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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
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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
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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
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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
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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
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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)
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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)
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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
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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
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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]
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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) + . . .
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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] . . .
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Isolated molecules
• Isolated organic molecules, 1H and 13C chemical shifts
• Comparison with Gaussian 98 calculation,(converged basis set DFT/BLYP)
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σH[ppm] - exp
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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
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Example system: Water cluster
• Water cluster: water moleculesurrounded by 6 neighbors
• Strong hydrogen bonding,nonsymmetric geometry
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Example system: Water cluster
• Hydrogen bonding effectsstrongly affect the proton
chemical shieldings
• Large range ofindividual shieldings
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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
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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)
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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”
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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
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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
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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
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Crystalline Imidazole-PEO
• Imidazole – [Ethyleneoxide]2 – Imidazole• Strongly hydrogen bonded dimers,
complex packing structure
• Anisotropic proton conductivity (fuel cell membranes)
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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
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Chromophore crystal: yellow dye
• Material for photographic films
• Unusual CH· · ·O bondunusual packing effects
• 244 atoms / unit cell
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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
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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
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