Relativistic calculations of pNMR parameters Stanislav Komorovsk - - PowerPoint PPT Presentation

Relativistic calculations of pNMR parameters

Stanislav Komorovský

22.2. 2014

CTCC Department of chemistry

• Skip motivation
• Progress in relativistic pNMR calculations
• Why is relativity computationally so tough

Content

2 2 2 2

ˆ ˆ ˆ ˆ ( ) ( ) ( ) ( )

z x y x y z z x y x y z L L L L S S S S

c c V V V V p c p ip c c p ip cp cp c p ip c c p ip cp c E

       

                                                      

2

1 2 p E V          

1928 1925

Gaussian, Turbomole, Dalton, NWChem, Orca, Molcas, … ReSpect, Dirac, Bertha, REL4D, BDF, BAGEL

Quantum physics

k k

p i r    

• F. Rastrelli and A. Bagno, Mag. Res. in Chem. 48, S132 (2010).
• S. Komorovsky, M. Repisky, K. Ruud, O. L. Malkina, and V. G. Malkin
• J. Phys. Chem. A 117, 14209 (2013).

Four-component relativistic calculations of pNMR shielding for doublet systems:

Current progress in our group

 

iso iso

1 3

cs e M M M

S S g A kT     

 

 

ani dip

1 Tr 9

pc e M M M

S S g A kT     

• rb

cs pc M M M M

      

• rbital shift



:

• rb

M



contact shift



:

cs M



pseudocontact shift



:

pc M



• rb

ref

• rb

M M M

    

Traditional way of expressing pNMR shift

Calculated 1H NMR shielding for NAMI and its Ru(II) diamagnetic analogue:

Can we use chemical shift of diamagnetic molecule as

• rbital shift of its paramagnetic counterpart?

5 4 2 NH

Approximation for orbital shift ߜ𝑝𝑠𝑐

 

iso iso

1 3

cs e M M M

S S g A kT     

• System must obey Curie law (no spin-orbit coupling effects)
• Holds for single (multiple) electron(s) in an orbital which is well

separated from any other excited level

• I. Bertini, C. Luchinat, and G. Parigi Prog. Nucl. Magn. Reson. Spectrosc. 40, 249 (2002).

Limitations of expression for ߜܿs

 

iso iso

1 3

cs e M M M

S S g A kT     

= 9.5 − 3.1 𝜀𝑝𝑠𝑐 5.0 𝜀𝑑𝑡 6.4 𝜀𝑞𝑑

• 3.5

𝜀𝑢𝑝𝑢 7.9 −1.3

How large can be spin-orbit contribution to ߜܿs?

How far from metal center will ߜܿs vanish?

δcs CH3(a) 56.4 CH3(b) 73.8 CH3(c)

• 8.5

H-5a

• 7.5

H-6a

• 7.9

H-5b

• 14.7

H-6b 7.0 H-5c 10.2 H-6c 2.6

5 6

• System must obey Curie law (no spin-orbit coupling effects)
• The electron spin on the paramagnetic center can be considered

as a point dipole

• No spin-orbit effects  𝑕𝑏𝑜𝑗 = 0  𝜀𝑁

𝑞𝑑 = 0 𝜀𝑁 𝑞𝑑 = 𝜈𝑓 𝛿𝑁 𝑇 𝑇 + 1 9𝑙𝑈 Tr 𝑕𝑏𝑜𝑗𝐵𝑁

𝑒𝑗𝑞

Limitations of expression for ߜ݌ܿ

1 10 1 10000

How much of the spin density matter?

9.5

Theoretical considerations

2 ( ,

)

M uv u v

E B B       

NMR shielding tensor for closed-shell systems (singlets):

2

( , )

M uv u v

E B B       

NMR shielding tensor for open-shell systems (multiplets):

/ /

m m

W kT m m W kT m

E e E e

 

 



singlet doublet triplet NR R B

( , ) E B 

3 / 3 /

m m

W kT m m W kT m

E e e

 

 

2 / 2 /

m m

W kT m m W kT m

E e e

 

 

Paramagnetic NMR shielding tensor

2 M uv u v

E B       Paramagnetic NMR shielding tensor:

/ /

m m

W kt m m W kt m

E e E e

 

 



doublet NR R B

(0,0) (0,0)

E E

 



( , ) ( ,0) (0, )

1 E E E kT  

B μ B μ

σ

(0,0) (0,0)

/ /

k k

E kT k k E kT k

X e X e

 





 

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

/ / / /

1 2

E kT E kT E kT E kT

X e X e X X X e e

   

       

    

( ,0) ( ,0)

E E

 

 

B B (0, ) (0, )

E E

 

 

μ μ ( , ) ( , )

E E

 



B μ B μ

 

( , ) ( ,0) (0, )

1 E E E kT  

B μ B μ

σ

Paramagnetic NMR shielding tensor for doublet:

Paramagnetic NMR shielding tensor

( , ) ( ,0) (0, )

1 E E E kT  

B μ B μ

σ

Paramagnetic NMR shielding tensor for doublet:

T

• rb

4

e I N

kTg     σ σ gA

• rb

fc pc

   σ σ σ σ

Orbital, contact and pseudocontact contribution:

Paramagnetic NMR shielding tensor

T T fc iso pc ani

A   σ g σ gA

T

• rb

4

e I N

kTg     σ σ gA

T pc ani

 σ gA

Pseudocontact contribution:

How to split NMR shielding tensor into different contributions?

iso ani

A   A 1 A

5 3

3 a b

ab I a b

r r H I r r            

Contact contribution:

H A   S I

T fc iso

A  σ g

iso ani iso

1 Tr[ ] 3 A A    A A A 1

• I. Bertini, C. Luchinat, and G. Parigi Prog. Nucl. Magn. Reson. Spectrosc. 40, 249 (2002).

• F. Rastrelli and A. Bagno, Mag. Res. in Chem. 48, S132 (2010).
• S. Komorovsky, M. Repisky, K. Ruud, O. L. Malkina, and V. G. Malkin
• J. Phys. Chem. A 117, 14209 (2013).

Four-component relativistic calculations of pNMR shielding for doublet systems:

Results

NR and ZORA

δZORA δmDKS mer-Ru(ma)3 CH3 a 71.3 59.7 CH3 b 91.9 75.6 CH3 c

• 6.0
• 5.2

H-5a

• 1.8
• 3.0

H-6a

• 3.4
• 1.6

H-5b

• 10.4 -10.0

H-6b 12.7 13.1 H-5c 16.4 17.8 H-6c 8.6 10.5

Small relativistic effects in 1H shifts in mer-Ru(ma)3

Calculated and experimental

1H shifts in mer-Ru(ma)3

5 6

δZORA δmDKS δexp mer-Ru(ma)3 CH3 a 71.3 59.7 41.0 CH3 b 91.9 75.6 43.2 CH3 c

• 6.0
• 5.2

21.1 H-5a

• 1.8
• 3.0

11.8 H-6a

• 3.4
• 1.6

9.2 H-5b

• 10.4 -10.0
• 4.6

H-6b 12.7 13.1 3.4 H-5c 16.4 17.8

• 0.9

H-6c 8.6 10.5 0.9

𝜀𝑢ℎ𝑓𝑝 𝑝𝑠𝑐 𝜀𝑓𝑦𝑞 𝑝𝑠𝑐 CH3(a) 2.1 9.0 CH3(b) 1.5 10.0 CH3(c) 2.2 14.3 H-5a 5.8 4.9 H-6a 6.7 8.0 H-5b 6.1 6.7 H-6b 6.5 9.0 H-5c 7.2 7.5 H-6c 7.6 9.0

“the extrapolated values (𝑈 → ∞) in every case are within 1.5 ppm

• f the diamagnetic values measured for free maltol in CD2Cl2 [H(5)

at 𝜀 6.4, and H(6) at 𝜀 7.7]”

D.C. Kennedy, A. Wu, B.O. Patrick, and B.R. James Inorg. Chem. 44, 6529 (2005).

“The x intercepts for the resonances of these groups from the low- temperature 1H NMR data (𝜀 14.3, 𝜀 10.0 and 𝜀 9.0) do not, however, correlate well with the Me resonance of free maltol (𝜀 2.4).”

Experimental assignment of pNMR shifts

5a 6a 5b 6b 5c 6c FC

• 0.232
• 0.293
• 0.496

0.265 0.333 0.078 PSO

• 0.033

0.001

• 0.032
• 0.002

0.039 0.010 SD

• 0.012
• 0.001
• 0.014
• 0.004

0.004 0.009 SUM

• 0.277
• 0.293
• 0.541

0.259 0.376 0.097

δcs δpc CH3(a) 56.4 1.3 CH3(b) 73.8 0.3 CH3(c)

• 8.5

1.1 H-5a

• 7.5 -1.3

H-6a

• 7.9 -0.4

H-5b

• 14.7 -1.4

H-6b 7.0 -0.4 H-5c 10.2 0.5 H-6c 2.6 0.4 “the magnitude of the hyperfine shift for H(5) is always greater than that for H(6) as expected because H(5) is closer to the Ru(III) center.” D.C. Kennedy, A. Wu, B.O. Patrick, and B.R. James Inorg. Chem. 44, 6529 (2005). Isotropic hyperfine coupling constants for 1H in mer-Ru(ma)3 [MHz]

Experimental assignment of pNMR shifts

D.C. Kennedy, A. Wu, B.O. Patrick, and B.R. James Inorg. Chem. 44, 6529 (2005).

Couplings between H-5 and H-6 protons

Assignment of experimental data to calculated values.

5 6

δnr δZORA δmDKS δexp NAMI O=S(CH3)2

• 7.4
• 5.8 -12.1 -14.5

H-2 12.9 11.4 8.0

• 5.6

H-4 3.3 5.6

• 2.7
• 7.8

H-5 0.7 2.0

• 2.7
• 3.5

NH 6.9 5.8 3.5

5 4 2

NH Significant differences between different methods and experimental 1H shifts in NAMI.

σorb σfc σpc σsum NAMI O=S(CH3)2 ZORA 27.8 9.4 37.2 mDKS 32.8 8.0 2.2 43.0 H-2 ZORA 22.7

• 2.7

19.9 mDKS 25.8

• 6.4

3.5 22.8 H-4 ZORA 22.5 3.2 25.7 mDKS 26.0 4.3 3.2 33.5 H-5 ZORA 25.1 4.3 29.4 mDKS 27.5 4.7 1.3 33.6 NH ZORA 23.4 2.1 25.5 mDKS 24.8 1.3 1.3 27.3

Origin of differences between ZORA and mDKS method for 1H shifts in NAMI.

5 4 2

NH

Systematical underestimation of orbital NMR shielding by ZORA method

ZORA mDKS Hamiltonian ZORA Hamiltonian (SO included) full four-component Dirac-Coulomb Hamiltonian Basis STO: TZ2P GTO: Ru - Dyall vTZ; light elements - upcJ-2 g-tensor, HFCC SO restricted calculation SR spin-polarized calculation non-collinear DFT functional Nuclear shielding closed-shell analog (Ru(II))

• pen-shell compound

Fitting no fitting fitting of the electron density and spin densities potential/kernel BP86/BP86 BP86/SVWN5

Possible differences between ZORA and mDKS calculations

Why there is so big difference for H-2 shift anyway?

5 4 2

NH

First hint: stretching Ru-N bond helps

5 4 2

NH

Effect of stretching of Ru−N bond in the interval 1.9−2.3 Å. Closer to the experimental data are results with longer bonds.

Second hint: hybrid functionals

5 4 2

NH

• rb

M M cs pc M M

      

Using hybrid functionals in mer-Ru(ma)3 compound

5 6

• rb

M M cs pc M M

      

Why is relativity complicated?

• describes the motion of electron in

the external potential

• has the 4-component form
• describes the spin naturally
• gives the positive energy (E>0) and

negative energy (E<0) solutions

• P. A. M. Dirac, Proc. R. Soc London Ser. A 117, 610 (1928)

2 2

ˆ ˆ 2 ˆ ˆ ( ) ( ) ( ) ( 2 )

z x y x S S S S y z z x y L y z L L x L

V cp c p ip c p c V c V V ip cp E cp c p ip c p ip cp

       

                                                      

• 2mc2

Energy

Dirac equation

2

ˆ ˆ 2

L L S S

V c p E c c p V                             

2 2

ˆ ˆ 2 ˆ ˆ ( ) ( ) ( ) ( 2 )

z x y x S S S S y z z x y L y z L L x L

V cp c p ip c p c V c V V ip cp E cp c p ip c p ip cp

       

                                                      

1 1 1 1

x y z

i i                          

Pauli matrices:

More compact Dirac notation

energy states Positive (electron) Negative (positron)

1

L L S S

        

L S

       

L S

       

small 0.01 large 0.99 small large

• 2mc2

Energy

How small is the small component?

Elimination of the small component:

1 ˆ ˆ 2

L L

V p K p E               Non-relativistic limit of Dirac equation

2

ˆ ˆ 2

L S S L

c p c p c E V V                           

2

ˆ ˆ ( 2 )

L S L S L S

E c V V p p E c c                1 ˆ 2

S L

K p c     



1 2

ˆ ˆ 1 2 E V K c



        

c 

Non-relativistic limit:

 

2 1 2

ˆ p V E    

 

2 2

p p   

In QCh we are looking for expansion coefficients:

1 2 p p V E             χ χ χ χ χ C χ C

c 

1 N

C

  

 





Non-relativistic limit: Dirac equation: Should be Schrodinger equation:

 

2 2

p p   

2

1 2 p V E         χ χ χ χ C C



1  χ χ

Complete (infinite) basis:

  

Wrong non-relativistic limit in finite basis

2

ˆ ˆ 2

L L S S

V c p E c p V c                           χ χ χ χ C C C C χ χ χ χ

1s orbital in hydrogen atom:

1 2 V p p E             χ χ χ χ C C χ χ

s type functions only

Z X Y X Y Z

i p i i                           s s s s s s s s s s

1 1 N N

C C s

     

 

 

 

 

s 

Zero kinetic energy for s orbital!

1 N L L

C

 

 





1 N S S

C

 

 





The same basis functions for both components

Solution: don’t use the same basis for ߮𝑀 and ߮𝑇

2

ˆ ˆ 2

L S L S

V c p E c p V c                             

2

ˆ ˆ 2

S S L L

V c p E c p V c                           χ C C χ χ χ χ χ χ C C χ

2 2

cos s in

S r i c S i L r r i c L

e e e e

    

     

  

                            

(1 ) : s 

2 2 2 2 r i S c S r i c

z e r x y x y i e r r r

 

 

 

                            

2 1 2 2 2 Z X Y i c c X Y Z r i c r r r i c

z e e e r x y i e r r i p i 

   

                                                

Polar coordinates Cartesian coordinates

1 N L L

C

 

 





1 2 1 N S S c

C p

  

  



 



Use restricted kinetically balanced basis set:

Solution: don’t use the same basis for ߮𝑀 and ߮𝑇

1 L N L L

C

  

 





1 1 1 2 N N S S S c S

p C C

     

   

 

  

 

RKB basis for small component:

Solution: use restricted kinetically balanced basis for ߮𝑇

2

ˆ ˆ 2

L S L S

V c p E c p V c                             

2

ˆ ˆ 2

S S S S S S S L L L L L L L L S

V c p E c p V c                                   χ χ χ χ χ χ C C C C χ χ χ χ χ χ

2 1 1 2 2 S c L c

p c p p p         χ χ χ χ χ χ

1 N L L

C

 

 





1 2 1 N S S c

C p

  

  



 



Matrix Dirac equation

2

ˆ ˆ 2

L L S S

V c p E c p V c                             

2

1 2 L L S S c

E                          S V T C C T T W T C C

2 1 2

p  T χ χ ˆ V  V χ χ  S χ χ

2

1 4

ˆ

c

pV p      W χ χ

• P. A. M. Dirac, Proc. R. Soc London, Ser. A 117, 610 (1928)

 

2 2 2 2 2

ˆ ˆ1 ˆ ˆ 2 1

L L S S

V c p E c p V c      

 

                       

Dirac one-electron equation:

         

2 2 2 2 2 2 2 2

ˆ ˆ ˆ ˆ ˆ 1 1

nuc ee

E r V r V r V r r   

   

               

Dirac-Kohn-Sham equation:

2 2 2 2 2 2 2

ˆ ˆ ˆ 2 1

L L i i i S S i i

V c p c p V c       

  

                      

Dirac-Kohn-Sham equation

2 2 2 2 2 2 2

ˆ ˆ ˆ 2 1

L L i i i S S i i

V c p c p V c       

  

                      

matrix Dirac-Kohn-Sham (mDKS) equation

1 N L L

C

 

 





1 2 1 N S S c

C p

  

  



 



RKB basis:

2

1 2 L L S S c

E                         S V T C C T T W T C C

2 1 2

p  T χ χ

2 2

ˆ V   V χ χ  S χ χ

2

1 2 2 4

ˆ

c

pV p  



   W χ χ

Thank you