OEPOEP--BasedBased CalculationsCalculations of of MagneticMagnetic ResonanceResonance ParametersParameters

Martin Kaupp, Alexei V. Arbuznikov

Universität Würzburg

OEP-Workshop, Berlin, March 2005

OEPOEP--BasedBased CalculationsCalculations of of MagneticMagnetic ResonanceResonance ParametersParameters

Martin Kaupp, Alexei V. Arbuznikov

Universität Würzburg

-shortcomings of standard functionals in calculations of NMR and EPR parameters

-performance of „localized hybrid potentials“ for: nuclear shieldings of main-group systemselectronic g-tensors of TM complexeshyperfine coupling of phosphorus atom

Work in progress:-concept of „double local hybrid“ potentials (position-dependent EXX admixture)-initial study of „local mixing functions“ for position-dependent EXX admixture

OEP-Workshop, Berlin, March 2005

A Relativistic Density-Functional Machinery to Compute NMR and EPR Parameters

NMR EPR δ J NQCC g A D

- mainly DFT used to incorporate electron correlation

- include scalar relativistic effects (all-electron-DKH or PP approaches)

- include spin-orbit coupling

perturbationally, based on 1-comp. approach

variationally, in 2-comp. DKH approach

efficient approximations to SO integrals

Gaussian basis set property program ReSpect (Version 1.2), 2004.; V. G. Malkin, O. L. Malkina, R. Reviakine, A. V. Arbuznikov, M. Kaupp, B. Schimmelpfennig, I. Malkin, T. Helgaker, K. Ruud

Calculation of second-order magnetic response properties

Nuclear shielding tensors: Electronic g-tensors:

2

0, 01 E

u,v x, y,z

∂σµ ∂ ∂ = ==

=

µµ B Nuv BB N,u v

d pˆ ˆ ˆσ σ σ= +

3, 0 0 ,

p2 1

0 0 n

ˆ ˆ r1ˆ . .

2 E Eσ

−

≠

Ψ Ψ Ψ ⋅ Ψ∑ ∑= +∑

−

n O v N u N nk k

uvn

l lc c

c

– angular momentum operator

- PSO operator3,

ˆ −⋅N u Nl rul

2

0, 0

1 E

B B S

g

u,v x, y,z

∂µ ∂ ∂

= =

=

=

Buvu vS

, 2.002319...e eg g g= + =g 1 ∆

/ ∆ ∆∆∆ SO OZ GC RMCggg g= + +

(0) (0) (0) (0)2 0 , , 0

/ , (0) (0)0

ˆˆ.

2

SO v n n O ue k

SO OZ uvn n

Ψ H Ψ Ψ l Ψgg c c

S E Eα

⎡ ⎤⎢ ⎥∆ = +⎢ ⎥−⎢ ⎥⎣ ⎦

∑∑

– spatial part of spin-orbit operator,ˆ

SO uH

H2O+

∆g33

∆g c

alc.

(a.u

.×10

6 )

∆gexp. (a.u.×106)-10000 -5000 0 5000 10000 15000

-10000

-5000

0

5000

10000

15000

Performance of DFT for g-Shift Tensors of Light Main-Group Radicals

UDFT-BP86, IGLO-III basis; H2O+, CO+, HCO, C3H5, NO2, NF2, MgF, 2,4,6-tris-t-Bu-C6H2O, tyrosylO. L. Malkina, J. Vaara, B. Schimmelpfennig, V. G. Malkin, M. Kaupp J. Am. Chem. Soc. 2000, 122, 9206.

3200 3400 3600 3800 4000 420030003200340036003800400042004400460048005000

∆gx experimental /ppm (in 2-propanol)

∆g x

calc

ulat

ed/p

pm

DMNQ.- (iPrOH)4

ideal agreement

UQ-M.- (iPrOH)6

gx-Component of Semiquinone Radical Anions in 2-Propanol

BQ.- (iPrOH)4

NQ.- (iPrOH)4

DQ.- (iPrOH)4

DMEQ.- (iPrOH)4

DMBQ.- (iPrOH)4

∆gx

UDFT/BP86 results: J. Am. Chem. Soc. 2002, 124, 2709.High-field EPR data in 2-propanol: Stehlik et al., 1993, 1995.

Modelling g-Tensors for Semiquinones in their Protein Environment

UDFT/BP86, ∆gx scaled by 0.92.S. Kacprzak, M. Kaupp J. Phys. Chem. 2004, 108, 2464; EPR data: MacMillan et al., 1997; Feher et al., 1995.

model: DMNQ-.(NMF)(indole)

Photosystem I (QK

-. in Synechococcus Elongatus)

∆gx ∆gy ∆gz

calc. 3889 2707 14exp. 3930 2710 -49

model: UQ-M-.(NMF)(imidazole)(indole)

Purple Bacteria Reaction Center (QA

-. in Rhodobacter Sphaeroides R-26)

∆gx ∆gy ∆gz

calc. 4280 3012 -17exp. 4300 3100 -100

Purple Bacteria Reaction Center (QB

-. in Rhodobacter Sphaeroides R-26)

model: UQ-M-.(hist)(SIG)

∆gx ∆gy ∆gz

calc. 3866 2902 -67exp. 3940 2950 -190

g-Tensor Dynamics forAqueous Benzosemiquinone

average of snapshots for 6.3 ps trajectory(calc. 4.5 Å cluster, 300 K, exp. 80 K):

∆gx= 4488 ppm (scaled) exp. 4300 ppm

∆gy= 2992 ppm exp. 2980 ppm

g-shift tensor calc.,5.0 Å cluster

Car-Parrinello ab initio molecular dynamics simulations

J. Asher, N. Doltsinis, M. Kaupp J. Am. Chem. Soc. 2004, 126, 9854 .

Performance of GGA functionals in calculating g-shift components of 3d complexes

-200 -100 0 100 200 300-200

-100

0

100

200

300

BP86GGA

∆g

(cal

c.) /

ppt

∆g (exp.) /ppt

slope 0.40, R = 0.972

TiF3, CrOF4-, MnO3, Fe(CO)5

+, Co(CO)4, Ni(CO)3H, Cu(acac)2, Cu(NO3)2

J. Comput. Chem. 2002, 23, 794.

Performance of DFT for δ(Performance of DFT for δ(5757Fe) Chemical Shifts of Fe) Chemical Shifts of OrganoironOrganoiron Complexes /Complexes /ppmppm vs. Fe(CO)vs. Fe(CO)55

1 - Fe(C4H4)(CO)3

2 – Fe(CO)5

3 – Fe(CO)3(CH2CHCHCH2)4 - Fe(CO)4(CH2CHCN)5 – Fe(CO)2Cp(CH3)6 – Fe(CO)3(CH2CHCHO)7 – FeCp2

-750 -500 -250 0 250 500 750 1000 1250 1500 1750-1000

-750

-500

-250

0

250

500

750

1000

1250

1500

1750

2000

δ(57Fe), exptl. / ppm

δ(57

Fe),

calc

d. /

ppm

Slope RC SDSVWN 0.568 0.986 79BPW91 0.546 0.966 121B3LYP 0.945 0.993 93FT98 0.381 0.887 163PKZB 0.561 0.980 93SAOP 0.334 0.884 143ideal

Phys. Chem. Chem. Phys. 2002, 4, 5467.See also: M. Bühl Chem. Phys. Lett. 1997, 267, 251.

Performance of hybrid functionals in calculating g-shift components of 3d complexes

-200 -100 0 100 200 300

-150

-100

-50

0

50

100

150

200

250

300 B3PW91ca. 20% HF exchange

∆g (exp.) /ppt

∆g (c

alc.

) /pp

t

slope 0.58, R = 0.981

( )3 91

88 910.72 0.8

0.20

1

exactB PW LDAxc xc

B P

LDAx

x c

x

W

E E

E

E E

E

= +

+ ∆ + ∆

−

-200 -100 0 100 200 300-200

-100

0

100

200

300

∆g (c

alc.

) /pp

t BHPW9150% HF exchange

∆g (exp.) /ppt

slope 0.99, R = 0.956

improved slope byexact-exchange admixture

but: sometimes problemswith spin contamination

TiF3, CrOF4-, MnO3, Fe(CO)5

+, Co(CO)4, Ni(CO)3H, Cu(acac)2, Cu(NO3)2

J. Comput. Chem. 2002, 23, 794.

Calculated Isotropic Hyperfine Coupling Constant for theCalculated Isotropic Hyperfine Coupling Constant for the44P GS of the Phosphorus AtomP GS of the Phosphorus Atom

Uncontracted 20s15p4d2f Partridge basis set.(M. Straka, unpublished).

MethodBHLYPBHP86BHPW91B3LYPB3P86 B3PW91BLYP BP86 BPW91HF MP2 CCSD(T)Exp.

HFC /MHz-30.3 -67.3

-112.6-48.1-84.9

-113.5-56.7-94.6

-141.6-84.2+2.8

+58.8+55

hybrid 50% νx,HF

hybrid ca. 20% νx,HF

GGA

ρ Nα

−β/2

S (in

a.u

.)

BLYPBP86 B3LYP

BPW91 B3PW91BHLYP

BHP86BHPW91

Functional

-0.18

-0.08

-0.10

-0.12

-0.14

-0.16

[Mn(CN)4]2- calc.

[Mn(CN)4]2- exp.

[Cr(CO)4]+ exp.

[Cr(CO)4]+ calc.

Performance of DFT for the calculation of isotropic hyperfine coupling constantsin transition metal complexes

HF exchange admixture causesproblems with spin contamination!

improved core-shell spin polarizationwith HF exchange admixture⇒ more negative spin density

Spin density ρNα−β at the metal nuclei, normalized to the number of unpaired electrons.

M. Munzarová, M. Kaupp J. Phys. Chem. A 1999, 103, 9966.

Localized hybrid exchange-correlation potentials: theory and implementation

Motivation: search for better exchange-correlation potentials for the description of properties (in particular, NMR and EPR parameters): self-consistent potential needed!

Ex is ca. 85-95% of the entire Exc Ex = Exexact ?

• self-interaction free ( is cancelled completely

for one-electron systems)

1 ( ) ( ) d d2

ρ ρ∫ ∫

−r r' r r'r r'

• correct asymptotic behaviour (at )xc1, ( )r vr

→ ∞ −r ∼

Attractive properties of exact exchange:

(Exexact + Ec

DFT ):

• reasonable for atoms;• very poor for molecules (non-dynamical correlation is missing).

Localized Hybrid Potentials: A New Class of Exchange-Correlation Potentials

„classical“ hybrid functionals, e.g. B3LYP:

( )3 880.20 0.72 0.81B LYP LDA exact LDA B LYPxc xc x x x cE E E E E E= + − + ∆ + ∆

problem with self-consistent implementation: νx

exact (νxHF) is non-local

⇒ violation of Kohn-Sham framework!⇒ unnecessary coupling terms in magnetic property calculations

often too large linear response for main group species

solution: 1) require νxexact to be local and multiplicative

2) use optimized effective potential (OEP) framework (LHF/CEDA)⇒ localized hybrid potentials⇒ higher accuracy, no coupling terms

possible for any hybrid functional!

localized hybrid potentials in ReSpect:A. V. Arbuznikov, M. Kaupp Chem. Phys. Lett. 2004, 386, 8.A. V. Arbuznikov, M. Kaupp Chem. Phys. Lett. 2004, 391, 16 (open-shell).See also: W. Hieringer, F. Della Sala, A. Görling Chem. Phys. Lett. 2004, 383, 115.

A. M. Teale, D. J. Tozer Chem. Phys. Lett. 2004, 383, 109.

Localized Hybrid Potentials for Nuclear Shieldings(linear regression analysis for 22 main group compounds)

coefficient

local UDFTnonlocalCDFT

B-EXX(L)-PW91B3(L)-PW91

B-EXX-PW91a0=0.5

B3PW91 a0=0.2

B-PW91 (GGA)a0= 0.7a0= 0.6a0= 0.5

60.2

0.9923

1.176

-60.7

125.5

0.9753

1.350

-95.6

30.322.921.320.524.9standard deviation(ppm)

0.99770.99840.99870.99880.9983regression

1.0740.9901.0011.0131.051slope A(ppm)

-38.86.80.8-5.4-26.4intercept B

A. V. Arbuznikov, M. Kaupp Chem. Phys. Lett. 2004, 386, 8.

See also: W. Hieringer, F. Della Sala, A. Görling Chem. Phys. Lett. 2004, 383, 115.

A. M. Teale, D. J. Tozer Chem. Phys. Lett. 2004, 383, 109.

Localized Hybrid Potentials with 50% Exact ExchangeNuclear shieldings

-1400 -1000 -600 -200 200 600 1000-2400

-2000

-1600

-1200

-800

-400

0

400

800

1200

σ cal

cin

ppt

σexp in ppt

local

ideal

non-local

Chem. Phys. Lett. 2004, 386, 8.Nuclear shielding tensors for 22 main group molecules (32 values)

A. V. Arbuznikov, M. Kaupp Chem. Phys. Lett. 2004, 391, 16.

coefficient

local UDFTnonlocalCDFT

B-EXX(L)-PW91B3(L)-PW91

B-EXX-PW91, a0=0.5

B3-PW91 a0=0.2

B-PW91 (GGA)

a0= 0.6a0= 0.5a0= 0.4

10.4

0.978

0.606

5.9

30.8

0.932

0.988

17.0

9.620.111.19.29.4standard deviation(ppm)

0.9660.9610.9830.9850.977regression

0.4490.8640.7330.6490.537slope A

(ppm)

5.1-7.5-2.30.33.2intercept B

Localized Hybrid Potentials with 50% Exact Exchangeg-Tensors for 3d Transition Metal Complexes (19 values)

Localized Hybrid Potentials with 50% Exact Exchangeg-Tensors for 3d Transition Metal Complexes

-150 -100 -50 0 50 100 150 200 250 300-150

-100

-50

0

50

100

150

200

250

300∆

g cal

cin

ppt

local

idealnon-local

∆gexp in ppt

Chem. Phys. Lett. 2004, 391, 16.9 complexes, 19 values: Co(CO)4, CrOF4

-, CrOF4-, Cu(NO3)2, Fe(CO)5

+, Mn(CO)5, MnO3, Ni(CO)3H, TiF3

Calculated Isotropic Hyperfine Coupling Constant for theCalculated Isotropic Hyperfine Coupling Constant for the44P GS of the Phosphorus AtomP GS of the Phosphorus Atom

uncontracted 17s12p4d Partridge basis set.

100% νx

100% νx + νc(LDA)100% νx + νc(LYP)100% νx + νc(PW91)80% νx + νc(LDA)

+20% νx(B88)Exp.

νx = νx(HF)-83.7+25.2+6.5+15.9

+8.9+55.5

νx = νx(KLI)-84.3+6.4-9.7-92.5

-8.0

νx = νx(LHF)-125.1-18.8-36.9-116.0

-24.9

A preliminary theoretical justification of the found optimum a0 values …

Local hybrid functional: (J. Jaramillo, G. E. Scuseria, M. Ernzerhof J. Chem. Phys. 118, 1068 (2003) )

[ ]{ }loc.-hybr. exact DFT DFTxc x x c

( )( ) ( ) 1 ( ) ( ) ( ) d ; ( )( )

τε ε ετ

= + − + =∫rr r r r r r r

rWE g g g

Let us average to see what a constant a0 would be!

loc.-hybr. exact DFT DFTxc x x c( )d (1 ) ( )d ( )dE g gε ε ε= + − +∫ ∫ ∫r r r r r r

Cf.: exacthybrid DFT DFTxc 0 x0 cx (1 )E a Ea E E= + − +

0a g=

a0 is within 0.49 - 0.62 for both main-group molecules (NMR shieldings)and transition-metal complexes (g-tensors)

Average g(r):

(A. V. Arbuznikov, M. Kaupp Int. J. Quantum Chem. 2005, 105)

0( ) d( )

( );

dga ρ

ρ∫=

∫

rrr

rr

Averaged mixing coefficient a0 =[ ]( ) ( ) ( )d

( )dτ τ ρ

ρ∫

∫

r r r rr r

W

Closed-shell main-group molecules

a0

C2H2 0.627

C2H4 0.628

CH3F 0.609

CH4 0.661

CHF3 0.570

CO 0.614

CO2 0.577

a0

F2 0.585

H2CO 0.611

H2O 0.602

H2S 0.529

HCl 0.513

HCN 0.617

HF 0.595

a0

N2 0.604

N2O 0.571

NH3 0.618

O3 0.571

P2H2 0.520

PH3 0.561

PN 0.543

SO2 0.528

.

Open-shell 3d-transition-metal complexes (ρσ ,τσ andτW,σ are used; σ =α, β )

a0,α a0,β

Co(CO)4 0.507 0.516

CrOF4- 0.510 0.525

0.481CrOCl4- 0.469

a0,α a0,β

Cu(NO3)2 0.492 0.496

Fe(CO)5+ 0.520 0.529

0.527Mn(CO)5 0.518

a0,α a0,β

MnO3 0.476 0.506

Ni(CO)3H 0.498 0.503

0.526TiF3 0.514

„Double local hybrid“ exchange-correlation potentials?

application of the OEP method

replacement of a0=const by ø[g(r)]

Local hybrid functionals:

[ ]{ }loc.-hybr.xc

exact DFT DFTx x c( )( ) 1 ( ) ( ) ( ) dε ε ε

=

+ +∫ −r rr r r rgE

g

Traditional hybrid functionals and potentials:

- non-local and non-multiplicative

exact0 x

hybrid DFTxc xc= +E Ea E

hybrid DFTxc xc

exact0 xˆˆ = +a vv v

hybridhybrid xcxc

12i

i

Ev δϕδϕ

→

exactxv

Localized hybrid potentials:

- local and multiplicative

hybridloc.-hybr. xcxc

Ev δδρ

=

exact0

loc.-hybr. DFTxc xx c= + vvv aexactxv

Double local hybrid potentials:

1. Rigorous (planned):

2. Model potentials (preliminary work):

loc.-hybr.DLH xcxc

Ev δδρ

=

[ ]exa

DLH(M)xc

DFT DFTx c

ctx( ) ( ) 1 ( )

( )( ) ( )

=+ +−r r

rr rr

vg v g v v

Typical radial behaviour of the ratio τW / τ

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

1.2 Cl-

r, Bohr

τw(r)/τ4π2ρ(r)/N

… or for open-shell systems: τW,σ / τσ , (σ = α, β)

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

P atom

r, Bohr

τw(r)/τ4π2ρ(r)/N

(τw/τ)2 (τw/τ)3 (τw/τ)4

Spatial behavior of local mixing functions

τw/τ

For comparison:

( )

12

2 52 3 31 3 6

10

wELF τ τ

π ρ

−⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟−⎢ ⎥= + ⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

1.0

Example:O3 molecule0.5

0.0

Still other choices of local mixing functions, g(r), are possible

Normalization of the „projected“ exchange hole:A. D. Becke J. Phys. Chem. 2003, 119, 2972.suggested as part of a model of nondynamicalcorrelation.

proj( ) ( )σ σ=rg f N3 2

proj 5 2σ

8 2N27

xx eQ xσ

σπ ρ

⎛ ⎞−= ⎜ ⎟

⎝ ⎠

(inverted machinery of the BRx89 functionalA. D. Becke, M. R. Roussel Phys. Rev A 1989, 39, 3761).

( )21 46 WQσ σρ τ τ⎡ ⎤= ∇ + −⎣ ⎦ curvature of exchange hole

Sx,2 2

( 2) 31 v2 4

x Qx xex

σσ

σπ ρ− ⎡ ⎤− − = −⎢ ⎥⎣ ⎦

Example:O3 molecule

occS * NLx, x,

1 ˆv vi ii σ σσ

σ σσ

ϕ ϕρ

= ∑ Slater potential

Next things to do:

-self-consistent implementation of „double local hybrid potentials“

-self-consistent implementation of Becke‘s „coordinate-space“ model of nondynamical correlation

-combination of localized hybrid potentials with two-componentrelativistic calculations of MR parameters, based on theDouglas-Kroll-Hess Hamiltonian (non-collinear)

A. V. Arbuznikov

Funding

Deutsche Forschungsgemeinschaft

Graduiertenkolleg „Magnetic Resonance“ (Stuttgart)

Development of ReSpect:

V. G. Malkin (Bratislava)

O. L. Malkina (Bratislava)

R. Reviakine

Acknowledgments, Collaborations

EPR examples:J. Asher, N. Doltsinis (Bochum),

S. Kacprzak, M. Munzarova