Compu e simula ions o localized small pola ons in amo phous polye hylene
Da id Cube o and Nicholas Qui ke
Ci a ion: The Jou nal o Chemical Physics 120, 7772 (2004); doi: 10.1063/1.1667471
View online: h p://dx.doi.o g/10.1063/1.1667471
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Compu e simula ions o localized small pola ons in amo phous
polye hylene
Da id Cube o and Nicholas Qui ke
Depa men o Chemis y, Impe ial College, London, SW7 2AY, Uni ed Kingdom
共Recei ed 20 Oc obe 2003; accep ed 14 Janua y 2004兲
We use a simple mean ield scheme o compu e he pola iza ion ene gy o an excess elec on in
amo phous polye hylene ha allows us o s udy dynamical p ope ies. Nonadiaba ic simula ions o
an excess elec on in amo phous polye hylene a oom empe a u e show he spon aneous o ma ion
o localized small pola on s a es in which he elec on is con ined in a sphe ically shaped egion
wi h a ypical dimension o 5 Å. We compu e he sel - apping ene gy o be ⫺0.06⫾0.03 eV, wi h
a li e ime on he ime scale o a ew ens o picoseconds. © 2004 Ame ican Ins i u e o Physics.
关DOI: 10.1063/1.1667471兴
I. INTRODUCTION
Polye hylene is he simples o ganic insula o , playing a
e y impo an ole in a numbe o echnological applica ions
such as high ension insula ion. Despi e a as li e a u e1
conce ned wi h he expe imen al cha ac e iza ion o i s elec-
ical p ope ies, e y li le is known abou he de ails o
elec onic anspo a he molecula le el. An unde s anding
o he mechanisms o cha ge anspo in hese ma e ials is
impo an in de e mining he elec onic and op ical p ope ies
and in he de elopmen o new ma e ials wi h mo e eliable
insula ing and o he p ope ies.
Recen em osecond spec oscopy expe imen s ha e
shown he exis ence o shallow sel - apped pola ons in ul-
a hin alkane laye s on a sil e 共111兲su ace.2In addi ion,
ecen Ca -Pa inello simula ions3ha e shown he spon ane-
ous o ma ion o sel - apped pola ons in bulk c ys alline
polye hylene 共modeled using ou chains o se en me hylene
uni s in pe iodic bounda y condi ions, each one ini ially all-
ans兲. This shallow pola on has been linked o he o ma ion
o wo opposi e ans-gauche de ec s in a single chain, wi h
he elec on apped in he o a ed po ion o he chain. In
his pape , we will demons a e he o ma ion o localized
pola ons in amo phous polye hylene. As in bulk c ys alline
polye hylene we ind e y small pola ons, wi h sel - apping
ene gies compa able wi h he he mal ene gy kBT, bu he
geome y o he sel - apped s a e is e y di e en , being
essen ially iso opic, e lec ing he unde ling symme y o
he dielec ic phase. Howe e , wi h such small sel - apping
ene gies one could ques ion he physical meaning o such
localized pola on s a es. We will show ha he li e ime o
hese s a es is la ge enough o allow hem o play a e y
impo an ole in elec on anspo .
In ou p e ious wo k we de eloped a new pseudopo en-
ial o elec on–polye hylene in e ac ions, which has been
used o s udy elec onic s a es in c ys alline and amo phous
con igu a ions o polye hylene. These s a es a e ound o be
in good ag eemen wi h he expe imen al da a.4,5 We also
ha e been able o loca e he mobili y edge 共sepa a ing local-
ized and ex ended s a es兲in amo phous polye hylene a
abou he acuum le el. The localized s a es ex end o abou
⫺0.3 eV. All he simula ions p esen ed in his a icle s a
om le els below he mobili y edge.
The a icle is o ganized as ollows: in Sec. II we desc ibe
he me hods and simula ion de ails. In Sec. III we desc ibe
he mean- ield app oach we ha e used o ca y ou dynamical
simula ions. In Sec. IV we desc ibe he sel - apped pola on.
Finally, Sec. V p o ides a sho summa y and conclusions.
II. METHODS
We ha e used a mixed quan um-classical app oach o
s udy he dynamics o an excess elec on in amo phous poly-
e hylene a oom empe a u e. The elec on is ea ed
quan um-mechanically while each me hylene g oup is
ea ed as a classical pa icle. A as Fou ie ans o m block
Lanczos diagonaliza ion algo i hm6was used o compu e he
adiaba ic elec onic s a es o each classical con igu a ion e -
e y ime s ep. The ime e olu ion o he sys em was gene -
a ed on a Bo n–Oppenheime po en ial ene gy su ace using
he molecula dynamics code DLPOLY.7The Hamil onian o
he classical subsys em con ains all he s anda d ing edien s
used o simula e polye hylene 共bond s e ch, alence angles,
and dihed als e ms兲plus he in e ac ion ene gy wi h he ex-
cess elec on, which is compu ed using he Hellman–
Feynman heo em.8T ansi ions be ween di e en elec onic
ene gy su aces a e allowed and compu ed by means o he
Tully’s ewes swi ches su ace hopping algo i hm.9
We ha e simula ed polye hylene sys ems employing a
single chain in a cubic box wi h pe iodic bounda y condi-
ions. The chain was made o 360 CH2uni s, hough we ha e
also pe o med simula ions wi h a chain o 1215 CH2uni s
o compa ison pu poses. The ini ial equilib ium con igu a-
ions o amo phous polye hylene a oom empe a u e we e
gene a ed using he p ocedu e desc ibed in Re . 5. The elec-
onic wa e unc ions a e ep esen ed on a g id o 323o 463
poin s, depending on he sys em size. We es ima e he unce -
ain y in he esul s due o g id size as 0.02 eV. All in e ac-
ions we e unca ed a c⫽9 Å, wi h all elec onic ene gies
including a long- ange co ec ion based on he pola izabili y
in e ac ion.5
JOURNAL OF CHEMICAL PHYSICS VOLUME 120, NUMBER 16 22 APRIL 2004
77720021-9606/2004/120(16)/7772/7/$22.00 © 2004 Ame ican Ins i u e o Physics
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III. MEAN FIELD APPROACH
The in e ac ion ene gy be ween he excess elec on a
and he CH2uni s a Rican be decomposed in o epulsi e
and a ac i e pa s:
V共 ,Ri兲⫽兺
jVj
共
兩
⫺Rj
兩
兲)⫹Vp共 ,Ri兲,共1兲
whe e Vj
( ) is a sho - ange epulsi e pai po en ial5and Vp
is he elec os a ic ene gy ha accoun s o he poin cha ge-
induced-dipole pola iza ion in e ac ion:
Vp共 ,Ri兲⫽⫺ 1
2兺
jpj•Ej
共0兲Sj共
兩
⫺Rj
兩
兲,共2兲
wi h
Ej
共0兲⫽⫺eRj⫺
兩
Rj⫺
兩
3共3兲
he di ec elec ic ield due o he excess elec on, and
pj⫽
␣
jEj共4兲
he dipole momen o he uni ed a om jwi h a pola izabili y
enso
␣
j.Sj( ) is he swi ching unc ion ha anishes when
→0, accoun ing o he ini e size o he me hylene uni s.
The local elec ic ield a each a om Ejis he solu ion o he
se o equa ions
Ej⫽Ej
共0兲⫹兺
k⫽jT共Rj,Rk兲•
␣
kEk,共5兲
wi h T(Rj,Rk)⫽(3R
ˆjkR
ˆjk⫺1)/Rjk
3,Rjk⫽Rj⫺Rk, and R
ˆjk
⫽Rjk /Rjk . Knowledge o he local elec ic ield Ejallows
one o compu e he o ce exe ed on each a om due o he
pola iza ion in e ac ion as
具
␾
兩
Fj
p
兩
␾
典
, whe e
␾
is he elec-
onic wa e unc ion and 共Appendix A兲
Fj
p⫽eT共 ,Rj兲pj⫹兺
k⫽j
3
Rjk
4共共pj•pk兲Rjk⫹共pj•Rjk兲pk
⫹共pk•Rjk兲pj⫺5共pj•Rjk兲共pk•Rjk兲Rjk兲.共6兲
The i s e m in 共6兲is he o ce due o he di ec ield c ea ed
by he elec on 共which because o New on’s hi d law is mi-
nus he o ce exe ed by he dipole jon he elec on兲and he
second is he esul o he in e ac ion wi h all o he induced
dipoles. In Re s. 4 and 5 we sol ed sel -consis en ly he se
o equa ions 共5兲using an i e a i e app oach o se e al con-
igu a ions o bulk polye hylene. Howe e , he calcula ion is
e y expensi e and u e ly p ohibi i e in a simula ion o he
dynamics o an excess elec on in polye hylene since he
in e ac ion ene gy has o be compu ed e e y ime s ep.
In 1967 Lekne p oposed10 a mean ield app oach o
compu e he pola iza ion in e ac ion o a poin cha ge in a
sys em o iso opic dipoles 共wi h a scala pola izabili y
␣
兲.In
his app oxima ion, he local ield a each a om is eplaced by
Ej⫽Ej
共0兲 共
兩
Rj⫺
兩
兲,共7兲
whe e ( ) is a unc ion ha accoun s o he sc eening o
he di ec elec ic ield exe ed by he elec on a he dipole j
due o he o he induced dipoles. Assuming 共7兲and summing
he con ibu ion o all induced dipoles in a s a is ical way, we
ob ain an equa ion o ( ):10
共 兲⫽1⫺
␲
n
␣
冕
0
⬁dsg共s兲s⫺2
冕
兩
⫺s
兩
兩
⫹s
兩
d 共 兲 ⫺2
␪
共 ,s, 兲
共8兲
wi h
␪
共 ,s, 兲⫽3
2s2共s2⫹ 2⫺ 2兲共s2⫹ 2⫺ 2兲⫹共 2⫹ 2⫺s2兲.
共9兲
In Eq. 共8兲nis he numbe densi y and g( ) is he pai co -
ela ion unc ion o he dipole sys em, usually aken as he
equilib ium pai co ela ion unc ion in absence o he excess
elec on. S ic ly speaking g( ) should be modi ied due o
he p esence o he elec on; howe e , neglec ing his pe u -
ba ion has been shown o be e y success ul when applied o
simple pola izable luids.11 Wi h his app oach ( ) has o be
sol ed nume ically using an i e a i e sel -consis en me hod,
bu only once and as a esul we ob ain a pai wise addi i e
in e ac ion ene gy ha can be compu ed e icien ly e e y
ime s ep.
The i s di icul y in applying his mean- ield app oach
o polye hylene is ha he pola izabili y o he me hylene
uni s is no a scala bu a enso . In ac we will show la e
ha he mean- ield app oxima ion b eaks down o s ong
aniso opy in he pola izabili y enso . Ano he di icul y
a ises when he sys em is made o dipoles o di e en pola -
izabili y, which again is ou case, since ou pseudopo en ial
con ains wo di e en pola izabili y cen e s: one a he cen e
o he CH2uni s 共in Å3兲,
␣
CH2⫽
冉
0.5
1.12
1.64
冊
共10兲
共in a coo dina e sys em along he symme y axes o he me-
hylene uni 5兲, and ano he a he cen e o he C–C bonds
␣
C–C⫽
冉
2.103
0
0
冊
共11兲
共along he C–C bond兲.
These complica ions a e di icul o o e come heo e i-
cally. We show below, howe e , ha his is no necessa y,
since a nai e applica ion o Lekne ’s mean- ield me hod o
polye hylene al eady p o ides good esul s.
In Fig. 1 we ha e plo ed he sc eening unc ion ( )
共solid line兲 ha esul s when we use Eq. 共8兲wi h a o al
a e age pola izabili y o he me hylene uni s,
␣
¯
CH2⫽1
3关T 共
␣
CH2兲⫹T 共
␣
C–C兲兴⫽1.788 Å, 共12兲
and he pai co ela ion unc ion g( ) o he CH2uni s in
amo phous PE a oom empe a u e compu ed om a mo-
lecula dynamics simula ions in absence o he excess elec-
on. This adial dis ibu ion unc ion only con ains he pai s
connec ed by a an de Waals in e ac ions, which is consis-
7773J. Chem. Phys., Vol. 120, No. 16, 22 Ap il 2004 Localized pola ons in amo phous polye hylene
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en wi h he pseudopo en ial i sel , since he pola izabili ies
we e compu ed by ab ini io me hods o alkanes igno ing he
in amolecula dipole–dipole in e ac ion.5
In o de o p obe his app oxima ion we ha e compu ed
he e ec i e sc eening unc ions om simula ions o amo -
phous PE a oom empe a u e using he ull many-body sel -
consis en app oach 关Eq. 共5兲兴, de ined o each dipole species
as he a e age o e all dipoles o he quan i y pj•Ej
(0)/Ej
(0)
•
␣
j•Ej
(0) . The esul s co esponding o a single con igu a-
ion o a PE chain con aining 1215 CH2uni s a e shown in
Fig. 1. The sc eening unc ion o he
␣
CH2cen e s 共dashed
line in Fig. 1兲is e y smoo h, sugges ing ha a mean ield
app oach would succeed o his ype o dipole, whe e he
aniso opy o he pola izabili y enso is ela i ely weak.
Howe e , he esul s o C-C dipoles 共c osses兲a e no
smoo hly a ying and, e en hough he da a seem o be sca -
e ed a ound he CH2sc eening unc ion, he dispe sion is
la ge enough o ejec a mean ield ea men .
All sc eening unc ions sha e common ea u es. In he
limi →0, ( ) ends o uni y, since he di ec Coulomb
in e ac ion wi h he elec on becomes dominan , and in he
opposi e limi →⬁ he sc eening unc ion should app oach
共in iso opic sys ems like amo phous PE兲 he Lo en z local
ac o L⫽1/(1⫹(8/3)
␲
␣
¯
)共see, o example, Re . 12兲, since
a long dis ances he di ec elec ic ield is iewed locally as
a cons an ield. I can be p o en ha all solu ions o he
mean ield Eq. 共8兲 end o his limi 共see Appendix B兲,as
indeed does ou nume ical solu ion o he mean ield equa-
ion in Fig. 1 共solid line兲. Howe e , he sc eening unc ion
measu ed om he ull many-body simula ion appea s o
end o a highe alue 共dashed line兲. The eason o his
de ia ion is ha he local elec ic ield was compu ed sol ing
Eq. 共5兲 o all dipoles inside a cu o sphe e o adius c
⫽9 Å cen e ed a he excess elec on. Since he main con i-
bu ion o he sc eening unc ion a each dipole comes om
he su ounding dipoles, he sc eening o he di ec elec ic
ield due o he dielec ic i sel is no p ope ly accoun ed o
a dipoles close o he su ace o he cu o sphe e. A simu-
la ion o he same amo phous con igu a ion bu wi h a la ge
cu o adius c⫽9⫹3 Å shows ha he p ope Lo en z limi
is eco e ed 共do ed line in Fig. 1兲. This de ia ion is o li le
p ac ical impo ance, since he wa e unc ions o bo h sys-
ems a e p ac ically indis inguishable and he e o in he
ene gy le els is lowe han 0.01 eV 共p o ided he same long
ange co ec ion based on he Lo en z ac o Lis added,5
which in his case is as la ge as ⫺0.41 eV兲.
Apa om he la ge luc ua ions in he C–C-bond
sc eening unc ion, he ull-many body sc eening unc ions
a e in quali a i e ag eemen wi h he mean ield unc ion,
wi h he o me clea ly highe nea he i s minimum. This
de ia ion is esponsible o an inc ease in he absolu e alue
o he pola iza ion ene gy and hus a dec ease o he ene gy
le els. In Table I we p esen a compa ison be ween he en-
e gy le els ob ained wi h he ull-many body simula ion and
he mean- ield app oach using he Lekne ’s sc eening unc-
ion plo ed in Fig. 1 o bo h dipole species and he same
con igu a ion. I can be seen ha all ene gy le els a e shi ed
down by app oxima ely he same amoun o 0.15⫾0.02 eV
共wi h he e o aken om ou g id esolu ion e o 兲, and ha
he de ia ion in he i s six wa e unc ions is e y small 共 he
p ojec ion o one no malized eigens a e on he o he is close
o uni y兲. We ha e ound he same pa e n o di e en con-
igu a ions and di e en sys em sizes.
Since ou in e es is in he lowes ene gy le els, we will
use his mean- ield me hod in ou simula ions o he dynam-
ics, co ec ing all ene gy le els by ⫺0.15 eV, which can be
included in he long- ange co ec ion cons an . Then, he
o ce exe ed by he excess elec on in each a om due o he
pola iza ion in e ac ion is bes compu ed di ec ly om he
in e ac ion ene gy,
Vp⫽⫺ 1
2兺
jEj
共0兲•
␣
j•Ej
共0兲 共
兩
⫺Rj
兩
兲共13兲
关whe e he swi ching ac o S( ) has been included in he
sc eening ac o ( )]. Wi h Fj
p⫽⫺ⵜjVp, whe e ⵜjis he
g adien wi h espec o Rj, we ob ain
Fj
p⫽eT共 ,Rj兲pj⫹1
2共Ej
共0兲•
␣
j•Ej
共0兲兲ⵜj 共
兩
⫺Rj
兩
兲.共14兲
IV. THE LOCALIZED SMALL POLARON
An excess elec on in oduced in o amo phous polye h-
ylene is expec ed o in e ac wi h he dielec ic and be
apped in a dis o ed egion. In Fig. 2 we show he ime
e olu ion o he h ee componen s o he cen e o mass and
FIG. 1. Sc eening unc ion ( ). The solid line is he mean- ield esul . The
dash and do ed lines co espond o he sc eening unc ion o he elec on–
CH2pai compu ed om a ull many-body simula ion wi h a cu o adius o
c⫽9 and 12 Å, espec i ely. The c osses co espond o he elec on–共C–C
bond兲pai s.
TABLE I. Compa ison o he ene gy le els be ween a ull many-body sel -
consis en and a mean ield calcula ion.
nEn
MB En
MF⫺En
MB
具
␾
n
MB
兩
␾
n
MF
典
0⫺0.224 0.148 0.996
1⫺0.191 0.153 0.969
2⫺0.171 0.154 0.985
3⫺0.148 0.139 0.972
4⫺0.104 0.140 0.990
5⫺0.083 0.147 0.920
6⫺0.036 0.138 0.198
7⫺0.025 0.139 0.119
7774 J. Chem. Phys., Vol. 120, No. 16, 22 Ap il 2004 D. Cube o and N. Qui ke
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he localiza ion leng h o he excess elec on in a mic o-
canonical simula ion s a ing om he g ound s a e o an
amo phous PE sys em o leng h L⫽32.12 Å a oom em-
pe a u e unpe u bed by he p esence o he excess elec on.
These quan i ies we e compu ed unde pe iodic bounda y
condi ions ollowing he p ocedu e ske ched in Appendix C.
No e ha a he beginning he elec on is in a s a e below he
mobili y edge5and hus localized. Howe e , his s a e is no
s able and he elec on explo es pa o he sys em, ending up
a e abou 1 ps in a localized s a e wi h a smalle leng h. We
p esen in Fig. 3 he ime e olu ion o he ene gy spec um
o he same simula ion. In his case he elec on emains in
he g ound s a e, showing an adiaba ic elaxa ion. We ha e
obse ed simila beha io in all ou simula ions, hough in
some cases o i he ini ial s a e is no he g ound s a e we
wi nessed la ge elaxa ion imes, o he o de o 5–10 ps. In
Fig. 4 we show he ime e olu ion o he ene gy le els in a
simula ion whe e he excess elec on was ini ially in he i s
exci ed s a e. The elaxa ion in ol es a se ies o ene gy hops,
indica ing a s ong nonadiaba ic in e ac ion, e en ually o m-
ing he sel - apped s a e a e abou 7 ps.
We p esen in Fig. 5 he esul s wi h a smalle sys em
box o leng h L⫽21.41 Å s a ing om he g ound s a e. In
his case he ini ial elaxa ion is no op imally mimicked,
since he ini ial s a e, hough localized, ex ends h ough mos
o he sys em, bu he inal sel - apped s a e is well cha ac-
e ized, showing he same localiza ion leng h o abou 5 Å.
These smalle sys ems a e much easie o simula e, educing
d as ically he compu e ime used. In he ollowing we will
p esen esul s compu ed using his smalle sys em size.
I is clea om Fig. 3 o 4 ha , despi e he ac ha he
kine ic ene gy is inc eased due o a educ ion o he localiza-
ion leng h, he in e ac ion ene gy becomes mo e nega i e.
This can only be explained in e ms o a smalle epulsion
in e ac ion and/o a la ge pola iza ion in e ac ion Vp. This is
he eason hese s a es a e usually called sel - apping po-
la ons. Pa o he ene gy gained goes o he dielec ic o
c ea e he dis o ion ha makes possible his pola on s a e.
The di e ence be ween he ene gy gained and he dis o ion
ene gy 共bo h in absolu e alue兲should be s ill posi i e, since
FIG. 2. Componen s o he excess elec on cen e o mass 共lines, in Ang-
s oms兲as a unc ion o ime om an ini ially unpe u bed con igu a ion o
amo phous PE a oom empe a u e. C osses a e he localiza ion leng h o
he elec on a each ime. Sys em bounda y edges a e included as a e e ence
共ho izon al do ed lines兲, hough he ac ual posi ion is a bi a y because o
he pe iodic bounda y condi ions.
FIG. 3. Elec onic ene gy le els e sus ime o he same simula ion shown
in Fig. 2.
FIG. 4. Elec onic ene gy le els e sus ime o a simula ion s a ing om
an exci ed s a e showing a nonadiaba ic elaxa ion. The black squa es deno e
he cu en ene gy le el.
FIG. 5. The same as Fig. 2 bu o a simula ion wi h a smalle sys em size.
7775J. Chem. Phys., Vol. 120, No. 16, 22 Ap il 2004 Localized pola ons in amo phous polye hylene
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we obse e his sel - apped s a e in e e y simula ion. How-
e e , his ene gy 共o mo e p ecisely minus his ene gy兲, no -
mally called he sel - apping ene gy,13 is di icul o measu e
om a mic ocanonical simula ion, since he o al ene gy is
conse ed and hus i is ans e ed o he dielec ic in he
o m o he mal ene gy.
In o de o compu e he sel - apping ene gy i is mo e
con enien o pe o m a simula ion in he canonical en-
semble. We ha e used he Ande sen he mos a , which gen-
e a es ajec o ies in he canonical ensemble.14 In his case,
he quan i y ha is minimized in he sel - apping p ocess is
no he o al ene gy bu he ee ene gy ⌬Gs . We ha e com-
pu ed his ee ene gy using he accep ance a io me hod o
classical sys ems, which is desc ibed in de ail in Re . 15. To
sum up, we need o compu e he cons an C ha sa is ies he
equa ion
兺
mh共U0⫺U1⫹C兲⫽兺
m⬘
h共U1⫺U0⫺C兲,共15兲
whe e h(x)⫽(1⫹exp
␤
x)⫺1is he Fe mi unc ion (
␤
⫽1/kBT) and U0and U1a e he o al ene gies o he classi-
cal sys ems U0and U1a he con igu a ions m共in a simula-
ion o sys em 1兲o m⬘共equilib ium con igu a ions o sys-
em 0兲. The ee ene gy di e ence ⌬Fis hen gi en by
␤
⌬F⫽⫺ln共n1/n0兲⫹
␤
C,共16兲
whe e n1and n0a e he o al numbe o con igu a ions
sampled in each sys em. Since we a e in e es ed in he en-
e gy di e ence be ween he unpe u bed g ound s a e and he
sel - apped s a e, we chose U0as he Hamil onian o he
polye hylene sys em wi hou he excess elec on, and U1as
he ull Hamil onian wi h he elec on hold a he g ound
s a e. The e o e, he sel - apping ene gy is gi en by
␤
⌬Gs ⫽F1⫺共F0⫹
具
E0
典
0兲,共17兲
whe e
具
E0
典
0is he a e age o he elec onic g ound s a e
ene gy o e he equilib ium con igu a ions o he unpe -
u bed polye hylene sys em. A e aging o e 5⫻2 simula-
ions o 10 ps o adiaba ic dynamics we ob ained ⌬F
⫽⫺0.27⫾0.01 eV and
具
E0
典
0⫽⫺0.21⫾0.02 eV, which im-
plies ⌬Gs ⫽⫺0.06⫾0.03 eV. Mo eo e , he a e aged local-
iza ion leng h o he sel - apped pola on was 5.0⫾0.1 Å and
o sphe ical shape, in ag eemen wi h he mic ocanonical
simula ions.
I is wo h no ing ha a di ec calcula ion o he sel -
apping ene gy as he o al ene gy di e ence o he wo
s a es 共apa om neglec ing he en opy change兲does no
p oduce an accep able alue due o he la ge he mal luc ua-
ions o each a iable 共⬃0.1 eV兲. I is much mo e e icien o
use an app oach such as he accep ance a io me hod, in
which he ene gy di e ence o bo h sys ems is compu ed o
each con igu a ion o bo h simula ions.
We ha e obse ed a co ela ion be ween he sel -
apping pola on and he local densi y o he dielec ic. In all
cases we ound ha he cen e o mass o he excess elec on
was wi hin2Åo heglobal minimum o he local a omic
densi y 共de ined as he numbe o a omic cen e s in a sphe e
o adius equal o he elec on localiza ion leng h di ided by
he olume o he sphe e兲. Fu he mo e, his minimum local
densi y was always smalle in he p esence o he elec on
han he minimum local densi y in he unpe u bed sys em.
These changes, which in some sys ems amoun ed o a 50%
educ ion in local densi y, clea ly indica e he o ma ion o a
ca i ylike egion by means o which he elec on lowe s i s
ene gy.
In o de o compa e ou esul s in amo phous PE wi h
he sel - apped s a e ound in c ys alline PE3we ha e moni-
o ed he dihed al angle dis ibu ion and ound no signi ican
di e ence be ween he gauche popula ion be o e and a e
he pola on is o med, e lec ing he ac ha in he diso -
de ed phase he elec on has many mo e ways o dis o he
dielec ic han in he c ys alline phase, in which he elec on
can only c ea e a ca i ylike egion by ab up ly c ea ing wo
gauche de ec s in he o he wise all- ans chains.
In addi ion, we ha e obse ed ha he sel - apped po-
la on is sensi i e o he empe a u e o he sys em. Simula-
ions a 350 K show a la ge pola iza ion ene gy, wi h a
smalle localiza ion leng h o abou 4.2 Å, while a sys em a
200 K shows a mo e ex ended excess elec on wi h a ypical
leng h o 6.4 Å.
The calcula ions p esen ed abo e p edic ha he sel -
apping ene gy is o he same o de as he he mal ene gy a
oom empe a u e (kBT⫽0.026 eV), and as a esul we
would expec apid apping and de apping gi ing ise o
hopping conduc ion assis ed by phonons. This is in ac wha
we obse e in he simula ions. In Fig. 6 we p esen he com-
ponen s o he cen e o mass o he elec on as a unc ion o
ime o wo ypical long simula ions showing a hop be ween
sel - apping s a es a di e en posi ions. The li e ime o
each s a e is abou one o de o magni ude la ge han he
elaxa ion ime o he sel - apping s a es 共⬃ps兲, demons a -
ing he physical ele ance o hese sel - apped s a es, e-
ga dless o hei small sel - apping ene gies. The ime e o-
lu ion o he elec onic ene gy in hese simula ions shows
ha his is an adiaba ic p ocess, he elec on s aying in he
g ound s a e su ace du ing he hop. The obse ed li e ime is
in ag eemen wi h he widely accep ed model o adiaba ic
hopping conduc ion in pola on heo y,16 in which he li e ime
is es ima ed as
␶
⫽
␯
⫺1exp共Ea/kBT兲,共18兲
wi h
␯
being a ypical phonon equency esponsible o he
de apping, and Eaan ac i a ion ene gy, he e o be iden i ied
wi h he sel - apping ee ene gy ⌬Gs ⫽⫺0.06⫾0.03 eV.
Since he op ical equencies in polye hylene a e in he ange
700–1600 cm⫺1,17 he co esponding phonon ene gies h
␯
a e abou one o de o magni ude la ge han Eaand hus
only he acous ic o o sional 共 ans e sal兲phonons, wi h e-
quencies in he ange 0⭐
␯
⭐250 cm⫺1,18 can be esponsible
o he des uc ion o he sel - apping s a e. As a esul ,
␯
⫺1
is o he o de o a picosecond 共o la ge 兲, which is abou he
same ime scale o he elaxa ion o hese s a es, and he
exponen ial ac o in 共18兲p o ides he ac o o 10 obse ed
in he simula ions.
Fu he mo e, his analysis and he simula ion esul s
show ha he dynamic beha io a he empe a u e consid-
e ed is domina ed by he low-laying ene gy s a es and hus
7776 J. Chem. Phys., Vol. 120, No. 16, 22 Ap il 2004 D. Cube o and N. Qui ke
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p o iding a pos e io i jus i ica ion o he use o he Lekne ’s
mean ield me hod, which desc ibes co ec ly only he lowes
ene gy s a es.
V. CONCLUSIONS
We ha e shown ha e en when a complex nonpola di-
elec ic like amo phous polye hylene wi h wo species o an-
iso opic pola izable dipoles is conside ed, a simple mean
ield app oach neglec ing hese de ails p oduces good esul s
when one is only in e es ed in he lowes ene gy le els o he
excess elec on. This me hod makes dynamical simula ions
easible.
Nonadiaba ic simula ions o an excess elec on in amo -
phous PE a cons an ene gy show he o ma ion o a small
sel - apped pola on wi h a educed localiza ion leng h o 5
Å on picosecond imescales. The wa e unc ion is iso opic
and cen e ed a ound a small ca i ylike egion c ea ed by he
elec on. In o de o calcula e he sel - apping ene gy we
ha e pe o med adiaba ic simula ions in he canonical en-
semble using he Ande sen he mos a . The sel - apping en-
e gy is es ima ed o be ⌬Gs ⫽⫺0.06⫾0.03 eV om he ee
ene gy change be ween an unpe u bed dielec ic sys em and
a sys em pe u bed by he p esence o he elec on. This is
he ene gy equi ed o emo e he sel - apped s a e and is
small when compa ed o he ac i a ion ene gy o exci a ion
o ex ended s a es om unpe u bed con igu a ions, i.e.,
⬃0.3 eV, he eby p o iding a jus i ica ion o he me hod em-
ployed in Re . 5, in which he e ec o he excess elec on on
he dielec ic was neglec ed. F om Re . 5, we expec elec-
ons he mally exci ed o ene gy le els abo e he mobili y
edge o p o ide a con ibu ion o he ze o- ield mobili y o
o de o 10⫺3cm2/Vs, which is in good ag eemen wi h he
highes alues ound in expe imen s.
Howe e , he smallness o he sel - apping ene gy sug-
ges s a hopping mechanism assis ed by phonons, which is in
ac obse ed in he simula ions. The li e ime o each sel -
apping s a e is obse ed o be on he ime scale o a ew
ens o picoseconds. We show ha his imescale is consis-
en wi h an adiaba ic model o de apping and he com-
pu ed alue o he sel - apping ene gy.
Since he elec on is localized and nondegene a e, he
co esponding con ibu ion o he mobili y can be compu ed
using he Eins ein o mula13 by measu ing he di usion co-
e icien om e y long ime simula ions. P elimina y calcu-
la ions show ha he con ibu ion o he mobili y due o hop-
ping be ween hese sel - apped s a es may well be abou he
same o de o magni ude as he mobili y due o exci ed elec-
ons abo e he mobili y edge. These simula ions a e e-
po ed elsewhe e.19
APPENDIX A: POLARIZATION FORCES
In p inciple, he o ce on each dipole jcould be com-
pu ed om he g adien o he pola iza ion ene gy Vp. How-
e e , i is easie o calcula e i om he local elec ic ield.
Assume a simple dipole pa , hen he elec os a ic o ce on
he dipole will be
F⫽lim
d→0,qd→p
关⫺qE共 兲⫹qE共 ⫹d兲兴⫽共p•ⵜ兲E共 兲.共A1兲
I we now conside he ull sys em wi h dipoles a Rjand an
elec on a 0, he elec ic ield e e ywhe e is gi en by
E共 兲⫽E共0兲共 兲⫹兺
kT共 ,Rk兲•pk,共A2兲
wi h E(0)( )⫽⫺e( ⫺ 0)/
兩
⫺ 0
兩
3. Using hese equa ions,
oge he wi h
ⵜE共0兲共 兲⫽eT共 , 0兲共A3兲
and
⳵
T共 , ⬘兲m,n
⳵
xl
⫽3
兩
⫺ ⬘
兩
4
冉
共xl⫺xl
⬘兲
兩
⫺ ⬘
兩
␦
mn
⫹
共xm⫺xm
⬘兲
兩
⫺ ⬘
兩
␦
ln⫹
共xn⫺xn
⬘兲
兩
⫺ ⬘
兩
␦
lm
⫺5共xl⫺xl
⬘兲共xm⫺xm
⬘兲共xn⫺xn
⬘兲
兩
⫺ ⬘
兩
3
冊
,共A4兲
we ob ain 共6兲.
FIG. 6. The same as Fig. 2 bu o wo di e en long simula ions showing a
hopping mechanism be ween he sel - apped s a es.
7777J. Chem. Phys., Vol. 120, No. 16, 22 Ap il 2004 Localized pola ons in amo phous polye hylene
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APPENDIX B: LONG DISTANCE LIMIT OF THE
SCREENING FUNCTION
The i s s ep o ind he long dis ance limi o he mean
ield ( ) is o p o e
␦
共 ⫺ 兲⫽3
8 ⫺2
冕
兩
⫺
兩
⫹ dss⫺2
␪
共 ,s, 兲,共B1兲
whe e
␪
is gi en by Eq. 共9兲. Since o ⫽
冕
兩
⫺
兩
⫹ dss⫺2共 ,s, 兲
⫽共s⫺ ⫺ 兲共s⫹ ⫺ 兲共s⫺ ⫹ 兲共s⫹ ⫹ 兲
2s3
冏
兩
⫺
兩
⫹
⫽0, 共B2兲
i is only le o p o e ha i is p ope ly no malized. Re e s-
ing he o de o in eg a ion
冕
0
⫹␧d
冉
3
8 ⫺2
冕
兩
⫺
兩
⫹ dss⫺2
␪
冊
⫽
冕
0
␧ds
冕
兩
⫺s
兩
⫹sd 䊐
⫹
冕
␧
ds
冕
⫺s
⫹␧d 䊐⫹
冕
2 ⫹␧ds
冕
s⫺
⫹␧d 䊐
⫽3
8
冉
共␧⫺ 兲2共␧⫹5 兲
6 3⫺共␧⫹ 兲共␧2⫹2␧ ⫺11 2兲
6 3
冊
⫽1,
whe e 䊐⫽3 ⫺2s⫺2
␪
/8.
I we e e se he o de o in eg a ion in 共8兲, ake he
limi →⬁, and use 共B1兲and g(⬁)⫽1, we ob ain
共⬁兲⫽1⫺8
3
␲
n
␣
共⬁兲,共B3兲
and he e o e (⬁)⫽ L.
APPENDIX C: CALCULATION OF THE ELECTRONIC
CENTER OF MASS AND LOCALIZATION
LENGTH WITH PERIODIC BOUNDARY CONDITIONS
The cen e o mass o he excess elec on and he local-
iza ion leng h a e de ined om he wa e unc ion
␾
共 兲as
具
␾
兩 兩
␾
典and
冑
具
␾
兩
"
兩
␾
典
⫺
具
␾
兩
兩
␾
典
•
具
␾
兩
兩
␾
典
, espec i ely.
Howe e , unde pe iodic bounda y condi ions hese quan i-
ies a e no well de ined ma hema ically e en o localized
s a es, and di e en alues a e ob ained i he o igin o he
sys em is chosen so ha he bounda ies c oss a egion whe e
he elec on has a signi ican densi y p obabili y. Following
he localiza ion c i e ion p esen ed in Re . 5, we ha e o e -
come his p oblem by choosing each o igin componen a he
g id poin s so ha he absolu e alue o he lux ⌽ac oss he
co esponding bounda y is minimized, whe e
⌽⫽
冕
dS
␾
n•ⵜ
␾
共C1兲
and nis a uni ec o pe pendicula o he bounda y su ace.
This p ocedu e gua an ees ha wa e unc ion has decayed a
he bounda ies and he localiza ion leng h is well de ined.
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he ein.
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10J. Lekne , Phys. Re . 158, 130 共1967兲.
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12N. W. Ashc o and N. D. Me min, Solid S a e Physics 共Saunde s, Fo
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共1993兲and/o J. Appel. Solid S a . Phys. 21, 193 共1968兲.
14H. C. Ande sen, J. Chem. Phys. 72,2384共1980兲.
15D. F enkel and B. Smi , Unde s anding Molecula Simula ion 共Academic,
San Diego, 2002兲.
16N. F. Mo and E. A. Da is, Elec onic P ocesses in Non-c ys alline Ma-
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17S. F. Pa ke , J. Chem. Soc., Fa aday T ans. 92,1941共1996兲.
18A. V. Sa in and L. I. Mane i ch, Phys. Re . B 67, 144302 共2003兲.
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7778 J. Chem. Phys., Vol. 120, No. 16, 22 Ap il 2004 D. Cube o and N. Qui ke
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