Pho oinduced Deso p ion Dynamics o CO om Pd(111): A Neu al
Ne wo k App oach
Al edo Se ano Jiménez, Albe o P. Sánchez Muzas, Yaolong Zhang, Ju aj O ca , Bin Jiang,
I o Lonca ic,J.In
aki Jua is i,*and Mai e Alducin*
Ci e This: J. Chem. Theo y Compu . 2021, 17, 4648−4659
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ABSTRACT: Modeling he ul a as pho oinduced dynamics and
eac i i y o adso ba es on me als equi es including he effec o he
lase -exci ed elec ons and, in many cases, also he effec o he highly
exci ed su ace la ice. Al hough he ecen ab ini io molecula dynamics
wi h elec onic ic ion and he mos a s, (Te,Tl)-AIMDEF [Alducin,
M.;e al. Phys. Re . Le . 2019,123, 246802], enables such complex
modeling, i s compu a ional cos may limi i s applicabili y. He e, we use
he new embedded a om neu al ne wo k (EANN) me hod [Zhang,
Y.;e al. J. Phys. Chem. Le . 2019,10, 4962] o de elop an accu a e and
ex emely complex po en ial ene gy su ace (PES) ha allows us a de ailed and eliable desc ip ion o he pho oinduced deso p ion
o CO om he Pd(111) su ace wi h a co e age o 0.75 monolaye . Molecula dynamics simula ions pe o med on his EANN-PES
ep oduce he (Te,Tl)-AIMDEF esul s wi h a ema kable le el o accu acy. This demons a es he ou s anding pe o mance o he
ob ained EANN-PES ha is able o ep oduce a ailable densi y unc ional heo y (DFT) da a o an ex ensi e ange o su ace
empe a u es (90−1000 K); a la ge numbe o deg ees o eedom, hose co esponding o six CO adso ba es and 24 mo ing su ace
a oms; and he a ying CO co e age caused by he abundan deso p ion e en s.
1. INTRODUCTION
The use o in ense (∼1 mJ/cm2) em osecond ( s) lase pulses
in he ul a iole , isible, and nea -in a ed egime has been
shown o be a e y efficien way o p omo e eac ions a
adso ba e-co e ed me al su aces.
1−3
A hese wa eleng hs, a
la ge ac ion o he ligh is abso bed by he me al gi ing ise o
elec onic exci a ions. Subsequen ly, ene gy ans e o he
la ice a oms akes place ia elec on−phonon coupling. As a
esul , he adso ba es encoun e a combined elec onic and
phononic exci ed sys em om which hey can gain ene gy and
expe ience diffe en eac ions, diffusion, and e en deso p ion
om he su ace.
4−9
In e es ingly, his kind o exci a ion
mechanism can inc ease significan ly he c oss sec ion o
eac ions wi h espec o wha is obse ed unde o dina y
he mal exci a ion condi ions and e en open new eac ion
channels. Two pulse co ela ion expe imen s a e cus oma ily
employed o ob ain he ime scale o he ene gy ans e
be ween he adso ba e and he subs a e. In his way, in
p inciple, whe he a specific eac ion is mainly go e ned by he
exci ed elec ons o phonons can be disen angled expe -
imen ally. Howe e , in se e al cases, his in o ma ion is no
unequi ocally ob ained om he expe imen s and heo e ical
modeling is necessa y.
F om a heo e ical aspec , he modeling o hese expe imen s
equi es pe o ming molecula dynamics simula ions in an
exci ed en i onmen .
10−17
Fi s , he exci a ion gene a ed by he
lase pulse in he subs a e is desc ibed in e ms o ime-
dependen elec onic (Te) and phononic (Tl) empe a u es
ha a e ob ained using he wo- empe a u e model (2TM).
18
Subsequen ly, he mo ion o he adso ba es is de e mined by
sol ing Lange in equa ions o mo ion in he g ound-s a e
po en ial ene gy su ace (PES). In his way, he coupling o he
adso ba es o he elec onic sys em is modeled in e ms o
elec onic ic ion o ces and associa ed s ochas ic o ces ha
depend on Te. A nonempi ical and accu a e po en ial ene gy
su ace is ypically ob ained by cha ac e izing he adso ba e−
me al su ace in e ac ion a he le el o densi y unc ional
heo y (DFT). Un il e y ecen ly, due o he la ge
compu a ional cos in ol ed in he DFT calcula ions, po en ial
ene gy su aces o educed dimensionali y, only in ol ing he
deg ees o eedom (DOFs) o he adso ba e, we e used. As a
esul , he effec o he hea ed phonon sys em was ei he no
included a all in he dynamics o in a a he app oxima e way
using he gene alized Lange in oscilla o (GLO) model ha
does no accoun o independen su ace a om mo e-
Recei ed: Ap il 8, 2021
Published: July 19, 2021
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men .
13−17
Mo eo e , a maximum o wo a omic adso ba es
(o a single dia omic molecule) we e desc ibed wi h six-
dimensional po en ial ene gy su aces, which did no allow one
o model in e adso ba e ene gy exchange and s udy effec s
ela ed o he local educ ion o he co e age ha is caused by
sequen ial deso p ion e en s.
In his espec , only e y ecen ly ha e hese limi a ions been
o e come using ab ini io molecula dynamics wi h elec onic
ic ion (AIMDEF).
19−27
The model, he ea e deno ed as
(Te,Tl)-AIMDEF,
27
inco po a es bo h he elec onic and
phononic exci a ion channels: he o me by sol ing he
Lange in equa ion o he adso ba es using Teand he la e by
coupling he su ace a oms o a he mos a a a empe a u e Tl.
This me hodological app oach na u ally includes all o he
sys em’s deg ees o eedom as equi ed. Specifically, in
p inciple, any numbe o su ace a oms a e allowed o mo e
independen ly and mul iple adso ba es can be ea ed. Using
(Te,Tl)-AIMDEF, he impo ance o including no only Tebu
also Tl(pa icula ly o hose su aces ha may each a high
Tl), as well as he in e adso ba e in e ac ions, has been
demons a ed.
27
The main sho coming o he app oach is ha
AIMDEF is ex emely compu a ionally demanding. This
means ha , in p ac ice, a educed numbe o he o de o
ew hund eds o ajec o ies can be ealis ically compu ed o a
gi en se o expe imen al condi ions. This esul s in limi ed
s a is ics. Fo he same eason, wi h he equi ed ime s eps o
he o de o em oseconds, he in eg a ion ime is limi ed o
a ound 2−4 ps.
In he las ew yea s, he use o neu al ne wo k (NN)-
gene a ed mul idimensional PESs has become an accu a e
al e na i e o ab ini io molecula dynamics (AIMD) o
desc ibe he dynamics o di e se gas−su ace p ocesses
28−39
and also he dynamics a solid−liquid wa e in e aces.
40−43
In
pa icula , o hese s udies, he de elopmen o he a omis ic
neu al ne wo k (A NN) app oach has cons i u ed a majo
ad ancemen .
44−46
Wi hin A NN, he PES is cons uc ed in
e ms o a omis ic con ibu ions, which allows o ob aining
NN-PESs ha a e a unc ion o all he a omic posi ions in
sys ems o a bi a y size. Applica ion o his me hodology o
gas−su ace dynamics s udies has allowed o cons uc ing
PESs no only o dia omic
31−33,35,39
bu also o polya omic
molecules
29,30,34,36,37
in e ac ing wi h su aces. Mo eo e , since
he NN-PES can also be made a unc ion o he su ace a om
coo dina es, he ea men o bo h he independen su ace
a om mo emen and he su ace empe a u e effec s has also
been pe o med, which has allowed o accoun ing o ene gy
exchange be ween he molecule and he su ace along he
dynamics.
31−33,35−39
Howe e , i mus be emphasized ha he equi emen s
imposed on a NN-PES capable o desc ibing em osecond
lase -induced eac ions a e ex emely demanding as compa ed
o hose equi ed in usual elemen a y gas−su ace p ocesses.
Since i is necessa y o model he mo emen o mul iple
adso ba es and su ace a oms, he numbe o deg ees o
eedom o be accoun ed o is huge. In his espec , i is wo h
men ioning ha , o ou knowledge, all he NN-PESs gene a ed
up o now o gas−su ace s udies a e es ic ed o a single gas
molecule. Mo eo e , he PES mus be able o desc ibe
accu a ely he e y dis inc and changing adso ba e co e ages
ha exis du ing he pho oinduced dynamics. This means ha
i is necessa y o ensu e a p ecise desc ip ion o adso ba e−
subs a e and in e adso ba e in e ac ions unde e y diffe en
condi ions in ol ing local changes o he configu a ion space o
neighbo adso ba es and s ong la ice dis o ions. In his
espec , no e ha in hese expe imen s, he la ice empe a u es
(Tl) may a y apidly in he ange o 90−1000 K, which
implies ha he configu a ion space co esponding o he
su ace a oms is e y la ge. The e o e, al oge he he
equi emen s o he a omis ic NN-PES a e unp eceden edly
ex eme and demanding o hese kinds o p ocesses. He e, we
show ha he ecen ly de eloped embedded a om neu al
ne wo k (EANN) me hod,
47
which has al eady been
success ully applied o cons uc he PESs o dia omic and
polya omic molecules in e ac ing wi h mul iple me al ac-
e s,
48,49
is indeed imp essi ely accu a e and flexible o accoun
o all hese necessi ies.
Fo his pu pose, we s udy he em osecond lase -induced
deso p ion o CO om Pd(111).
9
In pa icula , we concen a e
on he CO sa u a ion co e age o 0.75 monolaye (ML), in
which he CO molecules adso b in a op, ace-cen e ed cubic
( cc), and hexagonal close-packed (hcp) si es. (Te,Tl)-
AIMDEF esul s o his sys em we e ecen ly p esen ed in
e 27. In he p esen wo k, we use he configu a ions
encoun e ed along hese dynamics as he inpu da a o
gene a e ou EANN-PES.
The wo k is o ganized as ollows. In Sec ion 2, he
p ocedu e used o cons uc EANN-PES is de ailed and
accu acy es s a e p esen ed. Nex , Sec ion 3 is de o ed o he
desc ip ion o he heo e ical amewo k used o pe o m
molecula dynamics simula ions o he lase -induced eac ions
a me al su aces. Also, in his sec ion, we p esen he me hod
used o ob ain he backg ound elec onic densi y a he
posi ion o he mo ing adso ba es a each ime s ep o he
dynamics, which is equi ed o ob ain he ic ion coefficien s
ha desc ibe he coupling o he adso ba es o he hea ed
elec onic sys em in he Lange in equa ion o mo ion.
Subsequen ly, in Sec ion 4, he molecula dynamics simu-
la ions pe o med in he p ecalcula ed 0.75 ML-CO/Pd(111)
EANN-PES o model he lase -induced deso p ion o CO om
hePd(111)su acea ep esen ed.The esul so he
simula ions a e compa ed o he (Te,Tl)-AIMDEF esul s o
e 27 showing, conclusi ely, he alidi y o ou EANN-PES o
pe o m molecula dynamics in hese unp eceden edly exigen
condi ions. Finally, in Sec ion 5, he main conclusions o he
wo k a e summa ized.
2. EANN-PES GENERATION AND QUALITY CHECK
The analy ical ep esen a ion o he adiaba ic PES E({ k})
uling he deso p ion o CO om Pd(111) wi h 0.75 ML
co e age is calcula ed wi h he ecen ly de eloped embedded
a om neu al ne wo k (EANN) me hod.
47
Simila o A NN by
Behle and Pa inello,
44
he o al ene gy o an Na om sys em
is exp essed as he sum o he ene gy o each a om ha
con o ms i , Ei({ k}). In he EANN amewo k, Ei({ k}) is
desc ibed in e ms o he elec onic embedding densi y, i.e.
∑∑
ρ{}= {}=
==
EE () () NN()
k
i
N
ik
i
N
i
i
11 (1)
whe e NNiis he species-dependen a omic neu al ne wo k o
he i h a om in he sys em ha depends on he embedding
densi y ec o ρiwhose componen s ep esen he local
elec on densi y p o ided by he su ounding nea a oms.
Specifically, he se o local densi y componen s defining ρia e
gi en by
Jou nal o Chemical Theo y and Compu a ion pubs.acs.o g/JCTC A icle
h ps://doi.o g/10.1021/acs.jc c.1c00347
J. Chem. Theo y Compu . 2021, 17, 4648−4659
4649
i
k
j
j
j
j
j
j
j
y
{
z
z
z
z
z
z
z
∑∑
ρ
φ=!
!!! −
α
α
++=
=
L
lll c () ( )
L
i
lll
lllL
xyz j
n
ij j lll
ij
,,
,, 1
c
,
2
xyz
xyz
xyzs
a
s
(2)
whe e
φ
α
−=− − −
×−|−|
αxxyyzz
()( )()( )
exp( )
lll
ij ij
l
ij
l
ij
l
ij
,
s2
xyz
xy
z
s
(3)
a e Gaussian- ype o bi als (GTOs) cen e ed a each o he na
a oms j ha a e loca ed wi hin a adius c om he embedded
a om i. In hese equa ions, i=(xi,yi,zi) and j=(xj,yj,zj) a e
he Ca esian posi ion ec o s o a oms iand j, espec i ely,
wi h ij =| i− j|;αand sde e mine he wid h and he cen e
o he Gaussian-like e m in eq 3, espec i ely (and hus
con ol he shape o he adial dis ibu ion ela ed o each
GTO), while lx,ly, and lza e he alues o he o bi al angula
momen um in each coo dina e, whose sum equals he o al
angula momen um L, i.e., L=lx+ly+lz.Ineq 2, c( ij) is he
commonly used cosine ype cu off unc ion ha makes he
in e ac ion o smoo hly decay o ze o as ij app oaches he
cu off adius c,
44
and cjis he elemen - and o bi al-dependen
weigh
50
adjus ed du ing he NNi aining. Each cjcan be
ega ded as he expansion coefficien o he o bi al φlxlylz
α, so
a om j.
The a omic configu a ions in he e e ence da a se a e
ex ac ed om he (Te,Tl)-AIMDEF simula ions o he
pho oinduced deso p ion o CO om he Pd(111) su ace
wi h 0.75 ML co e age pe o med in e 27. These DFT
calcula ions we e pe o med wi h he Vienna Ab ini io
Simula ion Package (VASP)
51,52
( e sion 5.4), using he an
de Waals densi y unc ional ( dW-DF) exchange−co ela ion
unc ional p oposed by Dion e al.,
53
and he AIMDEF
module
20−26
ha was ex ended o include exci ed elec ons
and exci ed phonons effec s h ough ime-dependen elec onic
and la ice empe a u es (see Sec ion 3).
27
A o al o 100
ajec o ies we e calcula ed in a ou -laye (4 ×2) supe cell
ha con ained six CO adso ba es equally dis ibu ed among
a op, hcp, and cc si es and 32 Pd a oms desc ibing he
Pd(111) su ace. No e in passing ha he use o such a la ge
cell ( wice he uni cell o 0.75 ML) was aimed o accoun o
ou -o -phase mo emen s o he coadso bed CO molecules,
p o iding a mo e ealis ic desc ip ion o he in e adso ba e
in e ac ions. Each ajec o y was in eg a ed up o 3.5−4ps
using a ime s ep o 1 s. Al oge he , he whole da a se consis s
o 352 505 configu a ions, each defined by he posi ion ec o s
o he 44 a oms in cell { i}, o which he co esponding DFT
po en ial ene gies EDFT and DFT o ces on each a om Fi
DFT a e
well cha ac e ized. The aining p ocess can be g ea ly
simplified i , ins ead o using such a huge amoun o da a,
we selec a smalle subse ha co ec ly ep esen s he ele an
configu a ional space o he sys em. The e a e diffe en
p ope ies ha can guide his selec ion, e.g., dis ance o he
gas species o he su ace,
31
o ces,
37
e c. In ou case, he
pho oinduced deso p ion dynamics is cha ac e ized by
ajec o ies ha yield none, one, o wo deso bing CO, each
concei ably p o iding in o ma ion on he in e ac ion a
a iable co e ages. Thus, an ini ial subse is cons uc ed wi h
4500, 6000, and 4500 configu a ions ha a e andomly
selec ed om he se o ajec o ies wi h ze o, one, and wo
deso p ion e en s, espec i ely. The ul e io analysis o he
co e ed po en ial ene gy ange se ed us o alida e ha he
configu a ional space p obed in he AIMDEF simula ions is in
p inciple well sampled. Figu e 1 shows ha heEDFT
dis ibu ions o he whole and o each o he h ee ypes o
deso bing ajec o ies ( op panel) a e accu a ely ep oduced by
he subse o 15 000 ajec o ies (bo om panel). I is wo h
men ioning ha he alues EDFT in Figu e 1 a e he ou pu o
he VASP p og am. As shown in his figu e, he ele an
po en ial ene gy a ia ion ange in he dynamical configu a-
ional space co e s a ound 12 eV, which co esponds o
a ound 0.3 eV pe mo ing a om (see below).
The analy ical EANN-PES o he 0.75 ML-CO/Pd(111)
sys em uses 60 densi y desc ip o s o each a omic species ha
co espond o ake L=0−3 combined wi h 15 Gaussian
unc ions wi h α= 0.93 Å−2and s a ying wi hin he in e al
[0, c] in inc emen s Δ s= 0.46 Å o ou chosen alue c= 6.5
Å. Diffe en a chi ec u es we e ied, bu an op imal balance
be ween small e o s in ene gy and he equi ed compu a ion
ime is achie ed using wo hidden laye s wi h 60 neu ons each
o e e y a omic NNi. The EANN code akes ad an age o he
usual andom sepa a ion o he e e ence da a se in o aining
and alida ion subse s o p oduce diffe en NNs du ing he
same un. In ou case, fi e EANN-PESs ha e been ained
using 90 and 10% o he configu a ions as aining and
alida ion subse s, espec i ely. An efficien ex eme machine
lea ning Le enbe g−Ma qua d algo i hm is used in he
op imiza ion o he fi ing pa ame e s (cj).
54
The cos unc ion
used o e alua e a each i e a ion he quali y o an EANN-PES
in ol es he ene gies and a omic o ces. As discussed in e 47,
con e gence is e y efficien in he EANN me hod. Fo ou
se ings, he equi ed accu acy is achie ed in he fi e aining
PESs in less han 50 i e a ions, being he oo -mean-squa e-
e o s (RMSEs) in he ene gy pe mo ing a om o 0.43−0.58
Figu e 1. No malized DFT po en ial ene gy dis ibu ions ex ac ed
om (Te,Tl)-AIMDEF simula ions
27
(filled blue ba s). The se s o
ene gies co esponding o ajec o ies wi h ze o, one, and wo CO
deso p ion e en s a e plo ed in yellow, g een, and ed emp y
his og am ba s, espec i ely. Top: dis ibu ions o he whole
AIMDEF da a se (352 505 configu a ions). Bo om: dis ibu ions
o he 15 000 configu a ions used o he fi s EANN-PES aining.
Jou nal o Chemical Theo y and Compu a ion pubs.acs.o g/JCTC A icle
h ps://doi.o g/10.1021/acs.jc c.1c00347
J. Chem. Theo y Compu . 2021, 17, 4648−4659
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and 0.90−1.05 meV o he aining and alida ion se s,
espec i ely.
Since he pho oinduced deso p ion dynamics o in e es
implies ha he sys em is exposed o ex eme condi ions
cha ac e ized by high su ace empe a u es (1000 K) and
highly exci ed adso ba es, we conside i impo an o e alua e
he accu acy o he PES in p edic ing no only he ene gy bu
also, especially, he a omic o ces. A new da a se o med by
87 382 configu a ions andomly aken om he AIMDEF da a
se no used in he fi ing p ocess and ep esen a i e o he
h ee ypes o deso bing ajec o ies is used o his pu pose
(p edic da a se ). The esul s show ha he ob ained PESs a e
excellen wi h he RMSE in ene gy o only 0.86−0.95 meV pe
mo ing a om and he RMSE in he Ca esian componen s o
he o ces o 0.05−0.06 eV/Å. Ne e heless, we obse e ha
he maximum e o s in he o ces a e in some cases la ge
(be ween 1.1 and 5.9 eV/Å depending on he coo dina e and
ained EANN-PES conside ed). Thus, new configu a ions a e
added o he ini ial se o 15 000 s uc u es o imp o e he
quali y o ou EANN-PES. The c i e ion o selec hese new
configu a ions is as ollows. Fi s , among he fi e ained
EANN-PESs, we selec he one wi h he smalles maximum
absolu e e o s in he a omic o ces, i.e., |ΔFβ|=|Fβ
EANN −
Fβ
DFT|, whe e β=x,y,z e e s o he o ce componen . F om
his analysis, we selec he 10 la ges e o s o each o he 36
mo ing a oms (i.e., 12 a oms o ming he six CO adso ba es
and 24 Pd a oms ha co espond o he mo ing h ee o he
ou laye s desc ibing he su ace). Figu e 2 shows he
dis ibu ion o hese maximum e o s o each o he o ce
componen s as ob ained wi h he EANN-PES wi h he smalles
e o s in he o ces (black his og am ba s). The co esponding
configu a ions (883 because e o s in diffe en o ce
coo dina es can occu o he same configu a ion) a e added
o he o iginal 15 000 da a inpu o de elop fi e new EANN-
PESs, using he same NN se ings (i.e., basis se and
a chi ec u e).
The quali y o he fi e newly de eloped EANN-PESs is
imp essi ely good. Since he accu acy in he o ces is also
c ucial o ep oducing he sys em dynamics and hus i s
physical beha io p ope ly, we selec he EANN-PES wi h he
smalles e o s in he a omic o ces. In his PES, he RMSE in
he ene gies pe mo ing a om is 0.41 and 0.87 meV o he
aining and alida ion se s, espec i ely. I s accu acy is u he
confi med by he ene gy esul s ob ained wi h he p edic da a
se . As shown in Figu e 3 (le panel), he maximum e o pe
mo ing a om is 8.27 meV bu he small RMSE o 0.85 meV
ma ks he mino e o in oduced in mos o he config-
u a ions. The e o dis ibu ion plo ed in Figu e 3 ( igh
panel) clea ly shows ha his is he case.
Rega ding i s accu acy in p edic ing he a omic o ces, he
RMSE is no g ea ly imp o ed in he e aining p ocess and i
is s ill a ound 0.05 eV/Å, whe eas he mean absolu e e o s a e
a ound 0.04 eV/Å. Howe e , a compa ison o he 10 maximum
absolu e e o s o each a omic o ce componen |ΔFβ|
ob ained wi h ou final EANN-PES ( ed ba s in Figu e 2)
wi h he ones o he fi s aining p ocess (black ba s) shows a
ema kable educ ion in he e o s. The educ ion in he
maximum e o s is as ollows: |ΔFx|≈1.13 →0.66 eV/Å, |ΔFy|
≈1.04 →0.58 eV/Å, and |ΔFz|≈2.56 →0.62 eV/Å. This
imp o emen is also no iceable when compa ing he mean
absolu e e o ( aken o e he 10 maximum e o s o each o
he 36 mo ing a oms) o Fz, plo ed in black and ed
his og ams, ha dec eases om
|
Δ|≈F0.510 o 0.345 eV/
Å
z.
The a e aged e o s we e al eady small o he wo o he
componen s o he o ces and in his imp o ed EANN-PES a e
jus sligh ly educed,
|
Δ|≈ →F0.390 0.317 eV/
Å
xand
|
Δ|≈ →F0.374 0.314 eV/
Å
y. As an addi ional s ingen es ,
we also show he maximum e o p edic ions o he final
EANN-PES on he 352 505 configu a ions o ming he ull
AIMDEF da a se (g een ba s o Figu e 2). In his case, we
ob ain |ΔFx|= 0.68 eV/Å, |ΔFy|= 0.58 eV/Å, and |ΔFz|= 0.63
eV/Å so ha he maximum absolu e- alued e o s show none
o a e y li le inc ease wi h espec o he p edic ions o he
ed his og am. The co esponding mean absolu e e o alues
Figu e 2. E o his og ams o each Ca esian componen o he
a omic o ces, |ΔFβ|=|Fβ
DFT −Fβ
EANN|(whe e β=x,y,z), calcula ed
using only he 10 maximum e o alues o each o he 36 mo ing
a oms o he 0.75 ML-CO/Pd(111) sys em. Black ( ed) his og am
ba s co espond o e o s o he p edic se in he bes EANN-PES o
he fi s (final) aining. G een his og am ba s show he maximum
e o s o he whole AIMDEF da a se o 352 505 configu a ions as
ob ained wi h he final EANN-PES. The mean alues
|
Δ|
β
Fo each
his og am a e also p o ided.
Figu e 3. Le : compa ison o he po en ial ene gies pe mo ing a om
compu ed wi h he (final) EANN-PES o a se o 87 382
configu a ions no used in he aining and he co esponding DFT
alues. Righ : his og am o he e o s in ene gy pe mo ing a om ΔE/
Nmo =(EDFT −EEANN)/Nmo o he same se o p edic
configu a ions.
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4651
a e sligh ly la ge :
|
Δ|≈F0.348 eV/Å,
x
|
Δ|≈F0.345 eV/
Å
y,
and
|
Δ|≈F0.378 eV/
Å
z. The esul s om all hese alida ion
es s imply ha we should also obse e good con e gence
ega ding any a om o he CO/Pd(111) cell. Figu es 4 and 5
show wo examples o hese good ag eemen s be ween Fβ
EANN
and Fβ
DFT,β=x,y,z, o wo ep esen a i e a oms o he
sys em: one C a om om a CO a a cc si e and one O om a
hcp si e-placed CO. As a esul , a e he ull s a ic analysis, we
a e confiden abou his final EANN-PES’s obus ness, which
we u he de e mine by ying o ep oduce he dynamics o
he CO/Pd(111) sys em as desc ibed in e 27.
3. PHOTOINDUCED MOLECULAR DYNAMICS ON
SURFACES
The deso p ion o CO om he Pd(111) su ace induced by
em osecond lase pulses is simula ed wi h molecula dynamics
wi h elec onic ic ion and he mos a [(Te,Tl)-MDEF]
calcula ions pe o med in ou de eloped EANN-PES. To do
so, we ha e modified ou implemen a ion o he (Te,Tl)-
AIMDEF me hodology
27
o compu e ajec o ies in which
o ces and elec onic ic ion coefficien s a e e alua ed a each
isi ed configu a ion om an a bi a y PES and an a bi a y
elec onic densi y gene a o unc ion (DGF), espec i ely. In
Figu e 6, we show a concise scheme o he (Te,Tl)-(AI)MDEF
model. Mac oscopically, a lase pulse hea s he elec ons o he
me al su ace, which subsequen ly exci e su ace phonons. This
esponse is modeled wi h he same 2TM as in he o iginal
(Te,Tl)-AIMDEF calcula ions. The mac oscopic esponse o
he sys em is hus ollowed mic oscopically by CO and Pd
deg ees o eedom (DOFs) wi h diffe en equa ions o
mo ion.
CO DOFs a e subjec ed o he ollowing Lange in equa ions
o mo ion
ηη=−∇ { } − + [
]m
V T
R
d
d() ()
d
d(), ()
i
i
mii
i
iii
2
2e, e, e e,
i
(4)
whe e mi, i, and ηe,ia e he mass, posi ion ec o , and he
elec onic ic ion coefficien o he i h a om con o ming he
se o adso ba es, espec i ely. The fi s e m on he igh -hand
side o he equa ion ep esen s he adiaba ic o ce ha depends
on he posi ion o all (adso ba es and su ace) a oms. In he
(Te,Tl)-AIMDEF simula ions o e 27, his o cewas
calcula ed on he fly using he Hellmann−Feynman heo em.
In he (Te,Tl)-MDEF calcula ions ha we p esen he e, his
o ce is calcula ed using he p ecalcula ed EANN-PES. The
hi d e m Re,iis he andom fluc ua ing o ce ha mimics he
effec o he ho me al elec ons on he adso ba es, and i is
esponsible o he exci a ion o he la e by he exci ed
elec onic sys em. This o ce is ela ed o he elec onic ic ion
o ce (second e m) h ough he fluc ua ion−dissipa ion
heo em. Specifically, Re,iis modeled by Gaussian whi e
noise wi h a iance
η
η
[]=
Δ
TkT
R
Va ( , ) 2()()
ii
ii
e, e e,
Be e,
(5)
whe e kBand Δ a e he Bol zmann cons an and he ime-
in eg a ion s ep, espec i ely.
Figu e 4. Plo s o Fβ
EANN e sus Fβ
DFT o a cc C a om o he 0.75 ML-
CO/Pd(111) sys em
27
showing a compa ison o each Ca esian
componen β=x( ed), y(blue), and z(g een). The le panels
co espond o o ces calcula ed wi h he fi s ained EANN-PES and
he igh panels o hose calcula ed wi h he final EANN-PES.
Maximum e o and RMSE o he o ces a e w i en in each plo .
Figu e 5. Same as Figu e 4 o a hcp O a om.
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The equa ions o mo ion ollowed by he Pd su ace a oms
inco po a e wo kinds o o ce e ms. The fi s e m consis s o
he adiaba ic o ce ha , as in he case o adso ba es, depends
on he posi ion o all a oms. Again, in he (Te,Tl)-AIMDEF
simula ions o e 27, his o ce is calcula ed on he fly using
he Hellmann−Feynman heo em, whe eas in he (Te,Tl)-
MDEF calcula ions, i is compu ed using ou EANN-PES. The
second e m accoun s o he hea ing o he su ace laye s due
o he elec onic exci a ion gene a ed by he lase pulse. As
done in e 27, his effec is simula ed by coupling he a oms o
he wo opmos su ace laye s o a he mal ba h desc ibed by a
Nose−Hoo e he mos a ,
55,56
in which he empe a u e is he
ime-dependen empe a u e Tl( ) ob ained om he 2TM.
Thus, he equa ions o mo ion o hese su ace a oms jwi h
mass mjand posi ion ec o ja e as ollows
ξ=−∇ { } −
m
Vm
d
d() d
d
j
j
mj
j
2
2j(6)
i
k
j
j
j
j
j
j
j
y
{
z
z
z
z
z
z
z
∑
ξ=−
Q m Nk T
d
d
1d
d3
j
j
j
2
Bl
(7)
whe e Nis he numbe o a oms o he fi s wo laye s in ou
simula ion cell, Qis a pa ame e wi h dimensions o ene gy ×
ime2 ha ac s as he mass o he dynamical a iable s, and ξ=
Q−1spsis he he modynamic ic ion coefficien .
57
Finally, he
hi d laye is used as a ansi ion egion be ween he ho
su ace and he inne (no hea ed) bulk. The e o e, he
mo emen o each a om kin he hi d su ace laye is desc ibed
by he classical New on equa ions o mo ion and he adiaba ic
app oxima ion
=−∇ { }
m
V
d
d()
k
k
m
2
2k(8)
whe e mkand ka e i s co esponding mass and posi ion
ec o , espec i ely. The ou h-laye a oms a e kep ozen in
ou simula ions.
Rega ding he coupling o he adso ba es o he elec onic
sys em ia he second and hi d e ms in he igh -hand side o
eq 4, he elec onic ic ion coefficien o each a om in each
adso ba e ηe,iis calcula ed wi hin he o iginal local densi y
ic ion app oxima ion (LDFA)
19,58
ha in he case o
molecula adso ba es ea s he molecule as o med by
independen a oms (independen a om app oxima ion, IAA).
This means ha ηe,idepends on he alue o he ba e su ace
elec onic densi y a he posi ion o he a om i o ming he
adso ba e, i.e., nsu ( i). In he (Te,Tl)-AIMDEF simula ions,
27
his is achie ed using he Hi sh eld pa i ioning scheme
59
o
sub ac he con ibu ion o he adso ba es om he sel -
consis en elec onic densi y ha is calcula ed a each ime
s ep.
22,23
In he case o (Te,Tl)-MDEF, apa om an EANN-
PES, he ic ion coefficien s mus also be ob ained om
p ecalcula ed esul s. Mo eo e , o use an EANN-PES
efficien ly, i is clea ha he ic ion coefficien s ha e o be
calcula ed a leas as as as he po en ial. One could use a
simila NN scheme o in e pola e he densi y o he elec onic
ic ion coefficien s based on he AIMDEF da a se .
60
Ac ually,
e en a mo e complica ed objec such as he elec onic ic ion
enso can be in e pola ed by NNs.
61−63
Howe e , he e we
choose o employ a mo e simple app oach ha we show o be
accu a e and compa ed o a NN in e pola ion, as e and mo e
s able.
We s a by w i ing he Pd(111) elec on densi y a he
posi ion o each a om o each CO adso ba e n( C,O) as a sum
o he elec onic densi ies con ibu ed by indi idual Pd a oms
a his posi ion
∑ρ=|−|
n
() ( )
C,O
Pd
C,O Pd (9)
We hen assume ha ρ(| C,O − Pd|) can be desc ibed by wo
exponen ially decaying unc ions
ρ
=−+− a b c d () exp( ) exp( )
(10)
Using AIMDEF da a poin s o n( C,O), one can fi he ou
pa ame e s: a,b,c, and d. We did also y o he unc ions such
as Gaussian unc ions and hei combina ion wi h he
exponen ial unc ion, bu wo decaying exponen ial unc ions
ga e he bes esul s e aining he simplici y and a small
numbe o pa ame e s. Since i is easy o e alua e such a
unc ion, we decided o use all a ailable (Te,Tl)-AIMDEF
su ace elec onic densi ies om which he co esponding
ic ion coefficien s we e calcula ed. The fi ing p ocedu e
esul s in he ollowing pa ame e s: a= 3.15975 au, b=
4.25214 Å−1,c= 0.29080 au, and d= 2.52252 Å−1(au s ands
o a omic uni s).
Once he densi y n( C,O) is known, he ic ion coefficien s
a e calcula ed wi hin he LDFA,
19,58
as i was also done in he
(Te,Tl)-AIMDEF simula ions. In pa icula , he C and O
ic ion coefficien s a e indi idually fi ed o he ollowing
analy ical unc ion
Figu e 6. Scheme o he dynamics model used o simula e lase -induced deso p ion o CO om Pd(111) wi h 0.75 ML co e age.
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4653
∑
η
=
=
A C () () exp( ())
i
i
B
i
1
2
ss
i
(11)
whe e s( ) = [3/(4πn( ))]1/3 is he mean elec on adius. In
eq 11, all he physical quan i ies a e gi en in au, being he
fi ing pa ame e s (A1= 22.654, B1= 2.004, C1=−3.134, A2=
2.497, B2=−2.061, C2= 0.0793) o C and (A1= 1.36513, B1
=−1.8284, C1=−0.0820301, A2= 50.342, B2= 0.490785, C2
=−2.70429) o O.
In Figu e 7, we compa e he LDFA ic ion coefficien s
ob ained wi h he fi ed elec on densi y n( C,O) o hei
co esponding (Te,Tl)-AIMDEF ic ion coefficien s. I can be
seen ha bo h he maximum e o o 0.0086 au and RMSE o
0.0017 au a e small, especially compa ed o he e o s
associa ed wi h diffe en calcula ions o embedding densi ies.
23
In addi ion o accu acy, he ad an age o his app oach is ha
i is ex emely as in e alua ing ic ion coefficien s compa ed
o he po en ial e alua ion. Las ly, by he cons uc ion o wo
exponen ially decaying unc ions, i is also ensu ed ha he e is
no p oblem o o e fi ing o la ge e o s in he ex apola ion
ange ha could occu wi h app oaches such as NN. All his
gua an ees ha eq 9 oge he wi h eq 10 cons i u es an
adequa e DGF o be used in he (Te,Tl)-MDEF simula ions o
calcula e, a each ime s ep, he ic ion and s ochas ic o ces in
eq 4.
4. RESULTS OF THE DYNAMICS SIMULATIONS
We ha e hi he o shown how ou bes de eloped EANN-PES
(Sec ion 2) and ou a omwise addi i e elec onic densi y
unc ion app oach (Sec ion 3) yield esul s in being close o
he o iginal 0.75 ML-CO/Pd(111) AIMDEF da a se . Despi e
his being a easonable quali y check, i is no sufficien o
p o e ha using bo h unc ions in MDEF calcula ions
p oduces exac ly he same esul s as in AIMDEF. This is due
o wo possible undesi able si ua ions.
46
Fi s , MDEF
ajec o ies may sample egions o he configu a ional space
ha , e en when close o he EANN aining se o he DGF
da a se , a e no ye co ec ly desc ibed by he op imized
pa ame e s, and second, hese ajec o ies may en e egions o
he configu a ional space a om he AIMDEF da a se whe e
bo h fi ed unc ions ha e o ex apola e in o ma ion. In
addi ion, he p obabili y o encoun e ing any o he desc ibed
p oblems is inc eased by he high numbe o DOFs, in ou case
108, ha play a ole du ing his ype o lase -induced
deso p ion dynamics simula ions.
To ule ou hese sou ces o e o and u he add ess he
accu acy o ou de eloped EANN-PES and addi i e DGF, we
ha e simula ed he deso p ion o CO om 0.75 ML-CO/
Pd(111) wi h (Te,Tl)-MDEF, assuming he same expe imen al
condi ions as in he o iginal (Te,Tl)-AIMDEF calcula ions.
27
Specifically, he Pd(111) being ini ially a 90 K is hea ed by a
lase pulse o 780 nm wa eleng h, 100 s o ull wid h a hal -
maximum (FWHM), and abso bed fluence F= 13 mJ/cm2.In
p ac ice, he hea ed su ace is modeled wi h he same ime-
dependen empe a u es Te( ) and Tl( ) used in he (Te,Tl)-
AIMDEF simula ions,
27
in which he maximum o he lase
pulse a i es a he ins an = 410 s.
Using hese condi ions, wo se s o dynamics calcula ions
ha e been un. In he fi s one, he eina e called (Te,Tl)-
MDEF-1, we ha e used he same se o 100 ini ial
configu a ions as u ilized in e 27. This allows us o compa e
s ep by s ep he ex en o which (Te,Tl)-MDEF and (Te,Tl)-
AIMDEF ajec o ies a e simila and de ec i he e a e a i ac s
in he po en ial o elec onic ic ion coefficien s ha could
push ajec o ies away om he EANN-PES confidence zone,
e en when s a ing om (Te,Tl)-AIMDEF ini ial configu-
a ions. In he second se o calcula ions, he eina e called
(Te,Tl)-MDEF-2, we ha e used 2000 configu a ions selec ed
andomly om a se o 10 000 s uc u es gene a ed by le ing
he 100 ini ial configu a ions o e 27 e ol e in 1 ps wi h a
cons an empe a u e o 90 K. This enables us o es how
obus (Te,Tl)-MDEF dynamics is when ajec o ies s a om
configu a ions no included in any o he da a se s used o fi
he EANN-PES o DGF.
We find ha (Te,Tl)-MDEF-1 ajec o ies lie close o hei
(Te,Tl)-AIMDEF coun e pa s. In ac , configu a ions isi ed
du ing he fi s 400−500 s a e p ac ically iden ical o hose
isi ed by he o iginal ajec o ies. To illus a e his finding, in
Figu e 8 we show how a ypical (Te,Tl)-MDEF-1 ajec o y
(blue line) compa es wi h i s co esponding (Te,Tl)-AIMDEF
ajec o y (g een line) in e ms o ime e olu ion o C a om
ic ion coefficien s du ing 1 ps. The ag eemen o imes
below 500 s is e y high, independen o he ini ial posi ion o
he C a oms. A e 500 s, ic ion coefficien s s a o di e ge
as ajec o ies commence o ollow diffe en pa hways due o
cumula i e diffe ences in he dynamics. This beha io is sha ed
among all (Te,Tl)-MDEF-1 simula ions and eflec s ha he
quali y o ou fi ed EANN-PES and DGF is good enough o
keep (Te,Tl)-MDEF ajec o ies close o hei (Te,Tl)-AIMDEF
analogues in a e y de ailed way o se e al hund ed
em oseconds. This is a ema kable quali y achie emen as
he ag eemen was a ained despi e he possible dynamical
ins abili y, he s ochas ic na u e o he Lange in equa ions o
mo ions, and he diffe en in eg a ion ime s eps used in he
simula ions (Δ = 0.2 s in MDEF, Δ = 1 s in AIMDEF) o
an amoun o ime in which he elec onic empe a u e o he
2TM changes om 90 K ( = 0 s) o 5500 K ( = 500 s).
In Figu e 9, we show (Te,Tl)-MDEF-1 (blue line), (Te,Tl)-
MDEF-2 ( ed line), and (Te,Tl)-AIMDEF (black line) esul s
o o al ( op panel) and ini ial si e-dependen (bo om panels)
CO deso p ion p obabili ies as a unc ion o ime. In each case,
he deso p ion p obabili y pCO is defined as he (cumula i e)
numbe o CO molecules deso bed a a gi en ime di ided by
he o al numbe o ajec o ies (N aj) and he o al numbe o
Figu e 7. Compa ison o he LDFA ic ion coefficien s ob ained
om he fi ed elec onic densi y n( C,O)(eqs 9 and 10) o all he
a ailable (Te,Tl)-AIMDEF ic ion coefficien da a. F ic ion coe -
ficien s gi en in a omic uni s (au).
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CO molecules in ou simula ion cell (six; see Figu e 6). Since
his defini ion is equi alen o conside ha he numbe o CO
deso p ion e en s dis ibu e as a binomial dis ibu ion wi h six
N aj independen ials and success p obabili y pCO,we
associa e o each e alua ion o pCO a confidence in e al
calcula ed wi h Wilson asymme ic sco e in e als
64
o he
confidenceo 99%.I isappa en ha alldeso p ion
p obabili ies calcula ed wi h (Te,Tl)-MDEF-1 and (Te,Tl)-
MDEF-2 a e in good ag eemen wi h he (Te,Tl)-AIMDEF
esul s, especially in he case o o al and cc-CO deso p ion
p obabili ies. On a close look, we can see ha si e-dependen
(Te,Tl)-MDEF-1 deso p ion p obabili ies p esen mo e dis-
c epancies wi h (Te,Tl)-AIMDEF han (Te,Tl)-MDEF-2. In
pa icula , (Te,Tl)-MDEF-1 op-CO deso p ion a ios end o
be highe han hose o (Te,Tl)-AIMDEF, whe eas hcp-CO
deso p ion a ios end o be lowe . These de ia ions a e mos ly
om he s a is ical a iabili y associa ed wi h hei espec i e
100 ajec o ies’ensembles, since all (Te,Tl)-MDEF-1 and
(Te,Tl)-AIMDEF deso p ion p obabili ies a e wi hin hei 99%
confidence in e als ecip ocally. In he case o (Te,Tl)-MDEF-
2, all deso p ion p obabili ies lie close o hose o (Te,Tl)-
AIMDEF despi e no sha ing exac ly he same ini ial
configu a ions. This suppo s ha ou EANN-PES and DGF
a e accu a e enough o desc ibe he dynamic ene gy ba ie s
ha CO molecules encoun e du ing he lase -induced
deso p ion p ocess on he same oo ing as he o iginal
AIMDEF calcula ions.
Compa ed o (Te,Tl)-AIMDEF, he imp o ed s a is ics o
he (Te,Tl)-MDEF-2 simula ions allow us o es ablish wi h
mo e p ecision he ime scale o he deso p ion p ocess. I
akes mo e han 0.6 ps since he a i al o he lase -pulse
maximum (occu ing a = 410 s) o obse e he fi s
deso p ion e en s. The si e- esol ed deso p ion p obabili ies
show ha hese ini ial e en s co espond o a op-CO
adso ba es. A ound an addi ional 500 s a e equi ed o
deso p ion om ei he he hcp o cc si es.
Ha ing demons a ed he accu acy o ou me hodology o
ep oduce he (Te,Tl)-AIMDEF ajec o ies o he fi s ew
hund ed em oseconds and he final ou come in e ms o CO
deso p ion p obabili ies, we now ocus on he ime e olu ion
o he kine ic ene gy o CO molecules, which is mo e sensi i e
o he fine de ails o he pa hs ollowed on he PES. In Figu e
10 (le panel), we plo he mean ansla ional (Ekin
ans, eddish,
ull hick lines) and mean o ib a ional (Ekin
o ib , bluish, ull hick
lines) kine ic ene gy o adso bed CO molecules a e aged o e
ajec o ies as a unc ion o ime. We define Ekin
ans as only he
mean ansla ional ene gy o he cen e o mass o CO
molecules, while Ekin
o ib is calcula ed as he mean o C and O
o al kine ic ene gy minus Ekin
ans. S a ing wi h Ekin
ans, we obse e
ha he (Te,Tl)-MDEF-1 ( ed) and (Te,Tl)-MDEF-2 (ligh
ed) esul s lie e y close o he (Te,Tl)-AIMDEF (da k ed)
Figu e 8. F ic ion coefficien s o C a oms as a unc ion o ime o a ypical ajec o y. G een lines, (Te,Tl)-AIMDEF esul s o a gi en ajec o y
ex ac ed om e 27. Blue lines, EANN esul s o a ajec o y ini ia ed wi h he same (Te,Tl)-AIMDEF condi ions. Each panel s ands o a
diffe en C a om in he model. Top, cc, and hcp si es e e o he C a oms’ini ial posi ions.
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4655
alues, showing ha he mean kine ic ansla ional ene gies a e
e y well ep oduced by ou EANN-PES. A simila conclusion
is ex ac ed when compa ing he esul s o he mean
o ib a ional kine ic ene gies, since bo h he (Te,Tl)-MDEF-1
(blue) and (Te,Tl)-MDEF-2 (ligh blue) esul s ag ee
ema kably well wi h he (Te,Tl)-AIMDEF Ekin
o ib alues (da k
blue). Bu oge he wi h he mean ene gy alues, i is
impo an o confi m whe he he ins an aneous ene gy
dis ibu ions o he adso ba es a e also well ep oduced by
ou EANN-PES. The la e can be es ima ed by compa ing, o
ins ance, he associa ed s anda d de ia ions o he ansla ional
ΔEkin
ans and o ib a ional ΔEkin
ans kine ic ene gy dis ibu ions.
Thus, he e a e wo addi ional ( hin do ed) cu es
ep esen ing he alues Ekin
ans( o ib ) ±ΔEkin
ans( o ib ) associa ed
o each mean kine ic ene gy cu e in Figu e 10. As shown in
he figu e, he e is also a good ag eemen be ween he (Te,Tl)-
AIMDEF ins an aneous s anda d de ia ions and hose
ob ained om he (Te,Tl)-MDEF-1 and (Te,Tl)-MDEF-2
simula ions, which ein o ces he high quali y o he EANN-
PES.
In Figu e 10 ( igh panels), we also show he empi ical
cumula i e dis ibu ion unc ion (ECDF) o ensemble-
a e aged mean ansla ional ( op- igh panel) and mean
o ib a ional (bo om- igh panel) kine ic ene gies o deso bed
CO molecules. These ene gy dis ibu ions a e defined in he
same way as in he p e ious panel, bu hey a e only calcula ed
when deso bed CO molecules a e a away om he su ace
(cen e -o -mass- o-su ace-plane dis ance g ea e han 6 Å) a
he end o he dynamics (3500 s). We ha e ma ked wi h
shaded a eas a ound each ECDF hei associa ed 99%
D o e zky−Kie e −Wol owi z confidence in e als.
65
These
a eas define o each empi ical dis ibu ion he egion inside
which hei exac cumula i e dis ibu ion unc ions lie wi h a
99% confidence. F om hese panels, we can see ha (Te,Tl)-
MDEF-1 (blue line) and (Te,Tl)-MDEF-2 ( ed line) e alua ed
mean ene gies ag ee conside ably wi h (Te,Tl)-AIMDEF
(black) esul s, as all MDEF cu es lie wi hin he confidence
in e al o he o iginal calcula ion. I is also appa en ha he
(Te,Tl)-MDEF-2 confidence in e als a e much smalle han
he o he s (and he e o e close o he exac dis ibu ion), due
o he highe numbe o ajec o ies sampled. These esul s
u he show ha bo h, o ib a ion and ansla ion o CO
molecules on op o he su ace and a e deso p ion, a e in
good consonance wi h he findings in e 27.
Taking ad an age o he imp o ed s a is ics p o ided by
(Te,Tl)-MDEF-2, we can eliably de e mine he o ib a ional
s a e o he deso bed CO molecules (ν ,j ). In pe o ming such
an analysis, he o a ional quan um numbe j is compu ed
om he classical angula momen um Lcl as he closes in ege
ha e ifies =−+ + ℏ
j
L(1 1 4 / )/2
cl
22 , while he ib a-
ional s a e ν is de e mined om he ib a ional ac ion α as
he nea es in ege ha e ifies ν =α /h−1/2, whe e his he
Planck cons an .
66
The esul s om his quasi-classical analysis
show ha mos o he deso bed molecules (83.3%) a e in he
ib a ional g ound s a e and sligh ly o a ionally exci ed (j ≤
20).
5. CONCLUSIONS
Using he ecen ly de eloped embedded a omic neu al
ne wo k (EANN)
47
amewo k, we gene a e an accu a e and
complex po en ial ene gy su ace ha is able o desc ibe he
dynamics o he em osecond lase -induced deso p ion o CO
om Pd(111) wi h a co e age o 0.75 ML. As he aining da a
se , we use a ound 16 000 configu a ions aken om ab ini io
molecula dynamics simula ions [(Te,Tl)-AIMDEF]
27
ha
inco po a e bo h he effec o he lase -exci ed elec ons and
he concomi an exci a ion o he su ace phonons. The
Figu e 9. Top: global CO deso p ion p obabili y as a unc ion o
ime. Bo om: CO deso p ion p obabili ies o de ed by he ini ial si e
as a unc ion o ime. Fo all panels, (Te,Tl)-AIMDEF ep esen s
esul s ex ac ed om e 27 calcula ions (black), (Te,Tl)-MDEF-1
ep esen s dynamics esul s ob ained om ou bes ained po en ial
using he same 100 (Te,Tl)-AIMDEF ini ial condi ions (blue), and
(Te,Tl)-MDEF-2 ep esen s dynamics esul s ob ained wi h he same
EANN po en ial using 2000 andom ini ial condi ions ( ed).
Figu e 10. Le : mean cen e o mass ansla ional kine ic ene gy,
Ekin
ans ( eddish, ull hick lines), and mean o ib a ional kine ic ene gy,
Ekin
o ib (bluish, ull hick lines), o adso bed CO molecules as a
unc ion o ime. The co esponding hin do ed cu es abo e and
below each Ekin
ans( o ib ) cu e show he mean alues plus and minus
associa ed s anda d de ia ions, espec i ely. Righ : cen e o mass
ansla ional ene gy ( op) and o ib a ional kine ic ene gy (bo om)
empi ical cumula i e dis ibu ion unc ions (ECDFs) o deso bed CO
molecules. Shaded a eas ma k 99% D o e zky−Kie e −Wol owi z
confidence in e als.
65
Fo all panels, (Te,Tl)-AIMDEF ep esen s
esul s ex ac ed om e 27 calcula ions, (Te,Tl)-MDEF-1 ep esen s
dynamics esul s ob ained om ou bes ained po en ial using he
same 100 (Te,Tl)-AIMDEF ini ial condi ions, and (Te,Tl)-MDEF-2
ep esen s dynamics esul s ob ained wi h he same EANN po en ial
using 2000 andom ini ial condi ions.
Jou nal o Chemical Theo y and Compu a ion pubs.acs.o g/JCTC A icle
h ps://doi.o g/10.1021/acs.jc c.1c00347
J. Chem. Theo y Compu . 2021, 17, 4648−4659
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