a) Rayleigh-Schrdinger perturbation theory b) choice of Hamiltonian - - PowerPoint PPT Presentation

pNMR-ITN Workshop Mariapfarr, Austria, February 23, 2014

A Short Introduction into Quantum Chemical Calculations

• f NMR and EPR Parameters

Martin Kaupp Institut für Chemie, TU Berlin

• Part I: Some basics on quantum chemical methods
• Part II: Basics on perturbation theory methods

for the quantum-chemical calculation of NMR and EPR parameters a) Rayleigh-Schrödinger perturbation theory b) choice of Hamiltonian c) nuclear shieldings (diamagnetic systems) d) electronic g-tensor e) hyperfine coupling f) (zero-field splitting)

Theory and Applications

A comprehensive treatment, Wiley-VCH 2004. With 36 chapters on methodology and applications.

spin Hamiltonian parameters

simulation

model structure

ˆ 



H = E

properties  

comparison with model and/or homologous systems (e.g. from literature, database)

Applications to:

• spectra assignment
• structure elucidation
• interpretation
• dynamics

The effective spin Hamiltonian

Introduction (1)

effective spin Hamiltonian provides link between magnetic resonance experiment and quantum mechanical treatment: in most cases, property may be expressed as second derivative:

) , , ; , ( ) , ( I S B b I S a b a b a E b a H X          

 appropriately treated by second-order perturbation theory:

 

      E b a H H ) , (

0 V

   

   

                 

M N M NM NM N N N N N N N

I K D I B I I q I S D S I A S B g S H

, ,

1 

    

• relation between effective-spin Hamiltonian and quantum-chemical

treatment requires appropriate Hamiltonian that takes care of spins and magnetic fields  starting point relativistic quantum mechanics (e.g. Dirac equation)

• ften transformation to quasi-nonrelativistic formalism

(e.g. Breit-Pauli Hamiltonian) we will start by ignoring relativistic effects and magnetic interactions initially nonrelativistic quantum mechanics for H0 relativistic effects and magnetic interactions will be introduced later, mainly by perturbation theory

Introduction (2)

Part I: Some Basics on Quantum Chemical Methods

The general (non-relativistic) Hamiltonian describing N electrons and M nuclei is given (in a.u.) by:

H0: Nonrelativistic Quantum Mechanics (1)

      

       

       

M A M A A B AB B A N i i j ij N i M A iA A A A N i i

R Z Z r r Z M H

, 1 , 1 , 1 , 1 , 1 2 , 1 2

1 2 1 2 1

after Born-Oppenheimer approximation (electronic Schrödinger equation):*

) ( ) ( ˆ 2 1 ) ( ˆ 1 2 1 ) ( ˆ ˆ ˆ

2

i V i i r i E

j i ij i

        

 



h h H H

electron-electron repulsion

electronic Hamiltonian electronic wave function

• ne-electron

Hamiltonian

• perator of kinetic energy
• perator of potential energy

(mainly el.-nuclear attraction)

still missing: electronic spin (Pauli principle), *and time dependence removed

H0: Nonrelativistic Quantum Mechanics (2)

     H E0

energy expectation value: variational principle:

~ ~ ~ ~ ~ E H E      

exact ground-state solution provides lower bound to energies of approximate solutions!

H0: Nonrelativistic Quantum Mechanics (3)

in the absence of electron-electron interactions:

) ( )..... 4 ( ) 3 ( ) 2 ( ) 1 ( ) ,....... 4 , 3 , 2 , 1 (

4 3 2 1

n n

n

      

Hartree product, independent-particle model, violates Pauli principle, no correlation.

) ( )..... 4 ( ) 3 ( ) 2 ( ) 1 ( ) ,....... 4 , 3 , 2 , 1 (

4 3 2 1

n n

n

      

Slater determinant (antisym. linear combination of product wavefunctions)

• i(i) : one-electron wave functions (molecular orbitals)
• still no Coulomb correlation but Pauli exchange

! ~ principle l variationa into as insert 

Hartee-Fock method

H0: Nonrelativistic Quantum Mechanics (4)

~ ~ ~ ~ ~                   H q q E

take trial wave function with free parameters q and minimize energy:* for example: single Slater determinant Hartree-Fock method

) 1 ( ) 1 ( ) 1 (

a a a

f      

) 1 ( ) 2 ( ) 2 ( ) 1 ( ) 1 ( ) 2 ( ) 1 ( ) 1 ( 2 1 ) 1 ( ) 1 ( ) 1 ( ) 1 (

1 12 * 2 1 12 2 2 2 1 b a b a b b b b b b HF A iA A HF

r dx K r dx J K J v r Z h v h f     

   

 

         

f depends on coordinates of all electrons (via vHF)  iterative solution “self-consistent field” (SCF) method i : molecular orbitals Ji: Coulomb op., Ki: exchange op. *Lagrange multipliers a to account for constant N(electrons), and for

• rthogonality of MOs

H0: Nonrelativistic Quantum Mechanics (5)

algebraic expansion of i into basis set: Roothan-Hall method  matrix operations, linear algebra





j j ij i

A c ) (  

: atomic orbital basis function with center on atom A  MO expanded in linear combination of AOs  MO-LCAO method typical basis sets:

2

) , , (

r

e r f A







  

r

e r g B







   ) , , (

Gaussian-type orbitals (GTO) integrals more convenient Slater-type orbitals (STO) fullfil cusp condition

• thers: plane waves, numerical AOs, muffin-tin orbitals, etc……*

*also for DFT etc……..

• ptimizable parameters!!!

H0: Nonrelativistic Quantum Mechanics (6)

Hartree-Fock method accounts for Pauli exchange but not for Coulomb correlation 

~ E E HF 

(even with infinite basis set!)

HF corr

E E E ~  

Löwdin definition of correlation energy better methods (e.g. post-Hartree-Fock or DFT), to account for Ecorr post-HF methods: perturbation theory (MPn) size-consistent, not variational configuration interaction (CI) not size-consistent,* variational coupled-cluster theory (CC) size-consistent, not variational

*in truncated form

H0: Nonrelativistic Quantum Mechanics (7)

Hartree-Fock (HF), SCF, MO-LCAO

~ ( ) ( )...... ( ) ( )   



   

1 2 1

1 2 1

n n

n n

n4

A Hierarchy of post-Hartree-Fock Theories

Density Functional Theory (n3-n4 )

Semi-Empirical MO-Theory Perturbation Theory (MBPT) (e.g. MP2: n5, MP3: n6, MP4: n7) Configuration Interaction (CI)

~ ( ......)        

 



C C C

a r r a r ab rs r s ab rs

(e.g. SDCI: n6) Coupled Cluster Methods (CC) (e.g. CCSD: n6, CCSD(T): n7)

~ (    ........)



      eT T T T

1 2

“Dressed CI Methods” CPF, ACPF, MCPF, CEPA

typically n6

Multiconfiguration SCF (MCSCF, CASSCF, GVB) Multireference Coupled Cluster Theory Multireference Configuration Interaction (e.g. MR-SDCI, MR-ACPF) Multireference Perturbation Theory (CASPT2, MR-MP2, MR-MPn?) nm: formal scaling factors relative to system size n. Note that linear pre-factors are also important

Density Functional Theory (1)

Hohenberg-Kohn functional

Hohenberg-Kohn Theorem (1963): E = E[r] E[r] = T[r] + Vne[r] + V

ee[r] =

 r(r)(r)dr + FHK[r] FHK[r] = T[r] + V

ee[r]

Kohn-Sham Method (1964):

exchange-correlation functional of the KS method

F [r] = Ts[r] + J[r] + Exc[r] Exc[r] = T[r] - Ts[r] + V

ee[r] - J[r]

basic idea: use r(r) instead of the more complicated y(r1,r2,….rn):

) 1 ( ) 1 ( ) 1 (

a a a

f    

) 1 ( ) 1 ( ) 1 (

xc

v h f  

Kohn-Sham equations: while formally similar to HF-SCF equations, KS method incorporates electron correlation via (unknown) vxc approximations to Exc and xc: Excr]: local density approximation, LDA (e.g. SVWN, X) Excr, r]: gen. gradient approximation, GGA (e.g. BLYP, BP86, PW91, PBE,….) Excr, r, 2r, t]: meta-GGA (e.g. PKZB, FT98,…..) hybrid functionals (with exact exchange) (e.g. B3LYP, BHPW91, mPW1,…) further: OEP, exact-exchange functionals….

Density Functional Theory (2)

See also: https://sites.google.com/site/markcasida/dft

1/3 4/3

LDA LDA x x

3 3 (r) 4

E ( ) d

• d

r 

 r

          

   r r r

Local Density Approximation (LDA) GGA LDA GGA xc xc xc

E ( ) ( ), ( ) d  r r r

       

   r r r r F

Generalized Gradient Approximation (GGA)

(different functionals have been constructed).

2

mGGA mGGA xc xc

E ( ), ( ), ( ), ( ) d  r r r t

   

    r r r r r

Meta-Generalized Gradient Approximation (mGGA)

• cc

2

1 ( ) ( ) 2 t     r r

i i Ec for homogeneous electron gas not known analytically but from very accurate fits to Quantum Monte Carlo simulations

Density Functional Theory (3)

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

 

3 88

0.20 0.72 0.81

B LYP LDA exact LDA B LYP xc xc x x x c

E E E E E E       

nonlocal, nonmultiplicative exchange is not an explicit density functional,  complications

Examples Variables Type of xc-functional Examples Variables Type of xc-functional SVWN, X SVWN, X



r local density appr. (LDA) SVWN, X SVWN, X



r local density appr. (LDA) PP86, BP86, BLYP PP86, BP86, BLYP PW91, PBE PW91, PBE r , |r| generalized gradient approximation (GGA) PP86, BP86, BLYP PP86, BP86, BLYP PW91, PBE PW91, PBE r , |r| generalized gradient approximation (GGA) FT98, PKZB, TPSS, FT98, PKZB, TPSS, BRx89, Bc88 BRx89, Bc88 r , |r|, 2r, t meta-GGA FT98, PKZB, TPSS, FT98, PKZB, TPSS, BRx89, Bc88 BRx89, Bc88 r , |r|, 2r, t meta-GGA B3LYP, B3PW91 B3LYP, B3PW91 BHPW91, PBE0PBE BHPW91, PBE0PBE GGA + exact (nonlocal) exchange hybrid functionals B3LYP, B3PW91 B3LYP, B3PW91 BHPW91, PBE0PBE BHPW91, PBE0PBE GGA + exact (nonlocal) exchange hybrid functionals

See also: https://sites.google.com/site/markcasida/dft

Density Functional Theory (4)

HEAVEN (chemical accuracy) + explicit dependence on unoccupied orbitals rung 5 fully nonlocal + explicit dependence on

• ccupied orbitals

rung 4 example: hybrid functionals + explicit dependence on kinetic energy density rung 3 meta-GGAs + explicit dependence on gradients of the density rung 2 GGAs local density only rung 1 LDA EARTH (Hartree theory) Hell?

r  

r

t r,  

• cc

 

virt

 

Jacob’s Ladder to the heaven of chemical accuracy

John P. Perdew and Karla Schmidt, in Density Functional Theory and Its Applications to Materials, edited by V.E. Van Doren, C. Van Alsenoy, and P. Geerlings, AIP Conference Proceedings, Vol. 577 (American Institute of Physics, 2001), pp. 1-20.

class ingredients examples general nonlocal functionals + virtual-orbital dependent contributions e.g. OEP2, B2PLYP

• ccupied-orbital

dependent functionals t-dependent meta- GGA, global hybrid, OEP, local hybrid, B05-NDC model explicit density functionals LDA, GGA, some meta-GGAs

2

, , ,.... r r r

, ,....

x

 t

A simpler hierarchy of exchange-correlation functionals local hybrid B05-NDC model

• A. V. Arbuznikov, M. Kaupp , H. Bahmann Z. Phys. Chem. 2010, 224, 545.

Hybrid Functionals: Variation of Exact-Exchange Admixture The most important parameter in „classical“ hybrid functionals is a0: LDA or GGA functionals: a0 = 0.0 B3LYP: a0 = 0.20 BHLYP: a0 = 0.50 etc……

 

exa hybrid DFT DFT DFT xc c xc xc xc t x

; (explicit) r    E E E E a E

The next natural step in DFT: local hybrid functionals

local hybrid

O C      

1     

 

global hybrid X exact DFT X X

E a r dr a r dr

       

1         

 

local hybrid X exact DFT X X

E a r r dr a r r dr

admixture of exact exchange within the CO molecule

a0a(r)

traditional (global) hybrid O C

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

(first suggestion of principle by: F. G. Cruz, K.-C. Lam, K. Burke J. Chem. Phys. 1998, 102, 4911).

Part II: Basics on perturbation theory methods for the quantum-chemical calculation of NMR and EPR parameters

Rayleigh-Schrödinger Perturbation Theory (1)

 

      E H H V

unknown eigenvalue problem: (solutions orthonormalized, i.e. <i

(0)| j (0)>=ij).

) ( ) ( ) (

y y E H 

with presumably known eigenvalue problem: V is small. We want to derive  and E from (0) and E(0). Introduce ordering parameter l, which will transform H0 into H:

V H H l  

Expansion into Taylor series:

....

) 2 ( 2 ) 1 ( ) (

E E E E l l   

....

) 2 ( 2 ) 1 ( ) (

y l ly y    

Task: express exact quantities only as a function of zero-order energies and matrix elements of the perturbation V between the unperturbed wavefunctions <i

(0)| V |j (0)>!

   

   

) 2 ( 2 ) 1 ( ) ( ) 2 ( 2 ) 1 ( ) ( ) 2 ( 2 ) 1 ( ) (

... .. y l ly y l l y l ly y l l               E E E H H H V V

choose intermediate normalization: <|1  <(0)|

(n)>=0 (for n  0)

Rayleigh-Schrödinger Perturbation Theory (2)

) ( ) 3 ( ) 1 ( ) 2 ( ) 2 ( ) 1 ( ) 3 ( ) ( ) 2 ( ) 3 ( ) ( ) 2 ( ) 1 ( ) 1 ( ) 2 ( ) ( ) 1 ( ) 2 ( ) ( ) 1 ( ) 1 ( ) ( ) ( ) 1 ( ) ( ) ( ) (

y y y y y y y y y y y y y y y y y E E E E H E E E H E E H E H              V V V

collect and equate coefficients for a given power n of ln:

) 2 ( ) ( ) 3 ( ) 1 ( ) ( ) 2 ( ) ( ) ( ) 1 ( ) ( ) ( ) (

y y y y y y y y V V V     E E E H E

multiply each of the equations by <

(0)| and recall orthogonality relations:

(1), remember!

Rayleigh-Schrödinger Perturbation Theory (3)

) ( ) 1 ( ) 1 ( k k k

c y y







) ( ) ( ) ( 1 ) (

y y y y V

k k k ) ( k k

c ) E (H   





For E(2) we need (1)  expand (1):

) ( ) 1 ( ) 1 ( ) ( ) ( ) 1 (

y y y y E E H   V

recall (1): multiply by <k

(0)| and expand |0 (1)> :

) ( ) ( ) 1 (

y y V

k k k

)c E (E   

with hermiticity of H and intermediate normalization:

) E (E c

k k k ) ( ) ( ) 1 (

    y y V

) ( ) ( ) 1 ( ) 1 ( ) ( ) 1 (

y y y y V     E E H

Rayleigh-Schrödinger Perturbation Theory (4)





  

) ( ) ( ) ( ) ( ) ( ) ( ) 2 ( n n n n

E E E y y y y V V

) 1 ( ) ( ) ( ) 1 ( ) ( ) 2 ( k k

c E





 

k

V V y y y y

) E (E c

k k k ) ( ) ( ) 1 (

   y y V

  

   

a ar r a ra ar a aa a

v v v E

) ( ) ( ) (

  

in case of one-electron operators and one-electron-type wavefunctions:

) ( r v a var 

a,r represent spin orbitals recall:

Rayleigh-Schrödinger Perturbation Theory (5)

   

   

                 

M N M NM NM N N N N N N N

I K D I B I I q I S D S I A S B g S H

, ,

1 

    

) , , ; , ( ) , ( I S B b I S a b a b a E b a H X          

Perturbation Theory Applied to MR Parameters (1)





            

) ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) 1 (

) , ( ) , ( ) , (

n n

• n

n ele ele

E E b a V b a V b a V E E E E E

                        



0 ) ( ) ( ) ( ) ( ) ( ) (

) , ( ) , ( ) , ( ) , (

n n

• n

n ele

E E b a V b a V b a V E b a b a b a E X

note that only V depends on a and b!

 

 

                         

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

) , ( ) , ( ) , ( ) , ( ) , (

n n

• n

n n n

• n

n

E E b a V a b a V b E E b a V b b a V a b a V b a X

              

0 n n

E E b ) b , a ( H n n a ) b , a ( H 2 b a ) b , a ( H b a ) b , a ( E X

carry out differentiation: simplification in case of “real” wavefunctions and operators:

• 1. select appropriate Hamiltonian for both H0 and V
• 2. work out second-order perturbation theory

Perturbation Theory Applied to MR Parameters (2)

    

 

    

              

NM NM M N NM NM M NM N M N N N NM NM M N N N N N N N N N N N i Ni N

R I I R R I R I g g c k H R Z Z k e H B I g H Z e H H H

5 2 2 2 4 2 3 2 1

3 2 2

eN N e

H H H H    V

 

     

 

    

 

 

            

                                                                

ij j i j i e e e ij ij j i ij j i ij j e e e ij ij j i ij ij j i ij e e ij ij j ij j j ij i e e e j i j i e e ij ij e i i e e i i i i i i i e e e e i e i e i i i e e e i i e i e i e i ei e

r r s s c k g H r s s r r s r s c k g H r π π r r r π r c k H r π r s π r s c k g H r r c k H r k e H divE c m e H π E s E π s c m g H c m π H B s c k m π g H e H m π H H H

2 2 2 12 5 2 2 2 2 11 3 2 2 2 10 3 2 2 9 , 2 8 2 7 2 6 2 5 2 3 2 4 2 2 3 2 2 1

2 4 3 2 2 2 1 2 4 4 8 2 1 2   

kinetic energy

• int. with external electric field

electron Zeeman interaction relativistic correction to kinetic energy 2- electr. spin-orbit interaction Darwin corr. to electric field int. Coulomb repulsion 2-elect. Darwin term 1-elect. spin-orbit interaction electron-spin dipolar interaction

• rbit-orbit interaction

electron spin-spin interaction

• int. with external electric field

nuclear Zeeman interaction nuclear-nuclear Coulomb interaction

• nucl. dipole-dipole int

    

 

          

                             

N i N i N e eN N i iN i iN i N e eN N i iN i iN N N N e eN N i N i N i N N e e eN N i iN N i iN iN N iN i N N e e eN N i iN N eN i eNi eN

R r Z c k H r π r s Z c k g H r π r I g c k H R r I s g c k g H r I s r r I r s g e k g H r Z k e H H H

, 2 2 6 , 3 2 2 2 5 , 3 2 4 , 2 3 , 5 2 2 2 , 2 1

2 2 2 3 8 3  

electron-nuclear Coulomb interaction dipolar hyperfine int. Fermi contact term

• rbital hyperfine interaction

electron-electron spin-orbit hyperfine int. electron-nuclear Darwin term

The Breit-Pauli Hamiltonian (incl. ext. fields)

• var. stable quasirelativistic Hamiltonians:

Douglas-Kroll-Hess, ZORA. Or fully relativistic: Dirac-Coulomb-Breit

NMR/EPR Parameter Ingredients for pNMR Shifts

• Nuclear shieldings (need to be generalized from closed- to
• pen-shell case)  orbital shielding
• Hyperfine couplings (we need both isotropic and anisotropic

contributions)

• g-tensor (needed for „dipolar shifts“)
• zero-field splitting (modifies energy levels for S > ½ )

Nuclear Shieldings (NMR Chemical Shifts) for Diamagnetic Systems

 N,uv

N,u v

    

2E

B ;u,v x,y,z

H(B, ) H N

Nu N N

     

10 01 11

H H H

u v Nu Nu Nuv v

B B  

                

 

 

. . E E 2 1 ) ( 2 1

n 1 3 , , 2 3 2

c c r L n n L c r r r c

n kN u N v O i iN iN,u iO,v uv N,uv

 

iN iOr

r

d (diamagn. shielding) p (paramagn. shielding)

Application of second-order perturbation theory to nuclear shielding: The nonrelativistic Ramsey equation

gauge problem for finite basis sets  need for specialized basis sets to expand the first-order wave function  GIAO, IGLO

• V. G. Malkin, O. L. Malkina, L. A. Eriksson, and
• D. R. Salahub, in Modern Density Functional

Theory: A Tool for Chemistry; Theoretical and Computational Chemistry, Vol. 2, edited by J. M. Seminario and P. Politzer (Elsevier, Amsterdam, 1995).

BP86 functional, SOS-DFPT correction Molecules: CH4, C2H2, C2H4, allene, C3H8, c-C3H6, c-C3H4, benzene, CO, H2CO, H2O, N2O, NH3, N2, HCN, CH3F, HF, F2, PN, PH3, PF3.

Shielding constants calculated with SOS-DFPT versus experimental data

• 400
• 300
• 200
• 100

100 200 300 400 500 600

• 400 -300 -200 -100

100 200 300 400 500 600

Experimental (ppm) Calculated (ppm)

DFT computations of transition metal NMR chemical shifts

calc expt

BP86 structures

Review: M. Bühl, in: M. Kaupp, M. Bühl, V. G. Malkin (Eds.), Calculation of NMR and EPR Parameters. Theory and Applications, Wiley-VCH, Weinheim, 2004.

M.K., O. L. Malkina, V. G. Malkin J. Chem. Phys. 1997, 106, 9201. QR-ECP//exp., IGLO-II on oxygen, exptl. shifts converted, UDFT with BP86 functional

• 3000
• 2500
• 2000
• 1500
• 1000
• 500

500 1000 1500 2000 2500 3000

HF-GIAO expt. BP86(UDFT)-GIAO MP2-GIAO 17O (ppm)

WO4

2- MoO4 2- ReO4

• TcO4
• OsO4 CrO4

2- RuO4 MnO4

• Importance of Nondynamical Correlation for Nuclear Shieldings
• L. Benda, MnO4
• :

CCSD(T)/cc-pxVQZ: -743 ppm CCSD/cc-pxVQZ: -2132 ppm

A few useful practical details on NMR parameter calculations

• NMR shifts: use GIAOs (or IGLO or ……) to deal with gauge problem
• flexible basis sets needed for shifts and particularly for couplings

(e.g. IGLO-III, tzp,……). 6-31G* is definitely not good enough!

• MP2 can be best for „moderate“ correlation cases,

CCSD or CCSD(T) are the most powerful (and expensive) ab initio methods

• DFT has the best cost/performance ratio and is often stable

even in cases of significant nondynamical correlation

• main-group shifts are less critical (GGA after corrections is OK)

TM shifts and spin-spin couplings are typically better reproduced with hybrid functionals

Sum over all singlet excited states. Difference in the energies of excited (n) and ground (0) states. Interaction of electronic orbital angular momentum with external magnetic field (Orbital Zeeman Term). Interaction of nuclear magnetic dipole with electronic orbital motion (PSO Term).

. .c c E E r L L

n n 1 n k 3 kN kN k k n 1 p

        

 

 



Analyses of the paramagnetic term

Interpretations depend on the choice of gauge and choice of MOs. Obviously, NMR chemical shifts are not a simple property!

• M. Kaupp, in: M. Kaupp, M. Bühl, V. G. Malkin (Eds.), Calculation of NMR and EPR Parameters.

Theory and Applications, Wiley-VCH, Weinheim, 2004.

Si Si Me Me Me Me 118.0 Si Si SiH3 SiH3 H3Si H3Si 159.5 Si Si Me Me H3Si H3Si 247.8 –31.1 A Simple Example: Counter-Intuitive 29Si Shifts in Unsymmetrically Substituted Disilenes

• D. Auer, C. Strohmann, A. V. Arbuznikov, M. Kaupp Organometallics 2003 22, 2442.
• Differences between symmetrical species may be understood from energy denominators.
• But what about the unsymmetrical one?

PSO term couples (Si-C) and *(Si=Si) MOs much more strongly for Si2!

„Spin-Orbit Chemical Shifts“

                

 

 

. . E E 2 1 ) ( 2 1

n 1 3 , , 2 3 2

c c r L n n L c r r r c

n kN u N v O i iN iN,u iO,v uv N,uv

 

iN iOr

r

d (diamagn. shielding) p (paramagn. shielding) nonrelativistic Ramsey Equation

              



   , n 1 m 3 ) ( , , 2 ,

) E (E ) E (E

n m B S S i SO T T SD FC v u N I SO uv N

ns permutatio H n n H m m H B  

SO (third-order spin-orbit corrections) dominant heavy-atom effect for neighbor atoms

Review: M. Kaupp, in Relativistic Electronic Structure Theory II (Ed. P. Schwerdtfeger), Elsevier, Amsterdam 2003.

NMR

• nucl. A

electronic spin polarization NMR

• nucl. B

Fermi-contact interaction Fermi-contact interaction

Fermi-Contact Spin-Spin Coupling A useful analogy....

spin-orbit coupling electronic spin polarization NMR

• nucl. B

Fermi-contact interaction B0 heavy atom A

Spin-Orbit Chemical Shifts

• M. Kaupp, O. L. Malkina, V. G. Malkin, P. Pyykkö Chem. Eur. J. 1998, 4, 118.

• Chem. Eur. J. 1998, 4, 118.

Correlation between 13C "SO Shifts" and Reduced Coupling Constants in C6H5I

• 5

5 10 15 20 25 30 35 50

• 50
• 100
• 150
• 200
• 250
• 300
• 350

C ortho C meta C para "SO shift" KFC(C,I) C ipso

KFC(C,I) (1019NA-2m-3) SO contribution to 13C shielding (ppm)

I

ipso para

• rtho

meta

A Karplus-Type Relation for Spin-Orbit Shifts in Iodoethane

1

H "SO shift"

• 0.40
• 0.35
• 0.30
• 0.25
• 0.20
• 0.15
• 0.10
• 0.05

0.00 0.05 0.10 0.15

SO contribution to

1H shielding /ppm 20 40 60 80 100 120 140 160 180

12.0 10.5 9.0 7.5 6.0 4.5 3.0 1.5 0.0

• 1.5
• 3.0
• 4.5

3 K FC (H,I)

3 K FC

(E,I) /10

19

NA

• 2

m

• 3

H-C-C-I dihedral angle /deg

• Chem. Eur. J. 1998, 4, 118.

H I

• Chem. Eur. J. 1998, 4, 118.
• 250
• 300
• 350
• 400
• 450
• 500
• 550
• 600
• 650
• 700

25 30 35 40 45 50 55 60 65 70 1-bond SO contribution to

13C shielding /ppm

"SO shift"

1KFC (C,I)

The Role of s-Character for "SO Shifts" and Reduced Coupling Constants

H3C-CH2-I H2C=CH-I HCC-I

1KFC(C,I)/(1019NA-2m-3

• M. Kaupp, C. Aubauer, G. Engelhardt, T. Klapötke, O. L. Malkina J. Chem. Phys. 1999, 110, 3897.

Confirmation of the Extreme High-Field 31P NMR Shift of the PI4

+ Cation

PF4

+

PCl4

+

PBr4

+

PI4

+

31P /in ppm vs. 85% H3PO4

exp.

• calc. nonrel.
• calc. + 1-el.-SO correction
• calc. + 1- +2-el.-SO
• 600
• 500
• 400
• 300
• 200
• 100

100 200

Predictions of 1H NMR shifts in valid target hydride complexes

PBE0: 105.8 ppm (SO 107.3) PBE0-40HF:125.0 ppm (SO 155.4)

PBE0: 164.8 ppm (SO 98.0) PBE0-40HF: 187.9 ppm (SO 196.2)

PBE0: 100.7 ppm (SO 90.6) PBE0-40HF:86.9 ppm (SO 77.5)

Possibly, uranium(VI) hydride complexes had been made but were not identified. The search is on!

• P. Hrobárik, V. Hrobáriková, A. H. Greif, M. Kaupp
• Angew. Chem. Int. Ed. Engl. 2012, 51, 10884.

g-Tensors

B s free electron: e E ~ B g S g = 2.002319

Phenomenological Picture of the g-Tensor: Effective Spin Hamiltonian

molecule: B s experiment: G = g  g„ due to spin-orbit coupling! e E ~ B g S g-matrix l

RMC GC OZ SO

g g g g Δ Δ Δ Δ

/

  

g ge Δ 1 g  

g

B B S uv u v

S  

 

1

2

    E B

  ,

u,v x,y,z

HSO(1-el.) HSO(2-el.) HSO (other orbit)

           



n n u OZ n n v SO e uv OZ SO

c c E E Ψ H Ψ Ψ H Ψ S g g . 2

) ( ) ( ) ( , ) ( ) ( , ) ( 2 , /



dominant term

) r 2 r r Z ( 4

3 2

  

 

  

i j ij j iu i j ij i ju N iN iNu N e iu SO

g

3 3 ,

L L L H 

spin-orbit operator

g-Tensor: 2nd-Order Perturbation Theory

• 200
• 100

100 200 300

• 150
• 100
• 50

50 100 150 200 250 300

B3PW91

g (exp.) /ppt 

g (calc.) /ppt

slope 0.58, R = 0.981

• 200
• 100

100 200 300

• 200
• 100

100 200 300



g (calc.) /ppt

BHPW91

g (exp.) /ppt

slope 0.99, R = 0.956

improved slope by exact-exchange admixture but: potential problems with spin contamination

• 200
• 100

100 200 300

• 200
• 100

100 200 300

BP86



g (calc.) /ppt

g (exp.) /ppt

slope 0.40, R = 0.972

• J. Comput. Chem. 2002, 23, 794.

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

Hyperfine-Tensors

A-Tensor: The nonrelativistic first-order approximation

anisotropic contribution: the dipolar term isotropic contribution: the Fermi contact term

 

1 3 8

2





i iz iN z N e N e N FC

s r S c g g A    

     

3 3 1

5 2 ,



 

i iz iN v iN u iN iN uv z N e N e N uv D

s r r r r S c g g A   

difficult: spin density at the nucleus depends frequently on spin polarization easier: less dependent on spin polarization

A-Tensor: 2nd-Order Spin-Orbit Effects

N PSO SO N SO HC N D N FC N

A A A A A

/

   



z y, x, v u, A

S I N

 

  , 2E  v u N, uv N,

I S   

 

            

 n n N PSO i e SO z N e N e N uv PSO SO

c c E E H n n H S c mk g eg A . . 1 2

) 2 1 ( 4 2 2 , /

  

dominant second-order term (SO/PSO cross term):

contributions to:

• Aiso (pseudocontact Term)
• Adip (second-order anisotropy)
• Aanti (antisymmetric Tensor)

an accurate treatment of spin-orbit operators is mandatory!

Hyperfine Coupling Constants in Transition Metal Complexes

• performance of density functionals for metal HFCCs
• analysis of the problems
• mechanisms of spin polarization:

core-shell and valence-shell contributions

• M. Munzarová, M. Kaupp J. Phys. Chem. A 1999, 103, 9966.
• M. Munzarová, Pavel Kubáček, M. Kaupp J. Am. Chem. Soc. 2000, 122, 11900.
• spin-orbit contributions
• valence-shell spin polarization: spin contamination
• C. Remenyi, A. V. Arbuznikov,. R. Reviakine, J. Vaara, M. Kaupp J. Phys. Chem. A 2004, 108, 5026.
• A. V. Arbuznikov, J. Vaara, M. Kaupp J. Chem. Phys. 2004, 120, 2127.

Spin density rN

 at the metal nuclei, normalized to the number of unpaired electrons.

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

Performance of DFT for the calculation of isotropic hyperfine coupling constants

0.75 1.00 1.75 1.50 1.25 2.00 2.25

BLYP BP86 B3LYP BPW91 B3PW9 BHLYP BHP86 BHPW91 Functional TiN calc. TiN exp. ScO exp. TiO exp. VN exp. ScO calc. VN calc. TiO calc.

rN

/2S (in a.u.)

Performance of DFT for the calculation of isotropic hyperfine coupling constants

Performance of DFT for the Calculation of Hyperfine Coupling Constants: Effect of Spin Contamination

BLYP BP86 B3LYP BPW91 B3PW91 BHLYP BHP86 BHPW91 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.

rN

/2S (in a.u.)

Performance of DFT for the calculation of isotropic hyperfine coupling constants

Spin density rN

 at the metal nuclei, normalized to the number of unpaired electrons.

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

core-shell spin polarization dominates, no problem with spin contamination!

2r2(a.u) r (a.u.)

2s 3s 3d Watson and Freeman 1961:

• negative spin density at the nucleus.
• negative 2s-contribution larger than positive 3s-contribution

Spin Polarization in Transition Metal Ions

Theoretical Contact-Term r and Individual s-Shell contributions Ion Mn2+(3d5) Fe3+(3d5) Fe2+(3d6) Ni2+(3d8) r(a.u.)

• 3.34
• 3.00
• 3.29
• 3.94

1s-Shell contribution to r

• 0.16
• 0.25
• 0.21
• 0.27

2s-Shell contribution to r

• 6.73
• 8.51
• 7.80
• 9.62

3s-Shell contribution to r

+3.55 +5.77 +4.72 5.95

2s 3d exchange interaction - „attraction of parallel spins“

Analysis of Spin Polarization in Manganese Complexes

Contributions to Isotropic hyperfine coupling constants (Aiso, in MHz)a contribution complex 1 2 3 valence SOMO(s) total Expt.

2[Mn(CO)5]

15.5

• 262.8

135.8 60.7 61.8

• 12.1
• 2.8... 5.6

2[Mn(CN)4N]-

0.0

• 474.7

247.3 22.1 0.0

• 275.0
• 276

6MnO

3.0

• 362.9

184.5

• 141.5

824.9 507.5 479.8

6MnF2

0.0

• 406.3

194.3

• 86.2

512.3 214.1 104... 134

6[Mn(CN)4]2-

• 9.1
• 407.2

187.9 94.7 0.0

• 132.0
• 199

6Mn0

• 6.0
• 433.0

196.7 208.0 0.0

• 31.4
• 72.4

6Mn2+

• 6.2
• 450.4

197.0 0.0 0.0

• 260.0
• 273...-168

7MnH

• 2.6
• 347.8

163.6

• 194.9

709.3 329.6 279.4

7MnF

2.4

• 347.8

169.1

• 44.9

666.1 443.6 442

7Mn+

7.5

• 358.4

173.7 0.0 936.2 759.2 757.8

aB3PW91 results using standard metal 9s7p4d basis.

Core-Shell Spin-Polarization Contributions in Different Manganese Complexes are Proportional to the 3d Spin Population!

1.0 2.0 3.0 4.0 5.0 0.0

• 2.0

1.0 0.0

• 1.0

0.5

• 0.5
• 1.5

rN

 (a.u.)*

2s 3s

S metal 3d gross orbital spin population

*normalized spin density at the metal nucleus

• M. Munzarová, Pavel Kubáček, M. Kaupp J. Am. Chem. Soc.2000, 122, 11900.

Zero-Field Splitting

Zero-Field Splitting

For S > ½, already in the absence of magnetic field, 2S + 1 energy states split, due to: a) spin-orbit coupling (dominant for transition-metal systems) b) spin-spin coupling (dominant for organic radicals) fine structure in spectra For pNMR this affects the Boltzmann distribution of occupied energy levels.

Thank you!