Lecture 6. A More Quantitative Description of Pulsed NMR: Product Operator Formalism. So far we ve been describing NMR phenomena using vectors, with a minimal of mathematics. Sure, we discussed the Bloch equations and performed some simulations using them, but we ve not actually calculated anything, primarily because the Bloch equations are cumbersome to use, and not very intuitive. Now you might ask why we have used such a formalism when the quantum mechanical description of NMR can generally be solved analytically for any two-spin system? We ve steered away from the QM description for two reasons: first, it also is rather cumbersome and un-intuitive, and second, I don t know how to do the calculations myself. We are not alone in this predicament, and a different formalism, based on the quantum mechanical rigor with the intuitive insights of the vector diagrams, has been developed. This formalism, the product operator formalism, is general, and quite easy to use. We ll use this formalism for the vast majority of experiments discussed in this course, and we ll introduce it here. 6.1. Review, or Re-introduction, of Product Operators. The introduction of product operators went a bit rough, so I ll back up a bit and introducce the quantum mechanics more thoroughly in an effort to describe the origin and basis for the product operators. The starting point for elementary descriptions of quantum mechanics is the time-dependent Schrödinger Equation, ih d Ψ() t ĤΨ() t. [6.1] dt In this equation, the state of the system (the wavefunction) is denoted by Ψ(t), which is timedependent if the system changes with time (as they generally do in NMR experiments). The evolution of the system is calculated using a mathematical description of the potential, known as the Hamiltonian, Ĥ. The Hamiltonian describes the forces acting on the system and generally contains terms for the kinetic and potential energy. Eqn 7 has the general solution Ψ() t ψτ ( )exp( iet) where the time-independent part of the system is represented by ψ(τ), or, in general by ψ. The ψ are a set of N functions, called basis functions or vectors, that make up the total wavefunction. The amount that each basis function contributes to the total wavefunction is indicated by a weighting coefficient, c n, N Ψ c n ψ n. [6.3] n 1 If the time-dependence of Ψ(t) is written explicitly, then it is introduced as the time-dependent variation of the value of the coefficients. A good example of this is the need for resonance structures in organic chemistry. Each molecule can be described by a wavefunction (admittedly [6.2] 1
a very complicated one for most molecules) that contains terms for all of the various structures the molecule can have in its ground excited states (these terms are the basis functions). For many molecules, a single structure adequately represents all of the chemistry and energetics of the molecule (an example is cyclohexane), and the weighting coefficients for the individual basis functions are very close to zero except for those in the ground state (consisting of chair and boat conformers). On the other hand, for some molecules, like an aromatic (or a peptide), the chemical and physical properties are not representable by just a few terms in the wavefunction, and so the weighting coefficients of many basis functions will be significant. In general, we do an experiment to measure some physical parameter of the system that is relevant to us. Mathematically, this interogation of the system is done using functions that operate on the wavefunction to provide the parameter of interest. This introduces the concepts of operators and expectation values, two concepts central to quantum (and classical, for that matter) mechanical calculations. An operator can be a function as simple as multiplication by a number or by x, or can be more complicated, like first or second derivatives, etc. Operators that operate on a wavefunction to provide a physically observable quantity are known as Hermitian operators. An example of an Hermitian operator would be to provide the position, or momentum of a particle in the system of interest. Another Hermitian operator provides the z-component of the total, orbital, or spin angular momentum. When these Hermitian operators are applied to the wavefunction, the mathematical expression A Ψ * ÂΨdτ [6.4] is used. In this expression, the wavefunction is operated on by the operator Â, then multipled by the complex conjugate of the wavefunction (complex comjugate means multiplied by i 1 ), and integrated over some volume in space. The result of this calculation,, is the expectation value, which directly corresponds to the expected value measured on a large number of similar systems. The Hamiltonian operator is one example of an Hermitian operator, and the expectation value reported by the Hamiltonian is the energy of the system, hence the familar equation E Ψ * ĤΨdτ, or [6.5] ĤΨ As an example of the ideas discussed above, we ll calculate the expectation value for the z component of the spin angular momentum of a spin-1/2 particle. The spin-part of the wavefunction for this system can be written in polar coordinates as A EΨ. [6.6] Ψ c α ψ α + c β ψ β [6.7] Ψ c α exp( im s φ) + c β exp( im s φ) c α exp 1 [6.8] 2 --iφ 1 + c β exp -- iφ 2 where the c α and c β are the weighting coefficients for the two terms of the spin system, and ms is the spin angular momentum quantum number; in this case, it is ±1/2. To obtain the z-component 2
of the angular momentum, we operate on Ψ using that operator, which is I z : I z ( h i) φ [6.9] which gives I z Ψ I z c α exp 1 2 --iφ 1 + I z c β exp -- iφ 2 ( h i)c 1 α φ exp --iφ 2 + ( h i)c β φ 1 exp -- iφ 2 or I z Ψ h. [6.10] 2 --c α exp 1 2 --iφ h -- 1 c 2 β exp -- iφ 2 To get the expectation value, we have to multiply each of these terms by their complex conjugate to give (for the first term; analogous expression for the second term can easily be obtained) Ψ*I z Ψ h 2 --c * 1 α exp -- iφ c 2 αexp 1 2 --iφ h 2 --c α * cα [6.11] This gives us the expected value when we realize that the product of the coefficients is unity when the wavefunctions are properly normalized. Thus, the expectaion value of the z-component of the angular momentum has two values: +1/2 and -1/2, and the probability of getting either component is proportional to the weighting coefficients of each state. A slightly different notation for these calculations is the Dirac notation. Here, the wavefunction is the sum of a set of N basis vectors, where the n Ψ c n n n 1 notation indicates that the function n is a vector representing the individual basis vectors (or states). In the Dirac notation, the expectation value is still given by Eqn. 6.4, but the exact form has changed to [6.12] A Ψ * * ÂΨdτ Ψ A Ψ c m cn m  n. [6.13] where <Ψ is the complex conjugate vector of the Ψ> vector. Similarly, N and c m * are the complex conjugates of n and c n, respectively. In the Dirac notation, the m A n terms are the constants and the value of the expectation value is determined by the products of the coefficients * c m cn. This provides the expectation value for a series of measurements performed on a single particle. To obtain the expectation value for a single measurement performed on an ensemble of particles, we take the average over the products of the coefficients, and it is this ensemble average that is the density matrix or density operator; mn m 3
σ mn c m * cn [6.14] (it is called a matrix because mathematically, it is a matrix (of dimensions m x n), with individual elements given by the product indicated in Eqn 6.14). In a system with multiple spins, the total wavefunction, Ψ Tot, can be represented by a linear combination of the wavefunctions for each of the K particles, K Ψ Tot c k Ψ k. [6.15] k 1 The density operator can also be represented by the sum of the density matrices for each of the K particles, σ Tot K b k σ, [6.16] k 1 and since the individual σs are related to the expectation values for some operators, the total density matrix can be represented by a weighted summation over some set of basis operators that will operate on the total wavefunction, K σ Tot b k ()B t k. [6.17] k 1 The basis operators can be the angular momentum operators in Cartersian coordinates (I x, I y, and I z ) or some other set of basis operators. These Cartesian operators, then, provide the projection of the total magnetization vector onto one of the three Cartesian axes (there is generally a fourth vector, the unity vector, represented as E, which is the sum of the squares of the projections on the three axes, and represents the total magnetization present). Therefore, in the product operator formalism, terms in the density matrix are given by the various operators that correspond to particular elements of the density matrix. Now, I ll tie this in with what we discussed in the last lecture. Last time, we said that the timedependent behavior of the density matrix is given by the Liouville-von Neumann equation, d σˆ () t [ Ĥ, σˆ ( 0) ] dt exp( iĥt)σˆ ( 0)exp( iĥt) [6.18] which can now be re-written as K d σˆ t dt b ( 0 )exp ( iĥt)bˆ kexp( iĥt), [6.19] k 1 where the B k are the individual basis operators (or vectors). The solution to this equation is given 4
by [6.20] (where θ is an angle that depends on the nature of the Hamiltonian), which is the equation we use to calculate the product operators. In other words, the operators that we calculate directly correspond to terms in the density matrix, and these terms describe an amplitude, or a probability, for a certain type of magnetization being present in an NMR experiment. This magnetization can be directly observable (like I x, I y and I x I z ), or not (like I x I y, etc). The product operator terms can sometimes be identical to the vectors that we used previously, but more often, will represent things that were not easily (if at all) representable by vectors. 6.2. Product Operators. σ() t Bˆ kcosθ + [ Ĥ, Bˆ k] sinθ The time-dependent response of an arbitrary spin system to a series of rf pulses and freeprecession delays is represented by the time-dependent evolution of the density operator, σ, d σ() t i[ Ĥ()σt t, ] dt [6.21] where σ is the density operator, H is the Hamiltonian for specific events (e.g., pulses versus delays, etc.), and the quantity in the square brackets is known as a commutator, and is given by [ AB, ] AB BA for two arbitrary operators A and B. Time-dependent solutions to Eqn 6.21 are of the form [6.22] and, in general, with solutions σ() t exp( iĥt)σ( 0)exp( iĥt) R() t exp( ibbˆ t)âexp( ibbt ˆ ) [6.23] [6.24] ( ibt) R() t A ( ibt) [ Bˆ 2, Â] ------------- [ Bˆ,[ Bˆ, Â] ] ( ibt) 3 + ------------- [ Bˆ,[ Bˆ,[ Bˆ, Â] ]] +, [6.25] 2 3 or, in simplified form, Rt () Âcosb + [ Bˆ, Â] sinb. [6.26] This calculation is performed for every distinct step of a pulse sequence: individual pulses are applied as Hamiltonians; chemical shift evolution is calculated using its Hamiltonian; scalar coupling has still another Hamiltonian, and etc. For a pulse sequence with several pulses and free-precession delays, the Hamiltonians are applied sequentially, as we ll see below. The density matrix is a mathematical description of the spin system, and is given by the sum of all spins and spins states weighted by time-dependent factors (b s (t)), σ() t b s ()B t s s [6.27] 5
The Bs are the set of basis vectors describing the different spin states, and are given by B s 2 q 1 [6.28] where N is the number of spin 1/2 nuclei, q ranges from 0,1,2,... N and represents the number of spin operators in a given product operator term, k an index for the nucleus number, and v the x, y, or z axis. (see what I mean about the non-intuitive nature of the QM version? Don t worry, we re almost to the easy part again). In the Product Operator Formalism (Sorensen et al. 1983. Progress in NMR Spectroscopy. v16, p 163-192. ), we will be particularly interested in the effect of the various Hamiltonians have on the individual basis vectors. In many cases, we ll see that the Product Operator and vector formalisms are identical (except for sign changes from time to time). For example, for 2 spin-1/2 nuclei, you would have the following basis vectors: Table 1: basis vectors for 2 spins (most common situation) N ( ) I kv k 1 q product, or basis operator 0 unity operator (invariant to any rotations). 1 single spin operators: I 1x, I 1y, I 1z, I 2x, I 2y, and I 2z. 2 two-spin operators: 2I 1x I 2z, 2I 1y I 2z, 2I 1z I 2z, 2I 1z I 2x, 2I 1z I 2y, 2I 1x I 2x, 2I 1x I 2y, 2I 1y I 2x, and 2I 1y I 2y. For N spins, you should end up with 4 N basis operators. The single spin operators I x, I y, and I z can be directly associated with the projection of the magnetization vector on the particular axes, e.g., operating on the density matrix with Ix spin operator gives the projection of the magnetization vector on the x aixs, e.g., M x. The two-spin operators can not be directly associated with specific vectors, but often can be associated with particular states and processes that are not easily (if at all) representable using vectors. The best way to see the utility of the Product Operator formalism is to use it. First we ll present the rules for determining the commutator relationships, then look at several different Hamiltonians applied to simple spin systems, and finish up with several examples. 6.3. Evaluating Commutator Relationships. Commutators are used extensively in calculations involving angular momentum and other aspects of physics. As mentioned above, they are a short-hand notation for describing the interacting effects of two operators on a particular spin system (Eqn. 6.22). At first glimpse, the solution to 6.22 would appear to be zero, however we need to recall that the commutator involves operators, and operators do not necessarily commute. For two operators to commute (i.e., for 6
their commutator to be zero), they need to correspond to physical observables. In many cases, two operators will commute, but in general they won t when they concern angular momenta except for the trivial cases of [A,A], etc. Fortunately, the commutator relationships involved in NMR calculations are quite simple, and if you can remember xyz you ll have no problems in solving the necessary commutator relationships. First, some general facts about commutator relationships (these are taken from Molecular Quantum Mechanics, by P.W. Atkins, an excellent text book on quantum theory for the advanced undergraduate / beginning graduate student in chemistry). Fact 1. Associative Rule. [ AB, ] [ BA, ]. This relationship tells us that if we know a certain commutator, then using the operators in the opposite sense results in the negative of the first commutator (i.e., they re not simply associative as if it were a simple multiplication). Fact 2. Distributive Rule. operator can be taken outside of the commutator. Fact 3. Product Rule. [ AbB, ] bab [, ]. Here, a constant (scalar) multiplying an [ ABC, ] ABC BAC + BAC BCA [ AB, ]C+ B[ AC, ]. If the commutator involves the product of two operators (BC), then the commutator can be expanded as indicated. Fact 4. Chain Rule. [ A, [ BC, ]] [ ABC, CB] [ ABC, ] [ ACB, ]. IF you have multiple commutators, they can be expanded as indicated using the Chain Rule, and then expanded again suing the Product Rule. Fact 5. Commutators in NMR. As I mentioned before, all you need to remember to do product operator calculations is xyz, that s because of the following three fundamental commutation relations in NMR:, [6.29], and [6.30]. [6.31] The xyz comes in the cyclic permutation of the indices for the different operators. We ll see applications of these in the examples to come. 6.4. Various Hamiltonian Operators. [ I x, I y ] I z [ I y, I z ] I x [ I z, I x ] I y Pulses. The Hamiltonian for a pulse is given by Ĥ βi kv, where the flip angle is β, the k can indicate a broad-band or selective pulse (i.e., if you specify one nucleus out of several), and v indicates the axis along which the pulse is applied. Therefore, the Hamiltonian for a 90 x- pulse is Ĥ π --I. [6.32] 2 x Other pulses can be specified in a similar manner. An example of a 90 x pulse applied to equilibrium magnetization is shown in Figure 6.1, comparing the vector diagram and product 7
Figure 6.1. 90 x-pulse I z π H βi I x z cos -- 2 π + [ I x, I z ] sin -- 2 I y operator formalisms directly. In the product operator calculation, the starting state (operator A) is the I z state. The Hamiltonian (operator B) representing the pulse of (arbitrary) flip angle β about the x axis is βi x. Application of Eqn 6.20 gives the top equation on the right of the Figure, which is simplified to -I y by making use of the following: substituting π/2 for the argument of the trigonometric functions (cos 0; sin1), and the commutator [I x,i z ] -I y (see the associative rule for commutators). Pulses of arbitrary phase can be generated by recognizing that the phase shift in the pulse corresponds to a rotation about the z axis. Therefore, for a pulse of β radians flip angle and φ radian phase, we have Ĥ β( I kx cosφ + I ky sinφ). [6.33] Chemical Shift Evolution. The Hamiltonian for chemical shift evolution describes a rotation (we ve said precession before) about the z-axis (the axis parallel to the external magnetic field): Ĥ Ω k τi kz. [6.34] If we continue with the example in Figure 6.12, then following the pulse, the magnetization will again precess about B 0. As described by the product operators, this looks like: where the starting state is indicated at the left of the arrow (I ve dropped the negative sign), and the Hamiltonian is placed above the arrow. The first equation on the right side results from the application of Eqn. 6.20 and the second equation (after the equals sign) is the solution of the commutation relation. Does this look like what we re used to from the vector diagrams? Yes, it does. Immediately after the pulse (t0), we have pure I y magnetization (cos0 1; sin00), while at any other time, t, precession away from the y axis gives projections onto the x- (I x ) and y-axes (I y ), with magnitudes given by the arguments of the trigonometric functions (the product of angular frequency and time is an angle). Isn t this fun? Let s continue! What if we have multiple spins? Our initial state would be the sum of all three spins, represented as ( I 1y + I 2y + I 3y ) ΩtI I z y I y cos( Ωt) I 1z + I 2z + I 3x + [ I z, I y ] sin( Ωt) I y cos( Ωt) I x sin( Ωt). Excitation with a 90 x pulse gives the initial state. Assuming that none of these three spins are scalar coupled (that s coming, 8
don t worry), they will evolve under chemical shift evolution as indicated below. We can represent a selective pulse by specifying the nucleus / nuclei that the pulse is applied to with a Hamiltonian that looks like Ĥ βi 2x, for instance. I 1y + I 2y + I 3y k Ω k ti kz + I 1y I 2y cos( Ω 1 t) I 1x sin( Ω 1 t) cos( Ω 2 t) I 2x sin( Ω 2 t) + I 3y cos( Ω 3 t) I 3x sin( Ω 3 t) Scalar Coupling. The scalar coupling interaction involves two spins, and the Hamiltonian needs to contain operators that refer to each of these spins: Ĥ 2I kz ( πj kl t). [6.35] Here the k and l indices indicate the two spins that are scalar coupled. For example, I kx cos( πjt) + [ 2I kz ] sin( πjt) I kx cos( πjt) + 2I ky sin( πjt). [6.36] There are two diffferent kinds of magnetization represented by the terms on the right hand side of Eqn 6.36. The first term, consists of a single operator and is termed in-phase magnetization. This operator corresponds exactly to the projection of the bulk magnetization of spin k onto the x axis. We could also easily write an in-phase operator that gives the projection along the y-axis. The second term, arising from scalar coupling term, contains two spins operators, I ky, and is called anti-phase magnetization. We ve run into this term before, when discussing polarization transfer in the INEPT experiment and we ll see it again when we discuss 2D / 3D NMR sequences that bring about coherence transfer, e.g., COSY, HSQC, etc. Using vectors, we couldn t really represent this term, nor was it obvious where it came from. Now, using product operators, we not only have a term to describe it, but we can see where it comes from and why such a state has maximal intensity after evolving for a time, τ 1/2J. Now, from our discussion of the INEPT sequence, we know that scalar couplings can generate an antiphase state, and that the anti-phase state can be refocused, as well. In the next example, we see how to use the product operators to describe this process. The details of the calculation get a bit involved, but they are straightforward. You should follow the calculation and learn the result. If we start from an antiphase state and evolve under scalar couplings, then: 2I kx cos( πjt) + 4[ I kz ] sin( πjt) 2I kx cos( πjt) + 4{ [ I kz ] + [ ] + [ I kx, I kz ] + [ I kx, ]} sin( πjt) 2I kx cos( πjt) + 4 {[ I kz ] + [ I kz, ] + [ ] + [, ] + [ I kz ] + [ ] + [ I kz, I kz ] +[, I kz ]} sin( πjt) 2I kx cos( πjt) + 4{ I ky + 0 + 0 + 0 + I ky + 0 + 0 + 0} sin( πjt) 2I kx cos( πjt) + 8I ky sin( πjt) 8I ky for τ 1/2J. Note that the constants in front of the operators in the last expression are frequently 9
dropped in papers that use product operators to describe pulse sequences. This is because the product operators are mostly used to indicate what the magnetization terms are, rather than used in a quantitative analysis of magnetization intensities. After a bit of experience with these calculations you ll become quite adept at knowing when to drop the signs, as well. 6.5. Pulse sequence building blocks. In the few examples, we introduced the major Hamiltonians we will be concerned with in this class, and also looked at some examples. Now, let s start to put the product operator formalism to work for us. The problem is, as you ve just seen, that the product operator calculations quickly become cumbersome and unwieldy. Fortunately, there are a some shortcuts that can (and will!) be used in our calculations. These shortcuts are introduced to simplify calculations of common pulse sequence building blocks. Spin echo sequence. The most important example of this is the case of a spin echo sequence, (τ - 180 - τ). Intuitively, we would break the calculation into three steps: 1. evolution of the spins during the first t period using the chemical shift and scalar coupling Hamiltonians, 2. application of the 180 pulse, and 3. evolution of the spin system under chemical shift and scalar coupling Hamiltonians to refocus the chemical shift and scalar couplings. However, this calculation is difficult to apply in this simple manner, and the following two-step approach is taken. First, apply the Hamiltonian for the 180 pulse(s) as called for in the sequence, then allow evolution of scalar coupling Hamiltonian for a period of 2τ, neglecting chemical shift evolution. An example is shown in Figure 6.2 below. Simultaneous 180 pulses, as in INEPT. A similar approach can be taken when you have two simultaneous 180 pulses applied to two different nuclei (say 1 H and 13 C) as shown in Figure 6.3. Here, the carbon spin operator is denoted by S. Again, we apply the 1H 180 pulse, evolve under scalar coupling for a period of 2t (net time equals 1/2J), and then apply the 13C 180 Figure 6.2. τ τ Proper Hamiltonian I 1y + I 2y Ω k τi kz πjτ 2I kz π I kx pulse sequence block Ω k τi kz πjτ 2I kz answer Shortcut: πi kx + I 1y cos( π) I 1y sin( π) + I 2y cos( π) I 2y sin( π) I 1y I 2y I 1y I 1y I 2y I 2y πj2τ 2I kz cos( πj 2J) + 2I 1x I 2z sin( πj 2J) I 1y I 1x I 2z + I 1z I 2x (simplification when t 1/4J as in INEPT) 10
H C τ τ I y πi x 2πJ( 2τ)I z S z I y apply scalar 2I x S z (at t1/4j) coupling Figure 6.3. 2I x S z πs z pulse. The net effect is to take proton in-phase magnetization present before the first t period and convert it into proton - carbon antiphase magnetization (this is typically stated as proton magnetization in anti-phase with respect to its attached carbon) at the end of the pulse sequence building block. Again, what we see is that the scalar coupling term is maintained for the entire 2t period, converting in-phase proton magnetization into anti-phase magnetization. 6.6. Product Operator Analysis of INEPT Sequence As a final exercise in this chapter, we ll use the product operators to analyze the refocused- INEPT sequence for a simple NH system. When you re doing this, review the vector diagrams presented earlier to strengthen the correspondence between the two formalisms. Using the points indicated in the sequence (Figure 6.4), we see the following: a. at equilibrium the density matrix contains terms that represent the z magnetization of the proton and nitrogen terms. There are no other terms active at this time point. b. the first 90 pulse generates in-phase proton magnetization, but has no effect on nitrogen. c. at point c, we have the proton magnetization in antiphase with respect to the nitrogen, and the nitrogen z magnetization is inverted by the nitrogen 180 pulse. d. following the simultaneous 90 pulses on proton and nitrogen, the proton - nitrogen antiphase term has become nitrogen magnetization in anti-phase with respect to proton, and the nitrogen z magnetization term generates in-phase nitrogen magnetization. e. during the last 2t period, scalar coupling refocuses the nitrogen anti-phase magnetization to become in-phase x magnetization. The nitrogen in-phase y magnetization becomes anti- Figure 6.4. ±y τ τ τ τ a b c d e dec a. b. c. σ( a) I z + N z σ( b) I y + N z σ( c) I x N z N z d. σ( d) 2I z N y + N y e.σ ( e) N x N x I z 11
phase with respect to its attached proton, but there typically is no nitrogen-nitrogen couplings that we need to worry about. A few comments: The effect of the phase cycling on the proton 90 pulse in the middle of the sequence subtracting the terms at point e obtained from the two scans. By doing this, we can cancel the effects of the nitrogen magnetization present at the beginning of the pulse sequence (why did we want to do this, again?). The antiphase state development is easier to see in the product operator formalism than using vectors. from the product operator formal, it is readily seen that simultaneous 90 pulses can change 1 H-15N antiphase magnetization into 15 N in-phase magnetization. This shows the value of the anti-phase magnetization terms - they are used to transfer magnetization or coherence from one spin to its scalar-coupled partners. 6.7. Appendix: Trigonometry Review Quite often it is necessary to simplify the terms obtained from the PO formalism using simple trigonometric identities and equalities (Note that simple trigonometric is an oxymoron). I ll present several here for your enjoyment and fascination. Addition Formulae: sin( u + v) sinucosv + cosusinv ; cos( u + v) cosucosv sinusinv sin( u v) sinucosv cosusinv ; cos( u v) cosucosv+ sinusinv Double Angle Formulae: Product Formulae: can readily be seen by performing this sequence again, using Ĥ sin2u 2sinucosu; cos2u π --I 2 y ( cosu) 2 ( sinu) 2 1 2( sinu) 2 2( cosu) 2 2sinucosv sin( u + v) + sin( u v) ; 2cosusinv sin( u+ v) sin( u v) 2cosucosv cos( u + v) + cos( u v) ; 2sinusinv cos( u v) cos( u + v) and 12