first principles calculations of nmr chemical shifts
TRANSCRIPT
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