JOURNAL OF T H E AMERICAN CHEMICAL SOCIETY Registered in U.S. Parent Offire.
0 Copyright. 1975. by (he American Chemical S o c f e f y
VOLUME97, NUMBER 14
JULY 9, 1975
The Density Matrix Theory of Nuclear Magnetic Double Resonance Line Shapes in Chemically Exchanging Systems Jerome I. Kaplan, Ping Pin Yang, and Gideon Fraenkel* Contribution from the Department of Chemistry, The Ohio State University, Columbus, Ohio 4321 0. Received October 28, 1974
Abstract: A general density matrix formulation for describing N M R line shapes of exchanging systems under conditions of double resonance (one weak, w2, and one strong, W I , field) is presented. I t is shown that the double resonance spectrum for the example under consideration is about an order of magnitude more sensitive to slow exchange rates than the equivalent low power single resonance spectrum. Examples of effects in double resonance spectra such as line splitting, line distortion, and varying peak intensities are illustrated. An additional weak “absorption” at the frequency 2wl - 0 2 is shown to exist.
N M R line shapes for chemically reorganizing ~ y s t e m s l - ~ system, the extra fine structure which appears because the large rf field causes a breakdown of the allowed Am = f l have been utilized to obtain rate data and activation paramselection rules now permits Am = 0, f 2 transitions. Also eters for a variety of intramolecular rearrangements such as we shall predict the existence of extra resonance peaks from rotation, pseudorotation, conformational interconversion, as well as a wide collection of chemical exchange p r o c e ~ s . ~ - ~a response to a mixed frequency, the so-called pseudoabsorption. Note that these kinetic parameters extricated from N M R The exact shape of the absorption carries in it informadata apply to systems a t equilibrium. tion on relaxations including exchange phenomena. In prinSo far N M R line-shape analysis has been widely applied ciple one expects that selective decoupling should make the only for conditions of cw low rf p ~ w e r . ~The - ~ theory has N M R line shape more sensitive to the exchange rate than in been generalized, using the Permutation of Indices Method single resonance N M R spectra. Two papers dealing with (PI) to cover all typical exchange steps as well as combinaspecific examples of these effects have appeared.I2,l4 tions of these.I0 Recently this theory has been extended to In this paper, we describe in section I the general density include conditions of high rf field (saturation) employing a linearization procedure called Selective Neglect of Bilinear matrix theory for N M D R line shapes of any exchanging system including those where several different exchange terms ( S N 0 B j . l l Now in this paper, we turn our attention to nuclear magprocesses take place a t the same time. The theory allows for netic double resonance ( N M D R ) line shapes of chemically individual nuclear relaxation times employing the relaxaexchanging The same S N O B procedure origition operator described by us before.l0 This theory is apnally developed to calculate the high rf response” will also plied to a model exchanging system. The density matrix equations are derived, and an expression is obtained for the be utilized in the course of the double resonance calculations. absorption. Section I1 contains an interpretation of double resonance spectra and section I11 describes some line Multiple resonance has been the subject of intensive investigation. The techniques of N M D R have been used to shapes. Appendix I consists of a glossary of symbols to ease unravel and simplify complicated spectra by means of Ovthe reader’s progress through this article. erhauser effects, by the introduction of extra fine structure I. Density Matrix Equations for an Exchange System Under into a spectrum as well as completely decoupling one set of Conditions of Double Resonance nuclei from a n ~ t h e r . ’ ~ -With ~ O a few exception^,'^-'^ most existing treatments of double resonance handle only line poThe appropriate density matrix equation for a system sitions. undergoing chemical exchange is It will be shown that double resonance line-shape theory p = i [ p , X] + R p + E p ( 1) permits a clearer qualitative understanding of the double resonance spectrum even without exchange. For instance it For double resonance in the laboratory frame, the Hamiltonian is given by is easier to understand, as shown below, how, in a two spin 388 1
3882
x=
wgizie
+
Ji,Zi0
c id,[Zi"
+
zj
I,Y s i n wit]
cos wit
+
i
i >f
i
+ C(42[1~x cos
+
w,t
i
z,Y s i n uzt] (2)
where W O , W I (w2), and (dl (92)stand for chemical shift, frequency, and strength of the irradiating fields, respectively and i,j . . . sums over the spins. The operator for intermolecular relaxation ( R ) is given aslo
+
[ ~ ~ + i e - i ( w 2 - u i ) t 1-ie + i c w Z - w ii)j, i
] + Rp + Ep
The density matrix equations for the exchanging system defined in (5) are
+
[ 1 + ~ ~ ~ - i ( ~ 2 -+~ i1) AB t ei(w2-wi)tl p A B ] + m A B
1
[ y , [y,p Ti i
-
1
+
p,,]]
- P,ll
[I,',P
-[lie, T ,i
5 '
-
pap
( 4)
D
where sp stands for species. As described by us previously, the exact form which E p takes is determined by the mechanism of the reorganization process and is different for each typical exchange step.1° Space does not allow a repetition of all these possibilities. Instead, in this section we treat one model example, that of an AB molecule exchanging fragment B with the outside world (B')
AB
+
AB B' species
( 6)
a b c wavefunction
( 7)
Use of the Permutation of Indices method yields p(after exchange), called pae as
(ab(pAB,ela'b') =
c c
(15) Since we want only the low power response to the rf field (&2), we solve (14) and (15) by writing ,.,AB = P'AB + P + A B e i ( W 2 - w i ) t + P - A B e - i ( W , - U i ) t (16) -B' = P ' B + p + B ' e i ( W z - ~ i ) t + pB'e-i(W2-Wi)t (17) P
where p' is the solution of (1) with (02 = 0, and the p* a r e linear in (62. W e also note that, as ji has to be Hermitian, it follows that
( 8)
btb'
0
(c IPBIae IC')
=
IABZP
e-i w 1t IAB8
(18)
Thus we only need to solve for pfAB and p+B' or for p-AB and paE'. What we will eventually want to evaluate is the magnetization in the laboratory system. The x component of this magnetization is given by
+
M, = ( m ) T r p A B I A B x (B')Trp'B'IB,X
(19)
where (AB) means concentration of AB. Substituting for p from (10) into (19) and using the trace identity
=
(20)
TrBCA
one obtains Mx = (~)TrpABeiwixABzt~,,Xe-iWiIABat
+
( B') T r p B e' 1IBZtIB,Xe-i
(9)
P A Bac,ac'
recalling from above the chemical identity of AB to AB' and of B to B'. The decoupling signal d, is taken to be strong, while that of the observing frequency c42 is weak. To treat dl to all orders in @ ] , we go into a coordinate system rotating at frequency P I .The density matrix ji in the rotating system is obtained as ei w l t
( P 3 + + P'
TrABC
B'
PABWt@$
and
=
+ I-B,e{(Wz-Wi)t, ?'I + R ~ B +' B ~ '
( 5)
k -2
%x
[z+B,e-i(uz-ui)t
kZ
B ' e AB'+ B
where A, B, and B' each contain one proton, and fragments B and B' are chemically identical. As discussed in ref 10, the product representation is defined as
TAB
- + [ I + ~ , + I-~,,PB'I -
= i[jSB',
$9
Es'p = paD(afterexchange)
i Ig' et
Next, substitute for I, in (21) using (16) and (17) and recall that =
eiwlPtr*e-iwi~zt
i*$iwit
(22)
This procedure yields the following expression for M,:
Mx = ' / 2 ( A B ) T r p ' A B ( I + A B e i w+i t Z'ABe-iWit)+
'4( B') T r p '"(I+B,e{ i + I
-B,
emi
It)
+
I,,(( ~ ~ ) ~ ~ ~ + A B , i ( w , ) -t i2 -AB w i' + p A B e - i ( w z - 2 w l ) t ~ +
IA.
=
+
(11)
IBZ
',,((B')TrCo+B'ei(wz-zwl)tl- B' +
(mi
- wl)Iiz +
J i , j Z i *I j
p-ABe'iw2tI'AB} i/2(B')TrCo+B'eiwztr+B, + p - B ' e - i w z t I - B , } (23)
The absorption, Ab, at frequency w2 is given as the x component of magnetization out of phase with the rf field in the x direction; that is, we collect the terms in (23) proportional to sin o2t noting that
+
1, j
and the density matrix equation for one species is given by Journal of the American Chemical Society
/
+
+
+
'A(AB)T r b + A Be i w z t P A B+
The Hamiltonian in the rotating coordinate system is given by i
AB}
-{ ( W ~ - Z W ~ ) I + ~ , }
A, B
x=
(2 1)
(10)
where IABZ
@AB
(14)
(3)
where eq stands for equilibrium and the reorganization term E p is
(13)
97.14
/
July 9, 1975
e-iwpt
= cos w2t
+
i sin w,t
= c o s w,t
-
i s i n wzt
3883
to give Ab( 0 2 ) = 2 (AB){iTrp*ABI*AB- iTrp-ABI-A,}
The equation for P - ~ ’is written equivalently to (36). Actually as discussed above, we shall not need p- since the absorption can be written entirely with p+ terms. Note that the relaxation terms in (34) to (36) a r e in just p+ or p- not p+ - peq. Finally as written in (8), E p A B is bilinear. It would appear to be a function of p+p’. This is not the case as will shortly be demonstrated.
+
(B)’ {iTrp+B’I+B,- iTrpd’I-,,} 2
(24)
T o illustrate the use of (24), we now apply it to the calculation of the absorption a t 92 for the exchanging system ( 5 ) . In the spin product basis representation, the states a r e AI3 states =
$1
f
f
~
Solutions of the Equations for an Exchanging System, Double Resonance The solutions of the set of equations for p’, (32, 33), the CW high power spectrum, have been described by us before.’ This procedure involves writing the coupled equations for all the density matrix elements, including those involving Am = 0 f 2. The equations are linearized by means of dropping certain contributions to the E p term, the Selective Neglect of Bilinear Terms (SNOB) approximation. S N O B linearization is accomplished in the following way. The diagonal elements of p‘ in the product representation are written as shown in (37), where N is the number of
B states f
f
~
4, = ffP
= (YB’ = P
$5
$6
’
(25)
= Pff *4 = PP Employing the relationships $3
T r p I * = zp*,jI*j,i it
=
prj,i
(26)
J
(27)
P*i,j*
P+i,j = a i d
(28)
ibid
+
( n ~ I p ’ , , l n ~ )=
then the nonzero terms in the absorption are =
Ab(@,)
~
(concn) 2
{ c ip+i,
-
jI+jti
i>j
1
- + ( ~ ~ m l A p ’ , , l ~ ~ m(37) ) NSD
c
i p * , i1-i.
states, and sp stands for species. The first term in (37) is the infinite temperature probability of the diagonal matrix element. In linearizing Ep, all terms which a r e products of offdiagonal matrix elements (eq 38) or a product of an off-di-
j }
i>j
(29) so using (27) and (28), we find that Ab( 02) = (concn)
C -bij
(38)
i>j
= (concn)
agonal matrix element and the diagonal element Ap’ABi,i a r e dropped (eq 39). Investigation shows that the terms
(30) -Imp+,,
i>j
Applying (30) to exchange system ( 5 ) , we obtain Ab(@,) =
which a r e kept are a factor of los to lo6 larger than those discarded. In similar fashion to the above S N O B approximation for p’, we can linearize the second-order terms in eq 34 and 35 which appear in E p + . The maximum value which p+ elements take, both diagonal and off-diagonal is ca. Then the following approximations are valid.
- ( A B ) z w [ ~ + +~ ~pfAB3,1 ~ , ~ + p+AB4,2 + p+AB4,3] - (B’)IWpfB’6,5 (31) Notice that, in the P + ~ , , summed in (31), m ( i ) - m G ) = -1 always.
Density Matrix Equations for p+, p-, and p‘ To find the density matrix equations for p’ and p i , substitute for p from (1 6) and (1 7) into (1 4) and ( 1 5 ) and collect all terms with the same time dependence into separate equations.
0 = i[p‘AB,Xo””] - 34 [I+,,
+ ZXB,
p’AB]
With this linearization in mind, consider the equation for p+AB1,2. As stated before, we require the p+ elements to
+
solve for the absorption. Also for brevity we can leave off the species label from the density matrix’elements since the states already indicate which molecule is referred to, (1 -4 for A B and 5.6 for B’). We obtain the equation for p+1.2 by taking the ( 1 112) matrix element of eq 34. The result is
R(p‘AB - poAB)+ Ep‘AB (32)
0 = i[p’B’,
X,B‘]
- ii(/l[z+B‘+ I - ” , , p ’ B ‘ ]
R ( p ’ ” ’ - Po”’)
+
+ Ep‘”’
(33)
{
0 = i [E, - E , %p’1,3
R p + A B+ E P + ~ ” (34)
i(0, -
oJp+B’
y
[ICB‘
= i[p+B’,
+ I-,,,
XOB’]
p+”‘] -
ul)p-A” =
+[ZfA”
+
9
ib-,JC,AB]
I-AB,p-]
(0, -
-$b+1,1 -
w1)]p+1,2 p*2,2
-
+
p+3,2 +
p+1,4)
+
-
[I-B,, p‘”’]
RP*B,
-ib2-
+
-
-
+
+
EpC,.
1
(35)
1
?[ZfA,,
p’]
where Ei is the diagonal element X O ~ , Using ~. the linearization scheme described above reduces the term in 1/ T A B to eq 42, which is independent of the state of
+
Rp-AB + E p m A B (36) Fraenkel et al.
/
Density Matrix Theory in Chemically Exchanging Systems
3884 1 /2P+5,6
-
(42)
P+1,2
This will be true for all 20 density matrix equations describing the p+ABand p+B' elements. Without further comment, we will now list in this order the equations for p+2,1, P'2.3, P+3,2, P'1.4, and p+4,1. As seen in eq 31, p+2,1 contributes directly to the absorption, the p+2,3 and p+3,2 are Am = 0 transitions, and ~ ' 1 . 4 p+4,1 , are Am = f 2 transitions. p'.
0 = i ( [ E l - E2 -
+
+
+[-P+l,l
(
~
-2 w1)IP+2,1 -
P+2,2
+
Pc2,3
1 + 1'T ~ B -[Ti*+?;;, 1
P+2,1
1
-
[%
1 1 + ++TIB
T2A
%
P3,1
P+4,3
'I
TIA
-+
B' 3 AB' -+ B
Relative shifts, Hz ( J A B = 1 Hz) A ( o f AB) +5 B (of AB) 0 B' (free) -5
All T , and T , values = 2.0 sec [AB] = 2[B'] T A B = 27B'
coupled exchanging systems as long as weak.
WI
is strong and
w2
11. Density Matrix Interpretation of Effects in Double Resonance There are three observed effects by which NMDR spectra differ from single resonance spectra: (a) there a r e changes in peak amplitude ratios; (b) additional splitting is sometimes observed; and (c) there may be changes in line shape because of exchange phenomena not seen in single resonance CW N M R . The changes in amplitude are best discussed by comparing eq 43 with the i 1 2 ~element describing the low power CW spectrum. This spectrum would result by letting 0 (eq 48). The principal difference thus seen between eq 43
- P14,11
+
Table I. NMR Parameters AB
-
(44 )
p+2,3
and 48 lies in the coupling to the rf field which is +
$-PeQl,l
(49)
Peq2,21
for the low power single frequency response and +
$ w l , i
P'2,2
+
(50)
P'2,31
for the double resonance absorption. The other difference between eq 42 and 48 lies in the additional term %[p+l,l
+ P+2,2 + P+2,3 - P+4,11
(51)
This contribution (eq 51) appears because we have chosen the product representation rather than one which diagonalizes X A B @ , I x . The principal effect of the excluded terms is one of splitting the first-order absorption peaks in the . ~ ~ ignoring the peak limit of very large g ~ Temporarily splittings, we conclude that the Overhauser like change in amplitude of the first-order double resonance peaks relative to the equivalent low power single resonance peaks is given by the ratio of the amplitude terms of eq 43 to 48. Now we return to the extra line splitting phenomena in double resonance spectra which in most cases will appear only as a line-shape distortion. To obtain a qualitative feeling for the effect assume no exchange (7 a) and J A B / ( ~ O A- WOB)
