Ci a ion: de-la-Hue a-Sainz, S.;
Balles e os, A.; Co de o, N.A.
Gaussian Cu a u e E ec s on
G aphene Quan um Do s.
Nanoma e ials 2023,13, 95.
h ps://doi.o g/10.3390/
nano13010095
Academic Edi o : Filippo Giubileo
Recei ed: 21 No embe 2022
Re ised: 20 Decembe 2022
Accep ed: 20 Decembe 2022
Published: 25 Decembe 2022
Copy igh : © 2022 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
nanoma e ials
A icle
Gaussian Cu a u e E ec s on G aphene Quan um Do s
Se gio de-la-Hue a-Sainz 1, Angel Balles e os 1and Nicolás A. Co de o 1,2,3,*
1Physics Depa men , Uni e sidad de Bu gos, E-09001 Bu gos, Spain
2In e na ional Resea ch Cen e in C i ical Raw Ma e ials o Ad anced Indus ial Technologies (ICCRAM),
Un e sidad de Bu gos, E-09001 Bu gos, Spain
3Ins i u e Ca los I o Theo e ical and Compu a ional Physics (IC1), E-18016 G anada, Spain
*Co espondence: nco de [email p o ec ed]
Abs ac :
In he las ew yea s, much a en ion has been paid o he exo ic p ope ies ha g aphene
nanos uc u es exhibi , especially hose eme ging upon de o ming he ma e ial. He e we p esen
a s udy o he mechanical and elec onic p ope ies o ben hexagonal g aphene quan um do s
employing densi y unc ional heo y. We explo e h ee di e en kinds o su aces wi h Gaussian
cu a u e exhibi ing di e en shapes—sphe ical, cylind ical, and one-shee hype boloid—used o
bend he ma e ial, and se e al bounda y condi ions ega ding wha a oms a e o ced o lay on
he chosen su ace. In each case, we s udy he cu a u e ene gy and wo quan um egene a ion
imes (classic and e i al) o di e en alues o he cu a u e adius. A s ong co ela ion be ween
Gaussian cu a u e and hese egene a ion imes is ound, and a special di e gence is obse ed o
he e i al ime o he hype boloid case, p obably ela ed o he pseudo-magne ic ield gene a ed by
his cu a u e being capable o causing a phase ansi ion.
Keywo ds:
g aphene; Gaussian cu a u e; quan um e i al; DFT; pseudo-magne ic ield; phase
ansi ion
1. In oduc ion
While ea ly s ages o g aphene esea ch we e cen e ed on i s heo e ical aspec s [
1
–
4
],
a e he success ul isola ion o a single shee o g aphi ic ma e ial by Geim and No oselo
in 2004 [
5
], he e has been a long end o ad ancemen s popula ed wi h expe imen-
al con i ma ion o p edic ed p ope ies, he disco e y o new and exo ic phenomena,
and imp o emen s o he syn hesis me hods o his ma e ial. The e e -g owing lis
o po en ial applica ions o g aphene sp eads ac oss many ields due o i s ou s anding
p ope ies and exo ic beha io s, such as enginee ing [
6
–
8
], medicine [
9
–
12
], senso ab-
ica ion
[13–16],
ca alysis [
17
–
20
], ene gy s o age and managemen [
21
–
23
], and lexible
and high-pe o mance elec onic de ices [
24
–
29
]. G aphene nanos uc u es ha e had e en
g ea e po en ial since he disco e y o supe conduc i i y in bilaye g aphene [30–32].
Quan um e i al— he pe iodic egene a ion o he ini ial s a e o a ime-dependen
quan um sys em—on he o he hand, is s ill a subjec mainly s udied om a heo e ical
poin o iew, hough i s expe imen al ealiza ion is possible and opens up in e es ing
esea ch di ec ions o in o ma ion ansmission and he ab ica ion o quan um de ices.
Fo ins ance, quan um e i als can be used o measu e ideli y in quan um in o ma ion
echnology, and his measu emen is c ucial in o de o unde s and he e ec s o deco-
he ence, dissipa ion, and impe ec ions in quan um in o ma ion de ices [
33
]. In addi ion,
e i als ha e been p oposed as a me hod o anspo ing in o ma ion wi h high e iciency
o gene a ing en anglemen [
34
]. On a wide iew, he empo al e olu ion o a sys em
has been qui e use ul o eal- ime sc eening o high-speed phenomena, such as chemical
eac ions [35–38].
Thus, we aim o s udy how di e en ac o s, such as shape and Gaussian cu a u e,
change he beha io o a g aphene-based sys em h ough he simula ion o i s quan um
Nanoma e ials 2023,13, 95. h ps://doi.o g/10.3390/nano13010095 h ps://www.mdpi.com/jou nal/nanoma e ials
Nanoma e ials 2023,13, 95 2 o 15
e i als. This is especially ele an , since g aphene, which is commonly concei ed as
a pe ec ly la and p is ine shee o ca bon a oms, has na u ally a a mo e complex
s uc u e, wi h ipples, w inkles, and many o he de ia ions om i s ideal la ness [
39
–
45
],
co obo a ing heo e ical p edic ions done many yea s be o e i s isola ion [
46
–
48
]. While he
ue o igins o hese de ia ions om la ness a e s ill up o deba e, hei in luences on he
ma e ial p ope ies, such as cha ge anspo , allow o ine- uning o i s beha io [49–52].
O he ca bon nanos uc u es can be classi ied based on hei Gaussian cu a u e:
ca bon nano ubes [
53
–
56
] and nanocones [
57
–
61
]—and ideal g aphene—ha e null Gaussian
cu a u e, as he ma e ial s ays la in a leas one o i s p incipal di ec ions. O he s, such as
ulle enes, o m closed s uc u es ha ing posi i e cu a u es. The case o nega i e Gaussian
cu a u e has been mo e elusi e, bu i has been ound in schwa zi es [
62
,
63
] as open o
e en pe iodic s uc u es in which each poin esembles a saddle.
In ac , he e ha e been ecen ad ancemen s on he syn hesis and design o ca bon
nanos uc u es and polycyclic a oma ic sys ems wi h unable cu a u e— ia inco po a ion
o pen agons o hep agons [
64
–
66
]—o e en modeling o hyb id sys ems be ween la
and cu ed s uc u es [
67
], wi h impo an applica ions o ba e ies de elopmen and
enginee ing. Nega i ely cu ed g aphene has also been used as an analog o g a i a-
ional sys ems [
68
,
69
], allowing di ec obse a ion o exo ic beha io s in a much mo e
app oachable ashion.
We ecen ly p esen ed esul s on sphe ically de o med g aphene quan um do s [
70
].
Those nanos uc u es ha e posi i e Gaussian cu a u e. We expand ou s udy in his
a icle o nega i e and null Gaussian cu a u es, p esen ing a compa a i e s udy o cu -
a u e e ec s in ene gy and elec onic s uc u e ob ained using densi y unc ional heo y,
on g aphene quan um do s wi h a ious Gaussian cu a u e alues.
2. Ma e ials and Me hods
F om he many possibili ies a ailable o he heo e ical s udy o g aphene, each o
hem wi h i s own ad an ages, eliabili y, and ange o applica ion, we chose he p ocedu e
desc ibed in ou p e ious wo k [
70
] and used densi y unc ional heo y
(DFT) [71–73]
as
he main ool o simula ing he p ope ies o g aphene quan um do s h ough use o he he
Gaussian 16 [
74
] package. Fo he exchange-co ela ion unc ional, local densi y app oxima-
ion (LDA) [
75
,
76
] was used, in i ue o i s be e pe o mance o g aphi ic sys ems and
highe calcula ion speed compa ed wi h gene al g adien app oxima ions (GGAs) o hyb id
unc ionals such as B3LYP [
77
–
81
], and because i has been success ully used o analyze
in e ac ions in ca bon
nanos uc u es [82–87].
The basis se was
6-31G** [88]
, wi h d- ype
and p- ype unc ions as pola iza ion aids o a be e desc ip ion o he chemical bond.
We s udied a hexagonal g aphene quan um do wi h 10 ca bon a oms on each edge
and hyd ogen passi a ed esul ing in a
C600H60
molecula o mula (see Figu e 1). All edges
we e o he zig-zag ype o a oid he appea ance o o he phenomena, such as asymme ic o
unbalanced magne ic s a es [
89
,
90
] o signi ican epulsi e in e ac ions be ween passi a ing
a oms. This sys em, wi h a dis ance be ween e ices o 46.4 Å, allowed us o achie e a
comp omise be ween he expe imen al size o g aphene na u al co uga ion [
39
,
41
] and
compu a ional cos .
The ocus o his pape is on he e ec s o di e en Gaussian cu a u e alues (null,
posi i e and nega i e) o he ma e ial, and o ha we employed a amily o su aces
o bending he do . All hese su aces ha e a common exp ession ha can be w i en in
Ca esian coo dina es as:
z=qR2−ax2−by2. (1)
F om Equa ion (1), di e en kinds o Gaussian cu a u e can be ob ained: (i) posi i e,
o he sphe e (
a=b=
1); (ii) ze o, o he cylinde (ei he
a
o
b
being 1, he o he being 0);
and (iii) nega i e, o he one-shee hype boloid— e e ed o simply as hype boloid om
now on (ei he
a
o
b
being 1, he o he being
−
1). Al hough he wo possible cylinde s a e
equi alen o a squa e do , we simula ed bo h as sepa a e su aces, as he hexagonal do ’s
inal geome y is di e en ; we call hem cylinde s x and y ega ding which componen has
Nanoma e ials 2023,13, 95 3 o 15
a non-ze o coe icien . Fo he hype boloid, howe e , exchanging hese coe icien alues
only p o ides a lipped (and hus equi alen ) s uc u e, so only one case was conside ed.
These su aces can be seen in Figu e 2.
Figu e 1.
Hexagonal, la g aphene quan um do used as a s a ing poin o de o ma ion. Image
gene a ed wi h Gauss iew 6 [91].
(a)z=pR2−x2−y2(b)z=pR2+x2−y2
(c)z=√R2−x2(d)z=pR2−y2
Figu e 2.
The ou di e en geome ies conside ed in his s udy o he g aphene do wi h
R=
50 Å
and hei espec i e equa ions: (
a
) sphe e; (
b
) one-shee hype boloid; (
c
) x-cylinde ; (
d
); y-cylinde .
Images gene a ed wi h GaussView 6 [91].
The bounda y condi ions o he do we e a second ac o in his s udy because hey a e
impo an o he expe imen al ealiza ion o hese sys ems. The ideal case, whe e all a oms
a e con ined o he ini ial su ace, would be imp ac ical o ep oduce a his scale; he e o e,
we conside wo addi ional possibili ies mo e easible o an expe imen al se up. We used
h ee cases o each su ace ega ding wha a oms we e es ained: (i) all 600 ca bon a oms
mus emain on he su ace; (ii) only he 60 edge ca bon a oms a e ixed; (iii) only he
12 e ex
ca bon a oms a e ixed. Figu e 3shows hese possibili ies o he sphe ical case
wi h
R=
40 Å. This se o dec easing es ic ions allows he cu ed do o elax u he in
an a emp o eco e a om he edges i s ini ial and op imal la shape.
Nanoma e ials 2023,13, 95 4 o 15
(a) (b) (c)
Figu e 3.
Bounda y condi ions’ e ec s on he op imized geome ies o an ini ially sphe ical quan um
do wi h
R=
40 Å. Images gene a ed wi h Gauss iew 6 [
91
]. (
a
) Fixed su ace; (
b
) ixed edges;
(c) ixed e ices.
3. Resul s and Discussion
In his s udy, we ha e ocused on he analysis o cu a u e ene gy and quan um
egene a ion imes o ou hexagonal do as i is de o med acco ding o he di e en
su aces desc ibed and conside ing each se o bounda y condi ions. The pa ame e 1/R2
is used in all g aphics as a measu e o he cu a u e o he do . While his is only ue
o he pe ec ly sphe ical case (ha ing he Gaussian cu a u e as ze o o he cylinde ,
and nega i e, non-cons an o he hype boloid), we use i o compa a i e pu poses o
quan i ica ion o he de o ma ion.
3.1. Cu a u e Ene gy
Cu a u e ene gy, calcula ed as he di e ence be ween he ene gy o a gi en do and
ha o he la one, has been calcula ed o all a ailable cases and plo ed agains he 1
/R2
pa ame e . This enables a quick inspec ion o he s abili y o he sys em om a mechanical
poin o iew, allowing us o check wha su aces and bounda y condi ions lead o mo e
s able sys ems.
In o de o analyze he e ec s o he wo ac o s conside ed in his s udy, we i s
conside he ype o su ace used, plo ing he cu a u e ene gies o all he ideal geome ies
( ha is, wi h all ca bon a oms lying on he su ace) in Figu e 4. As expec ed, cu a u e
ene gy o all su aces inc eases as cu a u e does; highe alues exis o hype boloidal
and sphe ical su aces han o cylind ical ones, which gi e nea ly iden ical esul s (p o ing
he nea equi alence o he wo cylind ical s uc u es om a mechanical poin o iew).
These esul s a e consis en wi h he ac ha o sphe ical and hype boloidal su aces,
he de o ma ion is applied along wo spa ial axes ins ead o only one, as in he cylinde .
The sligh ly highe ins abili y o he hype boloid case is de i ed om he inhe en gene al
s e ching o he s uc u e ha o ces a la ge de ia ion om he sp
2
hyb idiza ion o plana
g aphene han in he sphe ical case.
Figu e 4.
Cu a u e ene gy s. 1
/R2
o all ou ideal geome ies—all a oms o ced o lay on he
su ace—wi h he la do aken as ene gy o igin. Bo h cylind ical cases gi e almos iden ical ene gies.
Nanoma e ials 2023,13, 95 5 o 15
The e ec s o he bounda y condi ions applied o each do a e plo ed in Figu e 5 o
each su ace so ha ela i e changes in he gene al beha io can be easily obse ed. Fo all
ou su aces conside ed, he cases whe e only he e ices we e ixed a e he mos s able
ones, as expec ed, while he ideal geome ies ep esen a nea ly op imal s uc u e only
o small cu a u e alues. In he hype boloid case, he de ia ion om he ideal su ace
s a s om e y small cu a u es, and he ene gy gain when elaxing bounda y condi ions
is bigge . Ne e heless, i was no possible o ge esul s o high alues o 1
/R2
. This is
p obably due o he ac ha a big de ia ion om he plana case wi h opposi e signs in
di e en di ec ions leads o he b eaking o he nanos uc u e. A dynamical ( o ins ance,
molecula dynamics) calcula ion would be necessa y o con i m his hypo hesis.
Figu e 5.
Cu a u e ene gy s. 1
/R2
plo s o all geome ies, wi h he la do aken as ene gy o igin.
While he ixed-su ace cylind ical plo s a e essen ially s aigh lines, as he leas -
squa es linea i s plo ed in Figu e 5p o e—showing he linea dependence on 1
/R2
cha ac e is ic o he con inuum model applied o a nano ube [
92
]— he sphe ical and
hype boloidal ones a e no . In ou p e ious wo k [
70
], i was shown how his disc epancy
wi h he con inuum model could be connec ed in he sphe ical case o he small posi ion
changes de i ed om he use o quan um mechanics in he op imiza ion ins ead o a
classical o ce ield. Howe e , hese new esul s p o e ha he con inuum model is indeed
alid o he cylind ical cases— ha a e cu ed su aces—e en when quan um mechanics is
used o de e mine geome ies, sugges ing he e a e o he impo an con ibu ions besides
he heo y used o geome y op imiza ion. I seems ha s uc u es wi h non-ze o Gaussian
cu a u e su e addi ional s ain ha he con inuum model canno ake in o accoun .
Since he con inuum model is alid o he cylind ical case, i is possible o calcula e
he bending modulus o his g aphene quan um do by making use o he leas -squa es i s
depic ed in he lowe wo panels o Figu e 5. The bending ene gy o a nano ube o adius
R
can be w i en as [93,94]:
E=Cb
2R2, (2)
wi h
Cb
being he bending modulus (also known as lexu al igidi y). Ou g aphene
quan um do is no a ca bon nano ube, bu , aking in o accoun ha all a oms on i s bo de s
a e passi a ed, he e a e no dangling bonds, and he nanos uc u e can be conside ed as a
piece o he wall o a nano ube. Looking a Equa ion (2),
Cb
is jus wice he slope o he
Nanoma e ials 2023,13, 95 6 o 15
linea i . Fo he x-cylinde , his calcula ion leads o
Cb=
4.00 eVÅ
2
pe C a om, and
o he y-cylinde , i yields
Cb=
3.99 eVÅ
2
. Bo h esul s a e nea ly iden ical, in spi e o
he ac ha he x-cylinde could be conside ed as a piece o a zig-zag nano ube, while
he y-cylinde would co espond o a piece o an a mchai ube. The bending modulus o
ca bon nano ubes being independen o he bending di ec ion is a well-known ac [
95
] and
a consequence o he hexagonal symme y o he g aphene la ice ha makes his ma e ial
iso opic in he linea elas ic egime [
96
]. Ou esul s a e in excellen ag eemen wi h
hose ob ained o he bending modulus pe C a om by Kü i e al. (3.9
±
0.1 eVÅ
2
) [
97
],
Sánchez-Po al e al. (4.00 eVÅ
2
o a mchai ubes) [
98
] and Kudin e al. (3.9 eVÅ
2
) [
93
].
An addi ional piece o in o ma ion we can ex ac om he plo s co esponding o
bo h cylind ical cases in Figu e 5is abou he accu acy o ou calcula ions. Fo e y low
alues o 1
/R2
(below 10
−4Å−2
), he e a e some jumps when a oms a e allowed o elax
ou side he ideal cylind ical su ace. Some imes he e is a gain in ene gy, bu o he imes
he e is no gain, showing ha he geome y had no ully elaxed because he code had
no de ec ed he addi ional s abiliza ion due o b eaking he exac cylind ical shape. These
jumps a e abou 2 milliHa ee, and his alue can be aken as an es ima ion o he accu acy
o he me hod.
3.2. Regene a ion Times
The s udy o quan um e i al phenomena was ca ied ou using he eigen alue spec a
ob ained o each do , pe o ming an analysis o he elec onic p ope ies o ou sys em by
means o a homemade code buil wi hin he Ma hema ica en i onmen [
99
]. In o de o
calcula e hese e i al phenomena, we de ine, ollowing Robine [
100
], he ini ial s a e o a
ime-independen wa epacke as a linea combina ion o eigens a es
|uni
wi h weigh s
an
:
|Ψ(0)i=
∞
∑
n=0
an|uni, (3)
wi h i s empo al e olu ion ha ing he ollowing exp ession:
|Ψ( )i=
∞
∑
n=0
an|unie−i
¯hEn , (4)
whe e Enis he he eigenene gy o |uni.
Since we a e using he ene gy spec um calcula ed wi h DFT o build he wa epacke ,
we can ake one single le el as a cen al poin and pe o m a Taylo expansion a ound i o
ge an analy ical exp ession o he spec um:
En=En0+E0
n0(n−n0) + 1
2!E00
n0(n−n0)2+1
3!E000
n0(n−n0)3+. . . . (5)
A e subs i u ing his expansion in o Equa ion (4), he inal empo al e olu ion shows
se e al e ms inside he exponen ial, each o hem co esponding o one ime scale and
gi ing ise o di e en egene a ion imes (classical,
TCl
; e i al,
TRe
; supe e i al,
TSup
;
...):
|Ψ( )i=
∞
∑
n=0
an|unie−i
¯hhEn0+E0
n0(n−n0)+ 1
2! E00
n0(n−n0)2+1
3! E000
n0(n−n0)3+...i , (6)
TCl =2π¯h
|E0
n0|, (7)
TRe =2π¯h
|E00
n0|/2! , (8)
TSup =2π¯h
|E000
n0|/3! . (9)
Nanoma e ials 2023,13, 95 7 o 15
As o he wa epacke i sel , he coe icien s anwill ollow a Gaussian dis ibu ion,
an=1
σ√πe−(n−n0)2
2σ2(10)
cen e ed a ound he i h unoccupied o bi al (LUMO+4), he eby ha ing a alue
n0=
5,
and a wid h
σ=
0.7, ensu ing a small collec ion o i e s a es wi h a signi ican con ibu ion
(an>0.001).
Tempo al e olu ion was s udied by means o he squa ed modulus o he au oco ela-
ion unc ion,
|A( )|2
, de ined as he o e lap o he he wa epacke a e an a bi a y ime
and i s ini ial s a e:
|A( )|2=|hΨ(0)|Ψ( )i|2. (11)
Figu e 6shows he plo co esponding o a ypical example in which he oscilla o y pa e ns
o
|A( )|2
a e e iden . The pe iodici ies a di e en ime scales co espond o di e en
egene a ion imes, wi h classic ime
TCl
being he high- equency one, and e i al ime
TRe
he low- equency one. While e i al imes o highe o de (such as
TSup
) a e heo e ically
possible, none beyond
TRe
could be obse ed in any case due o he in e e ence among
di e en egene a ion imes.
Figu e 6.
View o
|A( )|2
as a unc ion o
o a sphe ical do wi h
R=
100 Å. Analy ical alues o
bo h egene a ion imes a e shown wi h do ed lines (o ange o classical ime, yellow o e i al ime).
Ob aining he alues o hese wo egene a ion imes is an easy ask:
TCl
co esponds
o he i s maximum o
|A( )|2
, and
TRe
comes om he i s maximum o he en eloping
cu e, which can be calcula ed using he local maxima o
|A( )|2
. The e can be some
di icul ies in hei de e mina ion, howe e , i
TRe
is no much la ge han
TCl
. In his
case, in e e ence be ween hose imes can occu , making he isual obse a ion o bo h,
especially
TCl
, ha de . Fo his eason, he analy ical exp essions o bo h imes, de i ed
om he Taylo expansion desc ibed, ha e been used as an al e na i e me hod and aid in
i s de e mina ion. We ha e, hen, analy ical ( om Taylo expansion) and nume ical ( om
empo al e olu ion) alues o each ime. A pa abolic cu e, i ed o he h ee cen al
le els o he wa epacke , has been used as he i ing unc ion o he eigen alue spec um
in o de o calcula e de i a i es.
In a simila ashion o he ene gy analysis, he esul s o his sec ion a e p esen ed in
wo s eps: i s a s udy o he e ec s o he kind o su ace, and second, an indi idual iew
o each o hem upon elaxing bounda y condi ions.
Nanoma e ials 2023,13, 95 8 o 15
3.2.1. Classical Time
The plo s o
TCl
agains 1
/R2
o all ou su aces conside ed and all ca bon a oms on
he do s con ined o hem a e depic ed in Figu e 7, showing he nume ical alues as poin s
and he analy ical ones as do ed lines jus o cla i y. Fo he cylind ical cases, no only do
bo h su aces gi e almos iden ical alues—in a simila ashion o wha happened wi h he
cu a u e ene gy—bu
TCl
also emains nea ly cons an o he whole ange o
R
s udied.
In con as , he sphe ical and hype boloidal cases exhibi opposi e beha io s:
TCl
inc eases
wi h cu a u e in he o me and dec eases in he la e . Conside ing he in e se ela ion
be ween
TCl
and he i s de i a i e o he spec um, hese esul s e lec ha ene gy le els
ge close as he cu a u e o he sphe e inc eases, ge spa se o he hype boloid and e-
main almos unchanged o he cylinde . This g oup o opposi e endencies and cons an
beha io aligns wi h he signs o he Gaussian cu a u e o he co esponding su aces.
Figu e 7.
Classical ime plo s as unc ions o 1
/R2
o all ou ideal geome ies, wi h nume ical alues
as ma ke s and analy ical ones as do ed lines. Bo h cylinde s show nea pe ec coincidence.
The e is a s ong de ia ion in nume ical
TCl
om i s analy ical coun e pa o he
sphe e a highe alues o 1
/R2
. This is due, as we commen ed ea lie , o he in e e ence
be ween classical and e i al imes. As hey app oach each o he , he en eloping cu e
shi s mo e he posi ion o he i s maximum, dis o ing he nume ical alue o
TCl
. Since
he analy ical app oach conside s only he local shape o he spec um, his in e e ence
canno be aken in o accoun , and he co esponding plo is nea ly a s aigh line wi h a
slope opposi e o ha in he hype boloid case.
When bounda y condi ions a e elaxed (see Figu e 8), a simila phenomenon o he
one obse ed o he ene gy can be seen. While ixed-su ace quan um do s gi e smoo h
plo s wi h mono onic ends, he changes in he op imal geome y o he o he wo se s o
condi ions in oduce b eaking poin s in o he alues o
TCl
, esul ing in agmen ed plo s
in which he o e all end is o he wise conse ed.
Again, he plo s co esponding o bo h cylind ical cases in Figu e 8can be used o
es ima e he accu acy o ou calcula ions. The jumps o alues o 1
/R2
below 10
−4Å−2
can
be aken as an indica ion o he accu acy o he classical egene a ion imes we calcula ed:
a ound 1 s.
Nanoma e ials 2023,13, 95 9 o 15
Figu e 8.
Classical ime as a unc ion o 1
/R2
o di e en bounda y condi ions wi hin each geome y,
wi h nume ical alues as poin s and analy ical ones as do ed lines.
3.2.2. Re i al Time
A compa ison o
TRe
o he di e en kinds o ideal su aces conside ed can be seen in
Figu e 9. While e i al ime shows again a nea ly cons an alue o he wo cylind ical
cases, i dec eases nonlinea ly o he sphe ical geome y, and exhibi s clea ly di e gen
beha io in he hype boloidal case o 1
/R2≃
10
−4Å−2
. Again, his con as o ends has
a one- o-one co espondence wi h he sign o he Gaussian cu a u e o each su ace.
Figu e 9.
Re i al ime as a unc ion o 1
/R2
o all ou ideal geome ies, wi h nume ical alues as
poin s and analy ical ones as do ed lines.
TRe
being in e sely p opo ional o he second de i a i e o he spec um gi es in o -
ma ion abou i s linea i y and ela i e sepa a ion be ween consecu i e le els. The di e -
gence obse ed in he hype boloid sugges s an ideally in ini e alue o
TRe
, due o a null
second de i a i e caused by he le els being equally spaced in ene gy o a special alue
o R.
