Quantum-Chemical Calculations of NMR Parametersfolk.uio.no/helgaker/talks/Warsaw_2014.pdf ·...

Quantum-Chemical Calculations of NMR Parameters

Trygve Helgaker

Centre for Theoretical and Computational Chemistry (CTCC),Department of Chemistry, University of Oslo, Norway

VIIIth Symposium onNuclear Magnetic Resonance in Chemistry, Physics and Biological Sciences

Institute of Organic Chemistry, Polish Academy of SciencesKasprzaka 44/52, 01-224 Warsaw, Poland

24–26 September 2014

T. Helgaker (CTCC, University of Oslo) Calculation of NMR parameters 25 September 2014 1 / 50

Overview

I NMR parameters

I NMR spin Hamiltonian,I shielding and indirect spin–spin coupling constants as derivativesI Zeeman and hyperfine coupling constants,I Ramsey’s expressions

I Quantum chemistry methods and their application to NMR parameters

I Hartree–Fock theoryI multiconfiguration self-consistent field (MCSCF) theoryI coupled-cluster theoryI Kohn–Sham density-functional theory


High-resolution NMR spin Hamiltonian

I Consider a molecule in an external magnetic field B along the z axis and with nuclearspins IK related to the nuclear magnetic moments MK as:

MK = γK~IK ≈ 10−4 a.u.

where γK is the magnetogyric ratio of the nucleus.

I Assuming free molecular rotation, the nuclear magnetic energy levels can be reproduced bythe following high-resolution NMR spin Hamiltonian:

HNMR = −∑K

γK~(1− σK )BIK z︸︷︷︸nuclear Zeeman interaction

+∑K>L

γKγL~2KKLIK · IL︸︷︷︸nuclear spin–spin interaction

where we have introduced

I the nuclear shielding constants σKI the (reduced) indirect nuclear spin–spin coupling constants KKL

I This is an effective nuclear spin Hamiltonian:

I it reproduces NMR spectra without considering the electrons explicitlyI the spin parameters σK and KKL are adjusted to fit the observed spectraI we shall consider their evaluation from molecular electronic-structure theory


Taylor expansion of the energy

I Expand the energy in the presence of an external magnetic field B and nuclear magneticmoments MK around zero field and zero moments:

E (B,M) = E0 +

perm. magnetic moments︷︸︸︷BTE(10) +

hyperfine coupling︷︸︸︷∑K

MTKE

(01)K

+1

2BTE(20)B︸︷︷︸

− magnetisability

+1

2

∑K

BTE(11)K MK︸︷︷︸

shieldings + 1

+1

2

∑KL

MTKE

(02)KL ML︸︷︷︸

spin–spin couplings

+ · · ·

I First-order terms vanish for closed-shell systems because of symmetry

I these are not considered here (important in paramagnetic NMR)

I Second-order terms are important for many molecular properties

I magnetisabilitiesI nuclear shielding constants of NMRI nuclear spin–spin coupling constants of NMRI electronic g tensors of EPR (not dealt with here)

I We must calculate how the electronic energy changes in the presence of B and MK

I the properties of interest to us are second derivativesI such derivatives are calculated using special techniques response theory


Quantum-mechanical digression: perturbation theory

I The nonrelativistic electronic Hamiltonian:

H = H0 + H(1) + H(2) = H0 + A (r) · p + B (r) · s +1

2A (r)2

I Vector potentials of the uniform external field and the nuclear magnetic moments:

A (r) =1

2B× rO, AK (r) = α2 MK × rK

r3K

, ∇× A (r) = B (r) , ∇ · A (r) = 0

I Orbital and spin Zeeman interactions with the external magnetic field:

H(1)Z =

1

2B · LO + B · s

I Orbital and spin hyperfine interactions with the nuclear magnetic moments:

H(1)hf = α2 MK · LK

r3K︸︷︷︸

PSO

+8πα2

3δ (rK )MK · s︸︷︷︸

FC

+α2 3(s · rK )(rK ·MK )− (MK · s)r2K

r5K︸︷︷︸

SD

I Second-order Rayleigh–Schrodinger perturbation theory:

E (2) =1

2

⟨0∣∣A2∣∣ 0⟩−∑n

〈0 |A · p + B · s| n〉〈n |A · p + B · s| 0〉En − E0


Zeeman and hyperfine interactions

PSO hyperfine

Zeeman

PSOFC+SDFC+SD

SS, SO, OO

SO SO


Ramsey’s expressions for NMR parameters

I Nuclear shielding tensors of a closed-shell system:

σK =d2Eel

dBdMK=α2

2

⟨0

∣∣∣∣∣ rTO rK I3 − rOrTK

r3K

∣∣∣∣∣ 0

⟩︸︷︷︸

diamagnetic term

−α2∑n

⟨0∣∣LO∣∣ n⟩ ⟨n ∣∣∣r−3

K LTK

∣∣∣ 0⟩

En − E0︸︷︷︸paramagnetic term

I Indirect nuclear spin–spin coupling tensors of a closed-shell system:

KKL =d2Eel

dMKdML= α4

⟨0

∣∣∣∣∣ rTK rLI3 − rK rTL

r3K r

3L

∣∣∣∣∣ 0

⟩︸︷︷︸

diamagnetic spin–orbit (DSO)

− 2α4∑n

⟨0∣∣∣r−3K LK

∣∣∣ n⟩⟨n ∣∣∣r−3L LTL

∣∣∣ 0⟩

En − E0︸︷︷︸paramagnetic spin–orbit (PSO)

− 2α4∑n

⟨0

∣∣∣∣ 8π3δ(rK )s +

3rK rTK −r2K I3

r5K

s

∣∣∣∣ n⟩⟨n ∣∣∣∣ 8π3δ(rL)sT+

3rLrTL −r2

L I3r5L

sT∣∣∣∣ 0

⟩En − E0︸︷︷︸

Fermi contact (FC) and spin–dipole (SD)

I In practice, we do not use perturbation theory for the evaluation of these constants

I solution of linear response equations rather than sum over statesI three linear equations for shieldings, ten linear equations for spin–spin couplingsI gauge-origin problem for shieldings requires special attention (London orbitals)


Relative importance of the contributions to spin–spin coupling constants

I The isotropic indirect spin–spin coupling constants can be decomposed as:

JKL = JDSOKL + JPSO

KL + JFCKL + JSD

KL

I The spin–spin coupling constants are often dominated by the FC term

I Since the FC term is relatively easy to calculate, it is tempting to ignore the other terms.

I However, none of the contributions can be a priori neglected (N2 and CO)!

H2 HF H2O

O-H

NH3

N-H

CH4

C-H

C2H4

C-C

HCN

N-C

N2 CO C2H2

C-C

-100

0

100

200

FC

FCFC FC FC FC

FCFC

FCFC

PSO

PSO

PSO

SD

SD

SD


Calculation of NMR spectra by electronic-structure methods

I 200 MHz NMR spectra of vinyllithium (C2H3Li)


Quantum chemistry: wave functions and densities

I There are two approaches to the electronic-structure problem

I wave-function theory (WFT) and density-functional theory (DFT)

I WFT attempts an explicit construction of the electronic wave function Ψ

I the main tool is the Rayleigh–Ritz variation principle

E(v) = minΨ

〈Ψ|H(v)|Ψ〉〈Ψ|Ψ〉

I expensive since Ψ(x1, x2, . . . xN) depends on all electron coordinatesI a difficult manybody problemI but systematically refinable towards the exact solution

I DFT avoids the manybody problem by concentrating on the electron density ρ

I the main tool is the Hohenberg–Kohn variation principle:

E(v) = minρ

(F (ρ) +

∫ρ(r)v(r) dr

)I less expensive since ρ(r) depends on only three spatial coordinatesI the explicit form of universal density functional F is unknownI DFT is not systematically refinable towards the exact solution


Hartree–Fock approximation: the starting point of WFT

I The Hartree–Fock model is the fundamental approximation of wave-function theory

I each electron moves in the mean field of all other electronsI provides an uncorrelated description: average rather than instantaneous interactionsI gives rise to the concept of molecular orbitals (MOs)

I HF wave function is a determinant of MOs (each with a spin factor):

Ψ = det |ψ1α, ψ1β , ψ1α · · · |

I each MO is a linear combination of atomic orbitals (AOs):

ψi (r) =∑

µcµi χµ(r) ← LCAO

I MO coefficients are determined by the Rayleigh–Ritz variation principle

I Standard collections of AOs called basis sets have been developed for each atom:

I the quality of calculation depends on the number of AOs in the basis setI single-zeta (SZ), double-zeta (DZ), triple-zeta (TZ)

I Typical errors in HF calculations

I 0.5% in energy, 1% in geometries, 5% or more for other propertiesI mean relative errors for shielding constants (%):

DZ TZ QZ14 24 26


Calculated Hartree–Fock NMR constants

I HF theory gives qualitatively correct shieldings but incorrect spin–spin couplings

HF exp.HF F 413.6 410± 6 (300K)

H 28.4 28.5± 0.2 (300K)H2O O 328.1 323.6± 6 (300K)

H 30.7 30.05± 0.02NH3 N 262.3 264.5

H 31.7 31.2± 1.0CH4 C 194.8 198.7

H 31.7 30.61F2 F −167.9 −192.8N2 N −112.4 −61.6± 0.2 (300K)CO C −25.5 3.0± 0.9 (eq)

O −87.7 −56.8± 6 (eq)

HF expHF 1KHF 59.2 47.6CO 1KCO 13.4 −38.3N2

1KNN 175.0 −19.3H2O 1KOH 63.7 52.8

2KHH −1.9 −0.7NH3

1KNH 61.4 50.82KHH −1.9 −0.9

C2H41KCC 1672.0 87.81KCH 249.7 50.02KCH −189.3 −0.42KHH −28.7 0.23Kcis 30.0 0.93Ktns 33.3 1.4


HF theory and triplet instabilities

I The correct description of triplet excitations is important for spin–spin coupling constants

I In restricted Hartree–Fock (RHF) theory, triplet excitations are often poorly described

I upon H2 dissociation, RHF does not describe the singlet ground state correctlyI but the lowest triplet state dissociates correctly, leading to triplet instabilities

2 4 6 R

-2

-1

1

1Sg+H1Σg

2L

1Sg+H1Σu

2L

1Sg+HFCIL

1Sg+HFCIL

3Su+H1Σg1ΣuL

1Su+H1Σg1ΣuL

covalent

ionic

cov-ion

1SH1s2L

3PH1s2pL

1PH1s2pL

1DH2p2L

I Near such instabilities, the RHF description of spin interactions becomes unphysical

C2H4/Hz 1JCC1JCH

2JCH2JHH

3Jcis3Jtrans

exp. 68 156 −2 2 12 19RHF 1270 755 −572 −344 360 400CAS 76 156 −6 −2 12 18





Helium atom

I The motion of each electron is correlated with that of all others: electron correlationI in HF theory, this correlation is neglectedI each electron moves in the mean field generated by the other electrons

I Wave function of the helium atom:I wave function for one electron, with one electron fixed in spaceI HF wave function to the left, the exact wave function to the right

!1.0!0.5

0.00.5

1.0

!0.50.0

0.5

!0.5

0.0

0.5

!1.0!0.5

0.00.5

.5

0.0

!1.0!0.5

0.00.5

1.0

!0.50.0

0.5

!0.5

0.0

0.5

!1.0!0.5

0.00.5

.5

0.0

I concentric Hartree–Fock contours, reflecting an uncorrelated descriptionI in reality, the electrons see each other and the contours becomes distorted


Coulomb hole

I Let us subtract the HF wave function from the exact wave function:

-1.0

-0.5

0.0

0.5

1.0

-0.5

0.0

0.5

-0.10

-0.05

0.00

-1.0

-0.5

0.0

0.5

0.5

0.0

-0

-

I The exact wave function has reduced probability when the electrons are close to each other

I Reduced probability caused by Coulomb repulsion: Coulomb hole


Electron correlation in H2

I For an improved description, we must describe the effects of electron correlation

I in real space, the electrons are constantly being scattered by collisionsI we represent such events by excitations from occupied to virtual spin orbitalsI the most important among these are the double excitations or pair excitations

I Consider the effect of a double excitation in H2:

|1σ2g〉 → |1σ2

g〉 − 0.11|1σ2u〉

I the one-electron density ρ(z) is hardly affected:

-2 -1 0 1 2 -2 -1 0 1 2

I the two-electron density ρ(z1, z2) changes dramatically:

-2

0

2

-2

0

2

0.00

0.04

2

0

2

-2

0

2

-2

0

2

-2

0

2

0.00

0.04

2

0

2

-2

0

2


Multiconfiguration self-consistent field (MCSCF) method

I Let us improve upon HF theory by adding a few important extra determinants

I for example, in H2, we include not only bonding but also antibonding orbitals:

|1σ2g〉 → cg|1σ2

g〉 + cu|1σ2u〉

I simultaneous optimisation of orbitals and expansion coefficientsI such MCSCF calculations are flexible but expensiveI the spin–spin coupling constants are now also fairly well obtained

HF MCSCF exp.HF F 413.6 419.6 410

H 28.4 28.5 28.5H2O O 328.1 335.3 323.6

H 30.7 30.2 30.0NH3 N 262.3 269.6 264.5

H 31.7 31.0 31.2CH4 C 194.8 200.4 198.7

H 31.7 31.2 30.6F2 F −167.9 −136.6 −192.8N2 N −112.4 −53.0 −61.6CO C −25.5 8.2 3.0

O −87.7 −38.9 −56.8

HF MCSCF exp

HF 1KHF 59.2 48.0 47.6CO 1KCO 13.4 −28.1 −38.3N2

1KNN 175.0 −5.7 −19.3H2O 1KOH 63.7 51.5 52.8

2KHH −1.9 −0.8 −0.7NH3

1KNH 61.4 48.7 50.82KHH −1.9 −0.8 −0.9

C2H41KCC 1672.0 99.6 87.81KCH 249.7 51.5 50.02KCH −189.3 −1.9 −0.42KHH −28.7 −0.2 0.23Kcis 30.0 1.0 0.93Ktns 33.3 1.5 1.4





Coupled-cluster theory and virtual excitations

I For high accuracy, we must consider all possible excitations form the HF determinantI in HF theory we have a set of occupied and a set of unoccupied (virtual) orbitalsI new determinants are generated by replacing occupied orbitals by virtual orbitalsI improved description obtained by combining the HF determinant with such ‘excited’ determinantsI unlike in MCSCF theory, no orbital reoptimisation is attempted

I This is done in coupled-cluster (CC) theory, yielding a hierarchy of wave functions:I CCS contains all determinants generated by (repeated) single excitationsI CCSD contains also all determinants generated by (repeated) double excitationsI CCSDT contains also all determinants generated by (repeated) triple excitations

I Convergence of atomisation energies (kJ/mol):

HF SD T Q rel. vib. total exp.CH2 531.1 218.3 9.5 0.4 −0.7 −43.2 715.4 714.8±1.8H2O 652.3 305.3 17.3 0.8 −2.1 −55.4 918.2 917.8±0.2HF 405.7 178.2 9.1 0.6 −2.5 −24.5 566.7 566.2±0.7N2 482.9 426.0 42.4 3.9 −0.6 −14.1 940.6 941.6±0.2F2 −155.3 283.3 31.6 3.3 −3.3 −5.5 154.1 154.6±0.6CO 730.1 322.2 32.1 2.3 −2.0 −12.9 1071.8 1071.8±0.5

I Computational cost increases dramatically with increasing excitation levelsI cost proportional to n6 for CCSD and to n8 for CCSDT where n is system size

I The gold standard of quantum chemistry: CCSD(T)I CCSDT with a simplified description of triples


The two-dimensional chart of quantum chemistry

I The quality of ab initio calculations is determined by the description of

1 the N-electron space (wave-function model);2 the one-electron space (basis set).

I Normal distributions of errors in AEs (kJ/mol)

-200 200

HFDZ

-200 200 -200 200

HFTZ

-200 200 -200 200

HFQZ

-200 200 -200 200

HF5Z

-200 200 -200 200

HF6Z

-200 200

-200 200

MP2DZ

-200 200 -200 200

MP2TZ

-200 200 -200 200

MP2QZ

-200 200 -200 200

MP25Z

-200 200 -200 200

MP26Z

-200 200

-200 200

CCSDDZ

-200 200 -200 200

CCSDTZ

-200 200 -200 200

CCSDQZ

-200 200 -200 200

CCSD5Z

-200 200 -200 200

CCSD6Z

-200 200

-200 200

CCSD(T)DZ

-200 200 -200 200

CCSD(T)TZ

-200 200 -200 200

CCSD(T)QZ

-200 200 -200 200

CCSD(T)5Z

-200 200 -200 200

CCSD(T)6Z

-200 200

I The errors are systematically reduced by going up in the hierarchies


Benchmark calculations of BH shieldings (ppm)

σ(11B) ∆σ(11B) σ(1H) ∆σ(1H)HF −261.3 690.1 24.21 14.15MP2 −220.7 629.9 24.12 14.24CCSD −166.6 549.4 24.74 13.53CCSD(T) −171.5 555.2 24.62 13.69CCSDT −171.8 557.3 24.59 13.72CCSDTQ −170.1 554.7 24.60 13.70FCI −170.1 554.7 24.60 13.70

I TZP+ basis, RBH = 123.24 pm, all electrons correlated

I J. Gauss and K. Ruud, Int. J. Quantum Chem. S29 (1995) 437

I M. Kallay and J. Gauss, J. Chem. Phys. 120 (2004) 6841


Coupled-cluster convergence of shielding constants in CO (ppm)

CCSD CCSD(T) CCSDT CCSDTQ CCSDTQ5 FCI

σ(13C) 32.23 35.91 35.66 36.10 36.14 36.15∆σ(13C) 361.30 356.10 356.47 355.85 355.80 355.79σ(17O) −13.93 −13.03 −13.16 −12.81 −12.91 −12.91

∆σ(17O) 636.01 634.55 634.75 634.22 634.52 634.35

I All calculations in the cc-pVDZ basis and with a frozen core.

I Kallay and Gauss, J. Chem. Phys. 120 (2004) 6841.


Coupled-cluster shielding constants

HF MCSCF CCSD CCSD(T) exp.HF F 413.6 419.6 418.1 418.6 410± 6

H 28.4 28.5 29.1 29.2 28.5± 0.2H2O O 328.1 335.3 336.9 337.9 323.6± 6

H 30.7 30.2 30.9 30.9 30.05± 0.02NH3 N 262.3 269.6 269.7 270.7 264.5

H 31.7 31.0 31.6 31.6 31.2± 1.0CH4 C 194.8 200.4 198.7 198.9 198.7

H 31.7 31.2 31.5 31.6 30.61F2 F −167.9 −136.6 −171.1 −186.5 −192.8N2 N −112.4 −53.0 −63.9 −58.1 −61.6± 0.2CO C −25.5 8.2 0.8 5.6 3.0± 0.9

O −87.7 −38.9 −56.0 −52.9 −56.8± 6


Coupled-cluster indirect spin–spin coupling constants

HF MCSCF CCSD CC3 exp.HF 1KHF 59.2 48.0 46.1 46.1 47.6CO 1KCO 13.4 −28.1 −38.3 −37.3 −38.3N2

1KNN 175.0 −5.7 −20.4 −20.4 −19.3H2O 1KOH 63.7 51.5 48.4 48.2 52.8

2KHH −1.9 −0.8 −0.6 −0.6 −0.7NH3

1KNH 61.4 48.7 48.1 50.82KHH −1.9 −0.8 −1.0 −0.9

C2H41KCC 1672.0 99.6 92.3 87.81KCH 249.7 51.5 50.7 50.02KCH −189.3 −1.9 −1.0 −0.42KHH −28.7 −0.2 0.0 0.23Kcis 30.0 1.0 1.0 0.93Ktns 33.3 1.5 1.5 1.4


Indirect spin–spin coupling constant of HD molecule

I T. Helgaker, M. Jaszunski, P. Garbacz, and K. Jackowski, Mol. Phys. 110 (2012) 2611I combined experimental and computational study of JHD

I Computational studyI electronic equilibrium value (FCI/aug-pV7Z):

41.2197 (total) = 40.2515 (FC) + 0.4479 (SD) + 0.8344 (PSO)− 0.3141 (DSO)

I zero-point vibrational correction:

1.8943 (total) = 1.9104 (FC)− 0.0147 (SD)− 0.0242 (PSO) + 0.0228 (DSO)

I temperature correction at 300 K:

0.1993 (total) = 0.2068 (FC)− 0.0025 (SD)− 0.0052 (PSO) + 0.0002 (DSO)

I total spin–spin coupling constant:

43.3132 (total) = 42.3687 (FC) + 0.4307 (SD) + 0.8050 (PSO)− 0.2911 (DSO)

I Calculated result with estimated uncertainty: 43.31(5) Hz

I Experimental studyI extrapolation of values measured in HD–He mixtures to zero density gives 43.26(6) HzI discrepancy between experimental and theoretical values 0.05 Hz

I New experimental investigation by Garbacz, Chem. Phys. 443 (2014) 1I revised experimental value at 300 K of 43.136(7) HzI discrepancy between experimental and theoretical values is now 0.17 Hz


Absolute shielding scale of 33S

I Best shieldings obtained from nuclear spin–rotation interactions in rotational spectra

CK = 2γK

(σK − σdia

K

)I−1 + Cnuc

K

I spin–rotation tensor CK accurately measured in rotational spectroscopyI inverse moment-of-inertia tensor I−1 and nuclear contribution Cnuc

K trivially obtainedI diamagnetic part of shielding σdia

K calculated to very high accuracy

I Recent benchmark study revealed discrepancies with experiment for sulfur shieldings

I scale based on an absolute shielding of 817(12) ppm in OCS by Jackowski et al. (1998,2002)I this shielding does not agree well with calculated CCSD(T) value of 788 ppm

I We decided to attempt a new shielding scale based on H2S rather than SO2

I H2S better suited since vibrational corrections can be accurately calculatedI T. Helgaker, J. Gauss, G. Cazzoli, C. Puzzarini, J. Chem. Phys. 139, 244308 (2013)

I Combination of spin–rotation constant measurements with CCSD(T) calculations:

I measured spin–rotation constant −35.4(5) kHZI subtract CCSD(T) vibrational contribution to yield equilibrium spin–rot. constant −33.5(5) kHzI convert to yield the paramagnetic part of shielding constant −329(5) ppmI add CCSD(T) diamagnetic contribution to yield equilibrium shielding constant 737(5) ppmI add CCSD(T) vibrational correction to yield shielding constant 300 K 716(5) ppm

I Absolute shielding constant of 33S in H2S at 300 K: 716(5) ppm

I agrees well with calculated CCSD(T) shielding of 719 ppm at 300 K


DFT: the workhorse of quantum chemistry

I The traditional wave-function methods of quantum chemistry are capable of high accuracy

I nevertheless, most calculations are performed using density-functional theory (DFT)


Kohn–Sham DFT

I The Hohenberg–Kohn variation principle is given by

E(v) = minρ

(F (ρ) +

∫v(r)ρ(r) dr

)I The central quantity is the universal density functional F

I it is a complicated function of the electron densityI its functional dependence on ρ is unknown

I In Kohn–Sham DFT, F is decomposed in the manner:

F (ρ) = Ts(ρ) + J(ρ) + Exc(ρ), ρ(r) =∑

i φi (r)∗φi (r)

I The noninteracting kinetic energy Ts(ρ) is calculated by introducing orbitals

Ts(ρ) = − 12

∑i

∫φi (r)∇2φi (r) dr, ρ(r) =

∑iφ∗i (r)φi (r)

I The Coulomb self-repulsion energy J(ρ) is trivial

J(ρ) =

∫∫ρ(r1)ρ(r2)

r12dr1dr2

I The exchange–correlation energy Exc(ρ) is the only unknown entity

I it contains all effects of electron exchange and correlationI a large number of approximate exchange–correlation functionals have been developed


Approximate exchange–correlation functionals

I Local-density approximation (LDA)

I XC functional modelled after the uniform electron gas (which is known exactly)

ELDAxc (ρ) =

∫f (ρ(r)) dr local dependence on density

I not sufficiently accurate to compete with traditional methods of quantum chemistry

I Generalised-gradient approximation (GGA)

I introduce a dependence also on the density gradient

EGGAxc (ρ) =

∫f (ρ(r,∇ρ(r)) dr local dependence on density and its gradient

I Becke’s gradient correction to exchange (1988) changed the situationI the accuracy became sufficient to compete in chemistry

I Hybrid Kohn–Sham theory

I include some proportion of exact exchange in the calculations (Becke, 1993)I 20% is good for energetics; for other properties, 100% may be a good thing

I Progress has to a large extent been semi-empirical

I empirical and non-empirical functionals


A plethora of exchange–correlation functionals

exchange, Slater local exchange, and the nonlocal gradientcorrection of Becke88. Thus,

ExcB3LYP ! a0Ex

exact " !1 # a0"ExSlater " ax#Ex

B88 " acEcVWN

" !1 # ac"EcLYP. [11]

Becke obtained the hybrid parameters {a0, ax, ac} $ {0.20, 0.72,0.19} (3) from a least-squares fit to 56 atomization energies, 42IPs, and 8 proton affinities (PAs) of the G2-1 set of atoms andmolecules (4). B3LYP leads to excellent thermochemistry (0.13eV MAD) and structures for covalently systems but does notaccount for London dispersion (all noble gas dimers are pre-dicted unstable).

Following B3LYP, we introduce the extended hybrid func-tional, denoted as X3LYP:

ExcX3LYP ! a0Ex

exact " !1 # a0"ExSlater " ax#Ex

X " acEcVWN

" !1 # ac"EcLYP. [12]

We determined the hybrid parameters {a0, ax, ac} $ {0.218,0.709, 0.129} in X3LYP just as for XLYP. Thus, we normalizedthe mixing parameters of Eq. 10 and redetermined {ax1, ax2} ${0.765, 0.235} for X3LYP. The FX(s) function of X3LYP (Fig.1) agrees with FGauss(s) for larger s.

Results and DiscussionWe tested the accuracy of XLYP and X3LYP for a broad rangeof systems and properties not used in fitting the parameters.Table 1 compares the overall performance of 17 different flavorsof DFT methods, showing that X3LYP is the best or nearly best

Table 1. MADs (all energies in eV) for various level of theory for the extended G2 set

Method

G2(MAD)

H-Ne, Etot TM #E He2, #E(Re) Ne2, #E(Re) (H2O)2, De(RO . . . O)#Hf IP EA PA

HF 6.47 1.036 1.158 0.15 4.49 1.09 Unbound Unbound 0.161 (3.048)G2 or best ab initio 0.07a 0.053b 0.057b 0.05b 1.59c 0.19d 0.0011 (2.993)e 0.0043 (3.125)e 0.218 (2.912)f

LDA (SVWN) 3.94a 0.665 0.749 0.27 6.67 0.54g 0.0109 (2.377) 0.0231 (2.595) 0.391 (2.710)GGA

BP86 0.88a 0.175 0.212 0.05 0.19 0.46 Unbound Unbound 0.194 (2.889)BLYP 0.31a 0.187 0.106 0.08 0.19 0.37g Unbound Unbound 0.181 (2.952)BPW91 0.34a 0.163 0.094 0.05 0.16 0.60 Unbound Unbound 0.156 (2.946)PW91PW91 0.77 0.164 0.141 0.06 0.35 0.52 0.0100 (2.645) 0.0137 (3.016) 0.235 (2.886)mPWPWh 0.65 0.161 0.122 0.05 0.16 0.38 0.0052 (2.823) 0.0076 (3.178) 0.194 (2.911)PBEPBEi 0.74i 0.156 0.101 0.06 1.25 0.34 0.0032 (2.752) 0.0048 (3.097) 0.222 (2.899)XLYPj 0.33 0.186 0.117 0.09 0.95 0.24 0.0010 (2.805) 0.0030 (3.126) 0.192 (2.953)

Hybrid methodsBH & HLYPk 0.94 0.207 0.247 0.07 0.08 0.72 Unbound Unbound 0.214 (2.905)B3P86l 0.78a 0.636 0.593 0.03 2.80 0.34 Unbound Unbound 0.206 (2.878)B3LYPm 0.13a 0.168 0.103 0.06 0.38 0.25g Unbound Unbound 0.198 (2.926)B3PW91n 0.15a 0.161 0.100 0.03 0.24 0.38 Unbound Unbound 0.175 (2.923)PW1PWo 0.23 0.160 0.114 0.04 0.30 0.30 0.0066 (2.660) 0.0095 (3.003) 0.227 (2.884)mPW1PWp 0.17 0.160 0.118 0.04 0.16 0.31 0.0020 (3.052) 0.0023 (3.254) 0.199 (2.898)PBE1PBEq 0.21i 0.162 0.126 0.04 1.09 0.30 0.0018 (2.818) 0.0026 (3.118) 0.216 (2.896)O3LYPr 0.18 0.139 0.107 0.05 0.06 0.49 0.0031 (2.860) 0.0047 (3.225) 0.139 (3.095)X3LYPs 0.12 0.154 0.087 0.07 0.11 0.22 0.0010 (2.726) 0.0028 (2.904) 0.216 (2.908)Experimental — — — — — — 0.0010 (2.970)t 0.0036 (3.091)t 0.236u (2.948)v

#Hf, heat of formation at 298 K; PA, proton affinity; Etot, total energies (H-Ne); TM #E, s to d excitation energy of nine first-row transition metal atoms andnine positive ions. Bonding properties [#E or De in eV and (Re) in Å] are given for He2, Ne2, and (H2O)2. The best DFT results are in boldface, as are the most accurateanswers [experiment except for (H2O)2].aRef. 5.bRef. 19.cRef. 4.dRef. 35.eRef. 38.fRef. 34.gRef. 37.hRef. 7.iRef. 10.j1.0 Ex (Slater) % 0.722 #Ex (B88) % 0.347 #Ex (PW91) % 1.0 Ec (LYP).k0.5 Ex (HF) % 0.5 Ex (Slater) % 0.5 #Ex (B88) % 1.0 Ec (LYP).l0.20 Ex (HF) % 0.80 Ex (Slater) % 0.72 #Ex (B88) % 1.0 Ec (VWN) % 0.81 #Ec (P86).m0.20 Ex (HF) % 0.80 Ex (Slater) % 0.72 #Ex (B88) % 0.19 Ec (VWN) % 0.81 Ec (LYP).n0.20 Ex (HF) % 0.80 Ex (Slater) % 0.72 #Ex (B88) % 1.0 Ec (PW91, local) % 0.81 #Ec (PW91, nonlocal).o0.25 Ex (HF) % 0.75 Ex (Slater) % 0.75 #Ex (PW91) % 1.0 Ec (PW91).p0.25 Ex (HF) % 0.75 Ex (Slater) % 0.75 #Ex (mPW) % 1.0 Ec (PW91).q0.25 Ex (HF) % 0.75 Ex (Slater) % 0.75 #Ex (PBE) % 1.0 Ec (PW91, local) % 1.0 #Ec (PBE, nonlocal).r0.1161 Ex (HF) % 0.9262 Ex (Slater) % 0.8133 #Ex (OPTX) % 0.19 Ec (VWN5) % 0.81 Ec (LYP).s0.218 Ex (HF) % 0.782 Ex (Slater) % 0.542 #Ex (B88) % 0.167 #Ex (PW91) % 0.129 Ec (VWN) % 0.871 Ec (LYP).tRef. 27.uRef. 33.vRef. 32.

Xu and Goddard PNAS ! March 2, 2004 ! vol. 101 ! no. 9 ! 2675

CHEM

ISTR

Y


Comparison of Kohn–Sham and coupled-cluster theories

I Reaction enthalpies (kJ/mol) calculated using the DFT/B3LYP and CCSD(T) models

B3LYP CCSD(T) exp.CH2 + H2 → CH4 −543 1 −543 1 −544(2)C2H2 + H2 → C2H4 −208 −5 −206 −3 −203(2)C2H2 + 3H2 → 2CH4 −450 −4 −447 −1 −446(2)CO + H2 → CH2O −34 −13 −23 −2 −21(1)N2 + 3H2 → 2NH2 −166 −2 −165 −1 −164(1)F2 + H2 → 2HF −540 23 −564 −1 −563(1)O3 + 3H2 → 3H2O −909 24 −946 −13 −933(2)CH2O + 2H2 → CH4 + H2O −234 17 −250 1 −251(1)H2O2 + H2 → 2H2O −346 19 −362 3 −365(2)CO + 3H2 → CH4 + H2O −268 4 −273 −1 −272(1)HCN + 3H2 → CH4 + NH2 −320 0 −321 −1 −320(3)HNO + 2H2 → H2O + NH2 −429 15 −446 −2 −444(1)CO2 + 4H2 → CH4 + 2H2O −211 33 −244 0 −244(1)2CH2 → C2H4 −845 −1 −845 −1 −844(3)





Kohn–Sham shielding constants (ppm)

HF LDA BLYP B3LYP KT2 CCSD(T) exp.HF F 413.6 416.2 401.0 408.1 411.4 418.6 410± 6H2O O 328.1 334.8 318.2 325.0 329.5 337.9 323.6± 6NH3 N 262.3 266.3 254.6 259.2 264.6 270.7 264.5CH4 C 194.8 193.1 184.2 188.1 195.1 198.9 198.7F2 F −167.9 −284.2 −336.7 −208.3 −211.0 −186.5 −192.8N2 N −112.4 −91.4 −89.8 −86.4 −59.7 −58.1 −61.6± 0.2CO C −25.5 −20.3 −19.3 −17.5 7.4 5.6 3.0± 0.9 (eq)

O −87.7 −87.5 −85.4 −78.1 −57.1 −52.9 −56.8± 6 (eq)


NMR shieldings: normal distributions of errors

-200 -100 0 100 200

0.02

0.04

0.06

LDA

-200 -100 0 100 200

0.02

0.04

0.06

BLYP

-200 -100 0 100 200

0.02

0.04

0.06

PBE

-200 -100 0 100 200

0.02

0.04

0.06

KT2

-200 -100 0 100 200

0.02

0.04

0.06

B3LYP

-200 -100 0 100 200

0.02

0.04

0.06

B97-2

-200 -100 0 100 200

0.02

0.04

0.06

B97-3

-200 -100 0 100 200

0.02

0.04

0.06

PBE0

-200 -100 0 100 200

0.02

0.04

0.06

CAM-B3LYP

-200 -100 0 100 200

0.02

0.04

0.06

O-B3LYP

-200 -100 0 100 200

0.02

0.04

0.06

O-B97-2

-200 -100 0 100 200

0.02

0.04

0.06

O-B97-3

-200 -100 0 100 200

0.02

0.04

0.06

O-PBE0

-200 -100 0 100 200

0.02

0.04

0.06

O-CAM-B3LYP

-200 -100 0 100 200

0.02

0.04

0.06

CCSD

I Normal distributions of the errors in NMR shielding constants calculated in the aug-cc-pCVQZ basis relative to theCCSD(T)/aug-cc-pCV[TQ]Z benchmark data set for 26 molecules (Teale et al., JCP)


Mean absolute NMR shielding errors relative to empirical equilibrium values

0

10

20

30

40

50

60

70 M

AE /

pp

m

H C N O F All

CC

LDA

GGA Hybrid

OEP-Hybrid

RHF

I Mean absolute errors (in ppm) for NMR shielding constants relative to empirical equilibrium values for H (white), C (grey),N (blue), O (red), and F (yellow). The total mean absolute errors over all nuclear types are shown by the purple bars. TheDFT methodologies are arranged in the categories LDA, GGA, hybrid and OEP-hybrid. (Teal et al., JCP)


NMR shieldings: mean absolute errors relative to experiment

MAE (Exp.)

MAE (Emp. Eq.)

0

5

10

15

20

25

30

MA

E /

pp

m

MAE (Exp.) MAE (Emp. Eq.)

I Mean absolute errors (in ppm) for NMR shielding constants relative to experimental (blue) and empirical equilibrium values(red). The inclusion of vibrational corrections in the empirical equilibrium values leads to a degradation of the quality ofthe RHF and DFT results but to a notable improvement for the CCSD and CCSD(T) methods. (Teale et al., JCP)


Reduced spin–spin coupling constants by density-functional theory

LDA BLYP B3LYP PBE B97-3 RAS exp∗ vib

HF 1KHF 35.0 34.5 38.9 32.6 40.5 48.1 47.6 −3.4CO 1KCO −65.4 −55.7 −47.4 −62.0 −43.4 −39.3 −38.3 −1.7N2

1KNN 32.9 −46.6 −20.4 −43.2 −12.5 −9.1 −19.3 −1.1H2O 1KOH 40.3 44.6 47.2 41.2 46.3 47.1 52.8 −3.3

2KHH −0.3 −0.9 −0.7 −0.5 −0.6 −0.6 −0.7 0.1NH3

1KNH 41.0 49.6 52.3 47.0 50.1 50.2 50.8 −0.32KHH −0.4 −0.7 −0.9 −0.7 −0.8 −0.9 −0.9 0.1

C2H41KCC 66.6 90.3 96.2 83.4 92.9 90.5 87.8 1.21KCH 42.5 55.3 55.0 50.0 51.4 50.2 50.0 1.72KCH 0.4 0.0 −0.5 −0.2 −0.3 −0.5 −0.4 −0.42KHH 0.4 0.4 0.3 0.3 0.3 0.1 0.2 0.03Kcis 0.8 1.1 1.1 1.0 1.0 1.0 0.9 0.13Ktns 1.2 1.7 1.7 1.6 1.5 1.5 1.4 0.2∣∣∆∣∣ abs. 11.2 5.9 3.1 6.4 2.6 1.6 ∗at Re

% 72 48 14 33 14 14


Comparison of density-functional and wave-function theory

I Normal distributions of errors for indirect nuclear spin–spin coupling constants

I for the same molecules as on the previous slides

-30 30

CAS

-30 30 -30 30

RAS

-30 30 -30 30

SOPPA

-30 30 -30 30

CCSD

-30 30

-30 30

HF

-30 30

LDA

-30 30 -30 30

BLYP

-30 30 -30 30

B3LYP

-30 30

I Some observations:

I LDA underestimates only slightly, but has a large standard deviationI BLYP reduces the LDA errors by a factor of twoI B3LYP errors are similar to those of CASSCFI The CCSD method is slightly better than the SOPPA method


The Karplus curve

I Vicinal (three-bond) spin–spin coupling constants depend critically on the dihedral angle:

I 3JHH in ethane as a function of the dihedral angle:

25 50 75 100 125 150 175

2

4

6

8

10

12

14

DFT

empirical

I Good agreement with the (empirically constructed) Karplus curve


Valinomycin C54H90N8O18

I DFT can be applied to large molecular systems such as valinomycin (168 atoms)

– there are a total of 7587 spin–spin couplings to the carbon atoms in valinomycin– below, we have plotted the magnitude of the reduced LDA/6-31G coupling constants

on a logarithmic scale, as a function of the internuclear distance:

500 1000 1500

1019

1016

1013

– the coupling constants decay in characteristic fashion, which we shall examine– most of the indirect couplings beyond 500 pm are small and cannot be detected


Valinomycin C54H90N8O18One-bond spin–spin couplings to CH, CO, CN, CC greater than 0.01 Hz

100 200 300 400 500 600

100

30

10

3

1

0.3

0.1

0.03


Valinomycin C54H90N8O18Two-bond spin–spin couplings to CH, CO, CN, CC greater than 0.01 Hz

100 200 300 400 500 600

100

30

10

3

1

0.3

0.1

0.03


Valinomycin C54H90N8O18Three-bond spin–spin couplings to CH, CO, CN, CC greater than 0.01 Hz

100 200 300 400 500 600

100

30

10

3

1

0.3

0.1

0.03


Valinomycin C54H90N8O18Four-bond spin–spin couplings to CH, CO, CN, CC greater than 0.01 Hz

100 200 300 400 500 600

100

30

10

3

1

0.3

0.1

0.03


Long-range decay of the Ramsey terms

I Decomposition of indirect spin–spin coupling constants:

JKL = JFCKL + JSDKL + JDSO

KL + JPSOKL

I Their decay with increasing internuclear separation is as follows:

500 1000 1500

1019

1016

1013

DSO decays as R-2

negative

500 1000 1500

1019

1016

1013

PSO decays as R-2

positive

500 1000 1500

1019

1016

1013

FC decays exponentially

mixed signs

500 1000 1500

1019

1016

1013

SD decays as R-3

mixed signs


Large long-range contributions in valinomycin

I Maximum, mean, and minimum couplings plotted against the number of intervening bonds

I There are peaks at 11, 13, and 15 intervening bonds (greater than 1 Hz in magnitude).

I All peaks correspond to distances less than 350 pm, respectively.

5 10 15 201

10-3

10-6


Dirac vector model in valinomycin

I The Dirac vector model predicts sign alternation with increasing number of intervening bonds

I The plot shows the distribution of signs in valinomycin as function of the number of intervening bonds

I In the limit of many bonds, all couplings become negative because of the dominant DSO term

5 10 15 20

25

50

75

positive

negative

5 10 15 20

25

50

75


Karplus relation in valinomycin

I There are 282 vicinal coupling constants among the 7578 carbon couplings in valinomycin

I As predicted by the Karplus relation, these couplings vanish for dihedral angles close to 90◦

I The Karplus curve arises from the FC term (blue); the other terms (red) do not contribute.

45 90 135 180

-2

2

4

6

8

10


Acknowledgements

I Funding agencies:

I Norwegian Research Council for Centre for Theoretical and Computational Chemistry (CTCC)I European Research Council (ERC) Advanced Grant

I Students and collaborators:

I K. Lange, O. B. Lutnæs, T. RudenI J. Gauss, M. Hoffmann, C. Puzzarini, K. Ruud, A. TealeI M. Jaszunski, M. Pecul, P. Garbacz, K. Jackowski


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