MolecularNMRShieldings,J-Couplings,and … · 2018-07-15 · Introduction Theory Results Summary...

Introduction Theory Results Summary

Molecular NMR Shieldings, J-Couplings, andMagnetizabilities from Numeric Atom-Centered

Orbital Based Density-Functional Theory

Raul Laasner1, William Huhn1, Johannes Colell2, Thomas Theis2,Victor Yu1, Warren Warren2,3, Volker Blum1,2

1Department of Mechanical Engineering and Materials Science, Duke University, DurhamNC

2Department of Chemistry, Duke University, Durham NC3Department of Physics, Duke University, Durham NC

July 10, 2018

Laasner, Huhn, Colell, Theis, Yu, Warren, BlumNMR implementation in FHI-aims


I Nuclear magnetic resonance (NMR) spectroscopy is anon-destructive technique for studying the local chemicalenvironment by exploiting magnetic properties of atomic nuclei.

I Central concept: applied magnetic field is modified at the nucleusdue to shielding by the electron cloud.



Primary motivation for this work

I The (weak) NMR signal is proportional to the population differenceof nuclear spin-up and spin-down levels:

N↑N↓

= e−∆E/kT

I Experimental technique knownas hyperpolarization booststhe signal by several orders ofmagnitude.

I Collaboration withexperimentalists for finding newways of improvinghyperpolarization.

Potential applications:millitesla imaging



Which properties are probes in NMR?



Effective spin Hamiltonian

∆H = −∑A

~γAB(1−←→σ A)IA +∑

A>B

~IA←→J ABIB −

12B←→ξ B

I ←→σ A: chemical shielding tensorI←→J AB: J-coupling (indirect spin-spin coupling) tensor

I←→ξ : magnetizability tensor

I B: is external magnetic fieldI IA: spin of nucleus AI γA: gyromagnetic ratio of nucleus A



Effective spin Hamiltonian

I Commonly evaluated as second total energy derivatives:

←→σ A = 1~γA

d2E

dBdIA

∣∣∣∣∣B=0;IA=0

,

←→J AB = 1

h

d2E

dIAdIB

∣∣∣∣∣IA=0IB=0

,

←→ξ =− d2E

dBdB

∣∣∣∣∣B=0

.

I Total energy evaluated with a wavefunction theory of densityfunctional theory (DFT).



Magnetic Response with DFTI Magnetic interactions are introduced via the minimal coupling

transformation:p→ p+A = −i∇ +A

I DFT Hamiltonian with magnetic interactions:

H =HKS + pA+ 12A

2 + S(∇×A),

HKS =12p

2 + Vext + VH + Vxc.

I Vector potential A in the presence of an external magnetic field andnuclear magnetic moments:

A = 12B × r + α2∑

A

µA × rA

r3A

,

rA = r −RA: the position vector relative to nucleus A;µA: magnetic dipole moment of nucleus A.



Implementation in FHI-aims

I Chemical shieldings, J-couplings, magnetizabilityI Nonrelativistic formalismI Semilocal functionalsI Gauge origin problem of uniform magnetic field handled by the

GIAO (gauge including atomic orbitals) formalism



I Shieldings, J-couplings, and magnetizabilities were calculated for aset of small molecules.

I We compare two types of NAOs (FHI-aims-09 [Blum et al. (2009)],NAO-VCC-nZ [Zhang et al. (2013)]) against commonly usedGaussian type (GTO) basis sets.

H2O Li2O HFCO H2C2O Al(OH)3

BeF2 MgCl2 MgF2 BCl3 SiF4

NaCH3 LiCH3 Be(CH3)2 NH3 C6H6

H2SO4 SO2−4 B2H6 CH3NO2 Al2Cl6

H3PO4 PClF2 CF4 CFCl3 TMS



NMR shielding constants

0.01

0.1

1

10

100

0 250 500 750 1000

Dev

iation

from

refe

renc

e∗,pp

m

Average number of basis functionsper molecule, Nave

FHI-aims-09NAO-VCC-nZ

aug-pcS-ncc-pVnZ

aug-cc-pVnZcc-pCVnZ

aug-cc-pCVnZ

* reference: u-aug-pcS-4

I The NAOs converge faster thanDunning’s correlationconsistent basis sets.

I The NAOs are on par withpolarization consistent(aug-pcS-n) basis sets.



Magnetizabilities

0.1

1

10

100

0 200 400 600 800 1000Dev

iation

from

refe

renc

e∗,10−30J/

T2

Average number of basis functionsper molecule, Nave


aug-pc-n

cc-pVnZaug-cc-pVnZ

* reference: aug-pc-4

I The NAOs converge faster thanDunning’s correlationconsistent basis sets.

I The NAOs are on par withpolarization consistent(aug-pc-n) basis sets.



J-couplings

I The general purpose NAOs struggle to converge.I Reason: small size, insufficient flexibility near the nucleus.I Problematic mainly due to the Fermi contact term:

HFCµAmn ∝ 〈φm|δ(RA)|φn〉 = φm(RA)φn(RA).

I New type of NAOs, NAO-J, weredeveloped by merging NAO-VCC-nZwith tight s orbitals from the u-pcJ-nbasis sets.

I NAO-J converge at a rate similar tothe highly accurate pcJ-n.

OH J-coupling in Al(OH)3

−110−100−90−80−70−60−50−40−30−20−10

0 100 200 300 400 500 600 700 800

J-co

uplin

g,H

z

Basis set size

NAO-J-npcJ-n

NAO-VCC-nZ



Medium size systems

I Gaussian type basis sets can be come ill-conditioned for largersystems (basis functions become linearly dependent on each other).

I NAOs generally remain stable.I We take the lowest eigenvalue of the overlap matrix as a measure of

ill-conditioning:minλS ,

whereSmn = 〈φm|φn〉 .

I If minλS becomes too small, numerical errors may affectobservables.



Medium size systems

−30

−20

−10

0

10

20

30

40

50

0 3000 6000 9000 12000

Che

mic

alsh

ield

ing,

ppm

Basis set size


pcS-naug-pcS-ncc-pCVnZ

aug-cc-pCVnZ

FHI-aims-09 NAO-VCC-nZ aug-pcS-n

tier Nbasis minλS n Nbasis minλS n Nbasis minλS

1 1240 6.4× 10−4 2 1618 1.6× 10−4 0 1069 1.0× 10−6

2 3395 7.3× 10−5 3 3135 1.5× 10−5 1 2232 3.5× 10−8

3 5283 4.7× 10−6 4 5608 2.0× 10−6 2 4469 3.5× 10−10

4 6908 6.3× 10−7 5 9273 2.7× 10−7 3 9085 2.1× 10−13

4 14739 1.4× 10−14



Medium size systems

74

76

78

80

82

84

86

88

0 500 1000 1500 2000 2500 3000

Shie

ldin

gco

nsta

nt,pp

m

Total basis set size

C shielding constant


pcS-naug-pcS-n

Tier Nbasis minλS Tier Nbasis minλSFHI-aims-09 NAO-VCC-nZ

1 241 1.1 × 10−3 2Z 320 1.7 × 10−4

2 681 7.0 × 10−5 3Z 625 1.5 × 10−5

3 1049 7.5 × 10−6 4Z 1119 2.6 × 10−6

4 1399 7.5 × 10−7 5Z 1848 4.4 × 10−7

pcS-n aug-pcS-n0 144 6.0 × 10−3 0 209 6.8 × 10−6

1 283 4.3 × 10−4 1 445 8.0 × 10−7

2 588 7.7 × 10−5 2 893 1.2 × 10−9

3 1328 1.8 × 10−6 3 1822 5.6 × 10−11

4 2219 6.9 × 10−9 4 2948 1.9 × 10−12

I For GTOs, the (long-range)augmentation functions are primarilyresponsible for ill-conditioning.



Large scale systems

Basis set Basis size JCH

NAO-VCC-2Z/NAO-J-5 15 108 145.66HzNAO-VCC-3Z/NAO-J-5 29 041 145.07HzNAO-VCC-4Z/NAO-J-5 51 742 144.98Hz

Basis set Basis size σC

NAO-VCC-2Z 14 972 96.0 ppmNAO-VCC-3Z 28 930 97.6 ppmNAO-VCC-4Z 51 672 97.2 ppm

I Shieldings and J-coupling of aDNA segment

I 1052 atomsI PBE functional

NAO-VCC-2Z/NAO-J-5: 10 min with 3840 cores on Edison



I NMR shieldings, J-couplings, and magnetizabilities wereimplemented using numeric atom-centered orbitals (NAOs) inFHI-aims.

I For small molecules, NAOs are as accurate or better than Gaussiantype orbitals (GTOs).

I For larger systems, the NAOs suffer less from basis setill-conditioning than the GTOs.

I Large-scale applicability has been demonstrated.

Acknowledgment is made to the Donors of the American ChemicalSociety Petroleum Research Fund for partial support of this research.This work was also supported by the National Science Foundation underGrant Number 1450280.


MolecularNMRShieldings,J-Couplings,and … · 2018-07-15 · Introduction Theory Results Summary...

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