Coherent methyl dynamics

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Physica B 202 (1994) 256-263

Coherent methyl dynamics S. C l o u g h Department of Physics, University of Nottingham, NG7 2RD, UK

Abstract Coherent methyl trajectories are superpositions of quantum states transporting proton spin states amongst three lattice sites. They are described by three complex amplitudes and a theory similar to spin dynamics. Experiments on methyl groups measure the modulation spectrum of spin-dependent interactions averaged over an ensemble of trajectories. We show that the Gallilean relativity principle, the Pauli exclusion principle and the correspondence principle are all satisfied.

1. Introduction

1.1. Methyl dynamics and spin dynamics N M R is unusual in spectroscopy in being described in terms of the coherent evolution of superpositions of quantum states rather than incoherent transitions. This is because it employs perfectly coherent oscillating fields and spin-lattice relaxation times are long. At low temperatures methyl rotational relaxation times are also long and rotation may be induced by oscillating fields acting on the nuclear moments and through the dipole-dipole interactions [1]. In fact N M R spin dynamics and methyl rotational dynamics are parts of a composite subject. The main novelty rotation brings to spin dynamics is inertia. While the precession frequency of a nuclear spin depends on the instantaneous field, the rotation frequency of a methyl group depends on an integral over previous rotational impulses. A vector potential plays the role of inertial memory I-2]. A methyl group has three orientational states so the group relevant to methyl dynamics is SU

(3). Instead of three spin operators which propagate spin ½ orientations there are eight operators or generators in SU (3). Two of them may be identified with differences in the depths of the three wells of the hindering potential and three pairs with the real and imaginary parts of the three overlap integrals of ground oscillator functions localised in different wells. Fluctuations of these quantities propagate the methyl state vector which consists of three complex amplitudes for the localised oscillator functions. When nuclear spin is included it is necessary to use a more complex description by distinguishing eight states rather than three. If only two of these are superposed there are still 28 different types of excitation. More complex types of wave packets are still more numerous. Fortunately the evolution of wave packets is generally characterised by a small number of dynamical features, namely wave-packet rotation, wave-packet shape change and proton spin precession. By considering a small number of simple cases, the way in which these features interact can be displayed. In the dipole-dipole interaction the SU(3) operators are coupled to spin

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operators and this provides the means of driving rotation magnetically. 1.2. The P a u l i principle

The perception that methyl rotation is similar to spin dynamics has been concealed hitherto by dynamical constraints which have been usually thought to be necessary to satisfy the Pauli principle. The hindering potential is usually limited to threefold symmetrical terms [3] and the wave numbers of methyl states have been assumed to be integers. These restrictions eliminate seven of the eight fields which couple to the SU(3) operators. They arise due to the misconception, dating from the early days of quantum mechanics [4] and repeated or implied by most texts on quantum mechanics, that the permutations of labels which occur in antisymmetrised wave functions can be interpreted as a motion of particles. This leads to constants of motion being associated with particle permutations. It is generally supposed that the constraints can be ignored if particles are not observed to exchange positions on the time scale of the experiment so they are ignored throughout physics except for a few examples, notably methyl rotation, where indistinguishable particles seem to exchange positions rapidly. The rotation of symmetrical molecules has thus been isolated from the rest of physics through the concept of spin symmetry species. Inspection of an antisymmetrised function shows that the labels do not have separate coordinates and are incapable of relative motion. They are global properties of the collectivity of particles and their permutation is completely divorced from dynamics. Each particle is represented in an antisymmetrlsed function by all the labels. The particles are therefore multi-labelled, identical and intrinsically indistinguishable [-5]. No dynamical constraints are required. The only observable consequence of the labels is to keep fermions apart by imposing a single occupancy rule so the Pauli principle is trivially satisfied for the methyl group by restricting each hydrogen atom to one proton. The rest of physics has been correct to ignore the constraints and molecular rotation may do the same. The constraints are incompatible with the most fundamental principle of physics, the relativity principle.

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This requires any theory to be form invariant to changes of reference frame of any kind and observables to be independent of the choice of reference frame. This powerful rule eliminates theories which attach physical significance to particles exchanging position, since one may transform to a reference frame in which the particles do not exchange position.

2. The Gallilean relativity principle 2.1. Gallilean transforms

Experiments on methyl groups measure the relative motion of methyl group (M) and the classical measuring equipment (S). Any reference frame may be used since observables are independent of the choice of reference frame. To illustrate the essential features of Gallilean transformation we first discuss the relative motion of S and a particle P in a box moving linearly with velocity v relative to S. In the reference frame in which S is stationary the operator for the relative angular momentum is -ih~/~x and the ground state function in the box is a superposition of the two waves exp(i(k+x - o~+t)) where k+= + (n/L) + my~h, L being the length of the box. The eigenvalues of the kinetic energy are given by ~o = hk2/2 m. The wave packet in the box moves with velocity (o~ + - to_ )/(k + - k _ ) = v. If we transform to a reference frame moving with velocity u the effect is to diminish the velocity of the box to v - u and to cause S to move with velocity - u relative to the new frame. In the new frame the boundary conditions of the box are satisfied by superposing waves for which k ± = + ( n / L ) + m(v - u)/h and the operator for the momentum of P as perceived by S is ( - ih~/~x) + mu. The vector potential term which appears here is the momentum of P due to the motion of S in the chosen frame. The Hamiltonian is H = (( -- i h ~ / ~ x ) + m u ) 2 / 2 m

(1)

with eigenvalues co = h(k + (mu/h))E/2m.

(2)

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The relative velocity ( o g + - o g _ ) / ( k + - k _ ) = v is unchanged by the transformation. Thus an arbitrary change in the vector potential in the momentum operator is compensated by a change in the average wave number in the wave function [6]. This is a gauge transformation and is illustrated for the case where there is no relative motion by Fig. l(a) and (b). In the general case we may choose either the vector potential to be zero (the S frame, Fig. l(c)) or k± to be +__r~/L (the wave-packet frame, Fig. l(d)) but not both. Fig. 2 shows the relationship (2) and the wave numbers k ± indicated by the circles for the first three states of the particle. The two wave number scales correspond to the two choices of reference frame in Fig. l(c) and (d). The parabola is symmetrically disposed with respect to one scale and the circles with respect to the other. There is an infinite number of other possible choices but these are the simple ones. The slope of the broken lines gives the relative velocity. A change in the relative velocity causes the pairs of circles in Fig. 2 to slide along the parabola, so changing the inclination of the broken line. By choosing the reference frame fixed in the box the wave numbers are maintained constant and the relative motion is described by a change in the vector potential. Thus m dv/dt, which is proportional to the rate of change of the vector potential, can be identified with the force acting on P as perceived by S.

Particle in a box (a)

(b)

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Fig. 1. A schematic representation of a particle in a box and measuring equipment. In (a) and (b) they are not in relative motion but are referred to two different frames. In (c) and (d) they are in relative motion and are referred to reference frames in which one of them is stationary.

2.2. Rotating wave packets To convert the above discussion to rotation on a closed space ~b we replace the coordinate x by an angle coordinate ~b¢ which extends from - oo to and take L = 2re. The wave function ~,(~b~,t) occupies a strip of width 2n on the infinite plane ~bc, t. This can be transferred to the cylindrical space ~b,t by the mapping ~k(~b,t) = ~,~k(~b + 2n~, t). The result is a single valued wave packet which circulates round the space ~b. Like the wave function of the particle in a box it has leading and trailing edges. It no longer needs to be confined by the walls of a box. The leading edge on circuit n follows immediately after the trailing edge on circuit n - 1. To relate the discussion to methyl rotation a hindering potential Vcos(3tk) is

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Fig. 2. The observed states of a particle in a box moving relative to the measuring equipment are formed by superposing pairs of running waves whose wave numbers differ by a multiple of the reciprocal length of the box. The broken lines indicate the relative velocity v. The effect of v is to displace either the t~, k curve or the k scale depending on the reference frame (see Fig, l(c) and (d)).

S. Clough/Physica B 202 (1994) 256-263

introduced. The wave function becomes a Bloch wave which is small except at the three values of ~b which correspond to the potential wells. The particle in a box function gives the envelope of a methyl wave packet formed from the superposition of two Bloch waves. There are again two simple choices of reference frame, one in which the wave packet is stationary and one in which the measuring equipment S is stationary. In the former the wave numbers of Bloch waves are fixed and may be defined to be integers. The relative angular velocity is described by a vector potential. A change in the vector potential is due to an external torque acting on the group as perceived by S. Fig. 3 shows the ~o, k relation for a particle in a periodic potential. The two wave number scales represent the two simple choices of reference frame and the circles represent the waves whose superposition forms the rotating wave packets. The dependence of co on k is symmetrically disposed with respect to the second scale and displaced with respect to the first. The broken line represents the angular velocity of the wave packet relative to S for the wave packet whose wave numbers are _+1 in the wave-packet frame. The slope indicates the lifting of the Kramers degeneracy due to the breaking of time reversal symmetry due to the rotational impulse which caused the relative motion at some time in the past. The persistence of the motion is due to inertia. A change in the relative angular velocity is represented by sliding the circles along the cosine curve.

m

-½

,

,

,

k

-Va

Fig. 3. The observed states of a methyl group rotating relative to the measuring equipment are formed by superposing Bloch waves indicated by the circles. The slope of the broken line is proportional to the relative angular velocity whose effect is a relative displacement of the o9, k curve and the k scale. The introduction of nuclear spin results in the Zeeman splittings shown on the right.

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2.3. The cyclic boundary condition Although the wave packet is composed of waves with integer k values in the wave-packet frame, in any arbitrary frame the values are not integers. It is often supposed that waves with non-integral k cannot exist on a rotational space 4- This is expressed by the cyclic boundary condition @(~b+ 2 n ) = @(~) which restricts k to integral values. Since the relativity principle shows that k can have a continuous range of values it is clear that the boundary condition is a special and not a general condition and that the picture of the space ~b on which it is based is similarly special rather than general. A Gallilean transform changes both the k values and the vector potential and both must be taken into account. The essential concepts are those of curved spaces. On a curved surface there are only local reference frames on tangent spaces and these are related to each other by connection coefficients which depend on the curvature. A small displacement involves two parts, a displacement on a tangent space and the transfer from one tangent space to another. Consequently the connection coefficients occur in the differential operator in the form of a vector potential. The vector potential in our equations means that the space ~b may be curved. The circle ~b is divided into small regions (tangent spaces). Within a region the gradient of the phase of the wave function is k, = (k+ + k_)/2. Between regions there is also a phase shift (the connection) due to the vector potential. Only the total phase shift round the circle ~b has physical significance since the contributions to the total can be changed arbitrarily by coordinate transformations. The total phase in one circuit of q~is independent of the reference frame and is related to the relative angular velocity of the wave packet and S. If this is zero the q~ space is flat and closed and a transformation can be made to remove the vector potential and restore the k values to integers. If there is relative angular velocity the phase associated with one circuit is non-zero and the space is curved and open. If the phase is treated as a coordinate perpendicular to the circle, the circle opens into a helix. The helix can be imagined to be inhabited by a family of possible wave packets, each confined to one turn, and having different winding numbers.

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S. Clough/Physica B 202 (1994) 256 263

A wave packet is chosen by specifying the winding number. A circuit of the 4, helix is equivalent to changing the winding number by unity. This more general picture of the space allows non-integral k. The cyclic boundary condition thus selects the special case of no relative angular velocity. This is relevant at low temperatures when frictional effects diminish the relative motion of wave packet and host lattice. An ensemble of methyl groups has wave vectors distributed through the Hilbert space at high temperatures and thus exhibits a range of relative angular velocities. At low temperatures there is a condensation into the small region of the Hilbert space selected by the cyclic boundary condition and the quantum effects become apparent. It should not be supposed that the reference frame in which the condensation occurs is inevitably the lattice frame. In the presence of a rotating magnetic field there are two frames competing to establish the rest frame or frame of quantisation of the methyl group. In N M R it is well known that spin equilibrium can be established in a rotating frame by a rotating field [7]. Through the dipoledipole interaction this influence is transferred to methyl rotation. It may therefore be possible to drag the methyl group round with oscillating fields as the methyl group finds a state in which it experiences equally compensating torques due to the applied field and due to its interaction with the lattice. The concepts of spin thermodynamics in the rotating frame [8] would then find new application in the combined topic of spin and rotational dynamics.

3. Introducing proton spin The important ways of studying methyl rotation depend on the modulation of spin-dependent interactions. If the three methyl orientations are characterised by the proton spin states at the three lattice sites then eight orthogonal states can be identified. Clockwise rotation through 2n/3 converts I~,~, fl> to 1/~,~,~,) and anticlockwise rotation to [~,/3,~). The order of the symbols refers to the three lattice sites 1-3. Each of these orientation states may be regarded as shorthand for a Slater determinant in

which three methyl protons occupy three spacespin states. They form a basis for the representation of wave-packet excitations. The fact that there is only one state [ct,~, ~> means that rotations which do not modulate spin-dependent interactions are neglected. A general state is specified by eight complex amplitudes but we discuss the dynamics of simpler wave packets which involve only two or three amplitudes. The low temperature stationary states are

~k =[a,b,c] =a[~,o~,fl) +blfl, 0t,~,> +cl~,fl, ct> (3) with [a, b, c] = [1, exp(ik2n/3), exp( - ik2n/3)] with k = 0 or ___1.The states Ict,~,ct) and Ifl, fl, fl> have k = 0 only. When a magnetic field is applied there is an additional nuclear Zeeman energy. This is shown on the right-hand side of Fig. 3, where the stationary states are labelled by their total magnetic quantum numbers m. A simple wave packet formed by superposing two of the eight states shown in Fig. 3 is now characterised by two numbers Ak and Am. If Ak ~ 0 the methyl wave packet rotates at a frequency given by the slope of the line joining the two circles and if Am ~ 0 there is precessing spin. In addition the dipole-dipole interaction between the protons couples the rotation and spin precession together. This is particularly important when the applied field is chosen to make the proton Larmor frequency roughly equal to the tunnel frequency. Variation of the magnetic field in this region causes a wave packet to experience a change in both Ak and Am with a conversion between rotational and magnetic angular momentum. The simplest wave packets are those having Am = 0. In this case there is only rotation. When m = ½ states are combined the fl spin state moves between the three lattice sites providing a quasi-particle whose motion may be conveniently plotted to illustrate the evolution of the wave packet. This is illustrated in Figs. 4 and 5.

4. Rotational trajectories We shall discuss here only the simplest kind of A m - - 0 wave-packet trajectories. A general

S. Clough / Physica B 202 (1994) 256-263 a

b

c

d

e

f

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position of the three circles in Fig. 3 and depend on the value of the vector potential. The effect of the vector potential is that a phase shift 7 = 2rctr/3 is associated with a clockwise rotation through 2~/3, and - T with an anticlockwise rotation. This is introduced through the overlap integral connecting functions localised in adjacent potential wells. The integral is A exp(iT) for a clockwise rotation and d e x p ( - iy) for an anticlockwise rotation. The state vector is a three-component column matrix v which obeys the time-dependent Schrodinger equation d v / d t = - iS(t)v where s(t) =

W1

- A exp(iy)

- A exp( -iy)~ !

Fig. 4. Observables X = (cos(~b)) and Y = (sin(~b)) show the evolution of three-component wave packets for different vector potentials. A point representing the mean position of a nuclear spin state moves on a triangle of lattice sites. Oscillation across the triangle is combined with rotation. The vector potential increases to the right and is chosen so the diagrams close after rotation through rt (top row) and 2n (bottom row). Below each trajectory is the intensity at one site as a function of time.

(a)

(b)

(e)

- A exp(-i7)

Wz

--Aexp(iT)

(e)

(0

--A exp(iy)

--A exp(--iv)

W3

.~

The eigenvalues of S(t) are given by E ( k + a) = - 2A cos (2rc(k + a)/3),

(5)

where k = _+ 1 or 0 and 3A = hoot. The effect of a rotational impulse is to change the phase 7 and hence to change the eigenvalues. F r o m the eigenvalues the frequency spectrum of wave packets formed by the superposition of two basis functions is obtained as a function of a. The wave-packet rotation frequencies are (6)

where ka = k + a is the average wave number of a wave packet and x = ___½ or 0. When tr = 0 this gives the low temperature spectrum of three frequencies __+cotand 0. Each of the three frequencies changes with tr. The energy of a wave packet is given by E = aa* W1 + bb*W2 + cc*W3.

Fig. 5. The velocity of the wave-packet centre of gravity on the X, Y plane is shown for the same six wave packets as in Fig. 4.

(4)

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to(ka) = (2m,/31/2) sin (2rcka/3),

(d)

|.

(7)

This m a y be written in terms of the observables X = (cos(q~)) = aa* -- (bb* + cc*)/2,

combination of m = ½ states can be written as [a, b, c] as in Eq. (3). We use the basis consisting of the three states [1,1,1], [1,e,e*] and [1,e*,e] where e = exp (2~i/3). These are the Bloch wave with k = 0 and + 1. Their energies are given by the

Y = (3/4) 1/2 (bb* - cc*),

(8)

and then the torque is obtained by taking moments h da/dt = X(OE/~ Y) --Y(~E/~X).

(9)

S. Clough / Physica B 202 (1994) 256-263

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The amplitudes may be formed into a density matrix

aa* ab* ac*] p(t) =

ha*

bb*

bc*[

ca*

cb*

cc*]

(10)

An observable R is represented by a 3 x 3 matrix and its value is Tr (pR). Examples of wave-packet trajectories are shown in Fig. 4. The wave packets are formed by superposition of all three basis functions and are propagated by a matrix S(t) with six different values of y and with W1 = W2 = W3 = 0. For such wave packets one finds [-2] X =cos (¢olt) +cos (co2t) +cos ((092 -- co1)t), Y = -sin(o91t) +sin(og/t) -sin((o92 - col)t),

(11)

where h~ol = E(1 + a) - E ( a ) and hr.02 = E( -- 1 + a) -- E (a). The evolution is represented by the motion of a point with coordinates X, Y. The three orientations [1, 0,0], [0, 1, 0] and [0, 0, 1] are represented by the vertices of an equilateral triangle and the point moves within the triangle. The trajectories also trace the path of the mean position of a fl spin state on the triangle of hydrogen sites. For a small value of V the path has a form similar to a Foucault pendulum, namely a nearly linear oscillation with a slow rotation of the direction of oscillation. The oscillation is due to tunnelling motion in which there is no bias towards a particular sense of rotation. It may be described as a change of shape of the wave packet. With increasing vector potential a bias develops in one sense or the other. The two frequencies which characterise the motion are derived from the splittings indicated by the three circles in Fig. 3. Their ratio has been chosen so that the trajectories in Fig. 4 close, those in the top row after the oscillation direction has rotated through n and those in the bottom row after the plane has rotated through 2n. The vector potential increases from left to right. The trajectory of Fig. 4(f) is particularly interesting since it describes a wave packet undergoing pure coherent rotation, the shape change being now adapted to allow a wave packet initially localised in one well to

develop smoothly into a wave packet localised in another well. This occurs when 7 = -+ n/2, the sign determining the direction of rotation. Below each trajectory in Fig. 4 is a plot of a diagonal element of the density matrix as a function of time. It shows how the weight of a particular orientation changes as the wave packet rotates. The six off-diagonal elements of the density matrix contain the currents between the potential wells. The current between wells i and 2 depends on the difference a*b - ba*. Matrices corresponding to the components of the velocity of the wave packet can be defined. The expectation values are Vx = I (a*b - c'a)~(2) 1/2,

(12)

Vy = I (2b*c - a*b - c'a)~(6) ~/2.

(13)

Fig. 5 shows plots of Vx versus Vy for the same values of N as in Fig. 4.

5. Conclusions By combining the concepts of methyl rotation and spin dynamics new perspectives are opened. Spin dynamics is a very mature subject with extremely sophisticated concepts and experimental traditions. These now become available to study coherent magnetically driven molecular rotation at low temperatures, taking lattice dynamics out of the more restricted conceptual framework of incoherent transitions. The removal of the confusion over permutation as a special kind of dynamics restores an important link, the compatibility of quantum and classical mechanics [9]. The literature of physics and chemistry has been ambiguous about whether quantum and classical mechanics are the same theory precisely because of the confusion. From the discussion of Gallilean relativity it is clear that all the concepts like torque and angular velocity which are associated with classical mechanics reappear in quantum mechanics. It is also clear that there is no such thing as a "pure" quantum theory since all measurements are relative to classical equipment. The isolation of methyl dynamics from the rest of physics can end and the topic can assume a more central role as an ideal

S. Clough/Physica B 202 (1994) 256-263

t o o l for e x p l o r i n g the e m e r g e n c e o f q u a n t i s a t i o n from incoherence. A l t h o u g h d y n a m i c a l c o n s t r a i n t s d o n o t a c c o u n t for spin s y m m e t r y species, there are l o n g lived states o f r o t a t i n g m o l e c u l e s at low temperatures. T h e i r stability is b o t h a challenge to u n d e r s t a n d i n g a n d an o p p o r t u n i t y u p o n which m a y be b a s e d a new c o h e r e n t s p e c t r o s c o p y .

Acknowledgement T h e a u t h o r s are grateful for the s u p p o r t of the B P V e n t u r e R e s e a r c h Unit.

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References [1] J.R. Sutcliffe and S. Clough, presented at Conference on Quantum Tunnelling, Windsor (1993). [2] S. Clough, A.J. Horsewill and M.R. Johnson, Phys. Rev. A 47 (1993) 3420. [3] A. Wurger and A. Huller, Z. Phys. B 78 (1990) 479. [4] P.A.M. Dirac, The Principles of Quantum Mechanics ( Oxford, 1935; 2nd ed.) ch.10. [5] S. Clough, A.J. Horsewill and M.R. Johnson, Chem. Phys. Lett. 208 (1993) 143. [6] S. Clough, J. Phys. C 18 (1985) L1. [7] A.G. Redfield, Phys. Rev. 98 (1955) 1787. [-8] M. Goldman, Spin Temperature and Nuclear Magnetic Resonance in Solids (Clarendon Press, Oxford, 1970). [9] W.H. Zurek, Physics Today 36 (1991) 43.