A u he ocus on pene a ion indices o misfi an
de Waals c us s
Jo ge Eche e ´
ıa
a
and San iago Al a ez
b
The ecen p oposal o he an de Waals c us as he sphe ical sec ion comp ised be ween he a omic
adius and he an de Waals adius o an elemen gene a ed a commen a y by G ochala poin ing ou
some supposed weaknesses o he app oach (h ps://doi.o g/10.1039/D3SC90191B). Ou ini al s udy is
complemen ed he ein by an analysis o he effec s o he diffe en size o he wo in e ac ing c us s,
which shows ha he pene a ion index defined ea lie can adequa ely handle he misfi c us in e ac ions.
In a ecen pape published in his jou -
nal,
1
we analysed in dep h he po en ial
applica ion o a pene a ion index
be ween he an de Waals c us s o wo
a oms A and B, p
AB
,
2
as an al e na i e o
he use o aw in e a omic dis ances o
he analysis o a a ie y o bonding si u-
a ions, om weak an de Waals in e -
ac ions all he way o s ong mul iple
co alen bonds. The only pieces o in o -
ma ion needed a e he co alen and an
de Waals adii o each o he wo
elemen s,
A,
B
,
A
and
B
, as well as he
in e a omic dis ance d
AB
(Fig. 1a).
The in e pene a ion o he wo c us s
is dened as
i
AB
=
A
+
B
−d
AB
(1)
Di iding he in e pene a ion by he
sum o he an de Waals c us wid hs (w
A
and w
B
, Fig. 1a and eqn (2)), and mul i-
plying by 100, we ob ain he pene a ion
index as a pe cen age o he sum o he
wo c us wid hs (eqn (3)).
w
A
=
A
−
A
;w
B
=
B
−
B
(2)
pAB ¼100 iAB
wAþwB
(3)
Simul aneously o he appea ance o
ha pape , W. G ochala published
a commen a y wa ning abou he p ob-
lems ha migh a ise when using such
a pene a ion index o pai s o a oms
wi h dissimila sizes and diffe en an
de Waals c us wid hs.
3
Al hough we had pinpoin ed such
a possible p oblem in he ex and gi en
a simplied p elimina y analysis in an
Appendix, we hink ha he conce n
exp essed by G ochala dese es a mo e
de ailed analysis ha we p esen he e.
Fo an AB a om pai , we can diffe en-
ia e he pene a ion o a om A in o he
an de Waals c us o a om B (p
A
, eqn (4))
om he pene a ion o a om B in he
c us o a om A (p
B
, eqn (5)). He ewi h we
conside a om A o ha e a wide c us
han a om B, ega dless o he alues o
hei co alen and an de Waals adii,
i.e.,w
A
$w
B
.
pA¼100 iAB
2wB
(4)
pB¼100 iAB
2wA
(5)
F om eqn (4) and (5) i ollows ha p
A
$p
B
in all cases. No ice also ha he wo
deg ees o pene a ion depend on he
c us wid hs and on he dis ance, since
i
AB
depends linea ly on d
AB
, no on he
a omic adii. Finally, i mus be no ed
ha p
A
and p
B
a e calcula ed ela i e o
he wid h o one c us each, whe eas p
AB
is calcula ed wi h espec o he sum o
he wo wid hs, and o ha eason we
in oduce a ac o 2 in he denomina o
o eqn (4) and (5).
To explo e he ela ionships be ween
he h ee pene a ion pa ame e s we
compa e in Fig. 2 hei alues a some
special in e a omic dis ances o he case
o wo c us s o he same wid h (le
Fig. 1 (a) Defini ion o he co alen ( ) and an
de Waals ( ) adii, he wid h o he an de
Waals c us (w) and he in e a omic dis ance
(d) o wo nea by a oms A and B. (b) Pa am-
e e s needed o define he pene a ion o
a om A in o he an de Waals c us o a om B
and ice e sa (eqn (4) and (5)).
a
Ins i u o de S´
ın esis Qu´
ımica y Ca alisis Homog´
enea
(ISQCH) and Depa men o de Qu´
ımica Ino g´
anica,
Facul ad de Ciencias, Uni e sidad de Za agoza, Ped o
Ce buna 12, 50009 Za agoza, Spain. E-mail: Jo ge.
[email p o ec ed]
b
Depa men de Qu´
ımica Ino g`
anica i O g`
anica, Secci´
o
de Qu´
ımica Ino g`
anica, and Ins i u de Qu´
ımica
Te`
o ica i Compu acional, Uni e si a de Ba celona,
Ma ´
ıi F anqu`
es 1-11, 08028 Ba celona, Spain.
E-mail: [email p o ec ed].es
Ci e his: Chem. Sci.,2024,15,12166
All publica ion cha ges o his a icle
ha e been paid o by he Royal Socie y
o Chemis y
DOI: 10.1039/d4sc03827d
sc.li/chemical-science
12166 |Chem. Sci.,2024,15,12166–12168 © 2024 The Au ho (s). Published by he Royal Socie y o Chemis y
Chemical
Science
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column) and o wo mis an de Waals
c us s ( igh column), deduced by in o-
ducing he chosen dis ances in o eqn
(3)–(5). No ice ha he exp essions in he
igh hand column can inco po a e he
w
A
=w
B
cases (lehand side) by simply
eplacing in all he exp essions he “<”
and “>”symbols by “#”and “$”,
espec i ely.
Fi s , we conside he case in which
wo an de Waals sphe es a e ba ely in
con ac (Fig. 2a). By deni ion, he h ee
pene a ion pa ame e s a e 0% bo h o
c us s wi h he same wid h (Fig. 2a, le)
o o mis c us s (Fig. 2a, igh ). In
a second case (Fig. 2b), he dW c us o
a om A eaches he alence sphe e o
a om B. The h ee pene a ion pa ame-
e s adop a alue o 50% i he wo c us
wid hs a e he same (Fig. 2b, le), bu o
mis c us s only p
A
has ha alue, while
p
B
is less han 50% (Fig. 2b, igh ).
Fo wo mis a oms, a u he
app oach is equi ed o ge he VdW
sphe e o a om B in con ac wi h he
alence sphe e o A (Fig. 2b0), gi ing p
A
>
50% and p
B
=50%, while p
AB
is s ill less
han 100%.
Finally, when he wo alence sphe es
es ablish con ac , i.e., when he in e -
a omic dis ance is iden ical o he co a-
len adii sum (Fig. 2c), he
in e pene a ion index p
AB
is 100%
ega dless o he wid hs o he wo an
de Waals c us s. The indi idual pene-
a ion p
A
eaches alues highe han
Fig. 2 Values adop ed by he indi idual and dia omic pene a ion indices a special in e a omic dis ances. The le column shows he case o wo
a oms o iden ical cu s wid hs bu diffe en a om sizes and he igh column a case o wo a oms wi h misfi c us wid hs.
© 2024 The Au ho (s). Published by he Royal Socie y o Chemis y Chem. Sci.,2024,15,12166–12168 | 12167
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100%, bu hose o p
B
a e smalle han
100%.
All in all, i is seen ha he e olu ion
o he h ee pene a ion pa ame e s as
he in e a omic dis ance dec eases
p esen each a clea ly p edic able beha -
iou . S ill mo e in e es ing, he indi idual
pene a ions p
A
and p
B
can be ob ained
om he in e pene a ion index p
AB
acco ding o eqn (6) and (7). Con e sely,
p
AB
can be ob ained om he indi idual
pene a ions by means o eqn (8).
pA¼pAB
21þwA
wB(6)
pB¼pAB
21þwB
wA(7)
pAB ¼2pApB
pAþpB
(8)
Those ela ionships a e illus a ed in
Fig. 3, whe e we plo he indi idual
pene a ion pa ame e s p
A
and p
B
as
a unc ion o he in e pene a ion index
p
AB
, o wo a om pai s: one wi h p ac i-
cally iden ical c us wid hs, Ta–O(w
Ta
=
0.83, w
O
=0.84 Å), and one wi h e y
diffe en wid hs, Zn–Te (w
Zn
=1.17 Å, w
Te
=0.61 Å). In he  s case (Fig. 2, le
column), he indi idual pene a ions p
A
and p
B
a e equal o he in e pene a ion
index p
AB
a any in e a omic dis ance,
whe eas o a om pai s wi h diffe en
c us wid hs (e.g., Zn and Te) p
A
inc eases
a a highe a e han p
B
as he in e a omic
dis ance dec eases (i.e.,asp
AB
inc eases).
Conclusion
E en o mis an de Waals c us s he
in e pene a ion index p
AB
ca ies in o -
ma ion on he diffe en le- igh (p
A
) and
igh -le(p
B
) pene a ions. Those diffe -
ences depend only on he ela i e wid hs
o he wo in e pene a ing c us s (w
A
and
w
B
), bu no on he a omic sizes (
A
,
B
,
A
and
B
), as seen in eqn (6)–(8).
pA
pB
¼wA
wB
(9)
The de ia ion o he p
A
and p
B
alues
om p
AB
o an AB a om pai wi h mis
an de Waals c us s can be exp essed by
hei quo ien , which is ound o be
iden ical o he quo ien o he c us
wid hs (eqn (9)).
Fo hose easons, we hink ha i is
ad isable o use p e e en ially he in e -
pene a ion index p
AB
, which is concep-
ually clea e , implies no loss o
in o ma ion on he indi idual pene a-
ion indices p
A
and p
B
, and is simple o
use.
Da a a ailabili y
No p ima y esea ch esul s, sowa e o
code ha e been included and no new da a
we e gene a ed o analysed as pa o his
commen .
Au ho con ibu ions
Bo h au ho s con ibu ed o he discus-
sion, w i ing and e ision o he o iginal
d a.
Conflic s o in e es
The e a e no conic s o decla e.
Acknowledgemen s
This wo k has been suppo ed by he
Spanish Minis y o Economy and
Compe i i eness (PGC2018-093863-B-C21
and PID2022-140244NB-I00), he Spanish
S uc u es o Excellence Ma ´
ıa de Maez u
p og am (g an MDM-2017-0767) and he
Gene ali a de Ca alunya –AGAUR (g an
2017-SGR-1289). The alloca ion o
compu e ime a CSUC is also acknowl-
edged. J. E. is indeb ed o he Spanish
MICCIN o a Ram´
on y Cajal esea ch
con ac (RYC-2017-22853), and o he
Gobie no de A ag´
on-ESF p og am
(Resea ch G oup E07_23R) o nancial
suppo .
Re e ences
1 J. Eche e ´
ıa and S. Al a ez, The
Bo de less Wo ld o Chemical
Bonding Ac oss he Van de Waals
C us , Chem. Sci., 2023, 14, 11647–
11688.
2 D. M. Gil, J. Eche e ´
ıa and S. Al a ez,
Te ame hylammonium Ca ion:
Di ec ionali y and Co alency in I s
In e ac ions wi h Halide Ions, Ino g.
Chem., 2022, 61, 9082–9095.
3 W. G ochala, A ocus on pene a ion
index –a new desc ip o o chemical
bonding, Chem. Sci., 2023, 14, 11597–
11600.
4 B. Co de o, V. G´
omez, A. E. Pla e o-
P a s, M. Re ´
es, J. Eche e ´
ıa,
E. C emades, F. Ba ag´
an and
S. Al a ez, Co alen Radii Re isi ed,
Dal on T ans., 2008, 2832–2838.
5 S. Al a ez, A Ca og aphy o he Van de
Waals Te i o y, Dal on T ans., 2013,
42, 8617–8636.
Fig. 3 Dependence o he indi idual pene a ion pa ame e s p
A
and p
B
on he pene a ion
index p
AB
o he Zn–Te and O–Ta a om pai s, wi h c us wid h a ios w
A
/w
B
=1.92 and 1.01,
espec i ely. The poin s a, b, b0and c co espond o he special dis ances specified in Fig. 2. All
alues calcula ed om s anda d co alen
4
and an de Waals
5
a omic adii a a ying A–B
dis ances.
12168 |Chem. Sci.,2024,15,12166–12168 © 2024 The Au ho (s). Published by he Royal Socie y o Chemis y
Chemical Science Commen a y
Open Access A icle. Published on 15 July 2024. Downloaded on 11/28/2024 1:01:31 PM.
This a icle is licensed unde a
C ea i e Commons A ibu ion-NonComme cial 3.0 Unpo ed Licence.
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