pages 2-33 of this are tycko's lecture slides ... · pages 2-33 of this are tycko's...

Pages 2-33 of this pdf file are Tycko's lecture slides from January 8, 2018. Pages 34-74 are notes about quantum mechanics, NMR, homonuclear recoupling, and related things, prepared originally in 2008 and subsequently corrected/amended.

1. How do we describe the state of a nuclear spin system?

2. Angular momentum and rotations

3. How do we represent the interactions of nuclear spins thatdetermine NMR signals?

4. How do we calculate NMR signals?

5. Nuclear spin interactions that affect solid state NMR (dependenceon structure, orientation, motion)

6. Manipulation of spin interactions by radio‐frequency pulses andsample spinning (Average Hamiltonian theory; recoupling)

Introduction to the (Quantum Mechanical) Theory of Solid State NMR

(R. Tycko, 2018 Winter School on Biomolecular Solid State NMR)

How do we describe the state of a nuclear spin system?

In quantum mechanics, the state of a system at time t can be represented by the abstract symbol |(t), which is called a state vector.

State vectors have "complex conjugates", which are written as (t)|.

State vectors have a "length", defined as the inner product (t)|(t).

The inner product | represents the projection of one state vector onto another state vector. If | = 0, the two states are orthogonal to one another. Note that | = |*.

For any system, there is a set of states that represents all possible conditions of the system. Such a set is called a basis. (In fact, there are many different bases for a given system.)

For a spin‐1/2 nucleus (e.g., 1H, 13C, 15N, 31P), the basis can be {|+,|}. In other words, the two states in which the spin is either "pointing up" or "pointing down".

This means that any state of the spin‐1/2 nucleus is some linear combination of |+ and |. In other words, |(t) = a(t)|+ + b(t)|, where a(t) and b(t) are complex numbers.

The basis states are usually chosen to be "orthonormal", meaning that +|+ = | = 1 and +| = 0. Also, |a|2 + |b|2 = 1, so that (t)|(t) = 1.


For a many‐spin system, we can choose the basis to be {|m1m2m3...mN}, where mj is the "direction" of the jth spin. If all of the nuclei are spin‐1/2, the basis contains 2N state vectors. This choice of basis is called the "direct product" basis.

The overall state of the many‐spin system can then be written as

1 2 N

1 2 N

s

m ,m ,...,m 1 2 Nm ,m ,...,m s

(t) a (t) m m ...m

In standard quantum mechanics, the squared magnitudes of the coefficients in such a linear combination represent the probabilities of being in each individual state. If you were to ask for the probability that the many‐spin system at time t was in the state where m1 = +1/2, m2= +1/2, m3 = ‐1/2, m4 = ‐1/2, etc., the answer would be |a+,+,‐,‐,...|2.

The relative phases of the coefficients are also important. For example, in a one‐spin system, the states |1 = a|+ + b| and |2 = a|+ + ib| are different. If a = b = 2‐1/2, then |1 is a state that "points" along +x, while |2 is a state that "points" along +y.

The description given so far applies when all of the coefficients have precisely defined magnitudes and phases, so that there is a unique linear combination of basis states that fully describes the overall state of the spin system. Such a state is called a "pure state".

This may happen in other areas of science, but it does not happen in NMR.


What happens in NMR is that we have many copies of nominally identical spin systems, all of which contribute simultaneously to our NMR signals, but each of which is in a different quantum mechanical state. The "many copies" could be the individual molecules within our sample, or they could be different nanometer‐scale pieces within a larger chunk of material.

The different copies have different values of their complex coefficients in the linear combination of basis states on the previous slide.

If we knew the probability of having each specific choice of complex coefficients, then we could calculate NMR signals from each specific choice, and add up the results from many, many such calculations to get the total NMR signal. This would be perfectly correct, but it would be very inconvenient and laborious.

A more convenient, concise, and tractable approach is to use a density operator (or density matrix) description of the state of the nuclear spin system.

The density operator approach allows us to average over the many copies of identical spin systems. But it is not the same as simply taking the average quantum mechanical state.

1 2 N 1 2 N

1 2 N 1 2 N

1 2 N

1 2 N

s s

m ,m ,...,m 1 2 N m ,m ,...,m 1 2 Nm ,m ,...,m s m ,m ,...,m s

s

m ,m ,...,m 1 2 Nm ,m ,...,m s

(t) a (t) m m ...m a (t) m m ...m

b (t) m m ...m

This is still a pure state.

Digression: what is an operator?

An operator O is something that transforms a state | to another state |', written as the equation O| = |'. In quantum mechanics and NMR, we are mostly concerned with linear operators, which have the property that O(a| + b|) = aO| + bO| = |' + |'.

Some, but not all, operators have inverses, represented by O‐1, such that OO‐1 = O‐1O = 1. Here, the symbol "1" means the "identity operator", which has the property that 1| = |for any state |.

Some operators have "eigenstates", which are special states that are transformed into themselves, multiplied by a number that is called the "eigenvalue". So for an eigenstate of O, O| = |. In general, is a complex number.

All linear operators have "adjoints", symbolized by O†. Adjoints are like complex conjugates. If O| = |', then |O† = '|. And (cO)† = c*O†.

Two different operators usually do not commute with one another, meaning that O1O2 O2O1. The operator [O1,O2] O1O2 O2O1 is called the "commutator of O1 with O2".

If two operators do commute, then they have the same eigenstates (unless there are "degenerate" eigenstates, i.e., groups of eigenstates with the same eigenvalue).

If two operators have the same eigenstates, then they commute (when operating on their eigenstates).

Digression: what is an operator?

In quantum mechanics and NMR, we are particularly interested in two special types of linear operators, namely "Hermitian" operators and "unitary" operators.

Hermitian operators have a complete set of eigenstates, so that the eigenstates can be used as a basis to represent any other state of the system. In other words, any state can be written as a linear combination of the eigenstates. Moreover, the eigenvalues are real numbers. Consequently, if A is Hermitian, then A† = A. A is called "self‐adjoint".

Hermitian operators are used to describe energies (i.e., terms in the nuclear spin Hamiltonian) and other observable quantities (e.g., angular momentum components). The fact that Hermitian operators have real eigenvalues implies that observable quantities are real numbers.

Unitary operators have the property that their adjoints are their inverses. In other words, if B is unitary, then B† = B‐1. And vice versa. Consequently, unitary operators preserve inner productsof pairs of states. If B| = |' and B| = |', then '|' = |.

Unitary operators are used to describe the evolution in time of a quantum mechanical state, as in |(t) = U(t)|(0), where U(t) is the "time evolution operator".

Unitary operators are also used to describe rotations and similar transformations. So quantum mechanically, an rf pulse is a unitary operator.

Digression: matrices and vectors

Sometimes we can calculate NMR signals by using the abstract symbols for operators and states discussed so far. More frequently, signals are calculated in computer programs by using matrices and vectors.

Converting the abstract symbols to matrices and vectors requires us to choose a specific basis set, for example the direct product basis {|m1m2m3...mN}.

If the basis states are written simply as |k, with k = 1,2,...,N: N

kk 1

a k

A state is represented by an N‐dimensional column vector, with elements ak.

| is represented by an N‐dimensional row vector, with elements ak*.

An operator O is represented by an NN matrix, with the number j|O|k in row j and column k.

A Hermitian matrix has real numbers along the diagonal (j = k). Off‐diagonal elements on opposite sides of the diagonal are complex conjugates of one another, because j|O|k = k|O|j* for a Hermitian operator.

For a unitary matrix, rows and columns are orthonormal:N N N

** † † 1qk rk q,r

k 1 k 1 k 1

O O q O k r O k q O k k O r q OO r q OO r q | r

N N N N

** † † †kq kr r,q

k 1 k 1 k 1 k 1

O O k O q k O r k O q r O k r O k k O q r O O q r | q


Now back to the density operator as a description of a "mixed state" in NMR, i.e., a sample that contains many copies of the same spin system, each in a different quantum mechanical state, and all contributing simultaneously to the NMR signals:

The density operator (t) is defined to be the average of |(t)(t)|, where |ab| is the outer product of states |a and |b.

(t) (t) (t)

An outer product of two states is an operator, because it can be applied to some other state according to (|ab|)| = |ab| = x|a, with x = b|. Such operators are sometimes called "projection operators".

N

kk 1

(t) a (t) k

If we express the state in a basis {|k} as , then the density operator is

N N N N* *

k k ' k k 'k 1 k ' 1 k 1 k ' 1

(t) a (t) k a (t) k ' a (t)a (t) k k '

In this basis, the density matrix has the number in row k and column k'.*kk ' k k '(t) a (t)a (t)

Diagonal elements of the density matrix (k=k') are the population of state |k, in other words the overall probability of finding the system in this state, averaged over the "many copies".Off‐diagonal elements (kk') are called "coherences". A non‐zero coherence means that there is a non‐random relationship between the amplitudes of states |k and |k'.


The density operator sounds complicated, but actually it makes calculations of signals relatively simple, especially when the initial "mixed state" of the system is something simple.

If the system has quantum mechanical energy levels Ek (k = 1,2,...,N) and if the system is initially at thermal equilibrium, then the relative populations of the energy levels are given by Boltzmann factors, so kk(0) exp(‐Ek), where = 1/kBT. At thermal equilibrium, coherences between different energy states are assumed to be zero, so kk'(0) = 0 for k k'.

In general, if the Hamiltonian of the system (i.e., the energy operator) is H, then at thermal equilibrium, (0) exp(‐H). The density matrix elements then have these properties.

In a high‐field NMR experiment, the largest part of the Hamiltonian is the Zeeman interaction with the large external field along z. For a homonuclear system with NMR frequency 0, this is HZ = 0Sz, where Sz is the z‐component of spin angular momentum. It is therefore a very good approximation to use (0) exp(‐HZ) = exp(‐0Sz).

It is also true that 0 << 1, except at very, very low temperatures. Therefore, (0) 1 ‐ 0Sz.

The "1" does not contribute to NMR signals. So in signal calculations, we can say (0) Sz

This corresponds to an initial state with the nuclear spins weakly polarized along the external field.

Angular momentum and rotations

The "spin" of a nucleus is the total angular momentum of the nucleus (in its ground state).

Angular momentum in quantum mechanics is represented by operators such as Sx, Sy, and Sz. These are the x, y, and z components of the angular momentum vector operator S.

Angular momentum components do not commute with one another, which means that spin states can be eigenstates of only one angular momentum component, most commonly Sz.

The commutators are [Sx,Sy] = iSz and [Sy,Sz] = iSx and [Sz,Sx] = iSy (with Planck's constant hbar = 1).

These commutation relations dictate the properties of angular momentum operators, and any other set of operators with the same commutation relations will behave in the same way as angular momentum operators.

Each angular momentum component does commute with S2 = Sx2+Sy2+Sz2. So NMR‐active nuclei have a quantum number s, which defines the eigenvalue of S2 to be s(s+1), and they can simultaneously have eigenvalues of Sz, ranging from ‐s to +s in increments of 1.

This can be proven from the commutation relations. It is convenient to define spin "raising" and "lowering" operators S+ = Sx + iSy and S = Sx iSy. If |m is an eigenstate of Sz with eigenvalue m (i.e., Sz|m = m|m), then S+|m and S|m are eigenstates of Sz with eigenvalues m+1 and m‐1.

Except that, if m = s, then S+|m = 0. And if m = ‐s, then S|m = 0.


S m s(s 1) m(m 1) m 1 and S m s(s 1) m(m 1) m 1

These relations tells us the matrix elements of S+ and S. In the basis |m, diagonal elements are zero. The only non‐zero elements are one space off the diagonal.

Using Sx = (S+ + S)/2 and Sy = (S+ S)/2i, we can then construct matrices for Sx and Sy.

For a spin‐1/2 nucleus, the "Pauli spin matrices" are:

In general:

x y z

0 1/ 2 0 i / 2 1/ 2 0 1 0S S S 1

1/ 2 0 i / 2 0 0 1/ 2 0 1

For a pair of spin‐1/2 nuclei, Sx = Sx1 + Sx2, Sy = Sy1 + Sy2, and Sz = Sz1 + Sz2. How can we write these as matrices in the 4‐dimensional direct‐product basis {|++,|+,|+,|}?

We can think of such things as Sx = Sx112 + 11Sx2, where the "tensor product" means:

aA bA aB bBa b A B cA dA cB dBc d C D aC bC aD bD

cC dC cD dD

In this way, matrices for Sx, Sy, and Sz can be constructed for arbitrarily large spin systems, for example in an NMR simulation program.


Angular momentum operators are "generators" of rotations. This means that the unitary operator R() = exp(‐iS) is a rotation operator. Specifically, R() rotates angular momenta around an axis given by the direction of the vector , by a rotation angle given by the magnitude of .

For example, Rx() = exp(‐iSx) is a rotation around x by . We apply this rotation to statesaccording to |' = Rx()|.

x xiS iSx y x y y zR ( )S R ( ) e S e S cos S sin

We apply this rotation to operators according to O' = Rx()ORx()‐1 = Rx()ORx(‐). It can be shown that:

and similar relations for rotations around y and z. This is what one expects for rotations in a right‐handed axis system. These relations can be proven (laboriously) by expanding exp(‐iSx) and exp(iSx) in Taylor series and using the commutation relations to rearrange the resulting expressions.

z

y

x

A few interesting facts about rotations:

‐‐Any product of rotation operators is equivalent to a single rotation operator for some vector . In other words, there is always some net rotation axis and net rotation angle.

‐‐Any rotation or product of rotations can be expressed as Rz()Ry()Rz() for some choice of "Euler angles" ,,.

‐‐Products of rotation operators can often by simplified or rearranged by tricks such as the following:


x z y z x z y z z z

x z y z z z

x x z z

R ( / 2)R ( / 2)R ( / 2)R ( / 2) R ( / 2)R ( / 2)R ( / 2)R ( / 2)R ( / 2)R ( / 2)

R ( / 2)R ( / 2)R ( / 2)R ( / 2)R ( / 2)R ( / 2)

R ( / 2)R ( / 2)R ( / 2)R (

z

/ 2)R ( )

The third line follows from the second line because of the general relation, for any unitary operator U and any operator A, that 1A 1 UAUUe U e

This is proven easily by expanding eA as a Taylor series.

In fact, for any function F, UF(A)U‐1 = F(UAU‐1).

How do we represent the interactions that determine NMR signals?

Interactions of nuclear spins with one another, with external fields, and with their chemical/material environment are represented by the Hamiltonian operator H(t), which is a Hermitian operator. The eigenvalues of H(t) are the instantaneous energy levels of the spin system. Because H(t) is Hermitian, it always has a complete set of eigenstates, which could therefore be used as a basis.

The Hamiltonian generally is a sum of terms, each representing one type of interaction. In high‐field NMR, the largest term is the Zeeman interaction with the external magnetic field along z, HZ = 0Sz, where 0 is the Larmor frequency in radians per second (again assuming Planck's constant hbar = 1, so that energies are measured in rad/s).

(If there are two different types of spins, called S and I for example, then HZ = 0SSz + 0I Iz.)

Interactions of spins with rf pulses are represented (in the laboratory frame) by terms of the form Hrf(t) = 21cos(rft‐)Sx, where 1 is the rf amplitude, is the rf phase, and rf is the rffrequency, or "carrier frequency". In the rotating frame, Hrf = 1(Sxcos + Sysin) as will be shown later.

Other terms, such as terms representing spin‐spin couplings, are contained in an "internal Hamiltonian" Hint(t), with a time‐dependence that may arise from sample spinning or molecular motions. Details of these terms will be discussed later.

So usually the Hamiltonian can be written as H(t) = HZ + Hrf(t) + Hint(t).


The Hamiltonian determines how the state of the system changes with time. In other words, the Hamiltonian is the generator of time evolution, just as angular momentum operators are generators of rotations.

Time evolution is described by a unitary time‐evolution operator U(t), which transforms states according to |(t)=U(t)|(0) and transforms density operators according to (t)=U(t)(0)U(t)‐1.

The Schrodinger equation of standard quantum mechanics saysdi (t) H(t) (t)dt

If |(t)=U(t)|(0) for any initial state |(0), then it must be true that di U(t) H(t)U(t)dt

This is the Schrodinger equation for U(t). In the special case that H(t) = H0 for all t, the solution is U(t) = exp(‐iH0t).

If H(t) is "piecewise constant", equal to H0, H1, ..., HM for successive time periods of length t0, t1, ..., tM, then the overall evolution operator up to time = t0+t1+...+tM is U() = exp(‐iHMtM) ...exp(‐iH1t1) exp(‐iH0t0)

Note that the order of terms in this product is very important, because in general [Hp,Hq] 0.

And note that the inverse evolution operator (for going backwards in time) has the opposite order: U()‐1 = exp(iH0t0) exp(iH1t1)...exp(iHMtM)


In the special case that the Hamiltonian does commute with itself at different times, it is OK to add the exponents: U() = exp[‐i(HMtM +...+H1t1 +H0t0)] and U()‐1 = exp[i(HMtM +...+H1t1 +H0t0)]

If H(t) is a general function of time, not necessarily piecewise‐constant, and [H(t),H(t')] = 0 for any two times t and t', this becomes andt

0U(t) exp[ i H(t ')dt ']

t1

0U(t) exp[i H(t ')dt ']

If H(t) is a general function of time and [H(t),H(t')] 0, we can writewhere "T with the arrow on top" is the "Dyson time‐ordering operator".

t

0U(t) Texp[ i H(t ')dt ']

In practice, this is just fancy notation for the idea of dividing t into many short time periods, during which H(t) is nearly constant, and representing U(t) by

0M M 1 M 2 2 1 iH tiH t iH t iH t iH t iH tU(t) e e e ...e e e

How do we calculate NMR signals?

In quantum mechanical terms, signals are "observables", and observable quantities are represented by Hermitian operators. The "expectation value" of an observable O in a purestate |(t) at time t is o(t) = (t)|O|(t).

For a mixed state, the expectation value becomes , where the bar over the top again means an average over "many copies". Using the fact that for anyorthonormal basis {|n}, we can express o(t) as:

o(t) (t) | O | (t) N

n 1

n n 1

N N N

n 1 n 1 n 1N

n 1

o(t) (t) n n O | (t) n O | (t) (t) n n O (t) (t) n

n O (t) | n Tr{O (t)}

where Tr{A} is the "trace" of A, equal to the sum of the diagonal elements of A (in any basis).

N

nn 1

(t) a (t) n

N N N

* * 2n ' n n ' n n n n

n,n ' 1 n,n ' 1 n 1

o(t) a (t) a (t) n ' O n a (t) a (t)x n ' | n | a (t) | x

So o(t) is a weighted average of the eigenvalues of O, weighted by the relative populations of the eigenstates.

If |(t) is a linear combination of eigenstates |n with eigenvalues xn, such as

, then

If |(t) happens to be an eigenstate of O with eigenvalue x, then o(t) = (t)|O|(t) = x(t)|(t) = x.


In NMR, signals arise from oscillating magnetic flux in the rf coil (antenna) of our NMR probe, due to precessing nuclear spin magnetization. Magnetization is proportional to spin angular momentum (through the gyromagnetic ratio). The NMR signals Flab(t) (in the laboratory frame) are then proportional to one of the transverse angular momentum components, Sx.

Therefore,1 1

lab x x x zF (t) Tr{S (t)} Tr{S U(t) (0)U(t) } Tr{S U(t)S U(t) } As you may know, NMR signals are apparently detected in the rotating frame, and we measure both x and y components of the spin magnetization, or "real" and "imaginary" components. Where does this come from?

The transformation into a rotating frame in NMR is a special case of a more general transformation of reference system, which is described in quantum mechanics by some unitary operator A(t). If states in the original reference system are |(t), then states in the new reference system are |'(t) = A(t)|(t). Similarly, '(t) = A(t)(t)A(t)‐1.

What are the Hamiltonian and the evolution operator in the new reference system?

1 1

d (t)d d dA(t)i '(t) i [A(t) (t) ] i (t) A(t)[i ]dt dt dt dt

dA(t)i (t) A(t)H(t) (t)dtdA(t)i A(t) A(t)H(t)A(t) '(t) H '(t) '(t)

dt

H'(t) is the transformed Hamiltonian. Note that it includes an extra term, sometimes called a fictitious field (or a gauge field). U'(t) is the evolution operator generated by H'(t).


The rotating frame in NMR is the transformation brought about A(t) = exp(iSzrft), in other words by a rotation of spin angular momenta about z at the rf carrier frequency (opposite to the direction of precession).

If H(t) = HZ + Hrf(t) + Hint(t) in the laboratory frame, with HZ = 0Sz and Hrf(t) = 21cos(rft‐)Sx, then

z rfz rf z rf z rf z rf z rf z rf z rf

iS tiS t iS t iS t iS t iS t iS t iS t

0 z 1 rf x int

rf z 0 z 1 rf x rf y rf int

0 rf z 1 rf

d(e )H '(t) i e e ( S )e e [2 cos( t )S ]e e H (t)edtS S 2 cos( t )(S cos t S sin t) H (t)

( )S 2 (cos t cos s

rf x rf y rf int

z 1 x y int

in t sin )(S cos t S sin t) H (t)

S (S cos S sin ) H (t)

The last line follows by taking the time averages of terms like cosrft cosrft and cosrft sinrft, which are 1/2 and zero, respectively. This is the "rotating frame approximation" or the "rotating wave approximation".

is the resonance offset. The rf term now looks the way we expect. More will be said about Hint(t) later.


1 1lab x z x z rf z z rf

1z rf x z rf z

1x rf y rf z

1 1x z rf y z rf

F (t) Tr{S U(t)S U(t) } Tr{S R ( t)U '(t)S U '(t) R ( t)}

Tr{R ( t)S R ( t)U '(t)S U '(t) }

Tr{(S cos t S sin t)U '(t)S U '(t) }

Tr{S U '(t)S U '(t) }cos t Tr{S U '(t)S U '(t) }sin t

F

real rf imag rf(t) cos t F (t)sin t

The evolution operator in the rotating frame is U'(t), generated by H'(t). But the evolution operator in the laboratory frame can now be written as U(t)=A(t)‐1U'(t)=exp(‐iSzrft)U'(t).

Returning to the expression for the NMR signals:

NMR spectrometers are constructed to measure separately the cosrft and sinrft terms, so in the end we get the expected real and imaginary signals in the rotating frame, where precession around z occurs at frequency , where rf pulses look like rotations around x or y for = 0 or = /2, and where the internal spin interactions are changed to . intH (t)

Nuclear spin interactions that affect solid state NMR

Hint(t) contains magnetic dipole‐dipole interactions (HD), anisotropic chemical shifts (HCSA), scalar couplings (HJ), and electric quadrupole couplings (HQ). Other interactions are important in solid state physics, but are not relevant to biomolecular solid state NMR of non‐paramagnetic samples. For now, we will focus on HD and HCSA.

spherical harmonic functions irreducible tensor operators

220

i2 1

2 2i2 2

1Y (3cos 1)6

Y sin cos e1Y sin e2

20 z1 z2 1 2

z1 z2 1 2 1 2

2 1 z1 2 z2 1

2 2 1 2

1T (3S S )6

1 1[2S S (S S S S )]26

1T (S S S S )2

1T S S2

S S

Dipole‐dipole interaction between two magnetic moments 1 and 2, with = S:

1 2 1 2 1 2 1 2D 1 25 3 3 2

2m1 2

2m 2 m3m 2

3( )( ) 3( )( )Hr r r r

3 ( 1) Y ( , )Tr

μ r μ r μ μ S r S r S S

Angles and specify the direction of the internuclear vector in the current axis system. The internuclear distance is r.


Irreducible tensor operators (and spherical harmonics) transform among themselves under rotations, which makes them useful for understanding how rotations affect the spin interactions.

(s) im' (s) imm'm m'mD ( , , ) e d ( ) e Also, , where dm'm

(2) is a "reduced Wigner rotationmatrix element".

In particular, if a new axis system x'y'z' is related to an original axis system xyz by Euler angles ,,, then T2m in the original axis system turns into Ř()T2m in the new axis system, with:

2(2)

2m m'm 2m'm' 2

R( )T D ( , , )T

where Dm'm(2) is a "Wigner rotation matrix element", defined by

yz ziSiS iS(s)m'mD ( , ) sm ' e e e sm for a spin‐s particle.

Irreducible tensors have the important property that z ziS iS im2m 2me T e e T

(Note: I use Ř above to represent a "passive" rotation, meaning that we change the orientation of the axis system and then ask "What does T2m look like in the new axis system?" This is the opposite of keeping the axis system fixed, rotating T2m, and then asking "What has T2m turned into?", which would be an "active" rotation.)


In a non‐spinning sample (and without molecular motions), HD is constant. If there are no other interactions except HZ, the full rotating‐frame Hamiltonian is

z rf z rf

z rf z rf

rf

z int

iS t iS tz D

2iS t iS tm1 2

z 2m 2 m3m 2

2im tm1 2

z 2m 2 m3m 2

21 2 1 2

z 20 20 z z1 z2 1 23 3

H S H (t)

S e H e3S ( 1) Y ( , )e T e

r3S ( 1) Y ( , )e T

r3 (3cos 1)S Y ( , )T S (3S S )

r r 2

S S

This is the familiar homonuclear dipole‐dipole coupling.

The dipole‐dipole coupling Hamiltonian is "truncated" by the large external field along z, so that only the part of HD that commutes with Sz affects NMR signals. In this treatment, truncation occurs because exp(‐imrft) averages to zero in the rotating frame unless m = 0. So only the T20 term survives.

In the heteronuclear case, spins 1 and 2 have very different values of , so

21 2

1 z1 2 z2 z1 z2 1 23

21 2 1 2 1 2

z1 z2 z1 z2 z1 z2 1 23

1 2 1 2z1 z2 z1 z2

21 2

z1 z2 1 23

(3cos 1)H S S (3S S )r 2

( ) ( ) (3cos 1)(S S ) (S S ) (3S S )2 2 r 2

( ) ( )(S S ) (S S )2 2

(3cos 1) 1[2S S (S S Sr 2 2

S S

S S

1 2S )]


If we transform to a frame of reference defined by A(t) = exp(iHdifft), where Hdiff is the term proportional to Sz1‐Sz2, we find that the "flip‐flop" term in the dipole‐dipole coupling (i.e., the term involving S+1S‐2 +S‐1S+2) looks rapidly oscillatory. This term then averages out. Transforming back, we get

21 21 z1 2 z2 z1 z23H S S (3cos 1)S S

r

This is the familiar heteronuclear dipole‐dipole coupling. It is appropriate whenever the difference between NMR frequencies of spins 1 and 2 greatly exceeds the strength of the dipole‐dipole coupling.


That was for a non‐spinning sample.

We saw that a large external field along z truncates the dipole‐dipole coupling. But for a spinning sample, the direction of the external field relative to the sample is changing with time, so we can not truncate HD right away.

For magic‐angle spinning (MAS) experiments, we must first determine how HD is affected by spinning, before truncating it.

To do this, we transform from a "molecule‐fixed" axis system, to a "rotor‐fixed" axis system, and then to the laboratory axis system (where the external field is along z). Then we can truncate the coupling.

x

y

zmolecule‐fixed

m zx

y

rotor‐fixed

z

y

x

laboratory

arbitrary

(powder pattern)tilted and spinning


R

2m1 2

D 2m 2 m3m 2

2m im im' (2)1 2

2m m', m 2m'3m',m 2

R m

im' tm im im' (2) (2)1 22m m', m m'',m' m 2m''3

3H ( 1) Y ( , )TrR( , , arbitrary)

3 ( 1) Y ( , )e e d ( )Tr

R( t, , 0)3 ( 1) Y ( , )e e d ( )e d ( )T

r

R

2

m'',m',m 2

2im'( t )m im1 2

2m m', m 0,m' m 203m',m 2(m' 0)

truncate3 ( 1) Y ( , )e e d ( )d ( )T

r

Note that d(2)0,0(±m) = 0. So HD contains only terms that oscillate at R and 2R, and therefore averages to zero at the magic angle.

(Unless we also apply rf pulse sequences)


Anisotropic chemical shifts have the form , where is a 3 X 3 CSA matrix.

This means that any component of the magnetic field B couples to any component of the spin angular momentum S.

CSA 0H B S

But in the principal axis system of the CSA matrix, where this matrix is purely diagonal:

CSA 0 x x x y y y z z z

0 1 x x 2 y y 3 z z 0 iso

H ( B S B S B S )

( B S B S B S )

B S

iso x y z( ) 3 is the isotropic shift (or shielding), and 1 2 3 0

HCSA in its principal axis system can then be rewritten as:1 21 1 1

CSA 0 3 z z x x y y x x y y 0 iso2 2 23

0 3 20 0 33 22 2 2 0 iso

H [(B S B S B S ) (B S B S )

6 T (T T )

B S

B S

Then MAS can be analyzed in the same way as for dipole‐dipole couplings to get HCSA in the laboratory frame. Again, the anisotropic part oscillates at R and 2R, but with an extra term due to the asymmetry parameter = (1‐2)/3. The isotropic part becomes 0isoBzSz.

Manipulation of spin interactions by rf pulses and sample spinning

Pulse sequences, together with sample rotation, can be used to change the form of spin interactions and to turn various interactions on and off.

As a simple example, consider a homonuclear two‐spin system with a dipole‐dipole coupling and with chemical shifts, subjected to the following pulse sequence:

180X 180‐X

/2 /2

The evolution operator for this sequence iswhere HD and HCSA are now taken to be in their high‐field, truncated forms.

D CSA D CSAi(H H ) /2 i(H H ) /2x xU( ) R ( )e R ( )e

This can be rewritten as x D CSA x D CSA

D CSA D CSA

i[R ( )(H H )R ( )] /2 i(H H ) /2

i(H H ) /2 i(H H ) /2

U( ) e e

e e

D CSAH H D CSAH H

which looks like theevolution operator for:


Since HD is proportional to (3Sz1Sz2‐S1S2) and HZ is proportional to 1Sz1+ 2Sz2:

D D Z ZH H and H H

Strictly speaking, [HD+HZ,HD‐HZ] 0, so it is not precisely true that

D CSA D CSA D CSA D CSA

D

i(H H ) /2 i(H H ) /2 i(H H H H ) /2

iH

e e ee U( )

However, this is a good approximation as long as is small (compared with the inverse of the dipole‐dipole coupling strength and the chemical shift difference).

So the net effect of this simple pulse sequence is to make the Hamiltonian in the second /2 period different from the Hamiltonian in the first /2 period. And the pulses themselves vanish. If is sufficiently small, the overall evolution operator is determined by an "effective" Hamiltonian, given (to a lowest‐order approximation) by the time‐average of the Hamiltonian.

This is essentially how average Hamiltonian theory (AHT) works. In this simple example, the effective Hamiltonian is just HD. HZ is averaged out by the pulses.


The more abstract description of AHT is as follows:

Consider a rotating frame Hamiltonian of the form H(t) = Hint(t)+HRF(t). H(t) generates the evolution operator U(t). HRF(t) by itself generates the evolution operator URF(t), which is a rotation operator, since rf pulses produce rotations.

Imagine that the rf pulse sequence consists of a block of pulses, with length c called the"cycle time", that is repeated many times. We are interested in the evolution operator for one complete cycle U(c).

We are also interested in pulse sequences where the net rotation over one block is zero (or a multiple of 2), so that the pulses alone would have no net effect at multiples of c.

Thus, URF(c) = 1.

We transform to a frame of reference defined by A(t) = URF(t)‐1 and recall the following result for the Hamiltonian in the transformed frame:

11RF

RF RF RFdU (t)H '(t) i U (t) U (t) H(t)U (t)

dt

It can be shown that 1

RFRF RF

dU (t)i U (t) H (t)dt

Therefore, 1RF int RF intH '(t) U (t) H (t)U (t) H '(t)


The overall evolution operator becomes U(t) = URF(t)U'int(t), so that U(c) = U'int(c).

c

c int0U( ) Texp[ i H '(t ')dt ']

Therefore,

If Hint'(t) commutes with itself at all times, then

c

c

c int ave c0

ave int0c

U( ) exp[ i H '(t ')dt '] exp( iH ' )

1H ' H '(t ')dt '

More generally, U(c) can be expressed as exp(‐iHeffc), with the effective Hamiltonian represented by a "Magnus expansion":

c

(0) (1) (2)eff(0)

ave

t '(1)int int0 0

c

H H H H ...

H H '1H dt ' [H ' (t '),H ' (t '')]

2


Things to remember:

AHT applies only to pulse sequences that are "periodic and cyclic", in other words repetitive and with U(c) = 1. So, for example, it is not used much in solution NMR, where such pulse sequences are rare.

AHT only provides information about the effective Hamiltonian, and hence the NMR signals, at multiples of the cycle time c. So, for example, it does not provide information about spinning sidebands, or other effects due to variations within the cycle time.

AHT works well only when 1/c is large compared to the size of Hint (or when Hint'(t)

commutes with itself at all times).

AHT has been very important in the development of modern solid state NMR.

1

Homonuclear Dipolar Recoupling in Solid State NMR:

Analysis with Average Hamiltonian Theory

(Lecture notes originally prepared for the first Winter School on Biomolecular Solid State NMR,

Stowe, Vermont, January 20-25, 2008. Subsequently extended in June 2008, and corrected and

extended in May 2012. Corrected again in January 2018.)

Robert Tycko

Building 5, Room 112

National Institutes of Health

Bethesda, MD 20892-0520

phone: 301-402-8272; e-mail: [email protected]

Definition of homonuclear recoupling

Pulse sequences that create non-zero effective (i.e., average) dipole-dipole couplings among like

spins (e.g., 13C-13C couplings) during magic-angle spinning (MAS).

Motivations for homonuclear recoupling (partial list)

(1) to measure distances between like nuclei

(2) to produce crosspeaks between like nuclei in 2D or 3D MAS NMR spectra

(3) to permit double-quantum filtering, for selective observation of NMR signals arising from

pairs or groups of dipole-coupled nuclei

(4) to permit spin polarization transfers, as required for various other structural techniques (e.g.,

"tensor correlation" techniques)

Why are pulse sequences necessary?

MAS is usually required for sufficient resolution and sensitivity in solid state NMR of unoriented

systems. MAS produces narrow lines by averaging out anisotropy of chemical shifts and

magnetic dipole-dipole couplings. Recoupling sequences are needed to restore these

interactions.

2

Outline

I. Relevant quantum mechanical principles

II. Useful mathematical identities and tricks

III. Nuclear spin interactions under MAS

IV. Average Hamiltonian Theory in simple terms

V. Homonuclear dipolar recoupling mechanisms

A. Delta-function pulse sequences

B. Continuous rf irradiation

C. Finite-pulse recoupling sequences

D. Chemical-shift-driven recoupling

VI. Symmetry principles for recoupling sequences

Appendix: Derivation of time-dependent dipole-dipole coupling

under MAS

13C NMR spectra of uniformly 15N,13C-labeled

L-valine powder, obtained in a 14.1 T magnetic

field at the indicated MAS frequencies.

3

DISCLAIMERS:

1. THE PRESENTATION OF DIPOLAR RECOUPLING TECHNIQUES AND OTHER

TOPICS IN THESE NOTES IS MOTIVATED SOLELY BY PEDAGOGICAL

CONSIDERATIONS. MANY USEFUL TECHNIQUES AND BRILLIANT IDEAS ARE

OMITTED, AND MANY IMPORTANT PAPERS ARE NOT CITED. THE GOAL OF THESE

NOTES IS SOLELY TO SUMMARIZE THE THEORETICAL/MATHEMATICAL

BACKGROUND REQUIRED FOR AN UNDERSTANDING OF DIPOLAR RECOUPLING

AND RELATED TECHNIQUES IN SOLID STATE NMR.

2. THESE NOTES MAY CONTAIN MISTAKES. PLEASE LET ME KNOW IF YOU

NOTICE ANYTHING THAT SEEMS TO BE INCORRECT.

4

I. Relevant quantum mechanical principles

If a spin system is in a single, well-defined state, that state is represented by a state vector

)t(| . For example, a system of three spin-1/2 nuclei could be in the state |)0(|

at time t = 0.

The evolution of )t(| with time is determined by the Schrodinger equation:

)t(|)t(H)t(|dt

di (I.1)

where H(t) is the Hamiltonian operator (in angular frequency units), which contains terms that

represent each of the nuclear spin interactions. If H(t) is constant (i.e., H(t) = H), then Eq. (I.1)

has the solution

)0(|e)t(| iHt (I.2)

If H(t) is not constant, the solution to Eq. (I.1) is

)0(|)t(U)t(| (I.3a)

})'t(H'dtiexp{T)t(Ut

0

(I.3b)

where T

is the Dyson time-ordering operator. U(t) is the evolution operator. If the time interval

from 0 to t is divided into N intervals with lengths j during which the Hamiltonian is Hj, Eq.

(I.3b) is short-hand for

11221N1NNN iHiHiHiHee...ee)t(U

(I.3c)

which is simply an extension of Eq. (I.2). Also, Eqs. (I.1) and (I.3a) imply

)t(U)t(H)t(Udt

di (I.4)

Signals in quantum mechanics are "expectation values" of Hermitian operators, evaluated

according to

)t(|A|)t()t(SA (I.5)

)t(| and |)t( are called "ket" and "bra" vectors.

5

In actual calculations, Eq. (I.5) would be evaluated by choosing a complete basis of states for the

system, {|n>}, that satisfies 'n,n'n|n . )t(| would be represented as a column vector

with elements )t(|n . |)t( would be represented by a row vector with elements

*)t(|n . A would be a matrix with elements m|A|n in the nth row and mth column.

In NMR, we don't usually have spin systems in single, well-defined states. Therefore, we use

density operators instead of state vectors. The density operator is defined as

|)t()t(|)t( (I.6)

where the bar represents a weighted average over the spin states that are present in the sample. It

can be shown that Eq. (I.1) implies that (t) satisfies the equation

)]t(),t(H[)t(dt

di (I.7)

where [A,B] AB - BA means the commutator of operator A and operator B. Eq. (I.7) implies

1)t(U)0()t(U)t( (I.8)

We usually assume that the initial condition )0( before applying our pulse sequence is

proportional to the sum of the z components of spin angular momentum for the relevant nuclei,

i.e.,

k

zkz II)0( . This is appropriate at normal temperatures when the spins are at

thermal equilibrium in a strong magnetic field along z.

Signals are

)}t(A{Tr)t(SA (I.9)

where Tr{B} is the trace of operator B, defined as

n

n|B|n}B{Tr if {|n>} is a basis of

states as discussed above. Conventional NMR signals are proportional to the transverse

components of spin angular momentum. In the rotating frame, the NMR signals have real and

imaginary parts, proportional to k

xkx II and k

yky II . These are usually combined into

one complex signal S(t), which is then

})t(UI)t(UI{Tr

)}t(I{Tr

)}t(I(iTr)}t(I{Tr

)t(iS)t(S)t(S

1z

yx

imagreal

(I.10)

6

where I = Ix iIy and U(t) is the evolution operator for the nuclear spin system, resulting from a

combination of interactions with rf pulses and internal spin interactions.

II. Useful mathematical identities and tricks

If A, B, and C are normal quantum mechanical operators, then

Tr{ABC} = Tr{CAB} (II.1)

If the operator A has an inverse A-1, such that AA-1 = 1, then

Tr{B} = Tr{ABA-1} (II.2)

1ABA1B eAAe

(II.3)

If Ix, Iy, and Iz are the operators for the x, y, and z components of spin angular momentum, then

rotations of spin angular momentum (for example, by rf pulses) are expressed mathematically by

equations such as

sinIcosIeIe zyiI

yiI xx (II.4a)

sinIcosIeIe yziI

ziI xx (II.4b)

The same equations hold if the following substitutions are made: {Ix Iy, Iy Iz, and Iz Ix}

or { Ix Iz, Iy Ix, and Iz Iy}. These are cyclic permutations of x, y, and z.

It is often useful to represent spin angular momenta by raising and lowering operators, which

have the properties

yx iIII (II.5a)

IeeIe iiIiI zz (II.5b)

When two operators A and B satisfy AB = BA, these operators are said to commute with one

another. In general, quantum mechanical operators do not commute with one another. In other

words, [A,B] AB – BA 0. This means they can not be permuted without changing the result.

However, if B has an inverse B-1, then permutations can be accomplished (i.e., the order or

grouping of A and B can be rearranged) by using the following trick:

'BAAB (II.6a)

ABB'A 1 (II.6b)

7

Therefore, if you have a set of N operators {Aj} and a set of N invertible operators {Bj}, by

applying Eqs. (II.6a,b) repeatedly, you can show that

'A'A'...A'A'ABB...BBB

BABA...BABABA

122N1NN122N1NN

11222N2N1N1NNN

(II.7a)

122k1kkk1

k1

1k1

2k1

21

1k BB...BBBABBB...BB'A

(II.7b)

Eqs. (II.7a,b) are one of the keys to understanding Average Hamiltonian Theory and recoupling

sequences, as shown below. These equations show that all of the operators {Bj} can be pulled to

one side, leaving the operators {Aj} on the other side, but in the altered form {Aj’}.

As mentioned above, actual calculations or numerical simulations are usually performed by

choosing a complete basis of states for the system, {|n>}, that satisfies 'n,n'n|n and

representing the density operator, Hamiltonian, and other operators as N X N matrices (where N

is now the number of states in the basis set). For an operator A, the number in the nth row and

mth column would be m|A|n . In general, m|A|n is a complex number. The adjoint

of A is another operator A† with matrix elements *n|A|mm|A|n † .

A Hermitian operator is one for which A = A†, or *n|A|mm|A|n . In quantum

mechanics, H, , and all operators that represent observable quantities are Hermitian.

A unitary operator is one for which A-1 = A†, which implies that

N

1m

'n,n*'n|A|mm|A|n . Quantum mechanical evolution operators U(t) are unitary.

In general, if A is Hermitian, then eiA and e-iA are unitary. Angular momentum operators Ix, Iy,

and Iz (but not I+ and I-) are Hermitian, so rotation operators (e.g., xiIe ) are unitary.

For non-commuting operators,

†††† ABC)ABC( (II.8a)

-1-1-1-1 ABC)ABC( (II.8b)

According to Eqs. (II.8), we must reverse the order of noncommuting operators when we take the

adjoint or inverse of a product of these operators.

In general, if [A,B] 0, then eAeB eA+B eBeA (or eiAeiB ei(A+B) eiBeiA if we are concerned

with making unitary operators from Hermitian operators A and B). However, if both A and B

are very small, then the following approximations can be made:

ABBABA eeeee (II.9)

8

On the other hand, if [A,B] = 0, Eqs. (II.9) are exactly true even if A and/or B are not small.

III. Nuclear spin interactions under MAS

NMR experiments are performed in the rotating frame. For present purposes, we assume that the

nuclear spin Hamiltonian in the rotating frame contains only four terms, representing

homonuclear dipole-dipole couplings, chemical shift anisotropy, isotropic chemical shifts, and

interactions with rf pulses:

)t(HH)t(H)t(H)t(H RFICSCSAD (III.1)

A. Magnetic dipole-dipole coupling under MAS

For a pair of spins I1 and I2, the dipole-dipole coupling under MAS can be expressed as:

)II3()]2t2sin(),(D

)2t2cos(),(C)tsin(),(B)tcos(),(A[)t(H

212z1zR

RRRD

II

(III.2)

This is the “truncated” dipole-dipole coupling, i.e., the part for which 0)]t(H,I[ Dz0 , where

0 is the NMR frequency (in rad/s) and 0Iz is the Zeeman interaction with the large external

static field along z (which vanishes in the rotating frame). The full dipole-dipole coupling

includes other terms (see the Appendix), which normally don’t affect high-field NMR spectra

directly because they do not commute with the very strong Zeeman interaction, but do contribute

to spin relaxation. Eq. (III.2) is one useful way to express HD(t), but there are others (see Section

VI, for example).

The MAS frequency is RR /2 . The angles ,, are “Euler angles” that relate the

orientation of a particular molecule within the MAS rotor to an axis system that is fixed with

respect to the rotor:

''

''

''

'

'

'

)(R)(R)(R ''z''y''z

z

y

x

z

y

x

(III.3)

In Eq. (III.3), {x’,y’,z’} are the molecule-fixed axes and {x’’,y’’,z’’} are the rotor-fixed axes.

Rz’’() and Ry’’() are rotations of the axes about z’’ and y’’ by angle . These axis systems are

depicted below:

9

Schematic depiction of an MAS rotor, showing rotor-fixed axes {x’’,y’’,z’’} and randomly

oriented molecules with molecule-fixed axes {x’,y’,z’}. The two axis systems are related by

rotations by Euler angles, as in Eq. (III.3), which are different for different molecules. The MAS

rotor rotates about its z’’ axis.

The coefficients A(,), B(,), C(,), and D(,) in Eq. (III.2) are proportional to I2/R3,

where I is the gyromagnetic ratio and R is the internuclear distance. We shall not worry about

the detailed functional form of these coefficients in the analyses of recoupling sequences

presented below. See the Appendix for a complete derivation of Eq. (III.2), including general

expressions for these coefficients. The maximum value of the total coefficient of

)II3( 212z1z II is 32I R/ . For 13C spin pairs, this is 27.59 kHz when R = 1.00 Å.

Note that the dependence on the Euler angle always appears as Rt + . This is because

molecules with different values of (but the same and ) differ in their orientations by a

rotation about z'', which is the MAS rotation axis. So, as the sample spins, molecules with

different values of are rotated into one another, thus having the same coefficient of

)II3( 212z1z II at different values of t.

Note that the time average of HD(t) is zero under MAS. Note also that HD(t) contains terms that

oscillate at R and terms that oscillate at 2R.

Finally, note that HD(t) is a “zero-quantum operator”. This means that HD(t) has non-zero matrix

elements only between states that have the same total z component of angular momentum. For a

system of two spin-1/2 nuclei, the only non-zero matrix elements are

2/1|)II3(||)II3(| 212z1z212z1z IIII (III.4a)

10

2/1|)II3(||)II3(| 212z1z212z1z IIII (III.4b)

The dipole-dipole coupling for a static sample is obtained by setting R to zero in Eq. (III.2).

Recoupled dipole-dipole interactions generally have different orientation dependences than static

couplings (i.e., different dependences on ,,), and can be zero-quantum, one-quantum, or two-

quantum operators (or a mixture of these).

B. Chemical shift anisotropy and isotropic chemical shift

For one spin I,

zRR

RRCSA

I)]2t2sin(),('D)2t2cos(),('C

)tsin(),('B)tcos(),('A[)t(H

(III.5a)

zIICS IH (III.5b)

The isotropic chemical shift I is defined here to be the time-independent part of the difference

between the actual NMR frequency of spin I and the rf carrier frequency rf. Under MAS, the

chemical shift anisotropy (CSA) has the same type of time-dependence as the dipole-dipole

coupling (i.e., terms that oscillate at R and 2R), but the operator part is simply Iz. An

important difference is that

)t(He)t(He DiI

DiI xx

(III.6a)

but

)t(He)t(He CSAiI

CSAiI xx

(III.6b)

This allows chemical shifts and dipole-dipole couplings to be affected differently by pulse

sequences, especially recoupling sequences.

C. Radio-frequency pulses

)]t(sinI)t(cosI)[t()t(H yx1RF (III.7)

The rf amplitude is 1(t). The rf phase is (t).

When a pulse sequence contains short pulses with amplitudes that greatly exceed the strengths of

dipole-dipole and chemical shift interactions and the MAS frequency, the pulses can be treated

as instantaneous rotations of spin angular momenta. This is the delta-function pulse limit. In

this limit, the rotation (i.e., the evolution operator) produced by a pulse of length tp is the

operator

11

)(R)(R)(R

)(Re)(R

e),(R

zxz

ziI

z

)sinIcosI(i

x

yx

(III.7)

where = 1tp is the pulse flip angle.

In general, the net rotation produced by a pulse sequence alone (i.e., ignoring HD(t), HCSA(t), and

HICS) is

t

0 yx1RF )]}'t(sinI)'t(cosI)['t('dtiexp{T)t(U

(III.8)

If all pulses in a pulse sequence are phase-shifted by , then the net rotation becomes

zz

zz

zz

iIRF

iI

iIt

0 yx1iI

iIt

0 yx1iI

t

0 yx1RF

e)0;t(Ue

e)]}'t(sinI)'t(cosI)['t('dtiexp{Te

]}e)]'t(sinI)'t(cosI)['t('dte[iexp{T

)]})'t(sin(I))'t(cos(I)['t('dtiexp{T);t(U

(III.9)

Similarly, if we include all four Hamiltonian terms, the evolution operator for the pulse sequence

is

t

0 ICSCSADyx1 ]}H)'t(H)'t(H)]'t(sinI)'t(cosI)['t(['dtiexp{T)t(U

(III.10)

and a phase shift changes the evolution operator to

zz

zz

iIiI

iIt

0 ICSCSADyx1iI

t

0 ICSCSADyx1

e)0;t(Ue

]}e]H)'t(H)'t(H)]'t(sinI)'t(cosI)['t(['dte[iexp{T

]}H)'t(H)'t(H)])'t(sin(I))'t(cos(I)['t(['dtiexp{T);t(U

(III.11)

Eq. (III.11) is valid because HD(t), HCSA(t), and HICS all commute with Iz. If this were not true

(i.e., if we were not in the high-field limit), the effect of an overall rf phase shift could be more

complicated.

Another useful operation is an rf phase reversal, meaning (t) -(t). The effect of a phase

reversal is to change the evolution operator to

12

xx

xx

xx

IiIi

Iit

0 ICSCSADyx1Ii

t

0 ICSCSADIi

yx1Ii

t

0 ICSCSADyx1

t

0 ICSCSADyx1

e)t(''Ue

e]]H)'t(H)'t(H)]'t(sinI)'t(cosI)['t([[e'dtiexp{T

]H)'t(H)'t(He)]'t(sinI)'t(cosI)['t([e'dtiexp{T

]H)'t(H)'t(H)]'t(sinI)'t(cosI)['t(['dtiexp{T

]H)'t(H)'t(H))]'t(sin(I))'t(cos(I)['t(['dtiexp{T)t('U

(III.12)

Thus, a phase reversal is equivalent to changing the sign of chemical shifts and rotating the

resulting evolution operator U’’(t) by 180 around x.

IV. Average Hamiltonian Theory in simple terms

AHT [Haeberlen and Waugh, Phys. Rev. 175, 453 (1968)] is a mathematical formalism that

allows us to analyze how pulse sequences affect internal spin interactions1,2. AHT is particularly

useful in the derivation and analysis of pulse sequences that consist of a block of rf irradiation

that is repeated many times. This is precisely the situation that arises in dipolar recoupling

experiments. If the rf block has length c (called the cycle time), then AHT is applicable when

the following conditions are met:

)t(H)t(H RFcRF (IV.1a)

)t(H)t(H DcD (IV.1b)

)t(H)t(H CSAcCSA (IV.1c)

1)(U cRF (IV.2)

1||H||,1||H||,1||H|| cICScCSAcD (IV.3)

Eqs. (IV.1) says that the rf pulse sequence is periodic and the internal spin interactions are also

periodic. In recoupling sequences, this means c should be a multiple of R. Eq. (IV.2) says that

the net rotation produced by the rf block is zero (or sometimes a multiple of 2). The rf block is

then called a “cycle”. Eq. (IV.3) says that c is short enough that the dipole-dipole and chemical

shift interactions can not produce a large change in the state of the spin system in one cycle.

This allows a “perturbation theory” approach such as AHT to be employed, based on Eq. (II.9).

AHT depends on changing the picture in which we view the evolution of the spin system from

the usual rotating frame (in which HRF, HD, HCSA, and HICS together determine the spin

evolution) to a new frame of reference in which the rf pulses no longer appear directly. Instead,

13

the rf pulses cause additional time dependences in HD, HCSA, and HICS. This new frame of

reference is called the interaction representation with respect to HRF.

Mathematically, the interaction representation works as follows: If U(t) is the evolution operator

for the entire rotating frame Hamiltonian, we define a new evolution operator )t(U~

that satisfies

)t(U~

)t(U)t(U RF . Then we ask, “What is the new Hamiltonian that corresponds to )t(U~

?”

This is easily calculated, using the general relations )t(U)t(H)t(Udt

di [see Eq. (I.4)] and

)t(H)t(U])t(U[dt

di 11 [see Eqs (II.8) and recall that U(t) is unitary and H(t) is Hermitian;

also, the adjoint of the number i is –i].

)t(U~

)t(U]H)t(H)t(H[)t(U

)t(U]H)t(H)t(H)t(H[)t(U)t(U)t(H)t(U

)t(Udt

di)t(U)t(U])t(U[

dt

di

)]t(U)t(U[dt

di)t(U

~

dt

di

RFICSCSAD1

RF

ICSCSADRF1

RFRF1

RF

1RF

1RF

1RF

(IV.4)

Therefore,

ICSCSAD H~

)t(H~

)t(H~

)t(H~

(IV.5)

with

)t(U)t(H)t(U)t(H~

RFD1

RFD (IV.6a)

)t(U)t(H)t(U)t(H~

RFCSA1

RFCSA (IV.6b)

)t(UH)t(UH~

RFICS1

RFICS (IV.6c)

So the dipole-dipole and chemical shift Hamiltonian terms in the interaction representation are

the same as in the usual rotating frame, but with their spin operator parts rotated by 1RF )t(U .

(Note that this rotation is the inverse of )t(URF , so the order of the pulses in a pulse sequence is

reversed and the sign of the flip angles is changed. This is very important in AHT calculations.)

1

RF )t(U acts on the spin operator parts of HD(t), HCSA(t), and HICS(t), making these spin

operator parts time-independent. In recoupling techniques, the time-dependence of the spin

operator parts induced by the rf pulses interferes with the spatial time dependence from MAS, in

general preventing )t(H~

D and/or )t(H~

CSA from averaging to zero.

14

Because of Eq. (IV.2), )(U~

)(U cc . Because of Eq. (IV.1), Ncc )(U

~)N(U . Because of

Eq. (IV.3), we can approximate )(U~

c by

cave

c

H~

i

0c

e

})t(H~

dtiexp{)(U~

(IV.7)

with

c

0c

ave )t(H~

dt1

H~

. Therefore, as long as we care only about the state of the spin system

at multiples of c (and not in the middle of the rf blocks), then it is sufficient to calculate the

average Hamiltonian in the interaction representation. To a good approximation, the NMR

signals will be determined by aveH~

alone.

V. Homonuclear dipolar recoupling mechanisms

A. Delta-function pulse sequences

Consider the following simple pulse sequence, called DRAMA [see R. Tycko and G. Dabbagh,

Chem. Phys. Lett. 173, 461 (1990)]3:

How do we use AHT to calculate what this does to homonuclear dipole-dipole couplings and

chemical shifts?

Because the sequence consists of delta-function pulses, )t(URF is piecewise-constant:

R2

212/iI

1

RF

t,1

t,e

t0,1

)t(U x (V.1)

Abbreviating )2t2sin(D)2t2cos(C)tsin(B)tcos(A RRRR from Eq.

(III.2) by (A,B,C,D), the interaction representation Hamiltonians are

15

R2212z1z

21212y1y

1212z1z

D

t),II3()D,C,B,A(

t),II3()D,C,B,A(

t0),II3()D,C,B,A(

)t(H~

II

II

II

(V.2)

R2z

21y

1z

CSA

t,I)'D,'C,'B,'A(

t,I)'D,'C,'B,'A(

t0,I)'D,'C,'B,'A(

)t(H~

(V.3a)

R2zI

21yI

1zI

ICS

t,I

t,I

t0,I

)t(H~

(V.3b)

The average Hamiltonians are

)]}22cos()22[cos(4

D)]22sin()22[sin(

4

C

)]cos()[cos(2

B)]sin()[sin(

2

A){IIII(3H

~

1R2R1R2R

1R2R1R2R2z1z2y1yave,D

(V.4)

)]}22cos()22[cos(4

'D)]22sin()22[sin(

4

'C

)]cos()[cos(2

'B)]sin()[sin(

2

'A){II(H

~

1R2R1R2R

1R2R1R2Rzyave,CSA

(V.5a)

]I)II[(H~

zR

12zyIave,ICS

(V.5b)

So both the dipole-dipole coupling and the CSA are recoupled, and the isotropic chemical shift is

altered. Note that only the 2z1z II3 part of HD(t) contributes to the recoupled dipole-dipole

Hamiltonian. This is a general rule, because the I1I2 part is not affected by the rf pulses.

Now consider a longer DRAMA pulse sequence with additional 180 pulses:

16

For this sequence

R2RiI

2R1R2/3iI

1RRiI

R2

212/iI

1

RF

2t,e

t,e

t,e

t,1

t,e

t0,1

)t(U

x

x

x

x

(V.6)

R2R212z1z

2R1R212y1y

1RR212z1z

R2212z1z

21212y1y

1212z1z

D

2t),II3()D,C,B,A(

t),II3()D,C,B,A(

t),II3()D,C,B,A(

t),II3()D,C,B,A(

t),II3()D,C,B,A(

t0),II3()D,C,B,A(

)t(H~

II

II

II

II

II

II

(V.7)

R2Rz

2R1Ry

1RRz

R2z

21y

1z

CSA

2t,I)'D,'C,'B,'A(

t,I)'D,'C,'B,'A(

t,I)'D,'C,'B,'A(

t,I)'D,'C,'B,'A(

t,I)'D,'C,'B,'A(

t0,I)'D,'C,'B,'A(

)t(H~

(V.8a)

R2RzI

2R1RyI

1RRzI

R2zI

21yI

1zI

ICS

2t,I

t,I

t,I

t,I

t,I

t0,I

)t(H~

(V.8b)

17

Now, ave,DH~

is the same as before (because )t(H~

D is not affected by the additional 180

pulses), but 0H~

H~

ave,ICSave,CSA (because the 180 pulses change the signs of )t(H~

CSA

and )t(H~

ICS in the second rotor period). Dipole-dipole couplings are now selectively recoupled.

Through the dependence on , , and , ave,DH~

depends on molecular orientation in the MAS

rotor. Dipolar lineshapes are then powder patterns. The lineshape depends on 2 - 1, because

the A, B, C, and D terms have different relative magnitudes for different values of 2 - 1. In the

limit that 1/)( R12 , the lineshape resembles the Pake doublet pattern of a static sample

for a two-spin system. Note that the operator part of ave,DH~

under DRAMA is different from

the operator part of HD(t). Under the DRAMA sequences depicted above, ave,DH~

contains both

zero-quantum and double-quantum terms.

Simulated dipolar powder patterns for

two-spin 13C system with maximum

splitting = 1.5 kHz (R = 2.48 Å)

Pulse sequences used in DRAMA experiments

[Chem. Phys. Lett. 173, 461 (1990)]

Experimental dipolar powder patterns

(Fourier transform with respect to t1) for

(13CH3)2C(OH)SO3Na powder, for which R =

2.51 Å.

18

B. Continuous rf irradiation

Consider the following pulse sequence, called 2Q-HORROR [see N.C. Nielsen, H. Bildsoe, H.J.

Jakobsen, and M.H. Levitt, J. Chem. Phys. 101, 1805 (1994)]4:

In other words, the cycle here is a 90y-360x-90-y sequence, with delta-function 90 pulses and a

very long, weak 360 pulse. For this sequence,

2/iI2/tiI

2/iItiIRF

yRx

y1x

ee

ee)t(U

(V.9)

for 0 < t < 2R. The interaction representation Hamiltonians are then

]))IIII(tsin)IIII(itcos)IIII((4

3[)D,C,B,A(

])tsin)IIII()IIII(tcos)IIII((2

3[)D,C,B,A(

]2

tcos

2

tsin)IIII(

2

tsinII

2

tcosII(3[)D,C,B,A(

])2

tsinI

2

tcosI)(

2

tsinI

2

tcosI(3[)D,C,B,A()t(H

~

212121R2121R2121

21R2y1x2x1y2y1y2x1xR2y1y2x1x

21RR

2y1x2x1yR2

2y1yR2

2x1x

21R

2yR

2xR

1yR

1xD

II

II

II

II

(V.10)

)2

tsinI

2

tcosI()'D,'C,'B,'A()t(H

~ Ry

RxCSA

(V.11a)

)2

tsinI

2

tcosI()t(H

~ Ry

RxIICS

(V.11b)

Eq. (V.10) makes use of the identities 2/)2cos1(cos2 and 2/)2cos1(sin 2 , as well

as I = Ix + iIy. Although )t(H~

D contains both double-quantum and zero-quantum operator

terms, only the double-quantum terms oscillate in time. Therefore, only the double-quantum

terms are recoupled. Using the relations

R2

0 n,kRRR

2

tncos

2

tkcosdt ,

19

R2

0 n,kRRR

2

tnsin

2

tksindt , and

R2

0

RR 02

tncos

2

tksindt for integers k and n, it

is straightforward to show that 0H~

H~

ave,ICSave,CSA and that

)]cosBsinA)(IIII(i)sinBcosA)(IIII[(8

3H~

21212121ave,D (V.12)

Thus, the 2Q-HORROR sequence creates a pure double-quantum recoupled Hamiltonian

(assuming that the AHT approximation is valid and that the pulses are perfect). The average

Hamiltonian contains two double-quantum terms, whose magnitudes depend on the Euler angles

,,. Interestingly, the overall magnitude of ave,DH~

is independent of the angle. Recoupling

sequences with this property are called “-encoded”. This property causes the dipolar powder

pattern lineshape under 2Q-HORROR and other -encoded sequences to have two sharp horns

(see below), which means that the signal decay under ave,DH~

is strongly oscillatory. This is

good for quantitative measurements of internuclear distances, and also for double-quantum

filtering efficiencies.

Simulated and experimental 2Q-

HORROR data for a 13C-13C pair

with R 1.54 Å [from Nielsen et al,

J. Chem. Phys. 101, 1805 (1994)].

20

C. Finite-pulse recoupling sequences

Consider the following sequence, called “finite-pulse Radio-Frequency-Driven Recoupling” or

fpRFDR [see Y. Ishii, J. Chem. Phys. 114, 8473 (2001) and A.E. Bennett, C.M. Rienstra, J.M.

Griffiths, W.G. Zhen, P.T. Lansbury, and R.G. Griffin, J. Chem. Phys. 108, 9463 (1998)]5,6:

The cycle time is 4R, there is one 180 pulse in each rotor period, and the pulse length is a

significant fraction of the rotor period. The rf amplitude during each pulse is p

1 . For this

pulse sequence,

RpRIiIiIiIi

pRRIiIiIi/)3t(Ii

RpRIiIiIi

pRRIiIi/)2t(Ii

RpRIiIi

pRRIi/)t(Ii

RpIi

p/tIi

RF

4t3,eeee

3t3,eeee

3t2,eee

2t2,eee

2t,ee

t,ee

t,e

t0,e

)t(U

xyxy

xyxpRy

xyx

xypRx

xy

xpRy

x

px

(V.13)

Using the identity zxy IiIiIieee

, Eq. (V.13) can be simplified:

21

RpR

pRRIiIi/)2t(Ii

RpRIiIi

pRRIi/)2t(Ii

RpRIi

pRRIi/)t(Ii

RpIi

p/tIi

RF

4t3,1

3t3,eee

3t2,ee

2t2,ee

2t,e

t,ee

t,e

t0,e

)t(U

zxpRy

zx

zpRx

z

xpRy

x

px

(V.14)

Ignoring the I1I2 term (which is not recoupled, as explained above), the interaction-

representation dipole-dipole Hamiltonian is then

RpR2z1z

pRRp

)R2x

p

)R2z

p

)R1x

p

)R1z

RpR2z1z

pRRp

R2y

p

R2z

p

R1y

p

R1z

RpR2z1z

pRRp

)R2x

p

)R2z

p

)R1x

p

)R1z

Rp2z1z

pp

2yp

2zp

1yp

1z

D

4t3),II3()D,C,B,A(

3t3)],3t(

sinI3t(

cosI)(3t(

sinI3t(

cosI(3[)D,C,B,A(

3t2),II3()D,C,B,A(

2t2)],)2t(

sinI)2t(

cosI)()2t(

sinI)2t(

cosI(3[)D,C,B,A(

2t),II3()D,C,B,A(

t)],t(

sinIt(

cosI)(t(

sinIt(

cosI(3[)D,C,B,A(

t),II3()D,C,B,A(

t0)],t

sinIt

cosI)(t

sinIt

cosI(3[)D,C,B,A(

)t(H~

(V.15a)

This can be rewritten as

RpR2z1z

pRRp

R

p

R2z1x2x1z

p

R22x1x

p

R22z1z

RpR2z1z

pRRp

R

p

R2z1y2y1z

p

R22y1y

p

R22z1z

RpR2z1z

pRRp

R

p

R2z1x2x1z

p

R22x1x

p

R22z1z

Rp2z1z

ppp

2z1y2y1zp

22y1y

p

22z1z

D

4t3),II3()D,C,B,A(

3t3)],)3t(

cos)3t(

sin)IIII()3t(

sinII)3t(

cosII(3[)D,C,B,A(

3t2),II3()D,C,B,A(

2t2)],)2t(

cos)2t(

sin)IIII()2t(

sinII)2t(

cosII(3[)D,C,B,A(

2t),II3()D,C,B,A(

t)],)t(

cos)t(

sin)IIII()t(

sinII)t(

cosII(3[)D,C,B,A(

t),II3()D,C,B,A(

t0)],t

cost

sin)IIII(t

sinIIt

cosII(3[)D,C,B,A(

)t(H~

(V.15b)

Note that the signs of Iz1Ix2+Ix1Iz2 and Iz1Iy2+Iy1Iz2 are reversed in the third and fourth rotor

periods, relative to the first and second rotor periods. This is a consequence of the choice of

22

phases in the fpRFDR sequence [called XY-4 phases7], and it causes these single-quantum terms

to cancel out in the average Hamiltonian. Evaluating the integrals in the average Hamiltonian,

we find (omitting many intermediate steps)

]II3[

}D]2cos)22[cos()(16

3C]2sin)22[sin(

)(16

3

B]cos)[cos()4(2

3A]sin)[sin(

)4(2

3{

}t

sin)II3II3()D,C,B,A(dt2

)t

cosII3()D,C,B,A(dt4)II3()D,C,B,A(dt4{4

1

})t(H~

dt)t(H~

dt)t(H~

dt)t(H~

dt)t(H~

dt4{4

1H~

212z1z

pR21

2R

21

pR21

2R

21

pR21

2R

21

pR21

2R

21

0p

22y1y2x1x

0p

22z1z2z1z

R

3

3 D

2

2 DD0 DDR

ave,D

p

pR

p

pR

R

pR

R

pR

R

pR

p

II

(V.16)

Miraculously (or as a consequence of symmetry, as described below), the average dipole-dipole

Hamiltonian under the fpRFDR sequence is a zero-quantum operator with the same operator

form as the dipole-dipole coupling in a non-spinning sample. This has useful consequences in

certain applications, for example by allowing ideas that were originally developed for NMR of

static solids to be applied in MAS experiments8,9.

Under fpRFDR, it is also true that 0H~

H~

ave,ICSave,CSA .

1 in Eq. (V.16) is the rf amplitude during the 180 pulses. Note that ave,DH~

vanishes as

1 and 0p (i.e., in the delta-function pulse limit). As shown below, a different

recoupling mechanism, leading to a different average dipole-dipole Hamiltonian, comes into play

when the coupled spins have large chemical shift differences. This chemical-shift-driven

recoupling mechanism does not disappear in the delta-function pulse limit.

23

D. Chemical-shift-driven recoupling

1. Rotational resonance

Now consider the case where two dipole-coupled spins have a large difference in their isotropic

chemical shifts. Ignore chemical shift anisotropy for now. The nuclear spin Hamiltonian under

MAS is then

)t(H)II()II(

)t(H)II)(()II)((

)t(HII)t(H

D2z1z21

2z1z21

D2z1z2I1I21

2z1z2I1I21

D2z2I1z1I

(V.17)

where and are the sum and difference of the two chemical shifts (actually, resonance

offsets). Now go into an interaction representation with respect to the chemical shift terms. If

U(t) is the evolution operator for H(t), this interaction representation is defined by

)t(U~

)t(U)t(U ICS (V.18a)

]t)II(exp[]t)II(exp[

}t)]II()II([iexp{)t(U

2z1z2i

2z1z2i

2z1z21

2z1z21

ICS

(V.18b)

})'t(H~

'dtiexp{T)t(U~ t

0 D

(V.18c)

]t)II(exp[)]IIII](t)II(exp[)D,C,B,A(

)II2()D,C,B,A(

]t)II(exp[)]IIII(II2][t)II(exp[)D,C,B,A(

]t)II(exp[)II3](t)II(exp[)D,C,B,A(

]t)II(exp[)t(H]t)II(exp[

)t(U)t(H)t(U)t(H~

2z1z2i

21212z1z2i

21

2z1z

2z1z2i

212121

2z1z2z1z2i

2z1z2i

212z1z2z1z2i

2z1z2i

D2z1z2i

ICSD1

ICSD

II

(V.19)

Eq. (V.19) uses the facts that [(Iz1+Iz2),HD(t)] = 0, so the part of the chemical shift does not

affect )t(H~

D , and that [(Iz1-Iz2),Iz1Iz2] = 0, so the Iz1Iz2 part of HD(t) is not affected by the

chemical shifts. Also, Eq. (V.19) uses the identity )IIII(II 212121

2z1z21 II . Now,

using the identity

IeeIe iiIiI zz , Eq. (V.19) becomes

]II)tsinit(cosII)tsinit[(cos)D,C,B,A()II2()D,C,B,A(

)IIeIIe()D,C,B,A()II2()D,C,B,A()t(H~

212121

2z1z

21ti

21ti

21

2z1zD

24

(V.20)

Eq. (V.20) shows that the Iz1Iz2 part of the dipole-dipole coupling is not recoupled by the

chemical shift difference, but the “flip-flop” part can be recoupled if R|| or R2|| .

These are the “n = 1” and “n = 2” rotational resonance conditions10-14. At the rotational

resonance conditions, )t(H~

D is periodic with period R and AHT can be applied. The average

dipole-dipole Hamiltonian at rotational resonance is

21n,2i2

n,1i

41

21n,2i2

n,1i

41

ave,D II]e)iDC(e)iBA[(II]e)iDC(e)iBA[(H~

(V.21)

for n = 1 or n = 2. Note that this is a zero-quantum operator, but it is not the same as the zero-

quantum operator created by the fpRFDR sequence.

In a system with many 13C-labeled sites, rotational resonance allows pairs of spins with

particular chemical shift differences to be recoupled selectively15,16, as shown below. However,

unless the MAS frequency is varied during the pulse sequence, the couplings can not be switched

on and off. Therefore, other approaches to frequency-selective homonuclear dipolar recoupling

that employ rf pulses and do not depend on being exactly on rotational resonance have been

developed by several groups17-22.

13C NMR spectra of U-15N,13C-alanine powder, recorded at 100.8 MHz, with MAS frequencies

indicated by dashed lines. At the rotational resonance conditions, the recoupled lines become

dipolar powder patterns, while other lines remain sharp. Close to rotational resonance conditions

(C and D), the main NMR lines of strongly coupled pairs exhibit apparent shifts on the order of

0.5 ppm, which can complicate accurate measurements of chemical shifts in uniformly labeled

samples.

25

2. Radio-Frequency-Driven Recoupling

Consider the Hamiltonian in Eq. (V.17), but with an additional HRF(t) term that corresponds to

the following pulse sequence, with m being an arbitrary positive integer:

This sequence, called SEDRA23 or more commonly RFDR6,24, can be analyzed by first

transforming the Hamiltonian into an interaction representation with respect to HRF(t), and then

into an interaction representation with respect to the chemical shifts. Following the same

principles as used above for other recoupling sequences, the Hamiltonian in the interaction

representation with respect to HRF(t) is

ccD2z1z21

2z1z21

ccD2z1z21

2z1z21

cD2z1z21

2z1z21

t4/3),t(H)II()II(

4/3t4/),t(H)II()II(

4/t0),t(H)II()II(

)t(H~

(V.21)

HD(t) is not affected directly by the delta-function 180 pulses, but the chemical shifts change

sign between the two 180 pulses. Therefore, the isotropic chemical shifts now appear to be

time-dependent. In the second interaction representation, the dipole-dipole coupling becomes

ccc2z1z2i

Dc2z1z2i

ccc

2z1z2i

Dc

2z1z2i

c2z1z2i

D2z1z2i

D

t4/3)],t)(II(exp[)t(H)]t)(II(exp[

4/3t4/)],t2

)(II(exp[)t(H)]t2

)(II(exp[

4/t0],t)II(exp[)t(H]t)II(exp[

)t(H~~

(V.22)

Ignoring the Iz1Iz2 part of HD(t), which as shown above is not recoupled, Eq. (V.22) can be

written as

cc21)t(i

21)t(i

21

cc21)t2/(i

21)t2/(i

21

c21ti

21ti

21

D

t4/3],IIeIIe[)D,C,B,A(

4/3t4/],IIeIIe[)D,C,B,A(

4/t0],IIeIIe[)D,C,B,A(

)t(H~~

cc

cc (V.23)

26

The average Hamiltonian is then

)IIII)](2sinD2cosC()4(m

)2/msin()sinBcosA(

)(m

)2/msin()1[(H

~~212122

RR

R22

RR

Rmave,D

(V.24)

Note that the average dipole-dipole Hamiltonian, evaluated in the double interaction

representation described above, is non-zero for nearly all values of the chemical shift difference

. This is because the 180 pulses, spaced mR apart, force the flip-flop term in )t(H~~

D to be

periodic with period 2mR. For nearly all values of , )t(H~~

D has non-zero Fourier components

at R and/or 2R. However, 0H~~

ave,D as 0 . Thus, the recoupling mechanism for

RFDR in the delta-function pulse limit (i.e., when the 180 pulses are very short compared with

R) is qualitatively different from the recoupling mechanism in the finite-pulse limit (i.e., when

the 180 pulses occupy a significant fraction of the rotor period).

In actual experiments, the phases of the 180 pulses are usually chosen to follow an XY-4 or

high XY-n phase pattern7, because XY-n phase patterns compensate for rf inhomogeneity,

resonance offsets, and other imperfections in the 180 pulses. This makes the RFDR technique

(and the fpRFDR version) quite robust and useful in many experimental situations.

VI. Symmetry principles for recoupling sequences

A. Levitt’s “C” sequences

Malcolm Levitt and his colleagues have developed an approach to the development of recoupling

sequences that relies on general symmetry properties of pulse sequences, which lead to

“selection rules” that reveal which types of interactions can be recoupled by a sequence with a

given symmetry. Sequences belonging to two distinct symmetry classes have been described.

The first class includes "C" sequences, comprised of rf blocks (called C elements) that produce

no net rotation of spin angular momenta25,26. The general form for a C sequence is:

In other words, the C sequence contains N repetitions of the C element, with overall rf phase

shifts that increase in units of , and with a total cycle time of n rotor periods. The phase

27

increment satisfies N/2 . N, n, and are positive integers. The symmetry is represented

by the symbol CNn.

To analyze the effect of a CNn sequence with AHT, one considers a general nuclear spin

Hamiltonian under MAS that is a sum of terms of the form 0tim

m0m TeA)t(H R

in the

rotating frame [i.e., before transforming to an interaction representation with respect to HRF(t)].

Am is a function of the Euler angles ,, discussed above. m is -2, -1, 1, or 2, and is another

positive integer.

T0 is the "element of an irreducible tensor operator of rank " that commutes with the total spin

angular momentum Iz. Without going into the details of irreducible tensor operators, this means

T0 is an operator that is a member of a set of 2+1 operators {T}, with being an integer that

satisfies - . For dipole-dipole couplings, = 2 and the relevant set of operators is

)II3(T

)IIII(T

IIT

212z1z6

120

2z121z21

12

2121

22

II

(VI.1)

Thus, T2 is a -quantum operator. For isotropic and anisotropic chemical shifts, the relevant

operators have = 1.

Important properties of irreducible tensor operators include:

TeeTe iiIiI zz (VI.2a)

T)1(eTe zx iIiI (VI.2b)

T)1(T†

(VI.2c)

Elements of the set of operators {T} are transformed into one another by rotations of spin

angular momentum (i.e., by rf pulses in NMR experiments). Therefore, in the interaction

representation with respect to the first C element, the nuclear spin Hamiltonian is a sum of terms

of the form

Te)t(A~

)t(H~ tim

mmR in the interval 0 < t < nR/N. There are 4(2+1)

such terms. The interaction representation Hamiltonian for the kth C element must then be a sum

of terms of the form

)eTe(e)t(A~

)t(H~ )1k(iI)1k(iItim

mmzzR

(VI.3)

in the interval (k-1)nR/N < t < knR/N, taking into account the effect of the overall rf phase shift

as in Eqs. (III.8-III.11). The average Hamiltonian for the kth C element will then be a sum of

terms of the form

28

timN/n

0 mN/n)1k(im)1k(i

R

N/kn

N/n)1k( mR

ave,m

RrRR

r

R

e)t(A~

dteen

NT

)t(H~

dtn

N)k(H

~

(VI.4)

Eq. (VI.4) makes use of Eq. (VI.2a) and the fact that )t(A~

m is periodic, with period equal to

nR/N (the length of one C element). The total average Hamiltonian, for the entire cycle time

nR, is obtained by summing over contributions from all C elements. The total average

Hamiltonian will then contain 4(2+1) terms of the form

N

1k

N/)mn)(1k(2itimN/n

0 mR

N

1k

N/n)1k(im)1k(itimN/n

0 mR

N

1k

timN/n

0 mN/n)1k(im)1k(i

Rave,m

ee)t(A~

dtn

NT

eee)t(A~

dtn

NT

}e)t(A~

dtee{n

NTH

~

Rr

RRRr

RrRR

(VI.5)

For such a term to be non-zero, the sum over k must be non-zero. But it turns out that the

following relation is always true, for any positive integer N:

NZq,0

NZq,Ne

N

1k

N/q)1k(2i (VI.6)

where Z is some other integer. So the quantity - mn must be zero or another integer multiple

of N for the average Hamiltonian under a C sequence to contain non-zero -quantum terms that

arise from MAS-induced oscillations at frequency mR. This is the selection rule for CNn

sequences.

The C7 and POST-C7 recoupling sequences27,28 are good examples. For these sequences, N = 7,

n = 2, and = 1, which imply the selection rule that -2m = 0, 7, 14, etc. Thus, terms with = 2

and m = 1 or = -2 and m = -1are recoupled (corresponding to double-quantum dipolar

recoupling). Terms with = 0 and 1 are not recoupled, corresponding to the absence of CSA

recoupling and the absence of both zero-quantum and one-quantum dipolar recoupling. (Recall

that m has only the values 1 and 2, but not 0, under MAS.) C7 and POST-C7 differ in the

choice of the C element itself, which is better compensated for resonance offsets in the POST-C7

case.

29

B. Levitt’s “R” sequences

The second symmetry class considered by Levitt and coworkers includes the "R" sequences29,30,

which have the general form

The sequence consists of N repetitions of an "R element" in a cycle time nR. N is an even

integer. The rf pulses in the R element produce a net rotation by 180 around x. The R version

alternates with the R’- version. For the R version, all rf pulses in the R element are phase-

shifted by N/ . For the R’- version, all rf pulses in the R element are first reversed in

sign, then phase-shifted by -. The symmetry is represented by the symbol RNn.

Analysis of R sequences is similar to analysis of C sequences, but the fact that one R element

produces a net 180 rotation means that T and T- terms must be treated together [see Eq.

(VI.2b)]. For the Hamiltonian to be Hermitian, these must occur in the interaction representation

as terms of the form

Te*)t(A~

)1(Te)t(A~

)t(H~ tim

mtim

mmRR [so that

†mm )t(H

~)t(H

~ ; see Eq. (VI.2c)].

Considering only one m, combination initially, the evolution operator for the R element,

starting at t = 0, is

N/n

0 0tim*

mtim

mRFRR RR ]}T)eAeA()t(H[dtiexp{TU

(VI.7a)

Then, in the AHT approximation and considering only one m,, combination,

N/n

0

tim*m

timm

iIR

R RRx )]Te)t(A~

)1(Te)t(A~

(dtiexp[eU (VI.7b)

In going from Eq. (VI.7a) to Eq. (VI.7b), the fact that HRF(t) produces a net rotation by 180

around x has been used. In the interaction representation with respect to HRF(t), T0 becomes a

sum of terms of the form c(t)T + (-1) c(t)*T-. In Eq. (VI.7b), )t(cA)t(A~

mm .

30

Terms proportional to

Tetim R and

Te

tim R are not shown explicitly, but of course are

also present.

The corresponding equations for the phase-reversed element R’, starting at t = 0, are [see Eq.

(III.12)]

N/n

0 0tim*

mtim

miI

RFiI

'RR RRxx ]}T)eAeA(e)t(He[dtiexp{TU

(VI.8a)

N/n

0

tim*m

timm

iI

iIN/n

0

tim*m

timm

iIiI'R

R RRx

xR RRxx

)]Te)t(A~

)1(Te)t(A~

(dtiexp[e

e})]Te)t(A~

)1(Te)t(A~

(dt)1(iexp[e{eU (VI.8b)

Eqs. (VI.7) and (VI.8) show that UR and UR’ differ by exchange of T and T- in the interaction

representation.

Including rotations about z to account for the phase shifts, the evolution operator for the first

RR’- pair is

N/n

0

itimm

itimm

N/n

0

i3timN/mn2im

i3timN/mn2im

iI4

N/n

0

iItimm

timm

iI2

N/n

0

timN/mn2im

timN/mn2im

iI

N/n

0

iItimm

timm

iI2iI

N/n

0

timN/mn2im

timN/mn2im

iIiI

N/n

0

iItimm

timm

iI2iI

N/n2

N/n

timm

timm

iIiI

N/n

0

iItimm

timm

iIiI

N/n2

N/n

iItimm

timm

iIiI0

R RR

R RRz

R zRRz

R RRz

R zRRzx

R RRxz

R zRRzx

R

R

RRxz

R zRRxz

R

R

zRRxz

]}Tee*)t(A~

)1(Tee)t(A~

[dtiexp{

}]Teee*)t(A~

)1(Teee)t(A~

[)1(dtiexp{e

e]}Te*)t(A~

)1(Te)t(A~

[dtiexp{e

}]Tee*)t(A~

)1(Tee)t(A~

[)1(dtiexp{e

e]}Te*)t(A~

)1(Te)t(A~

[dtiexp{ee

]}Tee*)t(A~

)1(Tee)t(A~

[dtiexp{ee

e]}Te*)t(A~

)1(Te)t(A~

[dtiexp{ee

]}Te*)t(A~

)1(Te)t(A~

[dtiexp{ee

e]}Te*)t(A~

)1(Te)t(A~

[dtiexp{ee

e]}Te*)t(A~

)1(Te)t(A~

[dtiexp{eeU

(VI.9)

The total evolution operator is

1)2/N(

0k

R'RNI2i

total )k(U~

)k(U~

eU z (VI.10)

with )k(U~

R and )k(U~

'R defined by

31

]}Teee*)t(A~

)1(

Teee)t(A~

[dtiexp{)k(U~

)1k4(itimN/mnk4im

N/n

0

)1k4(itimN/mnk4imR

R

R R

(VI. 11a)

]}Teee*)t(A~

)1(

Teee)t(A~

[)1(dtiexp{)k(U~

)3k4(itimN/)1k2(mn2im

N/n

0

)3k4(itimN/)1k2(mn2im'R

R

R R

(VI.11b)

Note that 2N in Eq. (VI.10) is a multiple of 2, so the operator zNI2ie has no effect. The

average Hamiltonian for the entire R sequence is obtained by summing the exponents in UR(k)

and UR’(k) over all values of k and dividing by nR. Therefore, the coefficient of T in the

average Hamiltonian arising from MAS-induced oscillations at mR is proportional to

]N/k)mn(2iexp[)1(e]e)t(A~

[dt

]N/k)mn(2iexp[)1(]N/k)mn(2iexp[e]e)t(A~

[dt

]N/)2/1k)(mn(4iexp[)1(]N/k)mn(4iexp[e]e)t(A~

[dt

1N

)evenandodd(0k

kN/iN/n

0

timm

1N

)odd(1k

2N

)even(0k

N/iN/n

0

timm

1)2/N(

0k

1)2/N(

0k

N/iN/n

0

timm

R R

R R

R R

(VI.12)

Eq. (VI.12) shows that if is even, the coefficient of T is zero (i.e., no recoupling) unless mn -

is an even multiple of N/2. If is odd, the coefficient of T is zero unless mn - is an odd

multiple of N/2. These are the selection rules for RNn sequences.

Many examples of RNn recoupling sequences have been reported. The fpRFDR sequence

described above is a very simple example, for which n = 4, N = 4, and = 1. The symmetry

selection rules indicate that dipole-dipole couplings ( = 2) can be recoupled in a zero-quantum

( = 0) form, with |m| = 1 or |m| = 2. Chemical shifts ( = 1; || = 0 or 1) can not be recoupled.

This is in agreement with the detailed calculations described above.

C. Cyclic time displacement symmetry and constant-time recoupling

Another useful symmetry property of recoupling sequences involves their behavior when all

pulses are cyclically displaced in time within the cycle time c = nR. In the picture below, a

block of pulses (or delays) of length 1, called P1, is displaced by 2, causing the remaining block

P2 to move from the end to the beginning of the cycle:

32

What is the effect of this cyclic displacement on a recoupled Hamiltonian? If the rotations

produced by rf pulse blocks P1 and P2 are U1(t) and U2(t), with time t measured from the

beginning of each block, then the total rotation before the cyclic displacement is URF(t):

R11112

11RF

nt),(U)t(U

t0),t(U)t(U (VI.13)

The total rotation after the cyclic displacement is UCD(t):

R22221

22CD

nt),(U)t(U

t0),t(U)t(U (VI.14)

In the interaction representation with respect to the original pulse sequence, a Hamiltonian term

of the form 0tim

mm TeA)t(H R

under MAS becomes (recalling that 1 + 2 = nR)

211201

21

11'timim

m

1101

1tim

m

R1111201

121

11tim

m

1101

1tim

mm

't0),(U)'t(UT)'t(U)(UeeA

t0),t(UT)t(UeA

nt),(U)t(UT)t(U)(UeA

t0),t(UT)t(UeA)t(H

~

R1R

R

R

R

(VI.15)

with t’ = t + 1. In the interaction representation with respect to the cyclically displaced pulse

sequence, the same Hamiltonian term becomes

33

122101

11

22''timim

m

2201

2tim

m

R2222101

211

22tim

m

2201

2tim

mm

''t0),(U)''t(UT)''t(U)(UeeA

t0),t(UT)t(UeA

nt),(U)t(UT)t(U)(UeA

t0),t(UT)t(UeA)t(H

R2R

R

R

R

(VI.16)

with t’’ = t + 2. In both cases, the average Hamiltonian is the integral of the interaction

representation Hamiltonian over time from 0 to nR, divided by nR. According to Eqs. (VI.15)

and (VI.16), the integral in both cases is the sum of two integrals, over time intervals of length 1

and 2. Recalling that 1ee 2R1R imim

and 1)(U)(U 1122 , it can be shown that the

integrands for the cyclically displaced pulse sequence are equal to the integrands for the original

pulse sequence after rotation by U2(2)-1 = U1(1) and multiplication by 2Rim

e . Thus,

)(UH~

)(UeH 22ave,m1

22im

ave,m2R

(VI.17)

Eq. (VI.17) summarizes the effect of a cyclic time displacement on the average Hamiltonian for

an arbitrary recoupling sequence.

Why is this useful? As an example, consider a general recoupling sequence that ends in two

periods of length R/3. During these final two periods, either no pulses are applied or the applied

pulses produce a net rotation of 2. For such a sequence, two successive cyclic displacements by

R/3 multiply the average Hamiltonian by 3/m2ie and 3/m4ie . For m = 1 and m = 2, the

total average Hamiltonian for the pulse sequence obtained by concatenating the original

sequence with the two displaced versions will be zero, because 1+ 3/m2ie + 0e 3/m4i . This

provides a simple means of creating a “constant-time” dipolar recoupling technique31, as shown

below using cyclically displaced versions of the fpRFDR sequence:

34

Construction of a constant-time dipolar recoupling sequence from the fpRFDR sequence.

During the “k2” period, dipolar recoupling by the A, B, and C blocks cancels due to Eq. (VI.17).

Only the “k3” period has a net recoupling effect. Thus, by decrementing k2 and incrementing k3

while keeping k2 + k3 constant, the effective recoupling period D’ can be increased from 0 to

12k1(k2 + k3)R. For all values of D’, the total fpRFDR period is 12k1(k2 + k3)R and the total

number of pulses is constant. Thus, signal decay due to spin relaxation, incomplete proton

decoupling, and pulse imperfections is minimized. The dependence of NMR signals on D’ is

due primarily to dipole-dipole couplings rather than these extraneous effects, allowing the

recoupling data to be analyzed in a quantitative manner. [from J. Chem. Phys. 126, 064506

(2007)]

35

Measurements of backbone 15N-15N distances (which constrain the torsion angle) in a model

helical peptide in lyophilized form using the PITHIRDS-CT recoupling technique. Both 15N-

detected data (b) and 13C-detected data (a) are shown, along with numerical simulations (d).

[from J. Chem. Phys. 126, 064506 (2007)]

36

Appendix: Derivation of time-dependent dipole-dipole coupling under MAS

In angular frequency units and in a molecule-fixed frame (i.e., in an axis system that has any

specific orientation relative to the molecular structure or crystallite unit cell), the full dipole-

dipole coupling can be written as

2

2m

m2m2m

3

2I

D TY)1(R

3H

(A.1)

with

)II3(T

)IIII(T

IIT

212z1z6

120

2z121z21

12

2121

22

II

(A.2)

)1cos3(Y

ecossinY

esinY

2

6

120

i12

i2221

22

(A.3)

In Eq. (A.3), and are angles that specify the direction of the internuclear vector in the

molecule-fixed frame. It is important to realize that the Y2m functions are simply (complex)

numbers for a given pair of coupled nuclei. These numbers do not change when the

transformations described below are carried out. It is the T2m operators that change under these

transformations.

Suppose the molecule-fixed frame (with axes x',y',z' as in Section III.A) is related to a MAS-

rotor-fixed frame (with axes x'',y'',z'') by Euler angles . The spinning axis is taken to be z''.

x'' and y'' are perpendicular to the spinning axis. In other words, the x',y',z' axes are transformed

to the x'',y'',z'' axes by three consecutive rotations: first, a rotation about z' by ; second, a

rotation about the intermediate y axis by ; third, a rotation about the z'' axis by . Then the

irreducible tensor operator components are transformed according to

2

2'm

'm2

)2(

m'mm2 T),,(DT (A.4a)

with

'im)2(

m'm

im)2(

m'm e)(de),,(D (A.4b)

and

37

2

41)2(

22

2

41)2(

22

21)2(

12

21)2(

12

21)2(

21

21)2(

21

21)2(

11

21)2(

11

2

4

6)2(20

)2(20

2

6)2(10

)2(10

2)2(00

)cos1()(d

)cos1()(d

)cos1(sin)(d

)cos1(sin)(d

)cos1(sin)(d

)cos1(sin)(d

)1cos2)(cos1()d

)1cos2)(cos1()d

sin)(d)(d

cossin)(d)(d

2/)1cos3()(d

(A.5)

{ ),,(D)s(

m'm } are Wigner rotation matrix elements, equal to { sm|eee|'sm zyz iSiSiS } for

a spin-s particle, with m and m' being quantum numbers for Sz. { )(d)s(

m'm } are reduced rotation

matrix elements, equal to { sm|e|'sm yiS }.

The order and signs of the Euler angles may look strange in Eq. (A.4a). This is because Eq.

(A.4a) indicates how an irreducible tensor operator defined in the initial axis system appears in

the final axis system. Eq. (A.4a) does not indicate what happens when the rotation

zyz iSiSiSeee),,(R is applied to the irreducible tensor operator itself. The former case

can be called a "passive rotation", and the latter case an "active rotation". The distinction can be

confusing, leading to a variety of errors, which may or may not be significant, depending on the

whether the signs of angles are important, etc. The following article is a useful review of

rotations, Euler angles, irreducible tensors, spherical harmonics, and Wigner rotation matrices:

A.A. Wolf, "Rotation Operators", Am. J. Phys. 37, 531-536 (1969).

In Eq. (A.1), the T2m operators are functions of spin angular momentum components in the

molecule-fixed frame. When written in terms of spin angular momentum components in the

rotor-fixed frame, HD becomes

2

2'm,m

'm2

im)2(

m'm

'im

m2

m

3

2

ID Te)(deY)1(

R

3H

(A.6)

In Eq. (A.6), the T2m' operators are functions of spin angular momentum components in the rotor-

fixed frame. Next, the rotor fixed frame is transformed to the laboratory-fixed frame (with axes

38

x,y,z) by a time-dependent rotation about z'' by Rt that makes y'' perpendicular to z and

coincident with y (because z'' is the spinning axis and because we define t = 0 to be a time when

y'' and y coincide), followed by a rotation about y by the magic angle M that brings z'' to z.

Under these rotations,

2

2''m

''m2RM

)2(

'm''m'm2 T)t,,0(DT (A.7)

2

2''m,'m,m

''m2

)t('im

M

)2(

'm''m

im)2(

m'mm2

m

3

2

I

2

2''m,'m,m

''m2

t'im

M

)2(

'm''m

im)2(

m'm

'im

m2

m

3

2

ID

Te)(de)(dY)1(R

3

Te)(de)(deY)1(R

3H

R

R

(A.8)

HD in Eq. (A.8) is the full dipole-dipole coupling as it appears in the laboratory frame under

MAS, written in terms of the angles and that specify the direction of the internuclear vector

relative to the molecule-fixed axes and the angles , , and that specify the orientation of the

molecule in the rotor. The T2m'' operators in Eq. (A.8) are functions of spin angular momentum

components in the laboratory frame.

In high field, terms in Eq. (A.8) with m'' 0 are truncated by the much larger Zeeman interaction

with the external magnetic field. In other words, only the m'' = 0 term affects the frequencies

and intensities of NMR signals or affects coherent spin dynamics in high field. Furthermore, at

the magic angle, the m' = 0 term vanishes because 0)(d M

)2(

00 . This what makes the magic

angle so magical. Then the remaining terms oscillate at R and 2R, and average to zero unless

recoupling techniques are employed.

2

0'm2'm,m

20

)t('im

M

)2(

'm0

im)2(

m'mm2

m

3

2

ID Te)(de)(dY)1(

R

3H R

(A.9)

Explicit expressions for the A, B, C, and D coefficients in Eq. (III.2) and the Am coefficients

used in Section VI can be derived from Eq. (A.9). In particular, if

)II3(eA)t(H 212z1z

2

)0'm(2'm

)t('im'mD

R II

, then

39

]sin)1cos3(2

e)cos1(sine2sin2e)cos1(sine2sin2

e)cos1(esine)cos1(e[sinR16

e)(dY)1(R2

A

22

iiii

i22i22i22i22

3

2

I

2

2m

im)2(

m,2m2

m

3

2

I2

(A.10a)

]2sin)1cos3(

e)1cos2)(cos1(sine2sine)1cos2)(cos1(sine2sin

e)cos1(sinesine)cos1(sine[sinR24

e)(dY)1(R2

A

2

iiii

i2i22i2i22

3

2

I

2

2m

im)2(

m,1m2

m

3

2

I1

(A.10b)

In the special case of a two-spin system, we can take = 0, which implies 3

22I

2R4

sinA

and 3

2

I1

R22

2sinA

. The coefficients in Eq. (III.2) are given by A = 2Re(A1), B =

-2Im(A1), C = 2Re(A2), and D = 2Im(A2).

40

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