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This article was downloaded by: [Georgetown University] On: 05 October 2014, At: 17:19 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Molecular Physics: An International Journal at the Interface Between Chemistry and Physics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tmph20 Graphical displays of the motion of spin systems LAWRENCE L LOHR Published online: 03 Dec 2010. To cite this article: LAWRENCE L LOHR (1996) Graphical displays of the motion of spin systems, Molecular Physics: An International Journal at the Interface Between Chemistry and Physics, 89:5, 1397-1408 To link to this article: http://dx.doi.org/10.1080/002689796173255 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.

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This article was downloaded by: [Georgetown University]On: 05 October 2014, At: 17:19Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number:1072954 Registered office: Mortimer House, 37-41 Mortimer Street,London W1T 3JH, UK

Molecular Physics: AnInternational Journal atthe Interface BetweenChemistry and PhysicsPublication details, including instructionsfor authors and subscription information:http://www.tandfonline.com/loi/tmph20

Graphical displays of themotion of spin systemsLAWRENCE L LOHRPublished online: 03 Dec 2010.

To cite this article: LAWRENCE L LOHR (1996) Graphical displays of themotion of spin systems, Molecular Physics: An International Journal at theInterface Between Chemistry and Physics, 89:5, 1397-1408

To link to this article: http://dx.doi.org/10.1080/002689796173255

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy ofall the information (the “Content”) contained in the publicationson our platform. However, Taylor & Francis, our agents, and ourlicensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication arethe opinions and views of the authors, and are not the views of orendorsed by Taylor & Francis. The accuracy of the Content shouldnot be relied upon and should be independently verified with primarysources of information. Taylor and Francis shall not be liable for anylosses, actions, claims, proceedings, demands, costs, expenses,damages, and other liabilities whatsoever or howsoever causedarising directly or indirectly in connection with, in relation to orarising out of the use of the Content.

Page 2: Graphical displays of the motion of spin systems

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M o l e c u l a r P h y s i c s , 1996, V o l . 89, N o . 5, 1397 ± 1408

Graphical displays of the motion of spin systems

By LAW RENCE L. LOHR

Department of Chemistry, University of Michigan, Ann Arbor, MI 48109-1055,

USA

(Recei Š ed 30 No Š ember 1995 ; re Š ised Š ersion accepted 15 April 1996)

The motion of electron spins in anisotropic systems described by spin

Hamiltonians containing zero- ® eld tensors is described in terms of semi-

classical trajectories constructed either in spin space from surfaces of constantenergy or in energy space from surfaces of constant magnitude of the spin

angular momentum. The trajectories are presented as time-independent

projections for a range of values of the non-axial orthorhombic spin-Hamiltonian parameter E. Analogies are made to the closely related semi-

classical description of the rotational motion of spherical, symmetric, and

asymmetric top molecules. The spin-Hamiltonian parameters of a number oforganic and inorganic systems are used to illustrate the results.

1. Introduction

In recent publications [1, 2] we presented analytic expressions, parametric in

centrifugal displacement coordinates, which provide exact classical descriptions of

rotational energy dispersions, that is, the dependence of rotational energy on the

magnitude and direction of rotational angular momentum for small non-rigid rare-gas

(Rg) clusters modelled by pairwise additive Lennard-Jones 6-12 potential energies. As

a complement to our analytic descriptions our studies also included angular

momentum-conserving classical simulations. Speci® c properties discussed included

quartic and higher-order spectroscopic constants for Rg$, Rg

%, and Rg

’, rotational

instabilities for Rg$, and `cubic ’ rotational anisotropies for the spherical tops Rg

%and

Rg’. We have also reported studies [3, 4] of rotating clusters Ar

n, with n as large as 147,

described by pairwise additive potential energy functions based on the highly accurate

® fteen parameter diatomic Ar#

function obtained by Aziz [5]. Our rare-gas cluster

studies are an extension of our earlier analytic study [6] of centrifugal distortions in

diatomic molecules.

In several of our studies cited above [1, 2] we presented rotational energy surfaces,

which are one of the constructions outlined by Harter [7, 8] for the visualization of

semi-classical motions of rotational angular momenta with respect to molecular axes.

Numerous applications of such surfaces to the description of molecular rotations have

by now been made [9]. In the present study we extend this and related constructions to

the graphical display of the motions of spin systems. The ® rst of two approaches

begins with the construction of a constant energy surface, that is, the locus of points

in spin space which have a constant energy equal to one of the eigenvalues of the spin

Hamiltonian. The space curve which represents the intersection of such a surface with

a spherical surface having as its radius the magnitude of the spin angular momentum

is then the spin-space projection of the semi-classical trajectory for that energy

eigenvalue. The second approach is similar but begins instead with the construction of

0026± 8976 } 96 $12 ± 00 ’ 1996 Taylor & Francis Ltd

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Page 4: Graphical displays of the motion of spin systems

1398 L. L. Lohr

a so-called rotational energy surface, that is, the locus of points in energy space which

have a constant value of the magnitude of the spin angular momentum. The space

curve which represents the intersection of such a surface with a spherical surface

having as its radius an energy equal to one of the eigenvalues of the spin Hamiltonian

is then the energy-space projection of the semi-classical trajectory for that energy

eigenvalue. The trajectories, obtainable by either of these procedures, will be shown to

provide insights in to the motion of spin systems, such as S ¯ 1 electron spin systems

in the presence of both zero-® eld tensor and external Zeeman interactions.

2. Outline of procedures

2.1. Constant energy surfaces and spin trajectories

The spin system in question is assumed to be described by a standard spin

Hamiltonian which may be taken (for S " 1 } 2) as

Hs¯ b B [ gS ­ D(S #

z® S(S ­ 1) } 3) ­ E(S #

x® S #

y) (1)

where b is the Bohr magneton, B the magnetic induction, g the magnetogyric tensor,

S the total electron spin angular momentum, and D and E are elements of the zero-

® eld tensor having Sx, S

y, and S

zas its principal axes. The ® rst method for constructing

begins by ® nding the eigenvalues of (1) for a given value of the spin quantum number

S and for a given set of the quantities g, B , D , and E . A set of constant energy surfaces

in spin space is then de® ned by equating each of the eigenvalues to the energy

expression (1) taken as a function of Sx, S

y, and S

z. (The factor S(S ­ 1) in the second

term of (1) is taken as S #x­ S #

y­ S #

zrather than its numerical value.) The space curves

which represent the intersections of such surfaces with the single spherical surface

having as its radius the magnitude of the spin angular momentum r S r ¯ [S(S ­ 1)] " / # ò

are then the spin-space projections of the semi-classical trajectories for the set of

energy eigenvalues of the spin Hamiltonian Hs. For example, a spin triplet (S ¯ 1) of

an axially symmetric system (E ¯ 0) in the absence of a magnetic induction (B ¯ 0) has

two distinct eigenvalues, namely ® 2D } 3 for MS

¯ 0 and D } 3 for MS

¯ ³ 1, so that

there are two diŒerent constant energy surfaces, namely

D(2S #z® S #

x® S #

y) } 3 ¯ ® 2D } 3 (2 a)

or

Sz¯ ³ ((S #

x­ S #

y® 2) } 2) " / # (2 b)

for MS

¯ 0 and

D(2S #z® S #

x® S #

y) } 3 ¯ D } 3 (3 a)

or

Sz¯ ³ ((S #

x­ S #

y­ 1) } 2) " / # (3 b)

for each of the states with MS

¯ ³ 1. Thus there are two surfaces with positive and

negative values of Sz, respectively, for each of the three eigenstates, although the

surfaces are identical for the two degenerate states. Note that D cancels from the

equations, so that the surfaces are independent of its numerical value. The form of

(2 a, b) is that of a hyperboloid of revolution of one sheet with Sz

as its transverse axis,

while the form of (3 a, b) is that of an hyperboloid of revolution of two sheets, again

with Sz

as its transverse axis. (The surfaces may be expressed as ellipsoids by adding a

su� ciently large spherical term proportional to S #x­ S #

y­ S #

z.) The spin-space

trajectories are de® ned by simultaneous solution of each of the above (2, 3) with the

equation S #x­ S #

y­ S #

z¯ 2 (the square of the magnitude of the spin angular momentum,

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Motion of spin systems 1399

in units of ò # , for a triplet state). These solutions are Sz¯ 0 with S #

x­ S #

y¯ 2 for

MS

¯ 0 and Sz¯ ³ 1 with S #

x­ S #

y¯ 1 for M

S¯ ³ 1, that is, a circle of radius 2 " / # in

the Sz¯ 0 plane for M

S¯ 0 and a pair of circles of radius 1 in the planes S

z¯ ³ 1

for MS

¯ ³ 1.

The system with S ¯ 1 is special in that the basis function r SM ª ¯ r 10 ª is an

eigenfunction with eigenvalue MS

¯ 0 even if E 1 0 in the Hamiltonian given in (1)

provided that B is either 0 or along z. (For integral S greater than 1, the basis function

r SM ª ¯ r S0 ª is no longer an eigenfunction if E 1 0, as the orthorhombic term mixes

the function r S0 ª with the functions r S ³ 2 ª .) However the existence of a quantum state

with well-de® ned MS

¯ 0 is not a su� cient condition for the semi-classical trajectory

to remain in the plane Sz¯ 0. That is, if a state is an eigenfunction of S

z, but S

zdoes

not commute with the Hamiltonian, so that not all of the eigenstates of the

Hamiltonian, as opposed to a subset, are eigenstates of this operator, the quantity Sz

may evolve with time. In a later publication [10] we shall develop these time

dependencies and use them to describe electron spin± lattice relaxation phenomena.

Constant energy surfaces for the more general case with B and } or E 1 0 may be

similarly constructed by equating each of the eigenvalues k"

¯ ® 2D } 3, k#

¯D } 3 ® [( b B

zg

z) # ­ E # ] " / # , and k

$¯ D } 3 ­ [( b B

zg

z) # ­ E # ] " / # , where B is assumed to be

parallel to Sz, to the energy expression (1). The spin-space constant energy surfaces are

given byb B

zg

zS

z­ D(2S #

z® S #

x® S #

y) } 3 ­ E(S #

x® S #

y) ¯ ® 2D } 3 (4)

for the eigenstate with MS

¯ 0 and

b Bzg

zS

z­ D(2S #

z® S #

x® S #

y) } 3 ­ E(S #

x® S #

y) ¯ D } 3 ³ [( b B

zg

z) # ­ E # ] " / # (5)

for the two eigenstates which are superpositions of the basis states with MS

¯ ³ 1.

Again there are two surfaces, with positive and negative values of Sz, respectively, for

each of the three eigenstates. Each of the three pairs of surfaces is a diŒerent

hyperboloid (but not of revolution) as there is no degeneracy if B or E 1 0.

Figure 1 shows the spin-space trajectories for D ¯ 1, B ¯ 0, and E ranging from 0

to 1 } 2. For small positive E the circular path in the upper hemisphere which obtains

for E ¯ 0 (® gure 1 (a)) separates into two paths (® gures 1 (b) ± 1 (d )), each with a non-

constant " which is nonetheless periodic in 2 u . Similar behaviour obtains for the

circular path in the lower hemisphere. The separation between the curves in a given

hemisphere corresponds to the `inversion ’ splitting, that is, to the frequency x ¯ 2E } ò

with which the motion ¯ ips between hemispheres. Each of the eigenvalues D } 3 ­ E ,

and D } 3 ® E is associated with a pair of such closed curves, meaning that the semi-

classical motion for a given quantum state is split between the pair of curves having the

same energy, corresponding to the fact that each of these eigenstates is a superposition

of the basis functions associated with MS

¯ ³ 1.

The circle in the Sz¯ 0 plane shown in ® gure 1 (a) for the state M

S¯ 0 with E ¯ 0

is not a trajectory for spin motion, for S is oriented along a ® xed but unspeci® ed

direction in Sz¯ 0 plane, in direct analogy to the K ¯ 0 case for the rotation of a

symmetric top ; the circle simply denotes the continuous set of possible directions for

S . For E " 0, the curves (® gures 1 (b) ± 1 (d )) for this state, with eigenvalue ® 2D } 3, do

correspond to trajectories. Breaks appear along the relatively high energy Sx

axis in the

circle originally in the Sz¯ plane (for E ¯ 0). One branch loops back, rising above

(and below) the Sy

axis, the other branch being identical but rising above (and below)

the ® Sy

axis. These branches intersect the Sz¯ 0 plane at the four points S

x¯ ³ 1,

Sy

¯ ³ 1 (in units of ò ) for all E " 0. (If S is greater than 1, these intersections for the

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Page 6: Graphical displays of the motion of spin systems

1400 L. L. Lohr

Figure 1. For legend see facing page.

state arising from the basis function r SM ª ¯ r S0 ª vary with E .) As E is increased a

second state, that with eigenvalue D } 3 ® E , behaves similarly, as eventually does the

third, that with eigenvalue D } 3 ­ E , so that for su� ciently large E relative to D the

states are quantized with respect to the Sy

axis.

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Motion of spin systems 1401

Figure 1. Spin trajectories obtained from constant energy surfaces for an S ¯ 1 (triplet state)

system with D ¯ 1 and B ¯ 0. The values of E are : (a) 0 ; (b) 1 } 6 ; (c) 1 } 3 ; and (d ) 1 } 2. One

pair of curves for E ¯ 1 } 3 coincides with the separatrix (the locus of points for which theenergy equals that for rotation about the intermediate axis). The S

zdirection is vertical,

Sy

horizontal, and Sx

out from the page. In each ® gure pairs of curves related by inversion

through the origin are associated with a common energy.

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Page 8: Graphical displays of the motion of spin systems

1402 L. L. Lohr

The set of curves in each of the ® gures 1 (a) ± 1 (d ) are obtained, as described, from

a set of constant energy surfaces, one for each distinct eigenvalue (two for each, if we

count separately the upper and lower hemisphere surfaces. However the surfaces

themselves need not be plotted (only the generating expressions (4, 5) are required), so

that the description of spin-space trajectories in terms of diŒerent surfaces for diŒerent

eigenvalues is not a serious drawback.

As a ® nal example in this section we show in ® gures 2 (a) and 2 (b) the spin-space

trajectories obtained with D ¯ 1, b Bx

gx

Sx

¯ 5 } 6, and E ¯ 0 and 1 } 3, respectively.

The linear Zeeman term raises the energy along ­ Sx

but lowers it along ® Sx, so

that the states tend toward quantization along ® Sx. Two of the trajectories in

® gure 2 (a) (E ¯ 0) correspond to precession about ® Sx, but with a non-constant

projection of S on that axis. The one corresponding to the lowest eigenvalue of

® D } 6 ® (D # ­ 8( b Bx

gx

Sx) # ) " / # } 2 has a nearly constant projection on ® S

x, while that

corresponding to the intermediate eigenvalue of ® D } 6 ­ (D # ­ 8( b Bxg

xS

x) # ) " / # } 2 has

a very large periodic variation of its projection on ® Sx, with oscillations of approxi-

mately ³ 1. Adding a positive E term as in ® gure 2 (b) (E ¯ 1 } 3) destabilizes both

the ­ Sx

and ® Sx

directions, causing the trajectory with the largest projection on ® Sx

in ® gure 2 (a) to break apart into two loops, each about an axis in the (Sx, S

y) plane.

How do we interpret these curves ? Taking as a simple example the case where both

B and E ¯ 0, the resulting spin system is analogous to a symmetric top rotor in the

absence of external ® elds. If the projection K of the angular momentum J on the top

axis is not zero, the top axis nutates about J while the angular velocity x rotates about

the top axis, with J being ® xed in the laboratory frame. A familiar picture of these

motions is that by Herzberg [11], namely of a cone having the top axis as its axis rolling

on a ® xed cone having J as its axis, x being the moving contact line between the cones.

In the molecular frame J nutates about the top axis while x rotates about the top axis,

the motion of J in this reference frame resulting from eŒective torques exerted by the

molecule in this dynamically unbalanced system. (If K ¯ 0, J and x are parallel, the

system is dynamically balanced, and the molecule undergoes simple end-over-end

rotation with no precession of J.) It is this force-free nutation (precession) pathway of

J with respect to the top axis which is represented by the trajectories for the analogous

axial spin system. If we constrain the molecular axes to coincide with laboratory axes,

as obtained for a paramagnetic molecule or ion in a crystal, then any precession of S

with respect to the laboratory } molecular axes results from torques exerted by the

crystal, for example through the anisotropy in the second-order spin± orbit interaction.

Angular momentum is conserved overall by the undetectable compensating motions

of the crystal.

2.2. Rotational energy surfaces and spin trajectories

Consider now the construction of rotational energy surfaces. Combination of the

spin Hamiltonian (1) with the constant total spin angular momentum constraint

S #x­ S #

y­ S #

z¯ S(S ­ 1) in units of ò # is made by introducing spherical polar

coordinates asS

x¯ r S r sin ( " ) cos ( u ) (6 a)

Sy

¯ r S r sin ( " ) sin ( u ) (6 b)

Sz¯ r S r cos ( " ) (6 c)

with the magnitude of the spin angular momentum r S r being ® xed at [S(S ­ 1)] " / # . The

resulting surface describes the variation of the energy with respect to the direction of

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Motion of spin systems 1403

Figure 2. Spin trajectories obtained from constant energy surfaces for an S ¯ 1 (triplet state)system with D ¯ 1 and b B

xg

xS

x¯ 5 } 6. The values of E are : (a) E ¯ 0 ; and (b) E ¯ 1 } 3.

The axes directions Sx

and Sy

are reversed from ® gure 1, the higher energy Sx

direction

being horizontal, and the lower energy Sy

direction out from the page. As in ® gure 1, pairsof curves related by inversion through the origin are associated with a common energy.

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1404 L. L. Lohr

S . (The energy is the scalar radius in `energy ’ space.) For plotting purposes it is

desirable to add an arbitrary but su� ciently large spherical term a (S #x­ S #

y­ S #

z) to the

energy, so that the radius is positive for all directions of S.) for B s Sz

and E ¯ 0 the

rotational energy surface is independent of u and is given for a spin triplet (S ¯ 1) by

% ( " ) ¯ 2 a ­ 2D(3 cos # ( " ) ® 1) } 3 ­ 2 " / # b Bzg

zcos ( " ) (7)

where the factor of 2 in the ® rst two terms represents the value of r S r 2 ¯ S(S ­ 1) in

units of ò # and the factor of 2 " / # in the third term represents the value of r S r .The intersections of the above rotational energy surfaces, such as the axially

symmetric surface (7), with the set of spherical surfaces with radii equal to the

eigenvalues 2 a ® 2D } 3 for MS

¯ 0 and 2 a ­ D } 3 ³ b Bzg

zfor M

S¯ ³ 1 yield the desired

trajectories. (Note the inclusion of the constant 2 a .) For the former this reduces, if

B ¯ 0, to cos ( " ) ¯ 0 or " ¯ p } 2, while for the latter it reduces to cos ( " ) ¯ ³ 2 Õ " / # or

" ¯ p } 4 and 3 p } 4. The circular paths de® ned by these constant " values are identical to

those on constant energy surfaces, namely a circle of radius 2 " / # in the Sz¯ 0 plane for

MS

¯ 0 and a pair of circles of unit radius in the planes Sz¯ ³ 1 for M

S¯ ³ 1.

The precessional motion of S is simply described if the Zeeman interaction is taken

with B along the Sz

axis. If D is zero, but Bz

non-zero, the familiar picture of Larmor

precession about the z axis is obtained, with the same frequency x ¯ b Bzg

z} ò and

sense of precession for each of the three states. If instead Bz

is zero, but D non-zero,

the states with MS

¯ ³ 1 have the same precessional frequency x ¯ D } ò but opposite

senses, while the state with MS

¯ 0 is non-precessional. If both D and Bz

are non-zero,

the frequencies are x ¯ ( b Bzg

z³ D) } ò for the states with M

S¯ ³ 1 and x ¯ b B

zg

z} ò

for the state with MS

¯ 0.

Inclusion of a non-zero E term produces major changes in trajectories describing

spin motion. Harter presents in detail [7, 8] the rotational analogue, namely the

transition from a prolate symmetric top (with respect to Jz) with rotational constants

A ¯ 0 ± 2, B ¯ 0 ± 2, and C ¯ 0 ± 6 to an oblate symmetric top (with respect to Jx) with

constants A ¯ 0 ± 2, B ¯ 0 ± 6, and C ¯ 0 ± 6. His variable is the intermediate constant B ;

as B increases from 0 ± 2 to 0 ± 4 the spherical term a (A ­ B ­ C ) } 3 increases from 1 } 3 to

7 } 15, D ¯ C ® (A ­ B) } 2 decreases from 0 ± 4 to 0 ± 2, and E ­ (B ® A) } 2 increases from 0

to 0 ± 2. (As noted below, the spin Hamiltonian Hs

(1) implies that Sx, not S

y, is the

intermediate axis. W e have thus interchanged Sx

and Sy

in relating Harter’ s constants

to our parameters.) For the case B s Sz

the eigenvalues of the spin Hamiltonian Hs

(1)

with a constant term 2 a added are

k"

¯ 2 a ® 2D } 3, (8 a)

k#

¯ 2 a ­ D } 3 ® [( b Bzg

z) # ­ E # ] " / # , (8 b)

and

k$

¯ 2 a ­ D } 3 ­ [( b Bzg

z) # ­ E # ] " / # . (8 c)

In the language of molecular rotations our Sx

axis, not Sy, is the intermediate axis, as

our spin Hamiltonian (1), with the addition of a spherical term a S # , may be rewritten

asH

s¯ b B [ gS ­ ( a ® D } 3 ­ E ) S #

x­ ( a ® D } 3 ® E ) S #

y­ ( a ­ 2D } 3) S #

z(9)

If the induction B is zero, the separatrix, assuming E ! D , is de® ned by setting the

energy equal to term in Sx, namely to ( a ® D } 3 ­ E ) S #

x, with S #

x¯ S # ¯ S(S ­ 1) in

units of ò # , as the separatrix passes through the points Sx

¯ ³ r S r ¯ ³ [S(S ­ 1)] " / # . One

of the eigenvalues, namely 2 a ­ D } 3 ­ E , lies above the separatrix energy for all

E ! D , one, namely 2 a ® 2D } 3, lies below, while the intermediate eigenvalue

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Motion of spin systems 1405

Figure 3. Spin trajectories obtained from rotational energy surfaces with constant r S r # ¯S(S ­ 1) ¯ 2 (in units of ò # ) for an S ¯ 1 (triplet state) system with D ¯ 1 and B ¯ 0. A

spherical term a (S #x­ S #

y­ S #

z) with a ¯ 2 has been included in the energy. The values of

E are : (a) 1 } 6 ; and (b) 1 } 2. The energy component Ez

direction is vertical ; closed pathsare formed around the low energy E

ydirection, avoiding the higher energy E

xdirection.

As in ® gures 1 and 2, pairs of curves related by inversion through the origin are associated

with a common energy.

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1406 L. L. Lohr

2 a ­ D } 3 ® E lies above if E ! D } 3, equals the separatrix energy for E ¯ D } 3, and lies

below for D } 3 ! E ! D . Figure 3 displays the spin trajectories for D ¯ 1, B ¯ 0, and

E values of 1 } 6 and 1 } 2. Each surface includes a spherical term a (S #x­ S #

y­ S #

z) with

a ¯ 2. Each set of trajectories lies on a single surface, in contrast to the use of diŒerent

surfaces for diŒerent eigenvalues in the constant energy surface approach. The

information content of these curves is identical to that in the curves (® gures 1 and 2)

obtained from constant energy surfaces, although the procedure for obtaining them is

diŒerent. The metric is also diŒerent for the two types of trajectories ; the curves

obtained from constant energy surfaces are in spin-space, with a constant radius equal

to S(S ­ 1), while the curves obtained from rotational energy surfaces are in energy

space, with a constant radius equal to the energy of a particular eigenstate.

The preceding relationships are modi® ed if the induction B is not zero. If B is along

the Szaxis, the separatrix energy is still 2( a ® D } 3 ­ E ), but the intermediate eigenvalue

is k#

¯ 2 a ­ D } 3 ® [( b Bzg

z) # ­ E # ] " / # (8 b) ; these are equal if D ¯ 2E ­ [( b B

zg

z) # ­ E # ] " / # ,

which reduces to D ¯ 3E if B is zero.

Our rotational energy surface trajectories and our spin-space trajectories described

in the previous section, diŒer in one particular from Harter’ s constructions. In each of

our procedures we elect to construct curves for a given magnitude of the angular

momentum and energy eigenvalue, where each energy eigenvalue corresponds to a

`delocalized ’ eigenstate having equal probabilities of circulation about some axis Sior

Ji, where i ¯ x, y or z, and its negative ® S

ior ® J

i. If we were to apply this procedure

to the rotational states with J ¯ 10, say, for an asymmetric rotor, there would result

not 2J ­ 1 ¯ 21 curves, but rather 21 pairs of curves, one pair per energy eigenvalue.

By contrast, Harter displays [7] for this same J ¯ 10 example 20 curves plus a

separatrix on the rotational energy surface, with the super® ne structure arising from

`inversion ’ splittings of localized rotational states being displayed separately. Our

constructions by contrast already include the `inversion ’ splittings and are preferable

for describing systems with small angular momenta such as those with S ¯ 1, 3 } 2, or

2 as no choice of deperturbed levels has to be made.

3. Applications

Some examples of zero-® eld splitting parameters (all values given in cm Õ " ) for spin

triplet states of small gas-phase molecules as reviewed by LanghoŒand Kern [12] are :

NH ( $ R Õ ), D ¯ 1 ± 856 ³ 0 ± 014 ; O#

( $ R Õ ), D ¯ 3 ± 965 ; CH#

( $ B"), D ¯ 0 ± 76 ³ 0 ± 02 and

E ¯ 0 ± 052 ³ 0 ± 017 ; and CH#O ( $ A

#), D ¯ 0 ± 42 and E ¯ 0 ± 04. Some values for spin

triplet states of oriented organic molecules as reviewed by McGlynn et al. [13] are

naphthalene (C" !

H)), D ¯ 0 ± 1008 ³ 0 ± 0007 and E ¯ ® 0 ± 0138 ³ 0 ± 0002 ; phenanthrene

(C" %

H" !

), D ¯ ³ 0 ± 10044 and E ¯ ­ 0 ± 046 57 ; and pyrene (C" ’

H" !

), D ¯ ³ 0 ± 0678

and E ¯ ­ 0 ± 0314. For the last two examples the ratio r E r } r D r is nearly 0 ± 5, while it

is of the order of 0 ± 1 for several of the other examples.

An especially interesting inorganic example [14] is the spin quintet (S ¯ 2) of Fe # +

in an orthorhombic site in tetragonal ZnF#

with D ¯ ® 7 ± 4 and E ¯ ® 0 ± 7. Gri� th [15]

points out that D for this system may be made positive by a reselection of the principal

axis Sz, while Abragam and Bleaney [16] present expressions for the transformation of

D and E under a cyclic permutation of axes. Application of their expressions to the

ZnF#

(Fe # +) parameters produces transformed values of D ¯ 4 ± 7 and E ¯ ® 3 ± 3,

corresponding to a large ratio r E r } r D r , namely 0 ± 70, or alternatively, D ¯ 2 ± 6 and

E ¯ 4 ± 0, corresponding to an even larger ratio r E r } r D r , namely 1 ± 54. It should be

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Motion of spin systems 1407

noted, however, that axes Sx, S

y, S

zmay always be chosen such that this ratio is not

greater than 1 } 3, although the D value in such an axis system may be negative, as in

the ® rst set of D and E values for ZnF#

(Fe # +). Thus the values of D and E quoted

above for the spin triplets of phenanthrene and pyrene, for which the ratio is nearly 0 ± 5,

could be re-expressed to satisfy the inequality r E r } r D r ! 1 } 3.

Spin analogues of rotating non-rigid spherical tops are provided by the examples

of S-state ions in cubic sites. A familiar [16, 17] spin Hamiltonian (omitting the

Zeeman term) for such systems is the quartic form

Hs¯ (a } 6) (S %

x­ S %

y­ S %

z® (1 } 5) S(S ­ 1) (3S # ­ 3S ­ 1)) (10)

Some representative a values [17] in units of 10 Õ % cm Õ " for S ¯ 5 } 2 ions are : CaF#

(Mn # +), 0 ± 6 ; MgO (Mn # +), 18 ± 6 ; ZnS (Mn # +), 8 ± 3 ; and MgO (Fe $ +), 205. The constant

energy surfaces, rotational energy surfaces, and spin trajectories for these systems are

identical in form to those which describe [7, 8] the rotations of spherical tops.

Speci® cally, positive values of a correspond to energy maxima along the six fourfold

directions ³ Sx, ³ S

y, and ³ S

z, with energy minima along the eight threefold

directions, in direct analogy with SF’, while negative a corresponds to the

complementary case with energy minima along the six fourfold and maxima along the

eight threefold directions, in direct analogy with CH%.

4. Summary

We have described the motion of electron spins in anisotropic systems char-

acterized by spin Hamiltonians containing zero-® eld tensors in terms of semiclassical

trajectories constructed either in spin space from surfaces of constant energy or in

energy space from surfaces of constant magnitude of the spin angular momentum. The

former procedure requires diŒerent constant energy surfaces for diŒerent energy

eigenvalues, but is nonetheless somewhat simpler than the latter procedure, which

does have the advantage of requiring only a single surface. Both types of trajectories

have been presented as time-independent projections for a range of values of the non-

axial orthorhombic spin-Hamiltonian parameter E . These trajectories are closely

analogous to those discussed by Harter [7, 8] in his semi-classical descriptions of the

rotational motion of symmetric and asymmetric top molecules. Our results are applied

to the triplet (S ¯ 1) and quintet (S ¯ 2) states of a number of organic and inorganic

systems. In our present treatment only the time-independent projections of the spin

trajectories are determined, but in a later publication [10] we shall use these as the

starting point for a description of the explicit time-dependence of the spin motions.

Finally we note that all of our graphs were generated using the Maple‡ V Release 3

computer algebra system [18].

The author wishes to thank his colleague Professor Robert R. Sharp for the

suggestion of this research problem and for many valuable discussions.

References

[1] L o h r , L. L., and H u b e n , C. H., 1993, J. chem . Phys., 99, 6369.

[2] L o h r , L. L., and H u b e n , C. H., 1993, Mathematical Computation with Maple V : Ideas and

Applications, edited by T. Lee (Boston : Birkha$ user), pp. 137± 143.

[3] L o h r , L. L., 1995, Molec. Phys., 85, 607.

[4] L o h r , L. L., 1996, Int. J. Quantum Chem ., 57, 707.

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Page 14: Graphical displays of the motion of spin systems

1408 L. L. Lohr

[5] A z i z , R. A., 1993, J. chem . Phys., 99, 4518.

[6] L o h r , L. L., 1992, J. molec. Spectrosc., 155, 205.

[7] H a r t e r , W. G., 1988, Computer Phys. Rep., 8, 319.[8] H a r t e r , W. G., 1993, Principles of Symmetry, Dynamics, and Spectroscopy (New York :

Wiley).

[9] For some applications of rotational energy surfaces see (a) S a d o v s k i i , D. A., Z h i l i n s k i i ,B. I., C h a m p i o n , J. P., and P i e r r e , G., 1990, J. chem . Phys., 92, 1523 ; (b) Z h i l i n s k i i ,

B. I., B r o d e r s e n , S., and M a d s e n , M., 1993, J. molec. Spectrosc., 160, 192 ; (c) P e t r o v ,

S. V., 1993, Spectrosc. Lett., 26, 47.[10] L o h r , L. L., and S h a r p , R. R., to be published.

[11] H e r z b e r g , G., 1945, Molecular Spectra and Molecular Structure. II. Infrared and Raman

Spectra of Polyatomic Molecules (Princeton, NJ : Van Nostrand Co.), pp. 22± 24.[12] L a n g h o f f , S. R., and K e r n , C. W., 1977, Applications of Electronic Structure Theory,

edited by H. F. Schaefer, III (New York : Plenum Press), pp. 381± 437.

[13] M c G l y n n , S. P., A z u m i , T., and K i n o s h i t a , M., 1969, Molecular Spectroscopy of the

Triplet State (Englewood CliŒs, NJ : Prentice-Hall).

[14] T i n k h a m , M., 1956, Proc. R. Soc. London, A236, 549.

[15] G r i f f i t h , J. W., 1961, The Theory of Transition Metal Ions (Cambridge : CambridgeUniversity Press), pp. 355± 360.

[16] A b r a g a m , A., and B l e a n e y , B., 1970, Electron Paramagnetic Resonance of Transition Ions

(Oxford : Clarendon Press), pp. 142± 156.[17] L o w , W., 1960, Paramagnetic Resonance in Solids (New York : Academic Press),

pp. 113± 121.

[18] Maple‡ V Release 3, 1994 (Waterloo, Ontario : Waterloo Software, Inc.).

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