˆC H = H + λ H + H[ρλ] + ...

1 E = E(0) + λ Eλ + λ2 E(2) ... [ϕ] [ϕ] [ϕ(0)] 2 [ϕ]

12 Perturbation theory in DFT

• unperturbed wavefunctions ϕ(0) known: h i min E[ϕ] ⇐⇒ min E(2) ϕ(0), ϕ(1) {ϕ} {ϕ(1)}

δ2E(0) δEλ E(2) = ϕ(1) ϕ(1) + ϕ(1) δϕ δϕ δϕ

(0) (1) • orthogonality hϕj |ϕk i = 0 ∀ j, k

13 Perturbation theory in DFT

Iterative calculation ˆ(0) (0) λ ˆC (0) ˆλ (0) H δij − εij ϕj + H[ρλ] ϕi = −H ϕi

Formal solution λ ˆ ˆλ (0) ϕi = Gij H ϕj

14 Magnetic ﬁeld perturbation

• Magnetic ﬁeld perturbation: vector potential A 1 A = − (r − R ) × B 2 g e Hˆλ = − ˆp · Aˆ m

¯he = i B · (ˆr − R ) × ∇ˆ 2m g

• Cyclic variable: gauge origin Rg

• Perturbation Hamiltonian purely imaginary =⇒ ρλ = 0

15 Magnetic ﬁeld perturbation

Resulting electronic current density:

Dia- and paramagnetic contributions: zero and ﬁrst order wavefunctions

16 The Gauge origin problem

• Gauge origin Rg theoretically not relevant

dia 0 2 • In practice: very important: j (r ) ∝ Rg

• GIAO: Gauge Including Atomic Orbitals

• IGLO: Individual Gauges for Localized Orbitals

0 • 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 ˆr not periodic

• non-periodic perturbation Hamiltonian Hˆλ

18 PBC: Individual rˆ-operators for localized orbitals

• Localized Wannier orbitals ϕi via unitary rotation:

ϕi = Uij ψj

orbital centers of charge di

• Idea: ^ ra (x) ϕ a(x) Individual

position ^ rb (x)

operators ϕ (x) b (x)

0 db d a L 2L

19 Magnetic ﬁelds in electronic structure

• Variational principle 7→ electronic response orbitals

ˆλ ˆ 1 • Perturbation Hamiltonian H : A = −2 (ˆr − 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

0 (0) ˆ (α) (β) (∆) jk(r ) = hϕk | jr0 |ϕk i − |ϕk i + |ϕk i

e h 0 0 0 0 i ˆj 0 = ˆp|r ihr | + |r ihr |ˆp r 2m

modulus of current |j|

B-ﬁeld along Oz

21 Current and induced magnetic ﬁeld in graphite

Electronic current density |j| Induced magnetic ﬁeld Bz Identiﬁcation 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)

32 200 C H H O 2 6 31 2 C H 180 30 2 6 CH4 C2H2 160 29 C H 2 2 CH4 140 28 120 27 C2H4 [ppm] - calc [ppm] - calc

H 100 C H C σ 26 6 6 C H Gaussian (DFT) σ 6 6 Gaussian (DFT) 80 25 this work this work MPL method MPL method 24 60 C2H4

23 40 23 24 25 26 27 28 29 30 31 32 40 60 80 100 120 140 160 180 200 H C σ [ppm] - exp σ [ppm] - exp

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

32 water molecules at • NMR experiment: average over ρ=1g/cm3, under periodic entire phase space boundary conditions

26 Supercritical water: hydrogen bond network

CPCHFT 110 (8) 643 – 724 (2002) · ISSN 1439-4235 · Vol. 3 · No. 8 · August 16, 2002 D55711 • 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., 8/2002 Phys.Rev.Lett. 78, 2573 (1997) Concept: Conductance Calculations for Real Nanosystems (F.Grossmann) NOBEL2001 LECTURE Physics Highlight: Terahertz Biosensing Technology in this issue (X.-C. Zhang) Conference Report: Femtochemistry V (M. Chergui) 27 Supercritical water: chemical shift distributions

45 65 60 80 40 55 70 35 50 60 30 45 40 50 25 35 20 30 40 25 30 15 20 10 15 20 10 5 10 5 0 0 0 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 -1 -2 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 -1 -2 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 -1 -2 H H H δ [ppm] δ [ppm] δ [ppm] 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 6 δ calculated liq (this work) supercritical water 5 δ calculated liq (MPL) experimental δ 4 liq • Excellent agreement with 3 [ppm]

experiment H δ 2

• Slight overestimation of 1 H-bond strength at T◦− 0 0 0.2 0.4 0.6 0.8 1 3 BLYP overbinding ? ρ [g / cm ] Insuﬃcient relaxation ? =⇒ 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

• Molecular hydrogen-bonded crystal

experimental

(a)

calculated (b) (crystal)

• Very good reproduction 18 14 10 6 2 0 −2 (molecule) of experimental spectrum [ppm]

• 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

• 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 top: experimental middle: calculated (crystal) of experimental spectrum bottom: calculated (molecule)

33 Chromophore crystal: yellow dye

• Material for photographic ﬁlms

• Unusual CH···O bond unusual packing eﬀects

• 244 atoms / unit cell

34 Chromophore NMR spectrum

• 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 top: experimental bottom: calculated • 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