# Electron spin polarization of the excited quartet state of strongly coupled triplet-doublet spin systems.

**ABSTRACT** The electron spin polarization associated with electronic relaxation in molecules with trip-quartet and trip-doublet excited states is calculated. Such molecules typically relax to the lowest trip-quartet state via intersystem crossing from the trip doublet, and it is shown that when spin-orbit coupling provides the main mechanism for this relaxation pathway it leads to spin polarization of the trip quartet. Analytical expressions for this polarization are derived using first- and second-order perturbation theory and are used to calculate powder spectra for typical sets of magnetic parameters. It is shown that both net and multiplet contributions to the polarization occur and that these can be separated in the spectrum as a result of the different orientation dependences of the +/-1/2<-->+/-3/2 and +1/2<-->-1/2 transitions. The net polarization is found to be localized primarily in the center of the spectrum, while the multiplet contribution dominates in the outer wings. Despite the fact that the multiplet polarization is much stronger than the net polarization for individual orientations of the spin system, the difference in orientation dependence of the transitions leads to comparable amplitudes for the two contributions in the powder spectrum. The influence of this difference on the line shape is investigated in simulations of partially ordered samples. Because the initial nonpolarized state of the spin system is not conserved for the proposed mechanism, the net polarization can survive in the doublet ground state following electronic relaxation of the triplet part of the system.

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**ABSTRACT:**The first observation of a spin polarized excited state of a paramagnetic metal-complex using time-resolved electron paramagnetic resonance (TREPR) spectroscopy is reported for octaethylporphinatooxovanadium(iv). The TREPR spectra show well resolved orientation dependent hyperfine splitting to the I = 7/2 vanadium nucleus. The reduction of the hyperfine splitting by a factor of 3 compared to the ground state and the observation of a multiplet pattern of spin polarization allow the TREPR spectra to be assigned to the excited quartet state of the complex. The spin polarization patterns evolve with time and it is postulated that this is a result of the equilibration between the lowest excited quartet and doublet states.Physical Chemistry Chemical Physics 06/2006; 8(18):2129-32. · 4.20 Impact Factor - Alberto Moscatelli, Elena Sartori, Marco Ruzzi, Steffen Jockusch, Xuegong Lei, Igor Khudyakov, Nicholas Turro[Show abstract] [Hide abstract]

**ABSTRACT:**Time-resolved electron paramagnetic resonance spectroscopy, transient absorption, and phosphorescence spectroscopy were used to investigate the spin polarization of a nitroxide free radical induced by interaction with singlet oxygen ((1)O2). The latter was generated by photolysis of endoperoxides of two anthracene derivatives. Although both anthracene endoperoxides are structurally similar, opposite spin polarization of the nitroxide was observed. Photolysis of one endoperoxide leads to absorptive nitroxide spin polarization due to interaction with the generated (1)O2. Photolysis of the other endoperoxide generated emissive nitroxide spin polarization, probably due to interaction of the endoperoxide triplet states with nitroxides.Photochemical and Photobiological Sciences 09/2013; · 2.92 Impact Factor

Page 1

Electron spin polarization of the excited quartet state

of strongly coupled triplet–doublet spin systems

Yuri Kandrashkin

Department of Chemistry, Brock University, 500 Glenridge Avenue, St. Catharines, Ontario, Canada L2S 3A1

and Kazan Physical-Technical Institute Russian Academy of Sciences, Kazan, Russian Federation

Art van der Est

Department of Chemistry, Brock University, 500 Glenridge Avenue, St. Catharines, Ontario, Canada L2S 3A1

?Received 10 October 2003; accepted 11 December 2003?

The electron spin polarization associated with electronic relaxation in molecules with trip-quartet

and trip-doublet excited states is calculated. Such molecules typically relax to the lowest trip-quartet

state via intersystem crossing from the trip doublet, and it is shown that when spin–orbit coupling

provides the main mechanism for this relaxation pathway it leads to spin polarization of the trip

quartet. Analytical expressions for this polarization are derived using first- and second-order

perturbation theory and are used to calculate powder spectra for typical sets of magnetic parameters.

It is shown that both net and multiplet contributions to the polarization occur and that these can be

separated in the spectrum as a result of the different orientation dependences of the ?1/2↔?3/2 and

?1/2↔?1/2 transitions. The net polarization is found to be localized primarily in the center of the

spectrum, while the multiplet contribution dominates in the outer wings. Despite the fact that the

multiplet polarization is much stronger than the net polarization for individual orientations of the

spin system, the difference in orientation dependence of the transitions leads to comparable

amplitudes for the two contributions in the powder spectrum. The influence of this difference on the

line shape is investigated in simulations of partially ordered samples. Because the initial

nonpolarized state of the spin system is not conserved for the proposed mechanism, the net

polarization can survive in the doublet ground state following electronic relaxation of the triplet part

of the system. © 2004 American Institute of Physics. ?DOI: 10.1063/1.1645773?

I. INTRODUCTION

Many light-induced processes generate electron spin po-

larization, which can be measured by transient electron para-

magnetic resonance ?TREPR?. The polarization arises be-

cause of the spin selectivity associated with electronic

relaxation and/or electron transfer and thus provides a way of

studying the dynamics of excited states. A wide variety of

systems with diamagnetic ground states have been studied

for many years and their spin polarization is well

understood.1More recently, there has been considerable in-

terest in molecular systems with paramagnetic ground

states,2–11and their spin-polarized TREPR spectra have not

yet been fully explained. These molecules usually consist of

a nitroxide radical or a paramagnetic metal bound to a chro-

mophore such as a porphyrin,2–4phthalocyanine,5–7or

fullerene,9–13which can be excited to produce an interacting

triplet–doublet spin pair. In systems in which the exchange

coupling between the triplet and doublet is stronger than the

difference of their Zeeman interactions, their TREPR spectra

typically show a broad pattern with multiplet polarization

and narrow components with net polarization near the center

of the spectrum. The broad features are readily assigned2,12

to the ?1/2↔?3/2 transitions of the so-called ‘‘trip-quartet’’

state14of the spin system, while the origin of the narrow

features is not always as clear.15Often they can be assigned

to transitions between the ?1/2 and ?1/2 sublevels of the

quartet state.2,12However, in some cases, contributions from

the doublet states have also been postulated.15–17The mul-

tiplet polarization of the trip quartet is easily explained as a

result of intersystem crossing.6,15,18Because this process is

governed by interactions internal to the molecule, it follows

the molecular symmetry in an analogous manner to singlet–

triplet intersystem crossing in systems with diamagnetic

ground states. Since the states are invariant to inversion of

the molecular axes, the ?3/2 sublevels are equally populated

by intersystem crossing, as are the ?1/2 sublevels. Hence the

?1/2↔?3/2 and ?1/2↔?3/2 transitions have equal and

opposite polarization ?i.e., multiplet polarization?. However,

if this were strictly true, no net polarization would be ob-

served. Thus some element of the system must break the

symmetry and produce net polarization, which is most appar-

ent at the center of the spectrum where the two oppositely

polarized contributions to the multiplet polarization cancel

each other. In principle, the symmetry breaking can be pro-

vided by the external magnetic field and the net polarization

of intramolecular origin can be predicted from the difference

between the internal molecular and external magnetic field

symmetries. Two limiting cases of net polarization in

strongly coupled systems have been mainly presented in the

literature. The first model, used to explain the net polariza-

tion of spin labeled fullerenes,10is based on the ‘‘triplet

mechanism’’1and uses perturbations of the high-field spin

eigenstates by the zero-field splitting. According to this

mechanism, the net polarization is inversely proportional to

JOURNAL OF CHEMICAL PHYSICSVOLUME 120, NUMBER 10 8 MARCH 2004

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Page 2

the strength of external magnetic field. The second model

was proposed to explain the net polarization of porphyrin-

based systems15and invokes mixing of the quartet with a

nearby state. Because the difference in Zeeman energy of the

quartet sublevels enters into the mixing coefficients, it breaks

their symmetry and can produce net polarization, which is

linearly proportional to the strength of magnetic field.

In practice, the factors responsible for the net polariza-

tion in strongly coupled triplet–doublet systems are often

unclear and complicated by possible signals from the doublet

state.6,15,18–21Recently, in attempting to understand the na-

ture of the spin polarization in such systems, we presented a

preliminary report of a model in which the influence of spin–

orbit coupling was treated using first-order perturbation

theory and shown to induce quartet–doublet mixing.22Both

net and multiplet spin polarization of the quartet state were

then shown to occur if the rate of doublet–quartet intersys-

tem crossing is proportional to the degree of mixing. Here

we extend this model to include second-order terms in the

spin–orbit coupling and we present a more detailed analysis

of the expected line shapes including those in partially ori-

ented samples. The results show that the first- and second-

order perturbation terms have different field dependences,

which encompass both the linear and inverse field depen-

dences predicted by previous models.

II. MODEL

We begin by introducing the basic postulates and com-

ponents of the model. Our system consists of a strongly ex-

change coupled triplet–doublet pair, which can be described

by the spin Hamiltonian

H???S?TgT?S?FgF?B??2JS?TS?F?D?S?,Tz

2

?1

3S?T

2?

?E?S?,Tx

2

?S?,Ty

2

?,

?1?

where the T refers to the triplet spin and orbitals involved,

and F refers to the doublet spin and to the orbital carrying

the unpaired electron. ?To simplify the expressions we use

the notation ??1.? J denotes the exchange interaction be-

tween the triplet and doublet, and gTand gFare their respec-

tive g tensors. D and E are the zero-field splitting of param-

eters of the triplet spin state. Here the dipolar coupling

between the triplet and doublet is much weaker than the

zero-field splitting of the triplet and is ignored in Hamil-

tonian ?1? as are hyperfine interactions. The exchange cou-

pling between the triplet and doublet is much larger than the

difference in their Zeeman energies,

J??g?B,

?2?

and the Zeeman energy is much larger than the zero-field

splitting,

??D,E,

?3?

gT?B?gF?B???,

where gTand gFare the values of the respective triplet and

doublet g tensors along the direction of the magnetic field

and ?g is their difference. Under conditions ?2? and, ?3? the

eigenfunctions of Hamiltonian ?1? are separated into trip–

doublet, D, and trip–quartet, Q, substrates:23

?D,?1/2??

1

?3??2T???T0??,

?4?

?D,?1/2??

1

?3??2T???T0??,

?Q,?3/2??T??,

?Q,?1/2??

1

?3?T????2T0??,

?5?

?Q,?1/2??

1

?3?T????2T0??,

?Q,?3/2??T??.

T?and T0are the triplet spin states, and ? and ? are eigen-

states of the doublet spin.

The spin polarization associated with these states de-

pends on the pathway by which they are populated. As

shown by Gouterman,14if the two electrons of the triplet

state have different exchange interactions with the third elec-

tron, the trip–doublet state ?4? acquires some singlet charac-

ter from the sing–doublet state ?see also Toyama et al.24?. As

a result, the transition from the sing doublet to the trip dou-

blet is partially allowed and here we assume that this is the

dominant decay pathway for the sing doublet. This assump-

tion is supported by the observation24–26that in systems such

as porphyrin dimers containing a paramagnetic center, this

effect leads to intersystem crossing ?ISC? rates which are

orders of magnitude faster than in corresponding diamag-

netic complexes. However, it is possible that in some sys-

tems spin–orbit coupling also plays a significant role in the

decay of the excited sing doublet. In such cases, an addi-

tional multiplet contribution to polarization of the trip–

quartet state would be observed. Here we assume that this

contribution is negligible and that following light excitation

to the sing doublet, the initial relaxation predominantly

populates thetrip–doublet

doublet–quartet intersystem crossing then populates the

quartet state ?5?. We propose that the main mechanism of the

latter process is spin–orbit coupling accompanied by fast

internal conversion. ?By the term ‘‘internal conversion’’ here

we describe the spin-independent intramolecular processes

leading to relaxation between different orbital configura-

tions.? Because the time scales of these two processes are

very different, the problem can be treated in terms of a static

spin Hamiltonian in which the spin–orbit coupling acts as a

perturbation. In the molecular frame ?, the spin–orbit con-

tribution has the form

HSO??TL??,TS??,T??FL??,FS??,F,

where ?Tand ?Fare the spin–orbit coupling parameters.

The vector operators L??,Tand L??,Fdescribe the angular

momentum for the orbitals in which the triplet and doublet

spins reside. ?Rigorously speaking this Hamiltonian is cor-

rect only for the one-center approximation of the spin–orbit

sublevels

?4?. Subsequent

?6?

4791 J. Chem. Phys., Vol. 120, No. 10, 8 March 2004 Electron spin polarization

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Page 3

interactions, but for multicenter systems we can use ‘‘effec-

tive’’ orbitals to represent the spin–orbit interaction in the

form used in Eq. ?6?.?

The basic postulates of the model can be summarized as

follows:

?i? The trip–quartet state ?5? is populated from trip–

doublet state ?4? due to intersystem crossing.

?ii? The triplet and doublet parts of the system are fixed

spatially with respect to one another, so that their spin–spin

interaction does not vary. Hence the quartet spin polarization

does not arise from level crossing ?i.e., it is not due to varia-

tion of the exchange interaction? or from fluctuations of the

dipolar interactions.

?iii? Because of the strong exchange coupling between

the triplet and doublet spins, the spin polarization is gener-

ated by internal processes rather than by intermolecular mo-

tion and interactions.

?iv? The rate of the doublet–quartet transitions is limited

by the degree of quartet–doublet mixing described by a static

spin–orbit coupling Hamiltonian ?6?.

These postulates correspond to the properties of most

strongly exchange-coupled triplet–doublet systems at low

temperature. Several additional approximations will be intro-

duced below to derive analytical expressions using perturba-

tion theory.

A. First-order perturbation treatment

The different orbital states involved in spin–orbit cou-

pling ?6? are well separated, so their mixing is comparatively

weak and HSOcan be treated as a perturbation. The mixing

coefficients are determined by the perturbation coefficient:

?T/?Tx,y,z?1,

?F/?Fx,y,z?1,

?7?

where ?Tx,y,zand ?Fx,y,zare energy gaps between the orbit-

als which are mixed by spin–orbit coupling.

The linear approximation of perturbation theory27gives

new states, which can be written in the laboratory coordinate

system, with spins quantized along external magnetic field as

?Ki? ,m???Ki,m?? ?

m?,j?i

?Kj,m??

?Kj,m??HSO?Ki,m?

E?Kj,m???E?Ki,m?,

?8?

where

eignestates defined with no spin–orbit coupling and ?K?,m?

??D?,m? or ?Q?,m? are the first-order eigenstates. m and m?

are spin projections along the axis, and the indices i and j

represent different orbital configurations, linked by spin–

orbit coupling. The zero-order wave functions are given in

Eqs. ?4? and ?5?.

We consider the case in which spin–orbit coupling

mixes the lowest unoccupied molecular orbital ?LUMO? and

the next highest MO of the chromophore, while all other

contributions are negligibly small. This situation arises, for

example, in planar chromophores with extended ? conjuga-

tion and near axial symmetry. Such molecules often have a

pair of almost-degenerate orbitals originating from the

LUMO of the corresponding axially symmetric case. Be-

?K,m???D,m?

or

?Q,m?

are thezero-order

cause of the small energy gap between these two orbitals,

they are generally the ones which are mixed most strongly

by spin–orbit coupling. Thus, for the moment, we ignore the

contribution to the spin–orbit Hamiltonian ?6? from the dou-

blet spin. With the idea of a planar system with two nearly

degenerate in-plane orbitals in mind, we assume that the

mixing is most effective along the molecular z axis.

This assumption is equivalent to assuming that the spin–

orbit coupling has effective axial symmetry and is a conse-

quence of assuming that it only mixes the LUMO with the

next highest MO. We note that cases in which these two

orbitals are mixed by a different component of the spin–orbit

coupling can be treated by a redefinition of the molecular

axes.

We can express the traceless diagonal part of the quartet

density matrix, ??Q, describing the populations of quartet

state as22

??Q?p1sin2?SQz?p2?cos2??1

3??SQz

2?1

3S?Q

2?

?p3sin2?SQz

3,

?9?

where ? is the angle between the z directions in molecular

and laboratory coordinate systems and

S?Q?S?T?S?F

?10?

is the operator for the total spin of the quartet state.

It has been shown22that if the Zeeman energy is small

compared to the energy difference between the orbital con-

figurations 0 and 1, ?Tz,

?Tz?g?B,

?11?

in the first-order perturbation treatment the parameters p1,

p2, and p3can be derived as

p1?4g?B?J?J1?

????

p2,

p2??

?Tz

2???????2

6??

2??

2

,

?12?

p3?0,

where J1indicates the exchange integral in configuration 1.

The energy difference between the trip–quartet and trip–

doublet states of different orbital configurations can be writ-

ten

???E?D1??E?Q0???Tz?J?2J1,

?13?

???E?Q1??E?D0???Tz?2J?J1,

and the spin–orbit coupling parameter is

?Tz??T?1?L?,Tz?0?.

?14?

The first-order operator SQzcorresponds to the net polariza-

tion, while the second-order operator SQz

tiplet contribution and the third-order operator describes a

combination of both net and multiplet contributions. The

largest of the three amplitudes is p2associated with the mul-

tiplet polarization, whereas p3is zero to within our linear

approximation.

2, describes the mul-

4792J. Chem. Phys., Vol. 120, No. 10, 8 March 2004 Y. Kandrashkin and A. van der Est

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Page 4

Thus we find that in addition to the multiplet polariza-

tion, the presence of the magnetic field leads to a term with

net polarization along the field. This net polarization arises

because the different Zeeman energies of the spin sublevels

alter the mixing between doublet and quartet sublevels,

which results in different doublet–quartet ISC rates. This is

similar to the doublet–quartet mixing used by Corvaja et al.

to explain experimental results for the polarization of radi-

cals trapped in crystals of chloranil28and of a nitroxide co-

valently linked to C60in toluene solution.29However, time-

dependent spin–spin interactions rather than spin–orbit

coupling were proposed as the relaxation mechanism. This is

reasonable for interactions involving triplet excitons as sug-

gested for the chloranil crystals28and for time-dependent di-

polar interactions in liquid toluene solutions.29These pro-

cesses are not expected to be effective at low temperature

and in rigidly linked systems because of the fixed mutual

geometry between the triplet and doublet moieties. Thus, to

explain the observed polarization at low temperature in zinc

porphyrin systems with a nitroxide coordinated to the zinc,

Ishii and co-workers15proposed mixing between the trip–

doublet and a nearby quartet state. This is essentially the

same mechanism as outlined here; however, the interaction

responsible for the mixing was not specified in Ref. 15 and

no explicit expressions for the expected polarization were

given.

The importance of expressions ?9? and ?12? is that they

allow the dependence of the net polarization on various fac-

tors to be investigated. The strength of the net polarization is

given by the parameter p1, which is linear in the external

magnetic field provided inequality ?11? holds. This is in con-

trast to the multiplet polarization, which is independent of

the field. The dependence of the two contributions on the

exchange integrals, J and J1, between the doublet and triplet

spins is more complicated and not as obvious in Eqs. ?12?

because ??and ??also depend on J and J1. For values of

J ?or J1) comparable to the orbital energy gap, the net polar-

ization given by p1can become inversely proportional to J

?or J1). For relatively weak exchange coupling, on the other

hand, it is linearly proportional to the value of the exchange

interactions.

Another important consequence of net and multiplet po-

larizations is their different orientation dependences. The

multiplet polarization is proportional to (1?3 cos2?) and is

maximal when ??0—i.e., when the molecular symmetry

axis is parallel to the external field—while the net contribu-

tion is maximal when the symmetry axis is perpendicular to

the field. This effect can be easily understood for a spin–

orbit rotation around the molecular z axis. If the z direction is

parallel to the external magnetic field, the spin–orbit inter-

action couples spin states with equal projections along the

field direction. So the relaxation channels leading from the

doublet m??1/2 and m??1/2 states are symmetric and no

difference in rate occurs. However, for other orientations the

doublet–quartet channels become asymmetric, and the larger

the angle between the symmetry axis and the field, the larger

the asymmetry. As we will show in the next section, an im-

portant consequence of this difference in angular dependence

is that the net and multiplet contributions behave very differ-

ently in the EPR polarization patterns of partially oriented

samples.

B. EPR spectrum of the quartet state

The EPR spectrum of the quartet state is described by a

reduced Hamiltonian, the secular part of which in the labo-

ratory frame is given by30,31

HQ??Q?DQ?SQz

The quartet resonance frequency ?Qand effective zero-field

splitting DQare23

?Q?1

2?1

3S?Q

2?.

?15?

3?2gT?gF??B/?,

?16?

DQ?1

2?D?cos2??1

3??E cos2? sin2??,

where ? is the angle defining the projection of the magnetic

field direction in the molecular (xy) plane. Here we have

used the same notation: D and E for the zero-field splitting of

quartet ?16? and the triplet ?1? to indicate the fact that the

dipolar coupling between the triplet and doublet is assumed

to be negligible. However, in general, the quartet zero-field

splitting parameters are not the same as those of the triplet.

The quartet state has three one-quantum EPR transitions

whose resonance conditions for a microwave frequency ?0

are

?3/2↔?1/2,

?1/2↔?1/2,

?3/2↔?1/2,

Clearly the ?1/2↔?1/2 transition does not depend on the

molecular orientation. ?There are several important excep-

tions:

?i? when highly anisotropic Zeeman interactions ex-

ist either because the g-tensor anisotropy is large or because

the magnetic field is high and ?ii? or as a result of strong

hyperfine interactions, which have to be considered sepa-

rately.? Hence the powder spectrum consists of a central peak

due to the ?1/2↔?1/2 transition with broad wings due to

the ?3/2↔?1/2 and ?1/2↔?3/2 transitions. The central

part of the powder spectrum will be dominated by the net

contribution of the polarization because the oppositely polar-

ized multiplet contributions cancel each other. In the broad

wings, on the other hand, the multiplet contribution domi-

nates because p1/p2?1. According to Eqs. ?9? and ?17?, the

strongest multiplet polarization is expected for the extrema

of (3 cos2??1)—i.e., when the molecular symmetry axis is

either parallel ???0? or perpendicular to the field ???90°?.

In the observed spectrum the polarization is convoluted with

the Pake doublet pattern associated with the orientation de-

pendence of the zero-field splitting ?ZFS? tensor. For axially

symmetric molecules this line shape has two maxima sepa-

rated by

maxima of the multiplet polarization pattern are expected to

be associated with these features, which are well separated

from the maximum of the net polarization.

Numerical simulations verify the separation of the con-

tributions of the net and multiplet polarizations in the EPR

spectrum. Figure 1 demonstrates the main properties of the

EPR spectra of quartet states for typical parameters at the X

?Q??0??2DQ,

?Q??0?0,

?17?

?Q??0??2DQ.

2

3D due to molecules with ???90°?. Hence the

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Page 5

band (?0?9.5GHz). The solid line corresponds to the spec-

trum of the quartet state, populated according Eq. ?9? with

p1/p2?0.05. The dashed and dotted curves show the net

(p2?0) and multiplet (p1?0) contributions, respectively.

The spectra in the top part of Fig. 1 are calculated for an

axially symmetric ZFS tensor (E?0), while, in the bottom

of Fig. 1, the ZFS tensor has maximum anisotropy (D

?3E). As is evident from both sets of spectra, the net po-

larization ?dashed curves? occurs primarily in the center of

the spectra, while the multiplet contribution ?dotted curves?

dominates in the wings of the spectra. These two contribu-

tions also have different magnetic field behavior. The mul-

tiplet contribution is essentially independent of the external

field, and the line shape is determined primarily by the ZFS

parameters D and E. In contrast, the net polarization in-

creases with the Zeeman energy, so that the intensity of the

central peak is larger at higher field.

The resonance positions given by Eqs. ?17? with E?0

and the intensities of the EPR transitions given by the popu-

lation differences in expression ?9? allow an analytical ex-

pression for the spectra line shape to be derived. The angular

dependence of the population difference for the outer lines is

proportional to

dI?3/2↔?1/2????Q,11??Q,22?sin? d?

??2p2?cos2??1

3?sin? d?,

?18?

dI?1/2↔?3/2????Q,33??Q,44?sin? d?

?2p2?cos2??1

3?sin? d?.

The factor of 2 here comes from the definition of the opera-

tor SQz

the absorptive signal. Equations ?9?, ?16?, and ?17? are a set

of parametric equations dependent on the frequency differ-

ence between the outer lines of the quartet and central line.

We express this frequency difference as the dimensionless

parameter ?:

2–SQ

2/3. The sign is chosen to give a positive value for

????Q??0?/D.

The solution of Eqs. ?16?, ?17?, and ?9? for E?0 gives the

intensity of the outer lines:

?19?

I?3/2↔?1/2d??2p2

?

?1

3??

,

????2

3,1

3?,

?20?

I?1/2↔?3/2d??2p2

?

?1

3??

,

????1

3,2

3?,

where we have taken into account the fact that of the angles

? and ?–? give the same resonance field position.

The intensity of the central line can be found analo-

gously:

dI?1/2↔?1/2????Q,22??Q,33?sin? d???p1sin3? d?.

?21?

The position of the central transition does not depend on

molecular orientation so that for a stick spectrum its intensity

becomes proportional to the Dirac ? function:

I?1/2↔?1/2d???p1?????

0

?

sin3? d??4

3p1????d?.

?22?

Because functions ?20? and ?22? are defined for zero line-

widths, they give infinite intensity at ??0, ?1/3, which cor-

responds to the resonance positions ?Q??0?0, ? D/3. To

estimate the intensities at these singular points for a nonzero

linewidth A, we integrate functions ?20? and ?22? over a win-

dow of width A/D. For the net polarization, which domi-

nates the intensity of the central peak, integration of expres-

sion ?22? gives

An?

?A/2D

In,max?1

?A/2D

I?1/2↔?1/2d???4p1

3An,

?23?

whereas the maximum intensity of the outer lines, found by

integration of expression ?20? for the multiplet polarization,

yields

Am?

1/3?Am/D

?1

Im,max?2p2

1/3

?

?

3??

?

?

?1

3???d?

?

4p2

3?AmD.

?24?

In general, the linewidth associated with the net polarization

will be different than that of the multiplet polarization. Thus

we have introduced two different linewidths Anand Am,

respectively. According to Eqs. ?22? and ?23?, the overall net

polarization intensity is equal to ?4p1/3An. However, ex-

perimentally, the absolute intensity is difficult to determine.

FIG. 1. Spin-polarized EPR spectrum of a quartet state calculated according

to expressions ?9? and ?17? with p1/p2?0.05. The dotted curves are the

multiplet contribution p2, the dashed curve is the net polarization p1, and

the solid curves are sum of the net and multiplet contributions. Top: the

zero-field splitting parameters are taken as D?30 mT, E?0 and the inho-

mogeneous linewidth is taken as A?2 mT. From the calculated spectrum,

the ratio of the maxima of the central and outer features is found to be

In,max/Im,max?0.36. Bottom: the same set of parameters except E?10 mT.

The maxima of the central and outer lines is 1.06.

4794J. Chem. Phys., Vol. 120, No. 10, 8 March 2004 Y. Kandrashkin and A. van der Est

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Page 6

In contrast, the ratio of maximum intensities of the net ?23?

and multiplet ?24? polarization patterns is easily measured.

From Eqs. ?23? and ?24? we calculate this ratio as

?In,max/Im,max???p1/p2??AmDQ/An.

Expression ?25? describes the case of an axially symmetric

zero-field splitting tensor (E?0). The ratio In,max/Im,maxwill

be larger for asymmetric cases (E?0), because the range of

orientations which contribute to the maximum in the powder

pattern is smaller. The relative intensity of the net polariza-

tion is expected to be largest when ?D???3E? because the

zero-field splitting vanishes for one of the three canonical

orientations and the multiplet polarization is also zero for

this orientation. This situation is demonstrated in Fig. 1 ?bot-

tom? which shows a simulated spectrum with D?30mT and

E?10mT. All other parameters are the same as used for Fig.

1 ?top?. Comparison of Fig. 1 ?top? and ?bottom? shows that

the absolute intensity of the net contribution is the same in

both cases, but its relative intensity is larger when E?0 ?Fig.

1 ?bottom??.

The theoretical estimate of the intensity ratio ?25? does

not take into account spin relaxation, which is expected to be

faster for the outer lines because of the stronger orientation

dependence of the their resonance conditions ?17?. The

model presented here proposes that the time-dependent

populations of the trip–quartet sublevels given by ?Q,mmare

proportional to squares of mixing coefficients between trip–

doublet and trip–quartet states. As discussed previously in

Ref. 22, this leads to different absolute populations for the

four quartet sublevels, but with the same rise time deter-

mined by the lifetime of the trip doublet, ?. Thus, if the

relaxation of the multiplet polarization is sufficiently fast, the

maximum intensity of the two signal contributions will occur

at different times following the laser flash, whereas the two

signals have identical time dependences if only ISC from the

trip–doublet to the trip–quartet state is considered. However,

if the trip–doublet lifetime becomes long, a linear approxi-

mation of the time dependence shows that the behavior of

the sublevels is different:

?25?

?Q,mm?km??1?e?t/???1?e?kmt.

Here kmis the rate with which sublevel m of quartet state is

populated and ? is the average lifetime of the two trip–

doublet sublevels. The approximation on the right of Eq. ?26?

is valid for large values of the trip–doublet lifetime ? for

which the net polarization will be large. Thus, when the net

polarization is large, its rise time will approach that of the

multiplet polarization, km.

?26?

1. Partially ordered samples

Experimentally, liquid crystal solvents can be used to

provide macroscopic ordering of EPR samples and the order-

ing is maintained when the liquid crystal is frozen to the

glass phase or crystalline solid phase in the magnetic

field.32,33The ordered sample may then be rotated with re-

spect to the field to study the orientation dependence of the

polarization patterns. To investigate orientation dependence

of the net and multiplet contributions to the polarization pat-

terns of the strongly coupled triplet–doublet pairs, we have

calculated spectra for various anisotropic orientation distri-

butions. The form of the distribution function is described in

detail elsewhere34and has been used previously to simulate

the experimental spectra of porphyrin dimers. Here, for sim-

plicity, we assume that the orienting potential is axially sym-

metric, which allows the ordering of the molecule to be de-

scribed by the single order parameter Szz. Although the

potential will generally be anisotropic, the high degree of

ordering usually obtained experimentally leads to a nearly

axially symmetric order matrix. Thus the axial potential

gives spectra which are representative of those likely to be

observed. Figures 2 and 3 show a series of spectra calculated

for various values of Szzusing the same parameters as for the

top and bottom parts of Fig. 1, respectively. Clearly, the

spectra are very sensitive to the ordering, and in particular

the relative amplitudes of the net and multiplet contributions

vary strongly. The spectra demonstrate the sin2? dependence

of the net polarization and the (cos2??1

3) dependence of the

FIG. 2. Spin polarization patterns for partially oriented samples. The spectra

are calculated for three different anisotropic orientation distributions and the

same parameters as for Fig. 1, top. The orientation potential is chosen such

that the order matrix is axially symmetric and the order parameter of the

symmetry axis, Szz, is 0.7 ?top?, 0.3 ?middle?, and ?0.3 ?bottom?. The

director of the orientation ordering is assumed parallel to the magnetic field.

Note that the orientation dependence of the net polarization ?dashed curves?

and multiplet polarization ?dotted curves? are opposite such that the multip-

let polarization is strongest with Szz?0.7 ?top?, while the net polarization is

stronger with Szz??0.3 ?bottom?.

4795 J. Chem. Phys., Vol. 120, No. 10, 8 March 2004Electron spin polarization

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Page 7

multiplet contribution given in Eq. ?9?. For positive order

parameters close to 1, orientations with the molecular z axis

parallel to the field—i.e., small values of ?—are most prob-

able. Hence the net polarization is very weak and the mul-

tiplet polarization is very strong in the upper traces (Szz

?0.7) of Figs. 2 and 3. Conversely, for negative order pa-

rameters, orientations with ? near 90° are more probable and

the net polarization is stronger ?Figs. 2 and 3, bottom traces,

Szz??0.3). As can be seen by comparing Figs. 2 and 3, the

strength of the multiplet polarization for various order pa-

rameters depends on the value of the zero-field splitting pa-

rameter E. When E?0 ?Fig. 2? strong net polarization is

observed for both positive and negative order parameters, but

the spectral positions of the maxima change. For positive

order parameters ?Figs. 2 and 3, top? features corresponding

to the z component of the zero-field splitting tensor are domi-

nant whereas for negative order parameters, the features

from the x and y components of the tensor are most promi-

nent. When E?0 ?Fig. 3?, the x and y components appear at

different field positions and the multiplet polarization is

spread over a wide range of field when the order parameter is

negative. Which of these situations applies to a given com-

plex depends on its structure. For planar molecules, the plane

of the molecule typically orients parallel to the director of

the liquid crystal. Since the molecular z axis is typically per-

pendicular to the molecular plane, such molecules have

negative order parameters and spectra such as those in the

bottom traces of Figs. 2 and 3 can be expected. Figure 4

shows spectra corresponding to Fig. 3, but with the director

of the liquid crystal rotated by 90° with respect to the exter-

nal field. Under these conditions, the dependence of the spec-

tra on the order parameter is less pronounced. This is be-

cause, although the orientation distribution with respect to

the director remains the same, the distribution of orientations

with respect to the field becomes broader and more isotropic.

However, as can be seen by comparing Figs. 3 and 4, the

relative intensity of the net and multiplet contributions is

inverted when the sample is rotated, regardless of the order

parameter.

2. Variation of zero-field splitting axes

In the above discussion we have defined the molecular

axis about which effective spin–orbit coupling occurs to be

the molecular z axis. Although this direction will normally

correspond to one of the principal ZFS axes, it may not nec-

essarily be associated with the largest zero-field splitting.

Hence the z axis of the ZFS tensor, defined by the largest

FIG. 3. Spin polarization patterns for partially oriented samples and a zero-

field splitting tensor with D?3E. The orientation distribution is the same as

used for Fig. 2 and the other parameters are the same as for Fig. 1 ?top?.

FIG. 4. Partially ordered samples with the director of the orientation distri-

bution perpendicular to the magnetic field. All other parameters are the same

as for Fig. 3.

4796 J. Chem. Phys., Vol. 120, No. 10, 8 March 2004 Y. Kandrashkin and A. van der Est

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Page 8

principal value, may be parallel to the any one of the mo-

lecular axes x, y, or z. To demonstrate the effect of a different

possible ordering of the principal values of the ZFS tensor,

Fig. 5 shows a series of spectra in which two different cases

are compared. In both cases the labeling of the ZFS axes is

defined such that ?Dzz???Dyy???Dxx?. In the top part of Fig.

5, the ZFS axes and molecular axes have the same labels so

that the large principal value of ZFS tensor corresponds to

the molecular z axis. In the lower part of the figure the corr-

spondence of the two axes is chosen as xZFS→ymol, yZFS

→zmol, and zZFS→xmol. The solid spectra correspond to E

?D/3 and the dashed spectra have E?0. As can be seen, the

change in the ordering of the principal values of the ZFS

tensor leads to an inversion of the multiplet polarization,

while the net polarization is unaffected by this change. This

is because the sign of the multiplet contribution to the spec-

trum for a given orientation depends on both (cos2??1

the sign of the zero-field splitting, while the sign of net po-

larization is independent of these parameters. The latter is a

result of the fact that the sign of the net polarization is de-

termined by the sign of J, which is positive if the trip quartet

3) and

is lower in energy than the trip doublet. Thus a variety of

multiplet polarization patterns can be expected from mol-

ecules of different structure. In contrast, net polarization is

always predicted to be positive within a first-order perturba-

tion treatment.

In summary, the first-order ?linear? perturbation treat-

ment leads to the following conclusions about the net and

multiplet contributions to the spin polarization:

?i? The spectral positions, magnetic field dependence,

and kinetic behavior of the two contributions are different

and allow them to be separated.

?ii? The ratio of their maximal intensities is proportional

to the square of the spin–orbit coupling divided by the en-

ergy gap between the two lowest excited electronic configu-

rations.

?iii? The net polarization is predicted to be absorptive if

the exchange interaction is positive.

?iv? The orientation dependence of the multiplet and net

contributions are opposite.

In the following section we consider cases in which

higher-order terms of the perturbation must also be taken

into account.

C. Second-order perturbation terms

So far, we have ignored the off-diagonal matrix elements

of the spin Hamiltonians ?1? and ?6? written in the basis ?8?.

Although these terms are small for the conditions considered

above, they can become important, for example, in the limit

of weak exchange coupling. In this case, the mixing of the

doublet and quartet wave functions ?8? due to the spin–orbit

interaction ?6? is proportional to the square of the coupling:

?L,m?HSO?L,m????Tz

As above, we assume that the dominant effect is the mixing

of the molecular orbitals of the chromophore carrying the

triplet excitation. ?The effect of spin–orbit coupling of the

doublet spin will be considered below.? In addition to spin–

orbit coupling, the dipolar coupling of the triplet electrons

can also disturb the wave functions derived from first-order

perturbation treatment ?8?. Both of these effects ?i.e., off-

diagonal terms due to the spin–orbit coupling and dipolar

coupling of the triplet electrons? can modify the wave func-

tions of the doublet and quartet states and introduce addi-

tional terms in the expressions for the parameters p1and p3.

These terms represent a change in the net polarization, while

the effect on the multiplet polarization given by p2is negli-

gible. Another well-known consequence of the second-order

terms ?27? is that they cause a shift of the zero-field spin

energy levels.30As a result, the zero-field splitting of the

quartet state used in Eqs. ?15?, ?16?, and ?23?–?25? must be

replaced by a term which has both dipolar and spin–orbit

parts:

2/?Tz,

L?D?,Q?.

?27?

Deff?DQ?DSO.

To estimate the values of DSO, p1, and p3in the second-

order perturbation treatment, we suppose

?28?

DSO??Tz

2/?Tz??.

?29?

FIG. 5. Dependence of the polarization patterns on the relative magnitude of

the zero-field splitting parameters. Solid curves D?3E, dashed curves E

?0. All other parameters are as in Fig. 1. Top: the direction associated with

the largest zero-field splitting parameter, D, is assumed to correspond to the

molecular z axis which defines the direction of efficient spin–orbit coupling.

Bottom: the zero-field splitting tensor axes are redefined such that the largest

splitting corresponds to the molecular x axis and the x and y axes of the ZFS

tensor correspond to the molecular y and z axes. Note that the sign of the

multiplet polarization changes, while the net polarization is always absorp-

tive.

4797 J. Chem. Phys., Vol. 120, No. 10, 8 March 2004 Electron spin polarization

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Page 9

This requirement is completely analogous to the assumption

made in Eq. ?3? for the dipolar part of the zero-field splitting.

Expressions for p1, p2, and p3and DSOobtained using

second-order perturbation theory for various limiting cases

are presented in Table I. The expressions in the top part of

Table I are derived assuming spin–orbit coupling of the trip-

let dominates, while those in the bottom part assume that the

spin–orbit coupling of the doublet dominates. The top part of

Table I shows that, as expected, the first-order perturbation

result ?12? is obtained when the orbital energy gap is large

(?Tz??) and the zero-field splitting is weak (D?DSO

?0). However, second-order ?quadratic? terms are found for

two limiting situations: ?i? when the exchange interaction of

the higher excited states is negligible compared to the orbital

energy gap (J1??Tz) and ?ii? when the doublet–quartet

splitting of higher orbital states is significant (?Tz?J1). In

the latter case, ?????, so that higher doublet sublevels

can be neglected. Each of these situations is additionally

separated into the two cases for which J?? and J??. Un-

der all of these conditions, the term p3, Eq. ?9?, becomes

nonzero. This contribution to the polarization in the EPR

spectrum is similar to the net contribution p1, and because it

is small compared to the multiplet contribution p2, its influ-

ence mainly appears in the central peak. The most important

result of the second-order perturbation treatment is that the

polarizations p1and p3, which determine the net polariza-

tion, are both inversely proportionality to the Zeeman energy

? and are therefore inversely proportional to the magnetic

field strength:

p1?p3?Deff/g?B.

?30?

This result is based on not only the limiting cases, shown in

Table I, but also from an examination of the intermediate

cases J??. The dependence on the exchange coupling J is

much weaker except for at the level-crossing singularities

when perturbation treatment cannot be applied.

In some systems spin–orbit coupling can be expected to

influence the doublet spin—for example, if it resides in a d

orbital of a transition metal. Thus we have also calculated

initial populations and changes of zero-field splitting for the

case when the doublet spin–orbit coupling is dominant. The

terms shown in the lower part of Table I are very similar to

those induced by spin–orbit coupling of the triplet spin.

However, there are some important differences: First, the ef-

fect of spin–orbit coupling of the doublet spin depends cru-

cially on the value of the exchange integral J1. In fact, if the

energies of the higher excited doublet and quartet states are

close ?i.e., when J1is small?, then the second-order pertur-

bation due to spin–orbit has the form Sz

proportional to the identity operator for spin S?1

no net contribution due to DSOin the second-order terms of

p1and p3for J1??Fz.

2. Because Sz

2is

2, there is

TABLE I. Values of polarizations and DSOfound for different conditions. The notation of parameters when

spin–orbit coupling of doublet spin is dominant ?lower part of the table? is analogous the definitions for triplet

spin–orbit coupling case discussed in the text. For example, ?Fis a value of spin–orbit coupling of unpaired

electron of doublet state. First rows after Hamiltonians show the values in the first-order perturbation treatment

(D?DSO?0): other rows show expressions for the polarization when the second-order terms in p1and p3are

dominant. The second-order terms are calculated within assumption that zero-field splitting is much weaker than

other interactions.

Approximations

p2

?p1/p2

?p3/p2

DSO

HSO??TzL?,TzS?,Tz

?Tz??,

D?DSO?0

??T

2???????2

6??

2??

2

4?2?J?J1?

????

0.4(D?DSO)?

(17.2D?3.5DSO)cos2?

1.2(D?DSO)?

(45.2D?23.8DSO)cos2?

0

¯

?Tz?J1

?Tz?J??

?Tz?J1

?Tz???J

?

2?T

3?Tz

2

2

?0.1(D?DSO)?

(7.2D?1.9DSO)cos2?

?0.7(D?DSO)?

(22.0D?11.3DSO)cos2?

?

?T

2?Tz

2

?Tz?J1

?Tz?J??

?Tz?J1

?Tz???J

?

?T

2

6??Tz?J1?2

0.4(D?DSO)?

(17.2D?3.5DSO)cos2?

(1.9D?0.7DSO)?

(42.8D?9.2DSO)cos2?

?0.1(D?DSO)?

(7.2D?1.9DSO)cos2?

?(0.8D?0.3DSO)?

(22.1D?4.7DSO)cos2?

?

3?T

2

8??Tz?J1?

HSO??FL?,FzS?,Fz

?Fz??,

D?DSO?0

??T

2???????2

6??

2??

2

4?2?J?J1?

????

0

¯

?Fz?J1

?Fz?J??

?Fz?J1

?Fz???J

?Fz?J1

?Fz?J??

?Fz?J1

?Fz???J

?

2?F

3?Fz

2

2

(0.4?17.2 cos2?)D

(?0.1?7.2 cos2?)D

0

?14.7 cos2?D

5.3 cos2?D

?

?F

2

6??Fz?J1?2

0.4(D?DSO)?

(17.2D?3.5DSO)cos2?

(?0.4D?0.7DSO)?

(4.5D?9.2DSO)cos2?

?0.1(D?DSO)?

(7.2D?1.9DSO)cos2?

(0.2D?0.3DSO)?

(0.2D?4.7DSO)cos2?

?

3?F

2

2??Fz?J1?

4798J. Chem. Phys., Vol. 120, No. 10, 8 March 2004Y. Kandrashkin and A. van der Est

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Page 10

In Table I, we have assumed that the z component of the

spin–orbit coupling is dominant. However, if the spin–orbit

interactions in different planes are comparable, there are cru-

cial differences between the first- and second-order terms for

net polarization. The second-order terms mix doublet and

quartet states of the same orbital configuration; thus, they are

roughly proportional to the overall zero-field splitting param-

eter ?30?. Since Deffdepends on the anisotropy of the spin–

orbit coupling, the second-order terms of p1and p3will be

smaller if the spin–orbit coupling is equally effective in any

direction. However, under these conditions, the first-order

perturbation terms nonetheless contribute to p1because they

mix different orbital configurations, while both first- and

second-order terms of multiplet polarization will be effec-

tively canceled. Therefore, in this situation the ratio of p1/p2

will increase.

III. CONCLUSIONS

The described model shows the main features of electron

spin polarization of a strongly coupled structurally fixed

triplet–doublet system. From a practical point of view we

have shown that if the quartet state of such a system is popu-

lated by spin–orbit intersystem crossing from the trip dou-

blet, both net and multiplet contributions to the observed

polarization are expected. This is in contrast to the pure mul-

tiplet polarization predicted by the simple models often used

to describe triplet and quartet states in which a relative ISC

rate is assigned to each of the molecular symmetry axes and

the relative populations of the high-field states are calculated

as a linear combination of the rates ?see, e.g., Ref. 18?. Al-

though the net contribution to the polarization is much

weaker than the multiplet polarization for individual orienta-

tions of the molecule, the different orientation dependences

of the lines results in comparable amplitudes for the net and

multiplet contributions in the powder spectrum of the quartet

state. In general, the net polarization contributes to both the

central peak and the outer wings of the spectrum. However,

under the conditions described in this paper, one can con-

sider that the central peak comes purely from net polariza-

tion, while the outer wings of the spectrum show only mul-

tiplet polarization. As discussed in Ref. 15, spin polarization

of the excited doublet states can also occur. However, in the

model proposed here the initial spin state is not conserved so

that there is no requirement that the net polarization of the

excited quartet state be accompanied by an equal and oppo-

site polarization of the excited doublet state. Thus the net

polarization of quartet state can be generated via spin–orbit

coupling and can be observed while the excited doublet state

remains nonpolarized.

ACKNOWLEDGMENTS

This work was supported by the Natural Sciences and

Engineering Research Council in the form of a Discovery

Grant to A.v.d.E. and a NATO Science Fellowship to Y.K.

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4799J. Chem. Phys., Vol. 120, No. 10, 8 March 2004Electron spin polarization

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