Structure and motions of molecules in liquids as determined by selective proton relaxation time measurements

STRUCTURE AND MOTIONS OF MOLECULES

IN LIQUIDS AS DETERMINED BY

SELECTIVE PROTON RELAXATION

TIME MEASUREMENTS

/

A , , , . i J , , •

m ^ ' * " ' * " * .

U^

W.M.M.J. BOVEE

/ O ^ »V^

try

?/

g = ^ ^ ^ = - a i i i ^ ^ t s r V

^

C10088

60346

STRUCTURE AND MOTIONS OF MOLECULES

IN LIQUIDS AS DETERMINED BY

SELECTIVE PROTON RELAXATION

TIME MEASUREMENTS

P R O E F S C H R I F T

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE

HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS IR. H. B. BOEREMA, HOOGLERAAR IN DE AFDELING DER ELEKTROTECHNIEK, VOOR EEN

COMMIS-SIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN TE VERDEDIGEN OP WOENSDAG

25 JUNI 1975 te 14.00 UUR

DOOR

WILLEM IViATHIEU MARIE JOSEPH BOVEE doctorandus in de scheikunde geboren te Sterksel

RODOPl N.V.

AMSTERDAM 1975

/SO 2. i^^cfj

BIBLIOTHEEK TU Delft C 886034

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROIVIOTOR

PROF. DR. IR. J. SMIDT

)

structure and motions of molecules in liquids, as determined by-selective proton relaxation time measurements.

Contents '*

Concise list of symbols and abbreviations T

Chapter 1 Introduction If'

Chapter 2 Theory of relaxation in liquids 1^

2.1 The Redfield density matrix treatment of relaxation 1^

2.2 The solution of the Redfield equations 19

2.2.1 The longitudinal relaxation 19 2.2.2 The transverse relaxation 21 2.3 Relaxation mechanisms 22 2.it Molecular motions and the calculation of correlation

functions 25 2.U. 1 Molecular motions 25

2.1*.2 The temperature dependence of the correlation times 29

2.1*. 3 The calculation of the correlation functions 30 2.U.U The choice of the diffusion tensor principle axis

system 32

Chapter 3 Application of the Redfield theory to the strongly

coupled AB and AB systems 33

3.1 The AB system 33 3.1.1 Intramolecular dipolar interaction between the spins

A and B (internal relaxation) 3^

3.1.2 External relaxation 35 3.1.3 The combined intramolecular dipolar and external

relaxation ^0

3.1.U Scalar relaxation '*'*

coupled AX and AX systems 1*. 1 The AX system

it.2 The AXp system

i*.2.1 Intramolecular dipolar interactions between the three spins of the AX system

U.2.2 External relaxation in the AX system it.2.3 Scalar relaxation

itr 1*7 it7 ItS 51 53

Chapter 5 Spin systems without coupling 55 5.1 Summary of some relevant literature formulae

for intrajnolecular dipolar relaxation 55 5.2 Intermolecular dipolar relaxation 57

Chapter 6 I n t r a - and i n t e r m o l e c u l a r r e l a x a t i o n in a s e r i e s of AB systens

6.1 Introduction 6.2 Results and discussion

6.2.1 Evaluation of the spin lattice relaxation parameters

6.2.2 The information derived from the internal and external spin lattice relaxation parameters 6.2.3 Some remarks about the experimental determination

and the accuracy of the initial rat^ constants

/ - I I t

6.2.4 Determination of N-H coupling constants in 2-iodo-5-nitrothiophene 59 59 61 6l 63 73 76

Chapter 7 Molecular geometry, rotational diffusion and hidden nitrogen proton coupling constants m

2-amino-pyrimidine as an example of an AX„ system 78

7.1 I n t r o d u c t i o n 78 7.2 R e s u l t s and d i s c u s s i o n 79

7.2.1 The e v a l u a t i o n of t h e r o t a t i o n a l d i f f u s i o n

con-s t a n t con-s and the m t e r p r o t o n d i con-s t a n c e con-s 79 7.2.2 The l a t i o of t h e m t e r p r o t o n d i s t a n c e s 8l

7.2.3 The rotational diffusion

7.2.i| Determination of the N-H coupling constants

81*

Chapter 8 Overall and internal motions m substituted benzenes as examples of spin systemb without

coupling 89 8.1 Introduction 89 8.2 Experimental results 90

8.3 Discussion 9'*

8.3.1 lethods of evaluating rotational diffusion constants

and correlation times 9it 8.3.2 The rotational diffusion 98 8.3.3 Hydrogen bond formation m t h e phenolic compounds 101

8.3.ii Reorientation of methyl groups about their

C , axis lOlt 3v

8.3.5 Reorientation of the methoxy group about the

aryl oxygen bond 113

Chapter The potential of selectively determined relaxation

times as an aid in structure analysis 116

Chapter 10 Experimental

10.1 Apparatus and techniques 10.2 Chemicals

121 121 122

Appendix 1 Intramolecular dipolar interactions between an

aromatic ring proton and an ortho CH-, group 12it

Appendix 2 Intrajnolecular dipole-dipole relaxation of an aromatic ring proton by an ortho methoxy group and the dipolar relaxation withm the methoxy

group 13lt References Summary Samenvatting 137 I l t 3 Ilt5

7

Concise list of symbols and abbreviations

A ^ spin operator a molecular radius

a. . . component of the dipolar relaxation rate in coupled three spin systems

C correlation coefficient D diffusion constant

E experimental activation energy F lattice operator

G ( T ) correlation function AG free enthalpy of activation AH enthalpy of activation H magnetic field strength H Hamilton operator h Planck's constant (6.6256 x 10~^ J s ) -h h/2Ti {1.051+5 X 1 0 " ^ J s ) I s p i n o p e r a t o r I m o m e n t o f i n e r t i a J c o u p l i n g c o n s t a n t J(to) s p e c t r a l d e n s i t y at f r e q u e n c y to K e q u i l i b r i u m c o n s t a n t k B o l t z m a n n ' s c o n s t a n t 1, m, n d i r e c t i o n cosines M magnetization P ( T ) correlation function p((j)'+(ji,T ) c o n d i t i o n a l o c c u p a t i o n a l p r o b a b i l i t y Q_ i n i t i a l rate c o n s t a n t

8

ele-nent of relaxation matrix gas constant (8.311+3 J/K.mol) entropy of activation

absolute temperature

longitudinal relaxation time transverse relaxation time dipolar relaxation time

relaxation times for tie external relaxation height of n fold potential barrier

transition Brooability

line m the HR 1 IR spectrum, transition between level 1 and J .

normalized snherical harmonic spin functions

gyromagnetic ratio viscosity

magnetic or electric dipole loment

permeability of vacuum (if' x 10 kg mA s ) friction coefficient

density matrix operator

density matrix operator in the interaction representa-tion

shielding constant correlation t m e or time

ratio of experimental and free rotation correlation time

9 AP 2 - a m i n o p y r i m i d i n e HR NltR h i g h r e s o l u t i o n n u c l e a r m a g n e t i c r e s o n a n c e In t h e p r e s e n t t h e s i s t h e d e f i n i t i o n of t h e m a g n e t i c moment was chosen a c c o r d i n g t o t h e S . I . s y s t e m of K e n n e l l y . I n o r d e r t o f a c i l i t a t e the c o m p a r i s o n w i t h l i t e r a t u r e d a t a , i n t e r n u c l e a r d i s t a n c e s are e x p r e s s e d i n Angstroms ( l 8 = 10 m ) , and t h e thermodynamic d a t a i n k c a l / m o l e (1 k c a l = it.lSlt k J ) .

10 Chapter 1 Introduction

Nuclear magnetic resonance (rSffi) spectroscopy has become a powerful tool to get information about the structure of molecules. A high resolution NMi-i (HR NMR) spectrum of molecules in the liquid state generally consists of a series of lines. The positions and relative heights of the lines associated with a certain nucleus are in principle determined by the following parameters: 1) chemical shift 2) coupling constants and 3) relaxation times.

\ . . . .

->-1} Chemical shift. The presence of an external magnetic field H induces electronic currents in matter, giving rise to a secondary field at the nuclear site. As a result the effective field H' at the nucleus will be given by

H' = H ( 1-a) (1.1) o

0 is a dimensionless parameter, known as the shielding constant. It IS independent of H and a sensitive probe for the electronic and chemical environment of the nucleus. In fact a is a tensor, but in liquids only the average value is observed.

2) Coupling constants. There are two kinds of coupling between magnetic nuclei. The first one is the direct dipolar interaction. In liquids it normally averages to zero and contributes only to the relaxation, as will be shown in chapter 2. The second one is a scalar coupling, acting via the electrons in the molecule. The associated coupling constant J depends on the electronic structure of the molecule and on the gyromagnetic ratios of the nuclei involved. This so called J-coupling causes a splitting up of the chemically shifted lines in a HR HMR spectrum into multiplet structures. For a more detailed discussion of the parameters o and J, and the HR TM spectroscopy, the reader is referred to one of the text books on this subject.

magnetic field H , after some time it will be in thermal equi-librium with its surrounding (the lattice), resulting in a paramagnetic equilibrium magnetization M , parallel to H . Spin lattice relaxation returns M , the longitudinal component of the magnetization (the component parallel to H ) to its equilibrium value H , after any perturbation. Such a process requires a flow of energy between the spin system and the lattice in order to restore a thermodynamic equilibrium distribution of the spins over the Zeeman energy levels.

The spin-spin relaxation process returns any magnetization transverse to H to its equilibrium value of zero. This process disturbs the phase-relationship between the spins.

The relaxation of the longitudinal and transverse magnetization may under certain conditions be described as first order decays with time constants T and T , the longitudinal, resp. the transverse relaxation time. In these cases the T value of a certain line in the spectrum is related to the linewidth at half height, Atoj^, by the relation;

T = - ^ ( 1 2 )

2 AoJi ^^•'^'

while the height of the line is proportional to T„.

The relaxation is caused by fluctuating local magnetic (and electric) fields. The fluctuations arise from the random thermal motions of the molecules. Therefore T and T contain information about the rate of the fluctuations (molecular dynamics) and about the magnitude of the local fields (structural information).

The chemical shift and coupling constants determine the positions and relative heights of the lines associated with a certain nucleus in HR NiyiR spectra if the linewidth is zero. Longitudinal and transverse relaxation effect the linewidth and shape. The relaxation times can, however, seldom be determined from the HR NMR spectrum, as the line-width due to inhomogeneities in the static magnetic field generally

12

exceeds that due to the relaxation. Therefore in practice the "elaxatiop times ar^ determined oy special techniques.

..p to a few years ago the chemical shifts and tie coupling constants were used on a very larg° scale by tne cnemists to obtain structural information from H-IR spectra. Not much attention was paid to the r=l=LXation times T and T , while the information extracted from them was mainly associated with molecular motion, not with structure. This low interest in T and T as compared to the chemical shifts and coupling constants can be understood, because the relaxation time measurements up to then had a low resolution, only a single average relaxation time .^as generally found, no matter how complex the HR N'ffi spectrum «fas. In the last years techniques have oecome available, which allow the selective determination of the relaxation times of the individual lines m HR NI-IR spectra. Tiese selective measurements greatly enhance tne total number of experimental parameters that may be extracted from the HR N14R spectrum. The insight is growing that this improves the usefulness of 1^ and Tp as aids to get detailed information about molecular motion as well as structure. Selective T^ measurements have almost exclusively been performed on C HR NMR spectra, while selective Tp measurements have received almost no attention, due to the many experimental difficulties which are involved. As to protons only very few selective relaxation studies have been published.

The aim ol the research reported m this thesis is to g a m insight into the information that can be extracted about molecular structure and motion from selectively determined proton T and T values. To achieve this aim, the Redfield theory of nuclear magnetic relaxation, presented m cnapter 2, is applied to derive explicit expressions for the longitudinal and transverse relaxation of the individual lines m a number of types of HR -Ji4R spectra. These expressions are derived m chapters 3, it and 5. In chapters 6, 7 and 8, the results of the preceding chapters are used to determine from the experimental relaxation rates of the lines in HR NMR spectra of a number of

compounds rotational diffusion constants and correlation times. These give information about the rate and the anisotropy of the molecular motions, and about the rate of and the energy barrier opposing intra-molecular motions. Moreover information about the geometry of the molecules studied and about coupling constants not visible in the proton spectrum is obtained. In chapter 9 a discussion is given about the potential of using selectivily determined relaxation times as analytical tools in the way this is done with chemical shifts and coupling constants nowadays.

lit

Chapter 2 Theory of relaxation in liquids

2.1 The Redfield density matrix treatment of relaxation Consider an ensemble of identical molecules, containing; magnetic spins, in a static magnetic field H . Taking into account onlv magnetic interactions the Hamiltonian for each molecule can be written as:

H = h H + h H(t) (2.1) o

H consists of a Zeeman and a spin-spin coupling term. H is time o ^ ^ ^ " o independent, it is the same for all molecules of the ensemble, and it has eigenvalues and eigenstates denoted by to., u resp.

li>,

|J>

H(t) represents the interaction of the spin system with the lattice. H(t) differs from molecule to molecule. It is a stationary

perturbation which is random in time. The ensemble average of H(t) is zero. At the field strengths normally used in magnetic resonance experiments, the following relationship is valid:

H(t) << H (2.2) o

The fourier components of the matrix elements of H(t), at the various resonance frequencies of the system cause relaxation. The frequencies and relative intensities at infinite small line widths of the lines in HRNMR spectra of liquids are completely determined by H . To describe lineshapes and in order to describe how after a disturbance the spin system returns to equilibrium, due to the interaction H(t) with the lattice, a density matrix treatment is in general necessary. The density matrix theory of

1 1 2 3 magnetic resonance, developed by Wangsness , Bloch ' , Redfield

it

and Abragam enables one to calculate the spectrum as well as the longitudinal and transverse relaxation rates of the individual lines in the spectrum. From the equation of motion of the density

3

p.. = - i(o,.p.. + s' R..,^ (p,, - p | ° ^ (2.3) ij ij ij ijkl kl kl

K,l

P. . is an element of the density matrix P in the laboratory

"^•^ (o)

frame, p.. is its equilibrium value; to., (to.. = to. -to.) is the ij ij ij 1 J

transition frequency between levels i and j , which corresponds with a line in the spectrum. The prime in equation (2.3) means that the summation is limited to those terms for which

CO. . = to, , (2.it) ij kl

R. ., , is an element o f the relaxation m a t r i x : ijkl

R..,^ = l{2J...Au.^) - 5., S J (to , ) - 5. I J . ^(to J } ijkl ikjl jl jl n nkni nk ik n njnl nl

(2.5) with spectral densities given by

-110 T

J..^,(to ) = I <i|H(t)|j><k|n(t+T)|l>* e ""^ dT (2.6) 1jkl mn

The bar in the right hand side of equation (2.6) means an ensemble average.

In the derivation of equation (2.3) the lattice was treated classically. Its quantum mechanical treatment leads to the same results ' , if eq. (2.6) is replaced by

J..^Jto ) = f Z Z

_{ijkl m n L J ' iv vi i} < 9 | F ( ^ ' | J . > < ^ | F ( I ' V >

e ''^

Q'j) qq' -"

<1^ < i | A ( ^ ^ | j > < k | A ^ ^ ^ | l > e " ^ dT (2.6a)

L is the number of eigenstates of the lattice, |6> and |i|j> are

lattice eigenstates. The spin operators A and the lattice operators F are defined in section 2.3. T . is a correlation

16 time defining the motion of the lattice.

The diagonal element P.. of the density matrix Is the probability that state i is occupied. A.s the longitudinal magnetization, associated with a non degenerate line Y.. in the spectrum, is

t-J

proportional to the population difference between states i and j , the longitudinal relaxation of Y.. is determined by the time deoendence of (p.. - p . . ) , which can be determined by solving

11 JJ

eq. (2.3). According to eqs. (2.3), (2.5) and the restrictive condition (2.it) one finds:

P.. = J:' R..,^ (P^^ - p[°^) (2.7)

11 , iikk kk kk

k

iikk kkii ikik ik

This shows that in the case of non degenerate energy levels of H , as to the longitudinal relaxation, eq, (2.3) reduces to a simple master equation for populations. The element R--, , is equal to the transition probability W-^ tistween states i and k, induced by the interaction with the lattice. Comparing eq. (2.7) to a master equation for populations shows that R.... is the total probability that state i is depleted by transitions to other levels, which means

R = - S R,, . . (2.9)

1111 , ,. kkii kfi

Eq. (2.7) shows that the longitudinal relaxation must in general be described by a sum of exponential functions of the time. If H has degenerate energy levels, p, q, etc, the longitudinal relaxation is no longer described by a simple master equation, as owing to eq. (2.it), terms like R.. (p - p ) play a role.

iipq "^pq pq

The off-diagonal elements of p are zero at thermal equilibrium, A R.F. field with a frequency near or equal to co. . may stimulate

1J

created, wnich is proportional to p... The relaxation of this

magnetization is determined by the time dependence of p... Eqs. (2.3)

1J

(2.i+) and (2,5) give for a non degenerate line Y. .:

p. . = (- ico. . + R. .. .)p. . (2.10)

ij ij ijij iJ

R.... = g{2J...-(o)-J;j . . ( t o . ) - E J . .(to .)} (2.11) ijij iiJJ ^ nini ni ^ njnj nj

which shows that the transverse magnetization associated with Y..

ij

decays to zero in the form of a damped oscillation with frequency

ui. .. There is a single time constant R. .. ., the transverse relaxation

ij ^ ijij

rate of line Y. .:

ij

R. .. . = - T " L ' (2.12)

ijij 2ij

To gain some insight in the physical meaning of the transverse relaxation it is assumed for the sake of convenience that the following holds for the correlation function P.. , ( T ) in the right

ijkl

hand sight of eq. (2.6):

P..J^-^(T) S <i|H(t)|j><k|H(t+T)|l>*

-T/T

:i|H(o)|j><k|H(o)|l>'' e '^ (2.13)

which means that P. ., , ( T ) is independent of the time in the

ijkl

statistical sense and that it decays exponentially to zero, T is the correlation time, characterizing the molecular motion. It is also assumed that the extreme narrowing approximation holds, which means

to..T << 1 (2.llt) IJ c

Although both assumptions are frequently met in ;'IRR in liquids, in section 2.it.3 a more extensive treatment will be given. Under these assumptions the elements of the relaxation matrix become:

18

-^iikk = '"cl'^l^^°^l''''" (5.15)

-\jij " " '"2ij = - •^(,0<i|H(o)|i>-<j|H(o)|j>|' +

+ X R . . + S S . .} (2.16) ,. nnii ,. nnjj

n?i n^i '^^

The first term in the right hand side of eq. (2.l6) represents the fluctuations in the distance between the levels i and j , and consequently in the frequency to., of the spectral line Y... These

i j - i j

fluctuations are caused by the perturbation H(t), and result in a line broadening. The second and third terms, the transition probabilities from levels i and j to all the other levels in the system are related to the longitudinal relaxation. They determine the lifetimes of i and j , and cause a broadening in accordance with the Heisenberg uncertainty relations. This gives a second contri-bution to the linewidth of Y.•. From this discussion the relation

i j

(1.2) between the transverse relaxation time and the linewidth can be understood.

The validity of eq. (2.3) and the assumptions made in its derivation it . 5

are discussed by Abragam and Slichter . The reader is referred to these books for an extensive treatment of the density matrix theory of relaxation.

Two remarks can still be made with respect to the validity of the Redfield theory. It is a weak interaction theory. If relaxation takes place through strong interactions, each of which changes the state of the system by a large amount, a different approach is needed.

In the theory two time scales are present. One is connected with the relaxation of the macroscopic magnetization (the elements of the relaxation matrix), the other one with the molecular motion (T in

c eq. (2.13)). The theory is only valid under the following condition:

T,-« T^. Tg

In non viscous liquids one finds

10-^2 , , , ,,-8 ^ c

-It 3

10 < T^, T < 10 s

so condition (2.17) is easily met in these liquids.

2.2 The solution of the Redfield equations '^.2.1 The longitudinal relaxation

Eq. (2.3) consists of a set of coupled linear differential equations. The general solution of eq. (2.3) for the longitudinal relaxation of a line Y. . in the s-oectrum is given by

ij " o ..

a -X t

p.. - P • = >: {A, (i) - A, (j)} e -' (2.18) 11 JJ j^^i 1^ K

a = 2 -1 if only interactions between the n magnetic spins in a molecule are considered, else a is larger. The amplitudes A,, are determined by the initial conditions of the experiment, the '*'v's by the elements R..,, of the relaxation matrix. As will be sho-.m in

iikk

section 5.1 there are cases in which the longitudinal rela.xation is very well described by one time constant, since all of the A (i) except one, are negligible. In general, however, the recovery of the longitudinal magnetization is multi exponential, and it is very difficult to extract the several \, 's from the exDeriment,

k , b

This problem was tackled by Freeman e,a, , who showed that the initial section of the magnetization recovery curve can be

approximated by one time constant in the interval from 0 to t seconds if

(R..,. - R..^_,)^t^ « 1 • (2.19) iikk n i l

for all k.l ^ i . B.. . , and R. .,, are in the same order of iikk n i l

magnitude as T~ , so eq, (2.19) means that the measurements must be confined to such a short interval that no appreciable relaxation

20

;a;:en -olace. Freeman lefmed

(Y '' ) =

( 0 . . - P . . - p . . - p . . , 11 11 JJ JJ

(2.29) t=T

which is the initial recovery rate of the longitudinal magnetization of line f. ,, after its perturbation by a selective ^ pulse which inverts the s-cin -jonulation across line Y, , . Substituting (2.7)

• - kl into (2. 20) gives : 2. (Y. ., -o i j ' 5:(R.. - R.. )(p - p^°^) n 11nn j j nn nn nn kl (p.-JJ

(p5°^-P^.°))

11 JJ (2.21-) t = 0

The values of the diagonal elements of P at t = 0 (immediately after the TI pulse on line Y , which disturbs the equilibrium magnetization of Y..) depend on which line Y is inverted. This

ij kl

may be the line Y.. itself, or one or more of the other lines m the spectrum. For the details of this so called "initial rate approximation'', the reader is referred to the paper of Freeman . In section 6.2.3 a discussion will be given of errors which may be introduced by the initial rate approximation.

The physical meaning of (2.2J) and (2.21) is that by perturbing the lines in the spectrum in a suitable w a y , and determining the tangent at the magnetization recovery curve immediately after the perturbation, the several transition probabilities ^--j^v <^^i be found. The R..,, contain the information about the molecular

iikk motion and structure.

If the observed line Y is degenerate, e.g. if it consists of two overlacoing lines Y.• and Y , it can easily be shown with the

-' ^ ij m n •'

initial rate approximation theory thit the initial recovery rate is given bv

if the two terms on the right hand side of (2.22) do not differ too much.

2.2.2 The transverse relaxation

The transverse relaxation of a non degenerate line in the s'lectrum can oe described by one exponential function of the time, as immediately follows from eq, (2,10). The transverse relaxation rate in this case is defined in eq, (2,12).

If in the spectrum lines coincide or overlap, eqs. (2.3) and (2.it) must be modified. The resulting equation for the density matrix

is most elegantly written using the interaction representation. The equation of motion of 'p , the density matrix in the interaction

3 representation (in the rotating frame) is now given by

i ( w . • - CO, , )t , V

P**. = E e "J ^1 R.. (p* - p^^°') (2.23)

^ij ijkl^'^kl ^kl ' ^ ^ '

k,l

Instead of tne condition (2.it) the summation over k and 1 in (2.23) is restricted to those terms for which (co. . - co, ^ ) is less than a

ij kl

few times R. ., , .

ijkl

In section it.2.3 a situation will be met where two lines Y. . and

ij

y. , , having the same intensity, are exactly degenerate..From (2.23) it follows that the following equations must be solved:

p*'. = R p**. + R. ., , p!*, i j I J I J i j i j k l k l \ l = ^ k l i j Pij ^ \ l k l \ l ( 2 . 2 i t ; I n g e n e r a l R. ., , = R, , . . . The t i m e dependence of t h e t o t a l i j k l k l i j t r a n s v e r s e m a g n e t i z a t i o n of t h e two d e g e n e r a t e l i n e s f o l l o w s from e q . ( 2 , 2 i t ) : X t A t P i j ^ Pkl = ^1 ^ * S ^ ( 2 . 2 5 )

22 where

\,2

= ^(^ijij " «klkl) ± ^

'^^hiij -

\lkl)'

'

^«?jkl

With the initial conditions (after a 9^ nulse)

p**. + p"*^ = 2M ij kl o p'*. + p* - (R. .. . + R + 2R. . )M ij kl iJiJ klkl ijkl o (2.26) one finds M -2A^ , + R + H, „ , + 2R. ., ., C - + ° 5'^ ^'l^'l ^-^^^ ^•l'^^ (2 -^T) 2.3 Relaxation mechanisms

The information about molecular motion and structure, to be extracted from relaxation time measurements, enters the elements of the

relaxation matrix (see eqs. (2.5) and (2.6)) via the term H(t). H(t) represents the interaction between the spinsystem and the lattice or thermal bath. It is now appropriate to introduce the individual relaxation mechanisms which may be contained in the term H(t). Each mechanism can in general be described as

H(t) = I (-l)'l F(^'(t) A^"^^ (2,28)

with

p(ajs ^ (_i)%(-l) and

2(4)+ = (_l)^.4(-^)

The operators A only act on the spinsystem, the terms F (t) are stationary random lattice parameters. Due to the rapid molecular

motions in liquids they are time dependent, thereby providing a process for longitudinal and transverse relaxation (compare eqs. (2,15) and (2,16)), since by the fluctuations of the F '^ (t) energy exchange between the spinsystem and the lattice is possible. In table 2,3.1 the most important relaxation mechanisms in liquids except chemical exchange are summarized.

mechanism modulated by

i-'dipolar interaction A, B, C i

.spin rotation interaction A, B ;chemical shift anisotropy A, B quadrupole interaction A, B

jscalar coupling A, B i

Table 2.3.1 Relaxation mechanisms for TIR in liquids.

A, B and C define the following kind of molecular motions; A, intra-molecular; B, molecular rotational; C, molecular translational,

In this thesis only proton relaxation times are considered. The relaxation processes of importance in this case are dipolar and scalar interactions "between magnetic spins, Eq. (2.28) may then be written as:

H(t) = Z E (-l)'l F^'l^t) A^T^' (2.29)

i<j <1 '' ''

The summation i<j is over pairs of interacting spins i and j . Written in this way both F and A transform exactly like the normalized spherical harmonics.

For the dipolar interaction one finds:

Ff-^^t) = - / ^ ''-^ Y ^ ) (e.., *..) (2.30) •ij - 5 i^^^^3_ 2 ij' ^ij'

2it

(2,31:

l\°.^ =

3 I (i) T (j) - I(i).l(j)

l[l'^ =^~

{^(i) I,(J) + I,(i) I,(j)}

A;f' = ^ i , ( i ) i . ( j )

(q) . . .

The functions Y^ are the normalized spherical harmonics of the second order as defined b^' Rose . 9. ., ^. . and r. . are the polar

ij ij iJ

coordinates of the internuclear vector ?. • in the laboratory frame,

Intra and overall molecular motions make them time dependent,

In the case of scalar interactions between s'jins i and j , the H£imiltonian

H(t) = J.J I(i).l(j) (2.32)

can be made ti:ne dependent in two ways. Firstly the coupling constant J.• may be time dependent due to inter- or intra molecular exchange, with a time constant T much shorter than 1/J. . and the T, value

e ij 1

of either spin. Secondly, one of the spins, say i, has a relaxation t •-••.•- due to a relaxation mechanism other than the coupling with spin j, much shorter than 1/J^. . and the T value of spin j. Because of

1J 1

their short relaxation time, the spins i are after a disturbance very fast in equilibrium with the lattice again. In fact they are

• it ' , V

considered as a part of the lattice . The spmoperators I(i) fluctuate rapidly in time, making the coupling with l(j) time dependent. Only the second mechanism is met in this thesis. The F and A of eq. (2.29) can be written in this case as

F[°^(t) -lum) ; [ ; ' = j , . V j )

(2.33)

/2 ^J

F|r'(t) = . _ — - A^.- = . . V J )

""or an extensive description of the several relaxation mechanisms, It

•^. H Molecular motions and the calculation of correlation functions ^,U,1 Molecular motions

In order to determine the spectral densities (eq, (2.6)),the time depence of the correlation functions must be known. Substitution of (2,?Q) into (2,6) shows that the correlation functions

4i(^)=4f(^)-l^^''(-^) (2-3'*)

must be evaluated. They depend on the molecular motions which can be characterized by correlation times i , defined by

F[f{t) Fl^^*it^T)

di

^ ••vn = '^ (2.35) Cljkl —7 V 7—\

As translational motions do not play an important role in this thesis, the discussion will be restricted to molecular reorientations. These are often described by the Langevin equation

3to. -S.to. + N. (t) 1

CO. IS the angular rotational velocity of the molecule, E, • is the rotational friction coefficient, and I. is the moment of inertia,

1

all about axis i. N.(t) is a randomly fluctuating torque arising from interactions with the environment. Two limiting cases are often used in W'G to gain information about molecular dynamics from the relaxation times.

1. 5- ^^ N.(t), the inertial rotational model. In this model the motion of the molecule is essentially free as in the gas phase and it is modified only by collisions. This model is expected to be valid for

light, highly syiranetrical small molecules in a liquid where the intermolecular interactions are very weak. It gives a good description

Q

26 time, using eq, (2.35)

^fi=f4)'

^^-^'^

2. N.(t) << 5-;,the rotational diffusion model. It describes the rotation by random small angle jumps with many jumps required for the molecule to reorient by one radian. The motions are determined by frictional forces as opposed to inertial ones in the inertial model. The reorientations can often be described in terms of the three principal diffusion constants D , D and D of the

•^ xx' yy zz

rotational diffusion tensor in the diffusion principle axis system. The diffusion constants are defined by:

(Aa.)" = 2D..1 (2.33)

1 1 1

i = X, y, z. (A9.)^ is the net mean squared angular displacement in time x, about axis i. By means of eq, (2.35) one finds for the relation between the diffusion constants and the correlation times ^Di =

V = 1(1.1)D.. (2.39)

1 1

1 is the order of the spherical harmonic of interest in the ?.. (t) in eq, (2,29), For the diffusion model to be valid the relation must hold:

N^(t) « S- (2.it0)

Most times the condition (2.1t0) is met for larger molecules in solution. In cases where this relation does not apply strictly, the rotational diffusion model is used nevertheless (for reasons explained below), although then it gives only an approximate description of the .molecular reorientation. In the inertial model

Gaussian, in the diffusion model it is exponential. Because m non viscous liquids the extreme narrowing condition (2 lU) is alvays valid, one cannot extract from the N"IR spectral density (2.6) tie explicit time dependence of G ( T ) in this case. One only dete">-nines

G(T)dT and ootains no decisive answer about /lether one deals rfith o

the rotational diffusion model, the inertial model or some inter-mediate model. As the rotational diffusion model gives an adequate description of anisotropic molecular reorientations, taking into account interactions with the surroundings tnat are not included m the inertial model, the former model is most times used, oreo/er it can give useful information. The experimentally determined D depend on intermolecular interactions as jell as or tie moments of inertia of the molecule. Iltnough tneir absolute values nay be wrong, from the relative values of the experimental D for molecules under different conditions of solvent and temperature msignt can be gamed m the microscopic details of the liau_d state. \ery specific interactions, e.g. hjdrogen bonding, will

result in a relative change in one or miore oi tne diffusion constants. Attempts have been made to relate the diffusion coi^stants to

molecular parameters. In the hydrodynamic description oy Stokes of a fluid cpmposed of ellipsoidal molecules, t-ritn axeo a, o, L m a medium of viscosity n, D is related to tie stokes frictj.cn

1 1

coefficient C m t i e following .ray

C = Sirnx^ ( 2 . 1 t l )

X

D^^ = 1 ^ (2.it2)

X = a, 0, c. Tnis equation is very approximate, leading to diffusion constants which are 3 - 1 0 times too small, compared to the experimental ones. Corrections have been introduced ( m the case of isotropic reorientation) m the form of mut ^^ai viscosity

9 10 effects or of a microviscosity factor , whicn enlarge the diffusion constant by a factor of about 3, resp. 6. Calculai-ions using the mutual viscosity model of Hill give in general the best results. Another attempt to relate the diffusion constants to molecular

28

paraj'-cters was ;r.ade by 'tuntress , -'.''lo derived the follo'.'ing r e l a t i o n

( T . ) . k T

D... = ' - " ( 2 , i t 3 )

i = X, y, z, '• is the angular /elocity correlation time,

Solutions of eq. (2.36j for .models more general than the limiting rotational diffusion and the inertial model have been given

and the calculated correlation times were related to intermolecular potential functions and other microscopic quantities. "Jp to now these theories have found no practical applications. The reader is

referred for a review on this subject to references 11 and 1',

To examine whether the molecular motion is in the inertial or diffusion region, some tests may be done.

1. The X test ' compares the correlation time determined by the NMR experiment and the free rotation correlation time. If the rotational diffusion model applies this ratio must be much larger than 1. 'Jsing eqs. (2.37) and (2.39) one finds in this case

31(1+1)D. ( T - ) " (2.1*M

11

The experi.mentally determined D.. value is substituted in eq, (2.l|lt) and if x,- ^* 1 (x,- * 5 according to reference 13), many random jumps are needed for the molecule to reorient over 1 radian, and the motion is in the diffusion region,

2, The temperature dependence of the tv;o models is different as will be shown in the next section,

3, In the rotational diffusion region the following relation is found between the correlation times derived from NMR and dielectric

l i t , , ,,

relaxation time measurements (compare also (2.39))'

T . . = 3 T ('.it5) dielectric ^ T R ^-.t?;

The proportionality factor varies from 3 to 1 if the jump angle with which the molecule reorients varies from 0 to 120 , which is

equivalent to saying that the molecular motion goes from the diffusion to the inertial limit.

h. From infra-red band shape studies ' time correlation functions can be determined, which provides another test to discriminate between the two models.

If one wants to apply the tests 3 and it, one has to recourse to techniques other than the NMR mehtod.

2, it. 2 The temperature dependence of the correlation times From eqs, (2.39), (2,itl) and (2,lt2) follows for the rotational diffusion correlation time,for the sake of convenience denoted by

T = ^ (2,lt6) c T

For the inertial limit eq, (2,37) shows;

^rvr (2.it7)

so the rotational diffusion is much stronger temperature dependent than inertial motions,

In the case of diffusion the molecular motions, characterized by T , can be considered as a thermally activated process, and using the Eyring rate equation one finds :

AH^

(2.it8)

k is Boltzman's constant, f is the frequency factor (usually taken equal to unity), AG', AH and ASl" are the free enthalpy, the enthalpy, and the entropy of activation respectively, A plot of In 1/TT versus 1/T should give a straight line with slope and intercept AH' and AS' respectively. In practice the temperature dependence of the molecular motions is often described by the Arrhenius equation : 1 T

Aa+

kTf kT = e = _h hS^ ART kTf k kT = e e _h

30

V ^ ^ (2,it9)

T = T e

C CO

The activation energy E and the pre-exponential factor T can be obtained in a similar way as above by plotting In T versus 1/T, It should be noted that the pre-exponential factors in eqs. (2.lt8) and (2.it9) may be considered as rates, respectively correlation times of reorientation for zero barrier AG', resp. E .

2.it. 3 The calculation of the correlation functions

The measurements reported in this thesis were performed on medium sized polar molecules for which the rotational diffusion model is expected to be valid, so we restrict ourselves to this model in the calculation of the correlation functions. Papers about anisotropic rotational diffusion have been presented by several authors. Reference 11 gives a review on this subject. In this thesis the formulation of Ivanov and Hubbard is used, who treated the theory of rotational Brownian motion for molecules of arbitrary shape as a rotational random walk problem. They obtained as a special case, in the limit of small angles of rotation, the diffusion model.

The only correlation functions needed in the next sections are those concerned with dipolar interactions. Eqs. (2,29), (2.30),

(2.5) and (2.6) show that terms like

Y ( ^ ' ( 9 ^ t) Y'"^' '^(9 i t+T)

must be evaluated. From Ivanov's results the following expression can be derived;

P —X T

9. ., if. . and 9, , , ^, , are the polar coordinates 9 and C> of the 3 J iJ^. 1^1' kl

vectors r.. and r,, in the laboratory frame. The primed quantities are these coordinates with respect to a coordinate system fixed

in the molecule in such a way thaL the rotational diffusion tensor

"*• "*•

is diagonal. If no internal motions are present r. . and r, .,

•^ ij kl

are fixed in the molecular, but fluctuate in the laborarory coordinate system. The values of A are given by

2 ,^ ^2,0 = ^22 \ , t ^ = ^21 ' ^21

^2,t2= ^(^22 ^ ^ 2 0 ^ *

i^^^22-^20^'^^4o^'

in which 2(^2.o)mn= ^ ( ^ m 2 - V - 2 ' ( * n 2 - * n . - 2 ) 2(^0 4.1) = 5(6 , ± 6 J(<5 + « ,) _{2,±1 mn ml m,-1 nl n,-1} (2.51) b„„ = 2D_^ + itD a„„ = /6 D 22 + zz 22 -^21 = 5D^ ^ \ z -^21 = 3 D_ (2.52) ^20 = ^°+ ^20 = ^^ °-D, = ?(D + D ) ± ' XX yy

The matrix elements Z(A^ ) , are given by 2,r nn'

32

2,±2 mn 2 ' \i:^ tiio 20 :i ra,-2

•t('^22-\o,.2'^o-N/^2^V2^^'''%^

,, , x2, (2.53) ("22 - ^2.12^ ^

With the expressions ('\50) to (2.53) it is possible to calculate the necessary auto- and crosscorrelation functions (2.3't) in a convenient ".'ay.

In the special case of axiaJly symmetric rotational diffusion (D = D ) the sum over r in eq, {'^.3'^) reduces to

XX yy

2 -^T T -b^ T

E e -•'' -,(\-, ) , = .5 , e -" (2,5na) ^,r nn' nn'

r=-2

Eq, (2.50) will be used in section It,"^,!, eq. (?.50a) in appendix 1 and 2,

2. it.it The choice of the diffusion tensor principle axis system Huntress discussed this subject and his results will be

summ8,rized here. For a symmetric rotor the principal axis systems of the diffusion and of the moments of inertia tensors must by symmetry be coincident . ^^'or asymmetric rotors with sufficient symmetry this is also true. For asymmetric rotors with very low symmetry, the orientation of the diffusion tensor principle axis system is in general unknown, as it may be rotationally shifted from the principal inertial axes by intermolecular interactions. In such a case eq. (2.50) must be modified to take into account the orientation of the diffusion principal axis system with respect to the molecular reference frame, and the additional diffusion constants D , D and D enter the problem. An expression for the correlation

xy xz yz

functionL- in eq, (2.50) is in this case not readily soluble in closed form.

Chapter 3 Application of the Redfield theory to the strongly coupled A.B and AB system

In chapters 3, it and appendices 1 and 2 the results of sections 2.1 to 2.it.it will be used to derive explicit expressions for the proton longitudinal and transverse relaxation rates of the individual lines m HRNMP spectra, m terms of diffusion constants, correlation times, scalar coupling constants and internuclear distances. Spin systems will be considered with strong, weak and no coupling between the spins in chapters 3, it and both appendices respectively. If in a spin system the ratio of the coupling constants and the chemical shifts is of the order 1, the system is called strongly coupled, If this ratio is much smaller tnan 1, one deals with a

weakly coupled system. Strong coupling causes a mixing of the Zeeman states, the system would have if no coupling was present. This mixing influences the relaxation transition probabilities, lesultmg in different relaxation rates for the components of a multiplet of a certain nucleus m the spectrum. In the weak coupling limit these components have the same relaxation rates, which means that less information can be gained m this case. In weakly coupled spectra, however, the relaxation rates can be determined much easier.

3.1 The AB system

^ne hamiltonian H ol eq. (2.1) for an AB system m an external static magnetic iield H along the positive z-axis of a laboratory frame, may be written as

H = -^.I - to I + J I -I (3.1) o A z. ^ ^R AB A B

^A=^"o(^-^A^' - B = ^ " O ( ^ - " B ) (^-^^

Y IS the proton gyromagnetic ratio.

The eigen-functions and values of this system are given in table 3.1.1.

3lt , l e v e l

r

1

I

: 2 ' 3 It wave f u n c t i o n |aa> cos9|oiS> - s i n e ] Sa> s i n 9 | a 3 > + cos9 | Soi>

|ss>

e n e r g y ( s ) - \ ' ^AB/'* - ^AB/^ - = - ^AB/^ " = ^o ^ JAB/"*

Table 3,1.1 The wave functions and energy levels o f the AB system.

"o = '(co,^ + (^g).

(3.3) WA^ + J^

AB tg29 = J^j^/A

|a> and |B> are eigenfunctions of the one-soin operator I , if this spin is parallel or antiparallel to H respectively. The spectrum of the AB system is given in figure 3,1,1.

3-it

1-2

^1 ^2

2-1 it

Figure 3.1,1 The AB spectrum in the case that J._ and

A B are both positive

case that J._ and {a -a )

A B D A

3.1.1 Intramolecular dipolar interaction between the spins A and B (Internal relaxation)

The necessary elements of the relaxation m a t r i x in this case, determined by Shimizu and Fujiwara are given in table 3.1.1,1. It will appear that condition (2. lit) is always satisfied for the dipolar relaxation correlation time in this thesis.

Element x T" ' ' ^ 1 1 2 2 = =^22ltit ( l - s i n 2 9 ) / 1 0 « l 1 3 3 = ^33itU ( l . s i n 2 9 ) M 0 ^ , , 2 / 5 liitU R2233 ( c o s ^ 2 9 ) / 1 5 ^1212 = ^2it2U • ^ ^ / 3 ° - ( ^ ' ' 3 0 ) s i n 2 9 ^ R , 3 , 3 = R3,^3^ 17/30 + ( 7 / 3 0 ) s i n 2 9 ^

Table 3.1.1.1 The necessary elements of the intra.molecular dipolar relaxation matrix of the .A.J system,

The dipolar relaxation time T„ is defined by:

1 ^ 3

yV

( l^j

D 2 ,^ 2 2 6 c ^^ '

'^^ %^AB

In the AB system there is only the vector r.^» so from the dipolar -1 . .

relaxation ra;t,e, T , no data about anisotropic molecular reorientation can be gained. Therefore the molecular motion is characterized by one correlation time T in (".it).

3,1.2 External relaxation

The external relaxation comprises all interactions causing relaxation of proton A or B, other than the intramolecular ones bet'.'een spins A and B, External relaxation in 4B systems has been investigated by means of the so called random local field (RLE) model (reference 2, and references cited therein). The RLE hamiltonian is given by

H(t) = - E Y- I..H.(t) (3.5) i=A,B

-> / , .

H.(t) is the fluctuating local field at spin A or B, causing the relaxation. The assumptions made in the RLF model are:

36

2 ly

ik il -<1 ik

k,l = x,y,z, i = A.B

'-.'any interactions, e.g. the dipolar one (see section 2.3), can not be vrritten as a scalar interaction in the way this is done in eq, (3.5). loreover the assumptions (3.6) are questionable, while in the RLF .model cross relaxation effects (see section 5.1) with the spins producing the field H.(t), and particle identity effects are neglected. As will be shown furtheron, the RLF model leads to quite erroneous results if intermolecular dipolar interactions between identical A3 systems contribute significantly to the

3 . . external relaxation, Maclean and lackor went beyond the limits of the RLF model. For the intermolecular relaxation of an AX system they assumed a bimolecular complex. The density matrix of this svstem was expressed as a direct product of the two density

it

matrices of the separate molecules, Khazanovich e.a, obtained relaxation equations for a more general case by writing the many particle density matrix of the total system as a direct product of the density matrices of the separate molecules. They treated the AB system, and took into account all cross correlations. Considering the external relaxation of the AB system by intermolecular dipolar interactions with: 1) solvent molecules each containing x equivalent spins I (hetero relaxation); 2) the AB systems themselves (homo relaxation) they determined the elements of the relaxation matrix. These can be 'Written as

R = A.T (3.7)

The components of the column vector R are the relevant elements of the relaxation matrix, the column vector T contains the external relaxation rates of protons A and B and a cross correlation term petween them. R, A and T in the case of hetero external relaxation are given in table 3.1.2.1.

The components of the vector T of table 3.1.2.1 are defined by: ^ i = 2 x G ^ 3 ^ ^ 3 (T)

^ X B = 2 - S s , B S (^) (3.8)

^XC= 2^°AS,BS (^)

G. ., .(T) is defined by eq. (2.3it). T~ and T~ are the relaxation

IJKJ AA AB

rates of proton A , resp. B , due to the interaction with the solvent spins S. R 1122 ^1133 ^llitlt ^2233 ^22ltlt ^3itit ^1212 R 1313 '^2U2lt ^3lt3H 1 1 ,ss lti(,ss '^22,s& ~ "^^33,33 2 sin e 2 cos 9 0 5sin^26 2 cos 9 2 s m 9 1+sin 9 2 1+cos 9 2 1+cos 9 1+sin^9 1 2COs2e A 2 cos 9 9 sin"9 0 Jsin^29 2 sin 9 2 cos 9 1+cos^9 l+sin^9 1+sin^9 2 1+cos 9

-i

-5COS29 -sin29 sin29 0 -sin^29 -sin29 sin29 -sin29 sin26 -sin29 sin29 0 ' 0 ->-T T-^ o T-^ XB 2 M , - 1

Table 3.1.2,1 The vectors R and T and the matrix A of

eq. (3.7) for the external dipolar relaxation of the AB system by the solvent, the so called hetero relaxation, 9 is defined in eq, (3.3)

38

YI^'

(9 i t)

Y(I'*

(9 ^. t+T)

(

—

i^ i^

.

-= -|

^

) (3.9)

r.

r .

IS JS

(3.9) depends on rotational but much more on translational motions (r. and r. are time dependent too). Eq. (3.9) is verv hard to

IS JS - I ^ -.^ y t

calculate without doing assumptions. Mowever, eq. (3.o) shows that information can be drawn from the time constants T „, T and T

AA AB X O about relative distances of closest approach.

A correlation coefficient C is introduced by the relation T-1

^^ (3.10)

^

i TT

(T"

.

T"

)'

^ XA X B ^

When C = 1, the fluctuating dipolar fields of the spins S (causing the relaxation) at the nuclear sites A and B are completely correlated, whereas C = 0 for the completely uncorrelated case. The elements R.. in table 3.1.2.1 (i=l,2,3,it) are transition

11,ss

probabilities between the eigenstates of the AB system, and states of the set of equivalent spins S. The reverse transition probability is given by

•^1

R • .. = ^ R.. (3.11) ss,ii n„x 11,ss

where n.. and n are the number of AB systems and solvent molecules resp. per unit volume. The magnetizations of the four lines of the AB system are coupled to the longitudinal magnetization M of the solvent by the transition probabilities of eq. (3.11) as discussed in section 5.1, eqs. (5.3) and (5.it). If the solvent has a relaxation time much shorter than proton A and B, this coupling, the so called cross relaxation, is broken, as, on the time scale of the A and B relaxation, Tl may be considered to be zero. In the frame of the

z

initial rate approximation the cross relaxation may be neglected too if T„. and T,,^ are only a few times larger than R.. , while the

X.\ XB -^ 11,ss TT-pulses in the T experiment are so selective that M^ is not

39

will be derived for the initial rate constants are equal to those which would have been found if the RLF model was used. In the AB systems studied m chapter 6 the requirements mentioned above for the applicability of the RLF model are met.

K, A and T m the case of external homo relaxation are given in table 3.1.2.2. R A |- - - 1 r- — 1~ ; ~ i ^1122' •^•('^sin^g-l) -1(6003^6-1) 1 ! R,^33 ^(6cos^9_l) l(6sin^9-1) 1

1 R 1 i ' I

; " l l l t l t . It it 1 2 "" 1 - s i n 2 e s i n 2 9 0 T p - 1 "A

'';'

• ' - ^ ! R2233 • ^ ( s i n ^ 2 e + l ) ^ ( s i n ^ 2 9 + l ) ^ ( 3 s i n ^ 2 9 - l ) - s i n ^ 2 9 T^^ ! R22l,lt ^ ( 6 c o s ' 9 - l ) - ^ ( o s i n ^ 9 - l ) 1 - s i n 2 e ; ^RRltU |-5-(6sin^9-l) - ^ ( 6 c o s ^ 9 - l ) , 1 s i n 2 9 ^1212 '^ ""• 2 '^^^^^ ^ *" f ='='3^6 ^ ~ \ s i n 2 9 - s i n 2 9 i R i o , o I1 + T c o s ^ 9 I 1 + -I s i n ^ 9 3 + -^ s i n 2 9 i s i n 2 0 1 1 3 1 3 2 ( 2 2 ^2lt2lt 1^ "^ I "^""^^ '' •* I ''^"^'^ 3 " 2 ^^"^® -sin29 '

I ^3lt3lt i'' "^ f sin^9 ! 1 + | cos^9 , 3 + -^ sin29 , sin29

Table 3.1.2.2 The vectors R and T and the matrix A of eq. (3,7) for the external relaxation of the AB system by intermolecular dipolar interactions between the AB systems (homo relaxation)

The elements of the vector T of table 3,1.2.2 are defined by:

^l' ='^A.^,AA(^) ^ B ' = ° B B , B B ( ^ ) (3_^2J

ItO

T , T and T are autocorrelation functions, T is a cross corre-lation function involving three spins.

External hetero relaxation may in some cases be described by the RLF model, if the cross relaxation with the solvent can be neglected, as explained above. The RLF model can, however, not be used in the case of homo external relaxation (between identical AB systems), because: l) the cross relaxation acts now directly upon the magnetization under observation; 2) 5 resp. it elements of the vector T are needed in the case of homo, resp. hetero

relaxation (compare tables 3.1.2.2 and 3.1.2.1).

3.1.3 The combined intramolecular dipolar and external relaxation If it is assumed that the molecular motions determining the internal and external relaxation are statistically independent, the

transition probabilities for combined external and internal relaxation can be obtained by addition. With tables 3.1.1.1, 3,1,2.1 and 3.1,2.2 and equation ('^.21) the initial rate constants may be calculated. An expression analogous to (3.7) is found;

% = B,T (3.13)

The components of t'le vector 1 are the initial rate constants, The matrix B and the vectors T and 3 are given in table 3.1.3.1 for the case of tne combined internal and hetero external relaxation.

t '0 «o(^12 «o(^13 «o(^o^ «o(^12 Q.(Ylo ^o(^13

%^h2

«o(^13 ' ^ 2 ^ . Y^3)

'hk^

. ^ i t ) , Y,3) . ^3it' , T) , T) 3-'-°3^o sin 9 2 cos 9 D 3+ ^3^o cos 9 . 2„ 'it-\ + ^ I t -Jsin ?9 2D i t -\ + 0.5+D, 0.5+D 0.5+D 0.5+D 5+ 5-5+ "6 2D^ 2D 1 1 5+ T .,-1 \A ,-1 XB 1 XC 1 D

Jable 3.1.3,1 I'he vectors Q and T and the matrix 3 of eq, (3,13) for the combined internal and external hetero relaxation

The following identities hold for table 3.1.3.1; Q (Y.., Y ) = « o ( \ l ' ! i j ^ Qo(^12'Jl3^ = « o ( V ' '31*^' «o(^12'°^)'= %^hh' " ) ' Q ( Y , ^ , T ) = Q (Y^, , T ) . Q ( Y . . , T ) means the initial rate constant

o 13 o 24 o IJ

of line Y.., when all four lines in the spectrum are inverted to

ij

create the initial conditions. The pulses are so selective that M is not disturbed, so the same results would have been found if the

z

RLF model was used.

The following quantities are introduced in table 3.1.3.1: D^ = gcos^g + -J sin^29 + I D = gsin 9 + -^ sin^29 + 5 D = 1 ± ssin 29 D^^ = ± sin29- Jsin^29 D = ±0.2sin29 + ^ G O S ^ 2 9 D^ = 0.8 - 4r 003^26 _{6 15} (3.lit)

The matrix B and the vector T and Q for the case of the combined o

internal and external homo relaxation are given in table 3.1.3.2

=2 ^ 1

L

5+ ^ + "it+ "5+ "^5-2D It-2D _it+ 2D, 2D

n"-it+ s i n ^ 2 9 ItD, ItD 0 0 it+ 0.5+D^ 0.5+D 0.5+D 0.5+D,

°6

2D^ 5+ 5-5+ 2D 1 1 5+

I —

T' T. T' AB

Table 3.1.3.2 The matrix B and the vector T of eq. (3.13) for the combined internal and external homo relaxation.

it2

The vector Q and the identities are the same as in table 3.1.3.1. o

The following Quantities are introduced in table 3,1.3.2:

3 5 1 . 2 E, = T; sin^9 + -Q sin 29 + 1 1 2 o 3 2 1 . 2 E. = ^ cos 9 + -p- s m 29 + 1 2 2 o E = -^ sin'^29 (3,15) E, ^ = 2 ± 7!- sin^29 lt± it Ej^. = I (1 ± COS29)

Some more Q 's might have been added to tables 3.1.3,1 and 3.1.3.2. In the first place there are Q (Y^2' ^3lt' ^^'^ ^o('^13' ^2it^' "^^^ lines Y and Y , , like the lines Y and Y^, have no energy level in common. This means that i.mmediately after the disturbing pulse, which inverts Y , , resp, Y . , the equilibrium magnetizations of Y , resp, Y are not disturbed. This makes that Q (Y .„, Y,v) and Q ( Y , Y^, ) have no practical importance as they can only be determined very inaccurately. In the second place rate constants of a certain line, when two or three lines are inverted to create the initial conditions, are possible. They have not been added to the tables because they can only be determined with very specialized apparatus, if the line distances in the spectrum and the relaxation times meet certain conditions. Moreover it can be shown that these initial rate constants are linear combinations of those in tables 3.1.3,1 and 3.1.3,2, which could be expected as there are only it, resp, 5 unknoijns.

The transverse relaxation rates of the four lines in the spectrum can immediately be found bv adding the R.... of table 3,1,1,1 to those

ijij

of table 3.1,2.1 or 3.1.2.2, depending on the kind of external relaxation.

Inspection of the set of linear equations (3.13) and tables 3.1.3,1 and 3.1.3,2 shows that from the nine equations, corresponding with nine different experimentally to be determined initial rate constants,

only it, resp, 5 are linearly independent. The unknowns, it in the case of hetero and 5 in the case of homo relaxation can be found by determining It, resp, 5 suitable initial rate constants and performing the back transformation of eq. (3.13) ;

•^ 3 (3.16) o

However, the initial rate constants are subject to experimental

- > •

inaccuracies, which may accumulate in the components of T, if eq, (3.l6) is used, A better procedure is to determine the components of T by a least squares fit to all the experimentally determined initial rate constants,

. it . . . Withm the approach of Khazanovich , intermolecular interactions are additive. In samples in which homo, as well as hetero relaxation is important, the total relaxation can be found by combination of tables 3.1.3.1 and 3,1,3,2. There are now 8 unknowns, TyA' ^ Y R ' T^„, T., T^, T,„, T, and T^, while there are still only 5 linearly

At. A B AB t D

independent equations as to the longitudinal relaxation. If the transverse relaxation rates are determined by the same processes as the longitudinal ones (which not always is the case), they can be taken into account and one finds 7 independent equations, relating the 3 unknowns to the relaxation rates. This means that the 8 unknowns can not be determined now. So in practice one is restricted to cases in which one of the two kinds of external

relaxation mechanisms is negligible, compared to the other one, unless certain assumptions are made,

Pure homo external relaxation is only found in undiluted liquids containing no other magnetic spins than the A and B nuclei. In

ffllR experiments most times solutions are used. In a 10 molar percent solution of an AB system in a frequently used solvent like

acetone-D^, the external relaxation rate due to the solvent

deuterons is it to 10 times larger than the homo external relaxation rate. If moreover intramolecular external relaxation contributions are present, the homo relaxation can most times be neglected. So homo external relaxation is not of much practical importance.

It It 5.1.4 Scalar r^laxatTOn

In s e c t i o n 3.1.1 t o 3 , 1 . 3 t i c r e l a x a t i o n ol t h - .^ a d 3 proio is l i e t o dipoiar i n t e r a c t i o n s .•ras t r e a t e d , li t h i s s e c t i o . a t t e t^on " i l ^ be paid t o t h e r e l a x a t i o n of t ^ e A aid B protor s Sj. sca_ar

i i i t e r a c t i o n s . An A3 system xs cons-dered .^ v-n^ on ---e protons A a. J lit

B are ootr s c a l a r coupled to a '< lucleus. Tne coupling constants

lii

involved are J , , and J r e s p e c t i v e l y , - ' e 'I nucleus las a e..-; IA ^3

short r e l a x a t i o n time jecause i t i s r e l a x e i e ^ f - c i e x t l y ^/ quadrupolar i n t e r a c t i o n . Tn^s ma'-ies t i e s c a l a r coipl n^ i t ^ t^e protons time dependent, r e s u l t i n g m relaxat_o as d_ocassed i

section ' ' . 3 , In t h e molecules i n v e s t i g a t e d ±n t _3 t i°s_s t - i . following nolds .

o o

(co - (J ) " T~ >> 1 13,11 H N IN \->- ' I

( 2 - ^ A ^ I N ' ^ ' ( ' ^ S B V : ' ^ ^ ^ ' (3-^^)

tj„ and co„ are tne proton ana nitrogen resorance ^reoi^^c.es, -_,-, tne nitrogen longitudinal relaxation time is ol the orae" o

milliseconds, while the proton T and T^ values are seconds or larger. The protons can now oe considered as tne spxn 3;o^ei, tie

lit

N nuclei being lumped fith tie lattice .riti wh^c t ley are ass-med to be m equiliorium because of tneir s lort relaxation time, '-t^ eqs. (2.12), (2.5), (2.ba), (2.21) and (2.33) t-e necessar" el .-re its ot the relaxation i atrix can be calculated, Ihe results as to t'e transverse relaxation can be sunnarized in the folio m g for'^u-.ae

•^2' =f ^ \ N ^ | ( S A " \ B ) ^ (^A-JJB' - ^ MI^

(J - J ) " sm^-e

_ 1 A i 3 _ _ J (3_^^)

' ^ ^ 3 IN

The plus s gn m (3.1'^) lolis for tie elements of tne relaxation matrix '^13^3 anl R^j^o^s t^s minus sign for ^.lo-jo and 31^31 . In the

calculation of the spectral densities using eq, ('',6a), the terms

?.'. and X.~. ' o-f eo. (2,33) may be omitted. Their contributions

ij

are terms like

(3,20)

1 + (^jj - CO.)' TJ,J

"^ is of the same order of magnitude as T^ in eq, (3.19). Because of eq, (3.17) contributions like (3.20) may be neglected. This implies that the scalar relaxation contributes only to the transverse relaxation of the protons, and not to the longitudinal one. After 30.me rearrangement eq, (3,19) is identical to that derived by

.6 . .

-arris e.a. for band shape studies m A3 systems.

The experimental transverse relaxation rates consist of a dipolar and a scalar part. Because the scalar relaxation does not contribute to the longitudinal relaxation, eq, (3.19) in combination with the results of section 3,1.3 can be used to determine the coupling constants J and J^ from the differences between the longitudinal and transverse relaxation rates of the lines in the AB spectrum. These coupling constants cannot be determined from the proton HRN'IR spectrum, as the multiplet structures they would produce are averaged

lit

out by the fast relaxing quadrupolar N nucleus. 3.2 The AB system

JJuite analogous to the procedures followed in the preceding sections for the AB system, expressions can be found for the longitudinal and transverse relaxation rates of the eight lines in an AB system. An expression like eq. (3.13) results. The formulae are lengthy and difficult to manipulate. Therefore a computer program is written which determines after substitution of the values of J,_ and

=* . . .

(a, - a„) , the matrix B for the initial rate constants, and the

k D

transverse relaxation rates of the eight lines in the ABp spectrum. 'Ihe relaxation mechanisms considered are intramolecular dipolar interactions between the three nuclei and external hetero relaxation. Homo relaxation is not considered because one seldom deals with it in practice as pointed out in section 3.1.3. The results are very

1^6

analogous to those of the AX system, which will be treated in the next chapter. In the intramolecular dipolar relaxation the same vector a as in eq, (it.6) for the AX„ system occurs. In the

description of the hetero relaxation the vector T of table it.2,2,1 must be extended for the A3 system with two cross correlation terms,

-1 -1

T and T , The determination of the several unknowns from ^ r 2 . ^^l'^3

the experimental relaxation rates is performed m the same way as described in section 3.1.3.

Readers interested in the computer program for the AB system mentioned above can obtain it from the author.

Chapter i| Application of the Redfield theory to

the weakly coupled A X and AX systems

it. 1 The AX system

All results derived for the A B system in sections 3.1.1 to 3.1.it can be applied to the AX c a s e , if 9 (see eq. (3.3)) is taken equal to zero. From tables 3.1.3.1 and 3.1.3.2 it can be seen that all information about the correlation of the external relaxation of proton A and X (contained in the terms T and T ) is lost.

it.2 The A X system

The hamiltonian H of the AX system may be r^-itten as

»o = - V Z A , -

-x'-hx^ '

^2X3) ^

'AX\ • % ' \ ^ '

"• ' x x \ • ^X3 (it.D

" A = Y"o(^ - °A^ ^""^ "^X^ ^'^o(^ - ^x'

(it.2)

Proton A is given the index 1 and protons X the indices 2 and 3.

The eigen functions and eigen values are given in table it.2.1, the spectrum in fig. it.2.1

1 2 3 it 5 6 7 8 Eigenfunctions aaa -^ a(a3 + Ha) 3aa "Tj e(a6 + ga)

aee

638 -^ a(aS - Sci)

- ] ^ 6(a6 - Ba)

Eigenvalues

-H

-

^ ^ ^J^x

- ^ % '"A - "X - '^AX ^ ^ -^"A * \ - ^-^AX ^'-A ^ \ ^ ^"^M -^"A

^"A

1 ^ ' ^ X X

^ ^^xx

^^^xx

"^^xx

^^•^xx

^^v

" '-^xx

3T

It8 2-lt 7-3 1-3 A2 A3 1-2 3-it 2-5 it-6

Fig. it. 2,1 The AX,^ spectrum

-(f^

if J.., > 0 and AX

A X

•^1 h

The AX„ system has no degenerate energy levels (unless J „ = O ) , but the lines A , X and X,^ consist each of two degenerate transitions.

it,2,1 Intramolecular dipolar interactions between the three spins of the AX^ system

With eqs. ( 2 . 5 ) , ( 2 . 6 ) , ( 2 . 2 9 ) , ( 2 . 3 0 ) , (2.31) and the eigenfunctions in table it.2.1 the necessary elements of the dipolar relaxation matrix can be calculated. They may be written analogous to eq, (3,7) as

R = A. a (It,3)

The vectors R (containing the elements of the dipolar relaxation

. . •* . * . \

m a t r i x ) , a, and the matrix A are given m table 4,2.1.1, The

- > •

components of a are defined by :

Y -h T 1 = ^ i J J K — ijik , ^ 2 2 3 3 16iT u r. .r., o ij ik (lt.it)

T _ is defined by eq. (2,35) and calculated with eqs. (2.30) and ( 2 . 5 0 ) , In the case of isotropic molecular motion with correlation time T , the a. .., reduce to

c' ijik Y h Y h ^ 2 " i j i j i ^ " 2 2 6 ' c ' ° - i j i k , ^ 2 2^3 3 ( ^ ' ^ " ^ " i j i k 16-77 u r . . ^ 1677 u r . . r . , '' o i j o i j ik 1 ) T (it.5)

3. .., is the angle between the internuclear vectors r. . and r., , iJik ij ik

t

A

2233 1133 1122 2255 1155 "^2266 3377 "2277 ^2288 nf 1177 ' 3 3 8 8 l l f 3355 1313 '3U3it '^it6lt6 "3itlt6 1225 ltit55 ^5566 ' i t l t 6 6 "33itit ^3366 M i i t l t 5588 \ l t 8 8 ^itlt77 ^6677 5577 22itit ^7738 ^5656 •2525 1212 lt63it 2 5 1 2 _ , J 1 itO 2 0 6 0 3 12 3 0 12 2 it 6 12 3 0 - 3 i t - 2 5 -35 3 12 2lt 0 0 0 0 0 0 0 -36 - i t 8 - l t 8 - 1 2 3 0 12 — 2 - it - 6 - 1 2 - 3 0 - l i t - 1 0 - 1 0 3 - 6 t 0

0 1

0 0 0 0 0 0 0 9 - 9 0 ^a + a •^ ^^1212 ^ 1 3 1 3 2323 1213 1^2123 "" ^31321

T a b l e it, 2 , 1.1 The v e c t o r s R, a and t h e m a t r i x A of e q , (it, 3) f o r t h e i n t r a m o l e c u l a r d i p o l a r r e l a x a t i o n o f t h e AX s y s t e m .