NMR relaxometry of paramagnetic molecules Giacomo Parigi CERM - PowerPoint PPT Presentation

NMR relaxometry of paramagnetic molecules Giacomo Parigi CERM University of Florence

The paramagnetic contribution to relaxation t M t ’ M H t R H H t M H H t s O M e O O O H H + H H H H 2 O O t fast t M Bulk water H NH t D Bulk water t M t R m S m I 1 R 1 = R 1dia + R 1para m I 2 r 1 =( R 1 - R 1dia )/[Me] R 1para =[Me] r 1 m S =658.2 m I r 1 , relaxivity = paramagnetic relaxation rate due to 1 mmol/dm 3 paramagnetic centers

The paramagnetic contribution to relaxation Paramagnetic molecule Diamagnetic molecule Relaxivity 2 mM 2 mM (1 mM) 1.0 20 10 -1 ) 0.9 -1 ) 18 -1 ) 9 Proton relaxation rate (s Proton relaxation rate (s -1 mM 0.8 16 8 0.7 14 7 Proton relaxivity (s 0.6 12 6 - = 0.5 10 5 0.4 8 4 0.3 6 3 0.2 4 2 0.1 2 1 0.0 0 0 0.01 0.1 1 10 100 0.01 0.1 1 10 100 0.01 0.1 1 10 100 Proton Larmor frequency (MHz) Proton Larmor frequency (MHz) Proton Larmor frequency (MHz) m S m I 1 R 1 = R 1dia + R 1para m I 2 r 1 =( R 1 - R 1dia )/[Me] R 1para =[Me] r 1 m S =658.2 m I r 1 , relaxivity = paramagnetic relaxation rate due to 1 mmol/dm 3 paramagnetic centers

Nucleus-electron dipole-dipole coupling Fluctuations in the Hamiltonian couple each spin to the external world (lattice), thus allowing for energy exchanges - m +   r + + N m  S w S - - w =  B S S 0 w I - w =  B + through space I I 0 m S ,m I

Nucleus-electron dipole-dipole coupling - + R 1 M = w 0 +2 w 1 I + w 2 I w 1 w 0 + + The transition probabilities per unit time (for stochastic, stationary perturbations) are: w 2 w S - - - t t - t t t = =    / * / G ( ) G (0)e m | H | n n | H | m e w I c c mn mn 1 1 w 1 I - + If t c is the correlation time of the relaxation mechanism, MAGNETICALLY COUPLED for t < t c , there is a large correlation and G is large TWO-SPIN SYSTEM (DIPOLE-DIPOLE COUPLING) for t > t c , the correlation goes to zero t  0 0 E -  - w t - t t   2 i / H ( 0 ) c E w = t = J ( ) G ( 0 ) e d c   1 mn w = m n mn mn  w t 2 1 mn -  mn c  R 1  E dip 2 f ( t c , w )

For stationary perturbations: - t t - t t / / * t = =    G ( ) G (0)e m | H | n n | H | m e c c mn mn 1 1 and   - w t - t t i / e c  - w t - t t i / w = t = = J ( ) G ( 0 ) e d G ( 0 ) c mn mn mn - w - t i 1 / c -  -    t 1   = - = = c 2 G ( 0 ) 0 2 G ( 0 )   mn mn - w - t wt  i 1 / i 1   c c   2 t wt =   *   c c 2 m | H | n n | H | m i       1 1 2 2  wt  wt 1 1   c c

t 1  - w t i = t t W G ( ) e d mn mn mn 2  - t  w d -  - t i w = t t J ( ) G ( ) e mn mn    t wt w =   *   c c J ( ) 2 m | H | n n | H | m i       mn 1 1 2 2  wt  wt 1 1   c c   t 2 * =    c W m | H | n n | H | m     mn 1 1 2 2  w t 1    mn c

  1       dip = -      *   * H I S I S I S F I S I S F I S I S F I S F I S F    - -    - -   - - z z 0 z z 1 z z 1 2 2  4  A B C D E F - + 2 I w 1 F w 0 0 + + *   - -  - -  = w | H | | H | 0 1 1 16 w 2 2 F 1 - - *    -  -   = w | H | | H | 1 1 1 4 w 1 I - + 2 *    --  --   = w | H | | H | F 1 2 1 1 2  =   I | | z 2

3 3 2   = -  -i ( t ) = - 2  - 2 i ( t ) = -   F ( t ) k ( 1 3 cos ( t )) F ( t ) k sin ( t ) e F ( t ) k sin ( t )cos ( t ) e 0 1 2 2 4 2 m   I  = 0 S k  3 4 r 1 4 1  2 2 2 2 2 2 4 2  =  -  = -     = | F | k | 1 3 cos | k ( 1 6 cos 9 cos ) d (cos ) k 0 2 5 - 1 3 2 2 2  = = | F | | F | k 1 2 10 2 t 2 t 2 1 4 k k = c = c w 0 2 2 2 2 2 2  w - w t  w - w t 16 5 1 ( ) 10 1 ( )   S I c S I c 2   m 2  2  2 t t t   1 3 6  -   1 = 0 I S c  c  c T     1  6 2 2 2 2 2 2  w t  w - w t  w  w t 10 4   r 1 1 ( ) 1 ( )   I c I S c I S c

Nuclear relaxation due to the electron-nucleus dipolar coupling Solomon equation   2   m  m  t t 2 2 2   2 g S S 1 7 3 =   0  I e B c c R   1 M   w t  w t 6 2 2 2 2 15 4 r 1 1     S c I c 10 t c (s) 9 Spectral density ( t c units) 10 -10 8 10 -9 w =  B 10 -8 7 S S 0 7 J ( w S ) 6 w =  B 5 I I 0 4 3 2 Solomon, Phys. Rev. 99 (1955) 559 3 J ( w I ) 1 0 Bertini, Luchinat, Parigi, Ravera, 0.01 0.1 1 10 100 1000 NMR of paramagnetic molecules , Proton Larmor frequency (MHz) Elsevier, 2016

A B 0 Three times modulate the dipolar Hamiltonian: e N e N 1) Electron relaxation t s B B 0 e N e 2) Rotation t r N 3  4 a t = r 3 kT C B 0 3) Chemical exchange t M e e N N

Each time contributes to the decay of the correlation function: - - - - t - t - t = - t  t  t 1 1 1 exp( t / ) exp( t / ) exp( t / ) exp[ ( ) t ] s r M s r M - - - - 1 1 1 1  t = t  t  t c s r M t M t r t s s 10 -13 10 -7 10 -5 10 -11 10 -9

The paramagnetic contribution to solvent relaxation R 1 M If t M << 1/ R 1 M H H t M 1 = t s r f M R O M e O 1 M H H f M = mole fraction of ligand nuclei, in water: 0 . 001 q Bulk water t R f M = 55 . 6 q = number of coordinated water molecules r 1 , relaxivity = paramagnetic relaxation rate due to 1 mmol/dm 3 paramagnetic centers

Copper(II) aqua ion 3.0 298 K -1 ) -1 mM 2.5 Lorentzian dispersion 2.0 Proton relaxivity (s 1.5 1.0 R l.f. R l.f. =10/3 R h.f. 0.5 R h.f. 0.0 0.01 0.1 1 10 100 Proton Larmor frequency (MHz)

Copper(II) aqua ion 298 K 3.0 Best fit (with q =6) -1 ) -1 mM 2.5 r = 0.27 nm 2.0 Proton relaxivity (s t c = 2.6  10 -11 s 1.5 ( t s = 3  10 -10 s ) 1.0 0.5 0 . 001 q = 0.0 r R 0.01 0.1 1 10 100 1 1 M 55 . 6 Proton Larmor frequency (MHz)

Copper(II) aqua ion 4.0 Best fit 278 K -1 ) 3.5 -1 mM r = 0.27 nm 3.0 t c (278)= 4.0  10 -11 s Proton relaxivity (s 2.5 298 K 2.0 t c (298)= 2.6  10 -11 s 1.5 t c (338)= 0.9  10 -11 s 338 K 1.0 t M <1/ R 1 M  10 -5 s 0.5 3  4 a 0.0 t = 0.01 0.1 1 10 100 r 3 kT Proton Larmor frequency (MHz) B B 0 e N e N

The paramagnetic contribution to solvent relaxation R 1 M r 1 = f m (1/ R 1 M + t M ) -1 H H t M t s O M e O H H If temperature , times are faster: t r and t M Bulk water t R Since R 1 M  t c , R 1 M 35 298 K -1 ) - t M << 1/ R 1 M r 1 30 -1 mM 25 278 K Proton relaxivity (s r 1 - t M >> 1/ R 1 M 20 288 K 15 10 5 0 0.01 0.1 1 10 100 Proton Larmor frequency (MHz)

The paramagnetic contribution to solvent relaxation First and second-sphere contributions: 0 . 001 q t ’ M H =  -  t r i ( 1 / R ) H H t M 1 1 M , i M , i 55 . 6 t s O M e O i H H q = number of coordinated water molecules Bulk water t R - and t c,i = t c   2   m  m  t t   2 2 2 =  q 0 . 001 2 g S S 1 7 3  r i  0  I e B c c   1 6   w 2 t 2  w 2 t 2 r 55 . 6  4  15 1 1   i i S c I c - and t M,ss < t c,in       2 t t   m  2 2 m 2  t t 7 3 0 . 001   2 g S S 1 q 7 3 q     =   M , ss  M , ss     r 0 I e B is c c ss     1   w t  w t  w t  w t 6 2 2 2 2 6 2 2 2 2 55 . 6  4  15 r 1 1 r 1 1         is S c I c ss S M , ss I M , ss r 1 , relaxivity = paramagnetic relaxation rate due to 1 mmol/dm 3 paramagnetic centers

Effect of t M t M = 1 ns 0.1 ns 0.01 ns Proton Relaxivity (s -1 mM -1 ) 0.001 ns 0.0001 ns 0.00001 ns Proton Larmor frequency (MHz)

t s 0 for paramagnetic metal ions Cu(II) 300 ps VO(IV) 500 Ti(III) 40 Mn(II) 3500 t s < t r (30 ps) in aqua ions Fe(III) 90 Fe(II) 1 Cr(III) 400 Co(II) 3 Ni(II) 4 Gd(III) 120 (Low lying excited states make Ln(III) 0.1-1 Orbach process very efficient)