Time-Resol ed Chemical Bonding S uc u e E olu ion by Di ec -
Dynamics Chemical Simula ions
Ma io Pi is,*Xabie Lopez, and Jesus M. Ugalde
Ci e This: J. Phys. Chem. Le . 2024, 15, 12138−12143
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sı Suppo ing In o ma ion
ABSTRACT: Di ec -dynamics simula ions moni o a omic nuclei
ajec o ies du ing chemical eac ions, whe e chemical bonds a e
b oken and new ones a e o med. While hey p o ide aluable
in o ma ion abou he ongoing nuclea dynamics, he e olu ion o
he chemical bonds is cus oma ily o e looked, hus, hinde ing key
in o ma ion abou he eac ion mechanism. He e we examine he
e olu ion o he chemical bonds o he h ee main mechanisms o
he F−+ CH3CH2Cl eac ion using quasi-classical ajec o ies o
he nuclei, and global na u al o bi als o he elec ons. Key
indings include (i) bimolecula nucleophilic subs i u ion (SN2)
esembles a one-s ep bond b eaking and o ma ion p ocess; (ii) he
elimina ion mechanisms (syn- and an i-E2) ea u e a sequen ial
wo-s ep p ocess o p o on abs ac ion and Cl−elimina ion; and
(iii) he an i-E2 mechanism is slowe , exhibi s ebound e ec s, and ge s ac i a ed by speci ic ib a ional modes. This s udy highligh s
he impo ance o co ec ly desc ibing and ho oughly analyzing he dynamical e olu ion o chemical bonds o chemical eac ion
mechanis ic s udies.
Chemical eac ions cons i u e (complex) a e p ocesses
ha in ol e he conce ed ime e olu ion o bo h he
nuclei and he elec ons o he molecules engaged in he
eac ion om eac an s o p oduc s. F om a quan um
chemis y pe spec i e, he adiaba ic ime e olu ion o he
nuclei is cus oma ily ca ied ou in he classical domain, and
he elec ons a e assumed o espond quickly enough o he
nuclei’s e olu ion as o jus i y aking hem unde he auspices
o he Bo n−Oppenheime (BO) app oxima ion.
1
Non-
adiaba ic ime e olu ion dynamics, which equi es he
conside a ion o ansi ions be ween di e en elec onic s a es,
can be con enien ly handled by po en ial su ace hopping
me hods.
2
Consequen ly, he s udy o chemical a e p ocesses consis s
o (i) selec ing he app op ia e quan um mechanical me hod o
ea he elec ons; (ii) sol ing he co esponding Sch odinge
equa ion o he ene gy and es ima ing i s g adien a he
posi ion o each nucleus; (iii) making judicious choices o he
ini ial condi ions o he nuclei in he ensemble o ajec o ies
o be p opaga ed in ime; (i ) nume ically in eg a ing he
classical equa ions-o -mo ion o de e mine he ime e olu ion
o each nucleus in each calcula ed ajec o y; and ( )
ans o ming he ajec o ies’ inal nuclei coo dina es and
momen a in o p ope ies ha may be compa ed wi h
expe imen al esul s. As indica ed elsewhe e,
3
he p ope ies
a s ake include bond leng hs and angles, ib a ional,
o a ional, and ansla ion ene gies, quan um numbe s o he
ib a ional and o a ional deg ees o eedom, he amoun o
ene gy in indi idual molecula deg ees o eedom, and
sca e ing angles. These a e all p ope ies ha na u ally de i e
om he posi ions and momen a o he nuclei bu omi u he
conside a ions o he chemical bonding s uc u e e olu ion
along he ajec o y. The la e is simply no add essed by mos
chemical dynamics simula ion s udies.
Howe e , a chemical eac ion is concei ed as a p ocess in
which chemical componen s a e ans o med o o he chemical
componen s. Since chemis s iden i y componen s by hei
chemical bonds, i.e., hei elec onic s uc u e, he e olu ion o
hese bonds du ing he eac ion ca ies signi ican chemical
in o ma ion encoded in he concep o he eac ion
mechanism. This e e s o he ime-o de ed sequence o
o ming and/o b eaking chemical bonds, along wi h he
cha ac e iza ion o all ansien chemical componen s h ough-
ou he ajec o y om eac an s o p oduc s.
4
Elec onic s uc u e calcula ions en e in o chemical
dynamics simula ions in s eps (i) and (ii) men ioned abo e.
Howe e , hese a e demanding calcula ions aimed a no only
dealing wi h s a ic conside a ions o s uc u e bu also
Recei ed: Oc obe 17, 2024
Re ised: No embe 21, 2024
Accep ed: No embe 27, 2024
Published: No embe 28, 2024
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con on ing he dynamics o he eplacemen s o some
chemical bonds in o newly o med ones ha akes place in
he cou se o chemical eac ions. A wo d o cau ion abou hei
app op ia eness is in o de o mos such calcula ions do no
p oduce chemically accep able desc ip ions o he whole
e olu ion o he making and/o b eaking o chemical bonds.
5
Acco dingly, we ha e employed he ecen ly in oduced
Global Na u al O bi al Func ional (GNOF)
6
o e alua e he
BO ene gies, as p e ious s udies
7,8
ha e demons a ed ha
GNOF yields a a o able balance be ween s a ic and dynamic
elec on co ela ions, esul ing in chemically in o ma i e
desc ip ions and accu a e o al ene gies h oughou he en i e
ange o in e nuclea coo dina es, om s able s uc u es o
hei dissocia ed asymp o ic limi s. This is a c ucial ea u e o
he p esen in es iga ion, which ocuses on he analysis o
chemical bonds as hey o m and b eak. Addi ional in o ma ion
on GNOF and i s analy ic g adien s can be ound in Appendix.
This pape aims o communica e he easibili y o quasi-
classical ajec o y (QCT) di ec -dynamics simula ions,
suppo ed by GNOF calcula ions o he elec onic s uc u e,
o de e mine he ime e olu ion o he making and b eaking o
chemical bonds in eac i e p ocesses, which en iches he
al eady a ailable in o ma ion abou nuclea dynamics.
9
We
ha e selec ed o such a pu pose he eac ion be ween he
luo ide anion and e hyl chlo ide, an ex ensi ely s udied
canonical case.
10−18
Theo e ical simula ions conduc ed o
da e, along wi h hei compa ison o expe imen al da a, ha e
success ully add essed ene gy- and angle-di e en ial eac i e
sca e ing c oss sec ions and ha e e ec i ely dis inguished
be ween he compe ing nucleophilic subs i u ion (SN2) and
base-induced elimina ion (syn- and an i-E2) eac ions. Addi-
ionally, i has been es ablished ha he eac ion o F−+
CH3CH2Cl is p edominan ly go e ned by di ec dynamics,
excep a low collision ene gies. The an i-E2 mechanism has
eme ged as he dominan mechanism, al hough he SN2 and
syn-E2 mechanisms gain ele ance as he collision ene gy
inc eases.
I is wo h no ing ha he p esen s udy aims no a
eplica ing al eady explo ed aspec s o he i le eac ion bu a
p o iding a be e unde s anding o he ope a ing mechanisms.
This will be achie ed by a ho ough analysis o he ime
e olu ion o he chemically ac i e bonds along selec ed
ajec o ies. Thus, based on he p e ious esul s,
10−18
we will
choose impac pa ame e s and collision ene gies such ha we
can moni o eac i e ajec o ies and obse e he ime
e olu ion o he chemically ac i e na u al o bi als and,
consequen ly, he e olu ion o he chemical bonding s uc u e
along he selec ed eac i e ajec o ies.
Ea lie high-le el elec onic s uc u e calcula ions
19
ha e
iden i ied h ee p ima y eac ion channels: (i) bimolecula
base-induced elimina ion, which p oceeds ia bo h he an i-E2
and syn-E2 pa hways, esul ing in he o ma ion o CH2�CH2
+ HF + Cl−, and (ii) bimolecula nucleophilic subs i u ion
(SN2), whe e he incoming F−a acks he Cαca bon a om
om ei he he back o on side ela i e o Cl, p oducing
CH3CH2F + Cl−. Howe e , he on -side a ack has a oo-high
ene gy ba ie o ∼1.3 eV. Acco dingly, we adjus ed he impac
pa ame e and he di ec ion o he ini ial eloci y ec o o he
F−anion o a o he h ee p ima y mechanisms: (back-side)
SN2, an i-E2, and syn-E2.
All calcula ions we e pe o med using he molecula
dynamics module in eg a ed in o he DoNOF compu e
p og am,
20
which employs Beeman’s algo i hm
21
o nume i-
cally in eg a e he classical New on equa ions-o -mo ion o
he nuclea coo dina es, wi h o ces de e mined on- he- ly as
he g adien o he o al BO ene gy a each nuclea posi ion.
The ini ial condi ions o Beeman’s nume ical in eg a ion a e
selec ed o ep esen he quan um mechanical ib a ional and
o a ional ene gy le els o he eac an s in hei g ound s a e.
Consequen ly, ou calcula ions should be classi ied as quasi-
classical ajec o y (QCT) BO di ec -dynamics simula ions.
3
The equilib ium s uc u e o e hyl chlo ide was de e mined
ia a ull op imiza ion using he GNOF wi h he cc-pVDZ basis
se .
22,23
The molecule was posi ioned a he cen e o he
labo a o y coo dina e sys em, and he ini ial nuclea momen a
we e adjus ed acco ding o he equilib ium molecula no mal
ib a ional modes.
P e ious simula ions ha e iden i ied mul iple di ec and
indi ec mechanisms o SN2/E2 eac ions. I has been
es ablished ha bo h eac ions ansi ion om a highly indi ec
cha ac e o a di ec one as collision ene gy inc eases.
Schema ic po en ial ene gy diag ams o he h ee ele an
channels depic simila double-well po en ial ene gy su aces
(PESs), wi h a ansi ion s a e connec ing he eac an and
p oduc complexes.
15
The co esponding subme ged ba ie s
a e ex emely close, highligh ing he pi o al ole o dynamics in
he compe i ion be ween an i-E2, syn-E2, and SN2 pa hways.
Consequen ly, we selec ed a kine ic ene gy o Ek= 0.2 eV in
he labo a o y coo dina e sys em o F−, which co esponds o
a low collision ene gy o 0.154 eV in he cen e -o -mass
coo dina e sys em, whe e indi ec mechanisms domina e.
The ini ial sepa a ion be ween he luo ide anion and he
cen e -o -mass o e hyl chlo ide was se a 6 Å, and each
ajec o y was in eg a ed un il he sepa a ion be ween he inal
agmen s exceeded 6 Å. A ime s ep o 0.1 s was u ilized. The
po en ial ene gy p o iles o he h ee s udied eac i e
ajec o ies wi h F− ansla ional ene gy Ek= 0.2 eV and
CH3CH2Cl in he g ound s a e can be seen in Figu e 1, while a
mo ie o hese ajec o ies is a ailable in he Suppo ing
In o ma ion. Based on hese da a, we can conclude ha he
eac i e collisions occu oughly wi hin he ime ange o 130−
270 s.
The as luc ua ions o he o al po en ial ene gy o he
h ee men ioned pa hways, shown in Figu e 1 a bo h sho
and long imes, e lec he ene gy oscilla ions o he quasi-
Figu e 1. Po en ial ene gy p o iles o eac i e ajec o ies ela i e o
he ene gy o he eac an s, o F−wi h a ansla ional ene gy o 0.2 eV
and CH3CH2Cl in he g ound s a e. The ime s ep used was 0.1 s.
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12139
ha monic ib a ional modes o he eac an s and p oduc s,
espec i ely. Obse e ha , a in e media e imes, he oscilla ion
pa e n ma kedly di e s om quasi-ha monic, indica i e o
chemical bonding ea angemen being aking place. The
p ecise na u e o he occu ing ea angemen o each
pa hway will be discussed below by inspec ing he ime
e olu ion o he ele an bonding na u al o bi als.
The chemical-bonding s uc u e e olu ion is e lec ed in a
hand ul o alence na u al o bi als. These o bi als, he ea e
e med he “chemically ac i e o bi als”, will be discussed o
each o h ee main eac ion mechanisms. Full mo ies o bo h
he nuclea dynamics and he dynamical e olu ion o he
chemically ac i e o bi als may be ound in he Suppo ing
In o ma ion ideo iles.
SN2 Mechanism. Figu e 2 shows he ime e olu ion o he
chemically ac i e na u al o bi als o he SN2 mechanism o he
F−+ CH3CH2Cl →Cl−+ CH3CH2F eac ion.
No ice ha hese wo o bi als desc ibe he b eaking o he
Cl−Cαbond and he o ma ion o he Cα−F bond. Inspec ion
o hei ime e olu ion e eals ha hese wo p ocesses, Cl−Cα
bond b eaking and Cα−F bond o ma ion, occu synch o-
nously in a single s ep, which is comple ed in ∼100 s ime.
Ocula inspec ion o he ames shown in Figu e 2 sugges s
ha he ansi ion s a e has a li e ime o ∼8 s. No ice ha bo h
he σ(Cl−Cα) and σ(Cα−F) bonds begin hei b eaking and
o ming p ocesses, espec i ely, a = 190 s and inish a =
198 s.
syn-E2 Mechanism. Figu e 3 shows he o bi al ime
e olu ion o he chemically ac i e na u al o bi als o he syn-E2
mechanism o he F−+ CH3CH2Cl →FH + C2H4+ Cl−
eac ion.
He e we can see he ini ial p o on abs ac ion om Cβby
he inciden F−anion as e lec ed in he ime e olu ion o he
wo na u al o bi als shown in he bo om ows o Figu e 3,
be ween = 160 and 194 s. Thus, one can clea ly obse e he
o ma ion o he σ(F−H) bond (as seen in he bo om ow
ames) and he eme gence o a ca banion a Cβ, illus a ed in
he middle ow ames. The end esul is he o ma ion o a
ansien hyd ogen luo ine “s abilized” chlo ina ed ca banion
depic ed in he ame co esponding o = 194 s, whe e hese
h ee salien elemen s o i s elec onic s uc u e a e nea ly
shown. Namely, he op ame shows he σ(Cl−Cα) bond, he
middle ame shows he ClH2Cα−Cβ
⊖H2ca banion, and he
bo om ame shows he σ(H−F) bond.
F om his “c i ical poin ” onwa d, he e olu ion o he
elec on densi y, i.e., he chemical bonding s uc u e, mo es on
om he icini y o Cβ o Cα. Indeed, inspec ion o ames =
200 s −237 s e eals he synch onous o ma ion o he new
π(C−C) bond and he de achmen o he Cl−anion upon
b eaking o he σ(Cl−Cα) bond. This p ocess akes ∼43 s.
The main obse a ion he e is he sequen ial occu ence o
wo p ocesses. The o me cons i u es he p o on abs ac ion
by F− om he H−Cβσ-bond and he gene a ion o a
ca banion a Cβby e en ion o he bonding elec on pai o
he σ(H−Cβ) bond a he Cβca bon a om. The la e p ocess
consis s o he ea angemen o he bonding s uc u e a ound
he Cαca bon o yield a newly o med π(C−C) bond and a
chlo ine anion which akes he elec on pai o he σ(Cl−Cα)
bond as i b eaks apa .
These wo p ocesses a e decoupled in bo h ime- and eal-
space coo dina e domains. They occu a di e en eac ion
imes and in di e en loca ions. All in all, he abo e indings
Figu e 2. Time e olu ion o he wo chemically ac i e na u al o bi als o he SN2 mechanism. The bo om line indica es he ime s amps
( em oseconds) o he selec ed snapsho s. Con ou alue = 0.1 au. The op ow na u al-o bi als’ snapsho s show he adiaba ic e olu ion o he
σ(Cl−Cα) bond o he σ(Cα−F) bond. The bo om ow shows he adiaba ic e olu ion o he F−(2pz) o bi al o he Cl−(3pz) o bi al. The iny
o ange and g een do s ep esen he posi ions o F−and Cl−, espec i ely.
Figu e 3. Time e olu ion o he h ee chemically ac i e na u al o bi als o he syn-2 mechanism. The bo om line indica es he ime s amps ( s) o
he selec ed snapsho s. Con ou alue = 0.1 au. The op ow na u al-o bi als’ snapsho s show he adiaba ic e olu ion o he σ(Cl−Cα) bond o he
π(Cα−Cβ) bond. The middle ow shows he adiaba ic e olu ion o he F−(2pz) o bi al o he Cl−(3pz) o bi al, and he bo om ow shows he
adiaba ic e olu ion o he σ(H−Cβ) bond o he σ(F−H) bond. The iny o ange and g een do s ep esen he posi ions o F−and Cl−, espec i ely.
See ex o u he de ails.
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12140
sugges ha he syn-E2 mechanism esembles a wo-s ep
p ocess whe e bond b eaking and o ma ion p ocesses do no
ake place simul aneously bu sequen ially, opposi e o SN2
( ide sup a).
an i-E2 Mechanism. The inspec ion o he chemically
ac i e na u al o bi als o he an i-E2 mechanism, whose ime
e olu ion is shown in Figu e 4, highligh s no only some
simila i ies wi h syn-E2 bu also a numbe o p o ound and
consequen ial di e ences, which will be discussed below.
Fi s , we obse e ha he eac ion “s a s” ea lie han syn-
E2. The in e ac ion be ween he F− luo ine anion and he
p o on o σ(H−Cβ) is no iceable in he bo om ow ame a
= 140 s, whe eas he onse o his in e ac ion is 170 s o syn-
E2.
Second, i is obse ed ha he p o on abs ac ion akes
longe o comple e. Namely, he “c i ical poin ” is a i ed a =
233 s, which is o be compa ed wi h = 194 s o syn-E2.
Fu he inspec ion o he e olu ion o he na u al o bi al
desc ibing he o ming σ(H−F) bond (see bo om ow ames
om = 130 s o = 233 s) shows a ebound e ec , meaning
ha such an σ(H−F) o bi al e e ses i s de elopmen om =
145 s o = 190 s (see Figu e S1 in he Suppo ing
In o ma ion). Subsequen ly, om = 190 s onwa d he σ(H−
F) bond o ma ion esumes, and a = 233 s i ge s ully
o med and he Cβ−H is b oken. This is e y sugges i e o he
key ole played by he H−Cβs e ching ib a ional mode-
speci ic exci a ion o his mechanism ( ide in a).
Thi d, simila ly o he syn-E2 mechanism, he “c i ical poin ”
ma ks he poin o he e olu ion o he chemical bonding
s uc u e e olu ion o mo e on om he icini y o Cβ o Cα.
F om his c i ical poin onwa d, he synch onous o ma ion o
he π(C−C) bond, a he expense o he delocaliza ion o he
Cβca banion’s elec on pai o e bo h ca bon a oms, which
necessa ily u ns ou in o a π-symme y o bi al due o he
equi ed o hogonali y o he al eady exis ing σ-symme y C−
C bonding o bi al, and he de achmen o he Cl−chlo ine
anion, a he expense o he σ(Cl−Cα) bond b eaking, ake
place in a sho ime o ∼6 s. The di e ence is ha now he
“c i ical poin ” is a i ed a = 233 s, ∼40 s la e han in syn-
E2. This obse a ion sugges s ha an i-E2 shows a la e ba ie
( ide in a).
Ou line. Since he mechanism o a chemical eac ion e e s
o he o de in which chemical bonds a e made and/o
b oken,
4
i ollows na u ally ha he ime e olu ion o he
molecula na u al o bi als is he p ope “place” o look o
chemically insigh ul desc ip ions o eac ion mechanisms.
The p esen s udy epo s on he adiaba ically elaxed
molecula na u al o bi als along selec ed ajec o ies o he
canonical bimolecula (i) nucleophilic subs i u ion and (ii)
base-induced elimina ion eac ion mechanisms o F−+
CH3CH2Cl, e ealing how hese analyses yield signi ican
chemical insigh in o bo h compe ing eac ion mechanisms.
Ou s udy akes ad an age o he BO ab ini io molecula
dynamics whe e he adiaba ic elaxa ion o he elec onic
s uc u e is ca ied ou by GNOF calcula ions. Thus, he nuclei
a e p opaga ed ollowing he classical equa ions o mo ion, and
a each ime s ep, he quan um mechanical elec onic s uc u e
p oblem a he clamped nuclea posi ions is add essed using
GNOF, which has been epo ed o be chemically accu a e a
desc ibing molecula bond o ma ion and b eaking p ocesses.
Consequen ly, ou p ocedu e p o ides a de ailed and eliable
chemical desc ip ion o he ime e olu ion o he chemical
bonding s uc u e o he species in ol ed in he eac ion o
each selec ed mechanism.
We ha e shown ha he SN2 nucleophilic subs i u ion
p oceeds in one single s ep wi h he o ma ion o he σ(F−Cα)
chemical bond and he b eaking o he σ(Cα−Cl) bond aking
place synch onously. The chemical en i onmen a ound he Cβ
a om emains a spec a o du ing he comple ion o he halogen
subs i u ion.
The syn-E2 and he an i-E2 elimina ion mechanisms a e
subs an ially mo e complex because chemical ac ion akes
place on he wo ca bon a oms o CH3CH2Cl in a conce ed
bu sequen ial way. The ime-o de ed chemical e en s include
he abs ac ion o a p o on om a σ(H−Cβ) bond by he
incoming F−anion, and he subsequen o ma ion o a
ansien ca banion a Cβ. Du ing he comple ion o his i s
s ep, he Cαmoie y emains expec an . Nex , he second s ep
consis s o he delocaliza ion o he elec on pai o he Cβ
ca banion in o he Cα, o o m a π- ype co alen bond be ween
he wo ca bon a oms, and he Cl−de achmen om he
σ(Cl−Cα) bond. We obse e, he e o e, wo sequen ial s eps
aking place in his o de a he Cβand in he Cαa oms, which
ea u e wo decoupled bond b eaking/ o ma ion p ocesses
occu ing in di e en ime- and eal-space coo dina e domains,
nea ly di ided by he ba ie epi omized by he hyd ogen
luo ine “s abilized” chlo ina ed ca banion ansien s uc u e.
Fu he mo e, we obse e ha he p o on abs ac ion by he
incoming F−anion om he σ(H−Cβ) bond akes a longe
Figu e 4. Time e olu ion o he h ee chemically ac i e na u al o bi als o he an i-E2 mechanism. The bo om line indica es he ime s amps ( s)
o he selec ed snapsho s. Con ou alue = 0.1 au. The op ow na u al-o bi als’ snapsho s show he adiaba ic e olu ion o he σ(Cl−Cα) bond o
he π(Cα−Cβ) bond. The middle ow shows he adiaba ic e olu ion o he σ(H−Cβ) bond o he Cl−(3pz) o bi al, and he bo om ow shows he
adiaba ic e olu ion o he F−(2pz) o bi al o he σ(F−H) bond. The iny o ange and g een do s ep esen he posi ions o F−and Cl−, espec i ely.
See ex o u he de ails.
The Jou nal o Physical Chemis y Le e s pubs.acs.o g/JPCL Le e
h ps://doi.o g/10.1021/acs.jpcle .4c03010
J. Phys. Chem. Le . 2024, 15, 12138−12143
12141
ime in an i-E2 (233 s) han in syn-E2 (194 s) and exhibi s
“ ebound” e ec s in he o me , in he sense ha he p o on
mo es back and o h be ween Cβand F−. This is e y
sugges i e ha ib a ionally exci ing he H−Cβs e ching
mode should enhance he p o on abs ac ion and, e en ually,
should lead o an i-E2 dominance o e bo h syn-E2 and SN2
mechanisms, as sugges ed elsewhe e.
24,25
The second impo an implica ion o he longe ime aken
by he ini ial p o on abs ac ion in an i-E2, is ha he second
s ep, i.e., he o ma ion o he π(C−C) bond and de achmen
o Cl−, akes place a la e imes. Namely, he “ba ie ” o he
ansi ion om he i s s ep o he second comes a la e imes
in he an i-E2 bond ela i e o he SN2 mechanism. This is in
acco dance wi h he ex ension o he Polanyi ules
26
o
polya omic eac i e p ocesses.
25
Namely, inc easing collision
ene gy less e icien ly ac i a es la e-ba ie mechanisms han
ea ly ba ie mechanisms and ice e sa. Indeed, he
expe imen s
15
o Meye e al. con incingly show ha o he
he ein s udied eac ion, an i-E2 ceases o be he mos
impo an mechanism a inc eased collision ene gies.
■APPENDIX
Le us conside a mixed single s a e o N spin-pai ed elec ons,
speci ically 44 in he F−+ CH3CH2Cl eac ion. The GNOF is
an elec on-pai ing based app oach,
27
so he o bi al space Ωis
di ided in o N/2 mu ually disjoin subspaces Ωg. Each o bi al
exclusi ely belongs o one Ωg, housing one s ongly double-
occupied o bi al gand Ngweakly double-occupied o bi als.
This pa i ioning ensu es ha he sum o he occupa ion
numbe s equals 2 o each subspace, wi h he ace o he one-
pa icle educed densi y ma ix equa ing he o al numbe o
elec ons.
The econs uc ion unc ional o he wo-pa icle educed
densi y ma ix in e ms o he occupa ion numbe s leads o
GNOF, as illus a ed by he equa ion
E E E E E
el
in a
HF
in e
s a
in e
dyn
in e
= + + +
(1)
The in apai componen is o med by summing he ene gies
o elec on pai s, speci ically,
Ä
Ç
Å
Å
Å
Å
Å
Å
Å
Å
Å
Å
Å
Å
Å
É
Ö
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
E n H n n L2 ( , )
in a
g
N
p
p pp
p q
q p pq
1
/2
,
g g
= +
=
(2)
whe e Hpp a e he diagonal one-elec on ma ix elemen s o he
kine ic ene gy and ex e nal po en ial ope a o s, whe eas Lpq =
⟨pp|qq⟩a e he exchange- ime-in e sion in eg als.
28
The ma ix
elemen s Π(nq,np) = c(nq)c(np), whe e c(np) is de ined by he
squa e oo o he occupa ion numbe s acco ding o he
ollowing ule:
l
m
o
o
o
n
o
o
o
c n
n p N
n p N
( )
/2
/2
p
p
p
=
>
(3)
ha is, he phase ac o o c(np) is chosen o be +1 o he
s ongly occupied o bi al o a gi en subspace Ωg, and −1
o he wise. The in e -subspace Ha ee−Fock (HF) e m is
E n n J K(2 )
HF
in e
p q
q p pq pq
,
=
(4)
whe e Jpq =⟨pq|pq⟩and Kpq =⟨pq|qp⟩a e he Coulomb and
exchange in eg als, espec i ely. The p ime in he summa ion
indica es ha only he in e -subspace e ms a e aken in o
accoun . The in e -subspace s a ic componen is w i en as
E L(1 )
s a
in e
p q
q p q p pq
,
b b
=
(5)
whe e
n h
p p p
=
wi h he hole hp= 1−np. Finally, he in e -
subspace dynamic ene gy is
E n n n n L(1 ) ( , )
dyn
in e
p q
q p q
d
p
d
q
d
p
d
pq
,
b b
= [ + ]
(6)
In eqs 5 and 6,Ωbdeno es he subspace composed o o bi als
below he le el N/2, whe eas npdis he dynamic pa o he
occupa ion numbe npin acco dance wi h he Pulay c i e ion
ha es ablishes an occupancy de ia ion o app oxima ely 0.01
wi h espec o 1 o 0 o a na u al o bi al o con ibu e o he
dynamic co ela ion.
In he BO app oxima ion, he o al ene gy is cas as E=Eel +
Enuc, whe e Eel ep esen s he elec onic ene gy calcula ed using
(1), and Enuc is he nuclea ene gy
E Z Z R( / )
A B A Bnuc AB
=<
.
He e, ZAdeno es he a omic numbe o nucleus A, and RAB is
he dis ance be ween nuclei Aand B.
Conside ing ha all na u al o bi als a e expanded in a ixed
a omic basis se , he de i a i e o he o al ene gy wi h espec
o he coo dina e xo nucleus Ais gi en by
29
E
x
E
x
H
x
D
x
S
x
d
dA
nuc
A A A
A
= + + |
(7)
whe e Γμυ and Dμηυδ a e he GNOF one-pa icle and wo-
pa icle educed densi y ma ices, espec i ely, Sμυ =⟨μ|υ⟩is
he o e lap ma ix, and λμυ a e ob ained om he educed
densi y ma ices, all in he a omic o bi al ep esen a ion. The
de i a i es in eq 7 ha e an explici dependence on he nuclea
coo dina e xA, so he o ce ac ing on each nucleus A(FA=
−∇AE) can be ob ained by a single s a ic e alua ion a each
ime s ep o he ixed nuclea posi ions a ha ins an .
■ASSOCIATED CONTENT
*
sı Suppo ing In o ma ion
The Suppo ing In o ma ion is a ailable ee o cha ge a
h ps://pubs.acs.o g/doi/10.1021/acs.jpcle .4c03010.
SN2 eac ion (MP4)
syn-E2 eac ion (MP4)
an i-E2 eac ion (MP4)
Figu e o in e a omic dis ances e sus ime (PDF)
T anspa en Pee Re iew epo a ailable (PDF)
■AUTHOR INFORMATION
Co esponding Au ho
Ma io Pi is −Donos ia In e na ional Physics Cen e (DIPC)
&Kimika Fakul a ea, Euskal He iko Unibe si a ea (UPV/
EHU), 20018 Donos ia, Euskadi, Spain; Basque Founda ion
o Science (IKERBASQUE), 48009 Bilbao, Euskadi, Spain;
o cid.o g/0000-0003-0222-2953; Email: ma io.pi is@
ehu.eus
The Jou nal o Physical Chemis y Le e s pubs.acs.o g/JPCL Le e
h ps://doi.o g/10.1021/acs.jpcle .4c03010
J. Phys. Chem. Le . 2024, 15, 12138−12143
12142
Au ho s
Xabie Lopez −Donos ia In e na ional Physics Cen e
(DIPC) &Kimika Fakul a ea, Euskal He iko
Unibe si a ea (UPV/EHU), 20018 Donos ia, Euskadi,
Spain; o cid.o g/0000-0002-2711-3588
Jesus M. Ugalde −Donos ia In e na ional Physics Cen e
(DIPC) &Kimika Fakul a ea, Euskal He iko
Unibe si a ea (UPV/EHU), 20018 Donos ia, Euskadi,
Spain; o cid.o g/0000-0001-8980-9751
Comple e con ac in o ma ion is a ailable a :
h ps://pubs.acs.o g/10.1021/acs.jpcle .4c03010
No es
The au ho s decla e no compe ing inancial in e es .
■ACKNOWLEDGMENTS
Financial suppo comes om he Eusko Jau la i za (Basque
Go e nmen ), e .: IT1584-22 and om he G an No. PID
2021-126714NB-I00, unded by MCIN/AEI/10.13039/
501100011033. The au ho s a e hank ul o echnical and
human suppo p o ided by IZO-SGI SGIke o UPV/EHU
and DIPC.
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■NOTE ADDED AFTER ASAP PUBLICATION
This pape was published ASAP on No embe 28, 2024, wi h
an e o in he cap ion o Figu e 4. This was co ec ed in he
e sion published ASAP on Decembe 2, 2024.
The Jou nal o Physical Chemis y Le e s pubs.acs.o g/JPCL Le e
h ps://doi.o g/10.1021/acs.jpcle .4c03010
J. Phys. Chem. Le . 2024, 15, 12138−12143
12143
