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Mol
ecul
arQ
uant
umD
ynam
ics
Lect
ure
byR.M
arqu
ardt
–U
nive
rsité
deSt
rasb
ourg
–20
19–
180°
118.6
°
108.9
°
102
122.5
°
φ 1θ 2
r3
/ pm
70°
290°
1242 fs
145°
65°
90
135
1240 fs
1238 fs
hc
cm
−1
500
hc
cm
−1
4500
hc
cm
−1
2500
-6-4
-2 0
2 4
6
0 1 2 3 4 5 6
-6-4
-2 0
2 4
6
0 1 2 3 4 5 6
-6-4
-2 0
2 4
6
0 1 2 3 4 5 6
-6-4
-2 0
2 4
6
0 1 2 3 4 5 6
S 0
fs
440
fs
260
fs
600
fs
Pre
face
Thi
sye
ar’s
lect
ure
isin
tend
edto
bea
very
brie
fin
trod
uction
toes
sent
ialas
pect
sre
gard
ing
the
prim
ary
step
sof
chem
ical
reac
tion
s.It
also
intr
oduc
esst
uden
tsto
elem
enta
rym
etho
dsan
dco
mpu
tation
alal
gorit
hms
used
inth
eth
eore
tica
ltr
eatm
ent
ofm
olec
ular
quan
tum
dy-
nam
ics,
with
apa
rtic
ular
focu
son
the
tim
epr
opag
atio
nof
wav
epa
cket
s.T
hebo
oked
ited
byYea
zell
and
Uze
r[1
]ha
sco
ntrib
utio
nsfrom
man
ygr
oups
wor
king
onwav
epa
cket
dyna
mic
s.O
ther
book
sby
Tan
nor[2
],an
dGat
tian
dco
-aut
hors
[3–5
]co
nvey
addi
tion
alas
pect
sof
tim
ede
pend
ent
quan
tum
dyna
mic
s,th
eun
derly
ing
met
hods
and
mod
ern
appl
icat
ions
.Fi
nally
,th
ebo
oked
ited
byD
omck
e,Yar
kony
and
Köp
pel[
6]co
ntai
nsa
set
ofex
celle
ntco
ntrib
utio
nson
elec
tron
and
nucl
ear
dyna
mic
s,po
tent
iale
nerg
ysu
rfac
esan
dno
n-ad
iaba
tic
inte
ract
ions
.
Bib
liogr
aphy
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zell
and
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zer.
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iley,
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.J.
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oduc
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toQ
uant
umM
echa
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:A
tim
ede
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vers
itySc
ienc
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(Cal
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eyer
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orth
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uant
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ynam
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einh
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uant
umD
ynam
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omck
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ith
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ereo
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.
Tab
leof
cont
ents
1In
trod
uction
1
1.1
Histo
rical
aspe
cts
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.1
1.1.
1Fem
toch
emistr
y.
..
..
..
..
..
..
..
..
..
..
..
..
..
.1
1.1.
2M
olec
ular
Qua
ntum
Dyn
amic
s.
..
..
..
..
..
..
..
..
..
.6
1.2
One
exam
ple
ofa
typi
cala
pplic
atio
n.
..
..
..
..
..
..
..
..
..
..
12
2N
umer
ical
Met
hods
:Sol
ving
the
Tim
e-D
epen
dent
Sch
rödi
nger
Equ
atio
n13
2.1
Clo
sed
syst
ems
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
15
2.1.
1For
mal
inte
grat
ion
..
..
..
..
..
..
..
..
..
..
..
..
..
.15
2.1.
2Pra
gmat
ical
inte
grat
ion
inth
esp
ectr
alre
pres
enta
tion
..
..
..
..
.16
2.1.
3N
atur
alin
tegr
atio
n.
..
..
..
..
..
..
..
..
..
..
..
..
.21
2.1.
4Pra
gmat
ical
inte
grat
ion
inan
yba
sis
repr
esen
tation
:so
lution
bydi
agon
aliz
atio
n22
2.1.
5D
irec
tso
lution
byiter
atio
n(1
stan
d2n
dor
der)
..
..
..
..
..
..
30
2.2
Ope
nsy
stem
s.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
34
2.3
Mol
ecul
e-ra
diat
ion
inte
ract
ion
..
..
..
..
..
..
..
..
..
..
..
..
.35
2.3.
1T
heFlo
quet
-Lia
puno
ffm
etho
d.
..
..
..
..
..
..
..
..
..
.39
2.3.
2T
hequ
asi-re
sona
ntap
prox
imat
ion
for
period
icpr
oble
ms
..
..
..
.40
3Pot
ential
ener
gysu
rfac
es45
3.1
Brie
fhi
stor
ical
rem
arks
..
..
..
..
..
..
..
..
..
..
..
..
..
..
45
3.2
PES
from
“exp
erim
ent”
..
..
..
..
..
..
..
..
..
..
..
..
..
..
48
3.3
PES
from
“the
ory”
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
52
3.3.
1Adi
abat
icap
prox
imat
ion
..
..
..
..
..
..
..
..
..
..
..
..
53
3.3.
2Ste
p1:
“ele
ctro
nic
stru
ctur
e”,ad
iaba
tic
pote
ntia
len
ergy
surfac
es.
..
55
3.3.
3Ste
p2:
“vib
ration
alst
ruct
ure”
..
..
..
..
..
..
..
..
..
..
.59
3.3.
4Bor
n-O
ppen
heim
erex
pans
ion,
adia
batic
basis
..
..
..
..
..
..
.60
3.4
Non
-adi
abat
iceff
ects
,di
abat
icpo
tent
iale
nerg
ysu
rfac
es.
..
..
..
..
..
62
3.4.
1N
on-a
diab
atic
coup
ling
mat
rix
..
..
..
..
..
..
..
..
..
..
62
3.4.
2D
iaba
tic
base
s.
..
..
..
..
..
..
..
..
..
..
..
..
..
.66
3.4.
3D
iaba
tic
pote
ntia
len
ergy
surfac
es.
..
..
..
..
..
..
..
..
..
70
3.5
Exa
mpl
es.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
73
3.5.
1Am
mon
iadi
ssoc
iation
..
..
..
..
..
..
..
..
..
..
..
..
.73
3.5.
2M
etha
nest
ereo
mut
atio
npo
tent
ial
..
..
..
..
..
..
..
..
..
75
3.5.
3Vib
ration
alte
rmva
lues
inhy
drog
enflu
orid
e.
..
..
..
..
..
..
.76
AD
iago
naliz
atio
nof
a2×2
sym
met
ric
(her
mitia
n)m
atrix
77
A.1
Det
erm
inat
ion
ofth
eei
genv
alue
s(λ
1an
dλ2)
..
..
..
..
..
..
..
..
.77
A.2
Det
erm
inat
ion
ofth
eei
genv
ecto
rbe
long
ing
toλ1
..
..
..
..
..
..
..
.79
A.3
Det
erm
inat
ion
ofth
eei
genv
ecto
rbe
long
ing
toλ2
..
..
..
..
..
..
..
.84
A.4
Bas
istr
ansf
orm
atio
nm
atrix
..
..
..
..
..
..
..
..
..
..
..
..
..
88
BT
ight
bind
ing
ham
ilton
ian
89
B.1
Pro
of:
eige
nval
ues
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
91
B.2
Pro
of:
eige
nvec
tors
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
92
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t1
1In
trod
uction
1.1
Histo
rical
aspe
cts
Mol
ecul
arqu
antu
mdy
nam
ics
isof
ten
invo
ked
inco
nnec
tion
with
“fem
toch
emistr
y”,al
thou
ghth
isco
nnec
tion
isno
tm
anda
tory
and
non
hist
oric
al,we
intr
oduc
eth
isco
ncep
tfir
st.
1.1.
1Fem
toch
emistr
y
“Fem
toch
emistr
y”is
the
stud
yof
chem
ical
reac
tion
sat
real
tim
e.M
otiv
atio
n:W
hat
isth
eel
emen
tary
chem
ical
reac
tion
act?
Our
pres
ent
know
ledg
eis
base
don
:
A.ki
netic
data
B.tim
ere
solv
ed
C.tim
ein
depen
dent
spec
tros
copi
cda
ta
Histo
ry
µs
1950
Las
er19
60
ns19
66
ps19
70
fs19
85
as20
04
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t2
A.K
inet
ics
A+B→
AB
✲⑥
♠
∆r≈
10−10
m
AB
v rel
v rel
=√
8kT
πµ
µ≈
1uT
=300K
⇒v rel≈
2500m s
⇒∆t=
∆r
v rel≈
40·1
0−15
s
1fs
=10−15
s
Alter
native
estim
atio
nof
the
rele
vant
tim
esc
ales
from
:
-de
cay
cons
tant
sof
phot
oind
uced
diss
ocia
tion
s-flu
ores
cenc
equ
antu
myi
elds
-ab
sorp
tion
cros
sse
ctio
ns
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t3
B.T
ime
reso
lved
fs-s
pect
rosc
opy
”en
rout
e”ob
serv
atio
nof
the
chem
ical
reac
tion
act
(Joh
nPol
anyi
,19
79)
❀“t
rans
itio
nst
ate
spec
tros
copy
”(b
ette
rsp
ectr
osco
pyof
the
tran
stio
nst
ruct
ure
)
”che
mistr
yas
itac
tual
lyha
ppen
s”(R
icha
rdBer
nste
in)
Ess
ential
idea
:To
laun
cha
chem
ical
reac
tion
with
aph
ysic
alpe
rtur
bation
onth
etim
esc
ale
ofa
few
fem
-to
seco
nds
( zer
otim
etr
igge
r).
Pos
sibl
esinc
ear
ound
1985
byul
tras
hort
(fs-
)La
ser
pulses
.
The
oret
ical
unde
rsta
ndin
g:
Cal
cula
tion
ofth
ewav
epa
cket
dyna
mic
sre
late
dto
the
mot
ion
ofth
enu
lcei
LIF
r
V(r)
t
Exp
erim
enta
lpro
cedu
re:
pum
p︸︷︷︸
laun
ch
&pr
obe
︸︷︷︸
obse
rve
tim
ede
pend
ent
sign
alS(∆t)
-as
ympt
otic
ally
cons
tant
(rel
axat
ion
)-os
cilla
tory
(coh
eren
ce)
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t4
Exp
erim
enta
lset
up(s
chem
atic
ally
)
10-100 fs
t
Laser
Reaktions-
zelle
c.
1 2
Anregung (pump)
Nachweis (probe)
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t5
C.T
ime
inde
pend
ent
spec
tros
copy
Spec
tros
copi
cst
atesE
1,E
2,E
3,...
ν/cm
−1
Spec
trum
ofCHI 3
-vap
orfrom
Mar
quar
dtet
al,
J.Chem
.Phys
.103,83
91(1
995)
Supe
rpos
itio
n⇒
tim
ede
pend
ent
dyna
mic
s
Exa
mpl
e:
ψ(t,x)=c 1ψ1(x)exp(−
iE1t
2πh
)+c 2ψ2(x)exp(−
iE2t
2πh
)
∆E hc
≈1000
−3000
cm−1⇒
perio
dτ=
h ∆E
≈30
-10
fs
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t6
1.1.
2M
olec
ular
Qua
ntum
Dyn
amic
s
“Mol
ecul
arqu
antu
mdy
nam
ics”
iscu
rren
tly
bein
gus
edto
defin
eth
ere
sear
chfie
ldin
volv
ing
the
stud
yof
mol
ecul
arst
ruct
ure
and
its
tim
eev
olut
ion
inth
ere
alm
ofqu
antu
mm
echa
nics
.
Qui
teof
ten,
mol
ecul
arqu
antu
mdy
nam
icsde
alsw
ith
solu
tion
sof
the
tim
ede
pend
entSc
hröd
inge
req
uation
forth
enu
clea
rm
otio
nin
the
Bor
n-O
ppen
heim
er(a
diab
atic
)ap
prox
imat
ion,
ih 2π
∂ ∂tΨ
(t;...,r
j,...)=H
nuclΨ(t;...,r
j,...).
(1.1
)
Ψan
dH
nucl
are
the
wav
efu
nction
and
the
ham
ilton
ian
oper
ator
(the
ham
ilton
ian)
for
the
nucl
ear
mot
ion,
resp
ective
ly.
Hnucl
may
begi
ven
by Hnucl=−1 2
(h 2π
)2∑
j
∇2 j
mj
+Vne(...,rjk,...).
(1.2
)
InEq.
( 1.1
),h≈
6.62610
−34
Jsis
Pla
nck’
sco
nsta
ntan
diis
the
imag
inar
yun
it.
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t7
InEq.
(1.2
),th
eha
milt
onia
nfo
rth
enu
clea
rm
otio
nis
repr
esen
ted
ina
coor
dina
tesp
ace
span
ned
by(c
arte
sian
)nu
clea
rpo
sition
vect
orsrj
with
mas
sesmj;Vne
isth
epo
tent
ial
ener
gy(h
yper
-)su
rfac
ein
the
elec
tron
icst
ate
ofin
dexne,
whi
chis
basica
llya
func
tion
ofth
ein
tern
ucle
ardi
stan
cesr jk=|r
j−rk|.
Oth
erco
ordi
nate
syst
ems
exist
and
the
choi
ceof
aco
ordi
nate
syst
emfo
ran
appr
opria
tede
scrip
tion
ofm
olec
ular
stru
ctur
ean
dits
tim
eev
olut
ion
inco
nfigu
ration
spac
eis
cruc
ial,
not
only
for
tech
nica
lre
ason
s,bu
tal
sofo
rth
esa
keof
obta
inin
ga
prop
erin
terp
reta
tion
ofth
eth
eore
tica
lres
ults
.Coo
rdin
ate
syst
ems
will
bead
dres
sed
late
rin
this
lect
ure.
The
defin
itio
nsp
ace
ofm
olec
ular
stru
ctur
esis
calle
dco
nfigu
ration
spac
ein
mol
ecul
ardy
nam
-ic
s.
Che
mic
alre
action
sco
ncer
nth
etr
ansf
orm
atio
nof
mat
terby
rear
rang
emen
ts,lo
sses
and
gain
sof
atom
sin
mol
ecul
es,pa
rtic
les
forw
hich
the
law
sof
quan
tum
mec
hani
csap
ply
stric
tly.
Mol
ecul
arst
ruct
ure
and
dyna
mic
sis
thus
stro
ngly
rela
ted
toch
emic
alre
action
dyna
mic
s.
Withi
nth
epi
ctur
eth
atel
ectr
onsfo
llow
adia
batica
llyth
em
otio
nof
the
nucl
eidu
ring
ach
emic
alre
action
,m
olec
ular
quan
tum
dyna
mic
s,as
decr
ibed
abov
e,is
quite
anap
prop
riate
tool
for
unde
rsta
ndin
gex
perim
enta
ldat
afrom
mol
ecul
arsp
ectr
osco
py,b
oth
tim
ede
pend
entan
dtim
ein
depe
nden
t,sc
atte
ring
expe
rimen
tsan
dch
emic
alki
netics
.
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t8
Using
the
law
sof
quan
tum
mec
hani
csfo
rde
scrib
ing
chem
ical
reac
tion
dyna
mic
sis
not
easy
,de
spite
the
fact
that
Eq.
(1.1
)is
linea
r.T
his
fortw
ore
ason
s:
1.re
alistic
mod
els
requ
irehi
ghdi
men
sion
allin
ear
spac
es;
2.th
ean
alys
isof
the
theo
retica
lres
ults
invi
ewof
gain
ing
poss
ible
inte
rpre
tation
sof
mol
ecul
arst
ruct
ure
and
its
tim
eev
olut
ion
isco
mpl
icat
ed.
Eq.
(1.1
)is
asim
plifi
edve
rsio
nof
ase
tof
differ
ential
equa
tion
sth
atha
sac
tual
lyto
beso
lved
whe
nse
vera
lele
ctro
nic
stat
espa
rtic
ipat
eat
the
defin
itio
nof
the
mol
ecul
arst
ate.
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t9
Gen
eral
theo
retica
lap
proa
chto
nucl
ear
(fem
tose
cond
)dy
nam
ics
Bor
n-H
uang
(Dyn
amic
alT
heor
yof
Cry
stal
Latt
ices
,O
xfor
d,19
54):
Bor
n-O
ppen
heim
erex
pans
ion
Ψ(t,x
(n) ,y(e) )
=∑
k
ψ(n)
k(t,x
(n) )ψ(e)
k(y
(e) ;
x(n) )
Red
uced
prob
abili
tyde
nsity:
P(t,x
(n) )
=
∫
dy(e)|Ψ
(t,x
(n) ,y(e) )|2
=∑
k
|ψ(n)
k(t,x
(n) )|2
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t10
Nuc
lear
dyna
mic
son
mul
tipl
epo
tent
ialen
ergy
surfac
es
ih 2π
∂ ∂t
ψ(n)
1(t;x
(n) )
ψ(n)
2(t;x
(n) )
. . .
=
H(n)
11H
(n)
12···
H(n)
21H
(n)
22···
. . .. . .
. ..
ψ(n)
1(t;x
(n) )
ψ(n)
2(t;x
(n) )
. . .
H(n)
ik=<ψ(e)
i|H
|ψ(e)
k>
=
{
H(n)
kk
=T(x
(n) ,∂x(n))+Vk(x
(n) )
(i=k)
H(n)
ik(x
(n) ,∂x(n))
≈0
(i6=k)
րBor
n-O
ppen
heim
erad
iaba
tic
appr
oxim
atio
n
Boo
kon
non-
Bor
n-O
ppen
heim
erdy
nam
ics:
“Con
ical
Inte
rsec
tion
s”,D
omck
e,Yar
kony
,Köp
pel,
Wor
ldSc
ient
ific,
Lond
on,20
04[ 6
]
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t11
Mol
ecul
arqu
antu
mdy
nam
ics
isno
tye
ta
prop
erto
olfo
rth
ein
vest
igat
ion
ofla
rge
and
very
larg
em
olec
ules
such
asbi
olog
ical
lyre
leva
ntm
olec
ules
.M
olec
ular
dyna
mic
sca
nbe
stud
ied
withi
ncl
assica
lm
echa
nics
for
the
mot
ion
ofth
enu
clei
usin
gpot
ential
ener
gysu
rfac
esth
atar
eob
tain
edfrom
quan
tum
mec
hani
csin
the
Bor
n-O
ppen
heim
erap
prox
imat
ion.
Prim
ary
resu
lts
ofth
ese
met
hods
are
clas
sica
ltr
ajec
tories
ofth
enu
clei
,
whi
chca
nbe
used
withi
nst
atistica
lth
eories
for
the
sim
ulat
ion
ofqu
antu
mre
sults.
The
quan
tum
-cla
ssic
al
com
pariso
nis
expec
ted
tobe
poo
rfo
rlig
htnu
clei
.
Hyb
rid
clas
sica
l/qu
antu
m(“
sem
i-cl
assica
l”)
calc
ulat
ions
wer
ein
trod
uced
byH
elle
r[7
].
For
very
larg
esy
stem
s,“o
n-th
e-fly”
met
hods
ofm
olec
ular
dyna
mic
sha
vebee
nde
velo
pped
.T
heby
far
mos
t
succ
essf
ulm
etho
dwas
intr
oduc
edby
Car
and
Par
rine
llo[8
],in
whi
chth
epot
ential
ener
gysu
rfac
eis
dete
rmin
ed
from
DFT
calc
ulat
ions
atea
chpoi
ntof
acl
assica
ltr
ajec
tory
.
Ofal
lmet
hods
,on
lyth
eso
lution
ofEq.
(1.1
)co
rres
pon
dstr
uly
tom
olec
ular
quan
tum
dyna
mic
s.T
heco
mpl
ete
anal
ysis
ofth
etim
ede
pen
dent
wav
efu
nction
Ψ(t;...,r
j,...)
reve
als
the
tota
lin
form
atio
nre
leva
ntfo
rth
e
inte
rpre
tation
ofsp
ectr
osco
pic
and
kine
tic
data
.
Inde
ed,w
ith
the
upco
me
ofth
efirs
tfe
mto
seco
ndla
serpu
lses
and
the
cons
ider
able
impr
ovem
ents
inco
mpu
ta-
tion
alte
chno
logy
inth
e80
ies
ofth
ela
stce
ntur
y,m
olec
ular
quan
tum
dyna
mic
sha
sal
lowed
tofina
llyad
dres
s
the
cent
ralqu
estion
ofch
emic
alki
netics
:W
hat
isth
eel
emen
tary
chem
ical
act?
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t12
1.2
One
exam
ple
ofa
typi
cala
pplic
atio
n
Thi
sis
am
odel
mec
hani
smfo
rth
eco
ntro
lof
ast
ereo
mu-
tation
reac
tion
[9]fo
llow
ing
the
pum
p-du
mp-
sche
me
(Ric
e
&Tan
nor)
.Rad
iation
coup
ling
bet
wee
nel
ectr
onic
stat
esis
mod
elle
dw
ithi
nth
eFra
nck-
Con
don
appr
oxim
atio
n.
At
tim
et=
0,th
esy
stem
ison
the
left
hand
side
inth
e
elec
tron
icgr
ound
stat
e.At
tim
et 0
pop
ulat
ion
tran
sfer
into
the
exci
ted
elec
tron
icst
ate
isal
mos
tco
mpl
ete;
mod
el
para
met
ers
are
such
thatt 0≈
1fs
.
Fig
ure
1.1
:M
odel
pot
ential
sfo
ra
lase
rin
duce
dst
ereo
-
mut
atio
nre
action
.
Lef
t-ha
ndside
:Pot
ential
surfac
es,
vibr
atio
nal
leve
lsan
d
wav
epa
cket
satt=0
andt=t 0
.
Rig
ht-h
and
side
:Pop
ulat
ion
dist
ribu
tion
amon
gvi
brat
iona
l
leve
lsin
the
grou
ndan
dex
cite
del
ectr
onic
stat
esat
tim
et 0
;
exci
tation
resu
lt.
Bot
tom
:T
ime
evol
utio
nof
the
wav
epa
cket
fort≥t 0
;
mot
ion
esse
ntia
llyin
elec
tron
ical
lyex
cite
dst
ate.
t
r
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t13
2N
umer
ical
Met
hods
:Sol
ving
the
Tim
e-D
epen
dent
Sch
rödi
nger
Equ
atio
n
Solu
tion
sto
Eq.
(1.1
),at
tim
et 2
,m
aybe
give
nin
the
form
Ψ(t
2;...,r
j,...)=U(t
2,t
1)Ψ(t
1;...,r
j,...)
(2.1
)
for
the
wav
efu
nction
,in
case
ofco
here
nt,
pure
stat
edy
nam
ics,
i.e.
for
know
nph
ases
and
popu
lation
sat
anin
itia
ltim
et 1
.
Whe
nin
form
atio
nis
avai
labl
eon
the
popu
lation
dist
ribut
ion
only,at
tim
et 1
,i.e
.in
case
ofm
ixed
stat
edy
nam
ics,
solu
tion
sm
aybe
give
nin
form
ofth
eLi
ouvi
lle-v
on-N
eum
ann
equa
tion
ρ(t
2;...,r
j,...;...,r′ j,...)=U(t
2,t
1)ρ(t
1;...,r
j,...;...,r′ j,...)U
† (t 1,t
2),
(2.2
)
whe
reρ(t
1;...,r
j,...;...,r
′ j,...)
isth
ede
nsity
mat
rix,u
pon
aver
agin
gov
erra
ndom
initia
lph
ases
.
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t14
Uis
the
quan
tum
mec
hani
calt
ime
prop
agat
orop
erat
oran
dU
†itsad
join
tfo
rm.
With
resp
ect
toEq.
(1.1
),i.e
.in
mol
ecul
arqu
antu
mdy
nam
ics,U
isgi
ven
asth
eso
lution
ofth
eeq
uation
ih 2π
∂ ∂tU
(t,t
0)=H
nuclU(t,t
0),
(2.3
)
withU(t
0,t
0)=1.
We
shal
lus
et
tore
pres
ent
the
tim
eco
ordi
nate
,an
dr=
(...,r
j,...)
asa
shor
tcut
repr
e-se
ntat
ion
for
the
(mul
tidi
men
sion
al)
confi
gura
tion
spac
e.
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t15
2.1
Clo
sed
syst
ems
2.1.
1For
mal
inte
grat
ion
For
clos
edsy
stem
s,en
ergy
isa
cons
erve
dqu
antity
and
the
ham
ilton
ian
does
not
depe
ndex
plic
itel
yon
tim
e:H
6=H(t).
The
quan
tum
mec
hani
calt
ime
prop
agat
oris
then
give
nby
U(t,t
0)=exp
(
−i2π
H(t−t 0)
h
)
=∞∑ k=0
1 k!
(
−i2π
H(t−t 0)
h
)k.
(2.4
)
Pro
pert
ies:
-fo
rhe
rmitia
nha
milt
onia
nop
erat
ors,H
=H
† ,th
etim
epr
opag
ator
oper
ator
isun
itar
y,an
dvi
ce-v
ersa
:
H=H
†⇐⇒
U† (t 2,t
1)=U
−1(t
2,t
1)=U(t
1,t
2);
(2.5
)
-th
etim
epr
opag
ator
oper
ator
defin
esa
grou
p(a
Lie
grou
p,ac
tual
ly):
U(t
3,t
1)=U(t
3,t
2)U(t
2,t
1).
(2.6
)
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t16
2.1.
2Pra
gmat
ical
inte
grat
ion
inth
esp
ectr
alre
pres
enta
tion
To
solv
eth
eSc
hröd
inge
req
uation
,Eq.
(1.1
),on
ene
eds
two
action
s:fir
stly,de
fine
the
initia
lco
nditio
n,i.e
.Ψ(t
0,r),
atso
me
initia
ltim
et 0
;se
cond
,ca
lcul
ate
the
prop
agat
orU(t,t
0).
Inth
isan
dth
ene
xtse
ctio
nwe
show
how
toob
tain
prag
mat
icfo
rmul
aefo
rbo
thac
tion
s.
The
Schr
ödin
gere
quat
ion
islin
ear
.T
hism
eans
that
,ifχ
1(t,r)an
dχ2(t,r)ar
etw
oso
lution
sof
Eq.
(1.1
),so
willc 1χ1(t,r)+c 2χ2(t,r)al
sobe
aso
lution
,whe
rec 1
andc 2
are
any
cons
tant
(com
plex
)nu
mbe
rs.
Forcl
osed
syst
ems,
the
Schr
ödin
gereq
uation
,Eq.
(1.1
),is
sepa
rabl
ein
tim
ean
dsp
ace.
Thi
sm
eans
that
,th
ere
are
spec
ial
solu
tion
sto
it,
whi
chw
illha
veth
epr
oduc
tfo
rmχ(t,r)=c(t)f(r).
Witho
utlo
ssof
gene
ralit
y,on
em
ayas
sum
ec(t 0)=
1,he
ncef(r)=
χ(t
0,r).
Forsim
plic
ity,in
the
follo
win
gw
hene
ver
we
writ
eχ(r)
we
mea
nχ(t
0,r).
Inse
rtio
nof
the
prod
uct
form
ansa
tzin
toEq.
(1.1
)yi
elds
ihdc(t)
dt/c(t)=Hχ(r)/χ(r).
Div
isio
nby
bothc(t)
andχ(r)
isal
lowed
,asχ(t,r)
mus
tno
tva
nish
.Le
ftan
drig
htha
ndside
ofth
iseq
uation
are
inde
pend
entof
one
anot
her,
soth
eym
ustbe
cons
tant
,say
,eq
ualE
.
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t17
Hen
ceon
ede
duce
s:
Hχ(r)
=Eχ(r)
(2.7
)
and
id dtc(t)
=2π h
Ec(t)
(2.8
)
Eq.
(2.7
)is
anei
genv
alue
equa
tion
,whe
reE
isth
eei
genv
alue
,andχ(r)is
the
eige
nfun
ctio
n.O
nly
ina
few
case
s,Eq.
(2.7
)ca
nbe
solv
edan
alyt
ical
ly.
Nor
mal
ly,it
isso
lved
bynu
mer
ical
diag
onal
izat
ion,
ona
com
pute
r,as
disc
usse
din
one
ofth
efo
llow
ing
sect
ions
.
The
solu
tion
ofth
ese
cond
equa
tion
istr
ivia
l:
c(t)=c(t 0)exp
(
−i2π hE
(t−t 0))
(2.9
)
whe
rec(t 0)=
1in
the
spec
ialca
seas
sum
edab
ove,
but
gene
rally
this
cons
tant
coeffi
cien
tca
nta
keot
herva
lues
.
Aha
milt
onia
nde
fined
ina
clos
edsy
stem
isa
self-
adjo
int
oper
ator
.Su
chop
erat
ors
may
have
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t18
adi
scre
tesp
ectr
umE
1,E
2,...
ofei
genv
alue
san
dco
rres
pond
ingχ1,χ
2,...
eige
nfun
ctio
ns.
The
eige
nval
ueis
the
ener
gyof
the
(abs
trac
t)ei
gens
tate
ofth
esy
stem
,the
eige
nfun
ctio
nth
ere
pres
enta
tion
ofth
eei
gens
tate
inco
nfigu
ration
spac
e.
Bec
ause
for
prac
tica
lpu
rpos
es,
i.e.
for
solv
ing
Eq.
(1.1
)on
aco
mpu
ter,
adi
scre
tiza
tion
isne
cess
ary,
som
ech
oice
ofa
coun
tabl
e,in
prac
tice
finite
set
ofba
sis
func
tion
sm
ust
bem
ade.
Form
ally
we
may
call
the
(fini
te,b
utpo
ssib
lyve
ryla
rge)
subs
et{χ
1,..,χN}
ofei
genf
unct
ions
anei
genf
unct
ion
basis.
Due
toits
linea
rity,
the
gene
rals
olut
ion
ofEq.
(1.1
)ca
nbe
give
nin
the
form
Ψ(t,r)=∑
m
c m(t
0)exp(−
iωm(t−t 0))χm(r)
(2.1
0)
if
Ψ(t
0,r)=∑
m
c m(t
0)χm(r)
(2.1
1)
whe
reωm
=2πEm/h
.Eq.
(2.1
0)is
calle
dth
eei
gens
tate
repr
esen
tation
ofth
eso
lution
,m
ore
wid
ely
know
nas
wav
epac
ket
.
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t19
The
initia
lco
nditio
n,Eq.
(2.1
1),
isre
pres
ente
dby
the
vect
orc(t
0)=
(c1(t),c 2(t),...)T.
The
sym
bolT
mea
ns“t
rans
pose
d”,i.e
.c
isa
colu
mn
vect
or.
The
vect
orc
isca
lled
the
stat
eve
ctor
inth
eei
genf
unct
ion
basis.
Letth
em
atrixU(t,t
0)be
defin
edby
the
elem
entsUnm(t,t
0)=δ nmexp(−
iωn(t−t 0))
.T
his
mat
rixis
diag
onal
;it
isth
ere
pres
enta
tion
ofth
epr
opag
ator
inth
eei
genf
unct
ion
basis.
Ifon
lyth
ele
ftha
ndside
ofEq.
(2.1
1)is
know
n,on
em
ayca
lcul
ate
the
coeffi
cien
tsc m
(t0)
bypr
ojec
tion
ofth
efu
nction
Ψ(t
0,r)
onth
eba
sis
ofei
genf
unct
ions
.T
his
ispe
rfor
med
bym
ultipl
icat
ion
from
the
left
ofEq.
(2.1
1)by
asp
ecifi
cei
genf
unct
ion
func
tion
χm
and
subs
eque
ntin
tegr
atio
nov
erth
ede
finitio
nsp
ace
ofth
efu
nction
s:
〈χm|Ψ(t
0)〉
=N∑
n
c n(t
0)〈χ
m|χ
n〉
︸︷︷︸
δ nm
=c m
(t0)
(2.1
2)
Her
e,we
used
the
fact
that
eige
nfun
ctio
nsca
nal
way
sbe
chos
ensu
chth
atth
eyfo
rman
orth
onor
mal
set,
i.e.〈χ
n|χ
m〉=
δ nm
.
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t20
The
mea
ning
ofth
esy
mbo
l〈f|g〉i
s
〈f|g〉=
∫
dτf∗ (r)g(r),
(2.1
3)
whe
ref
andg
are
assu
med
tobe
squa
rein
tegr
able
func
tion
san
ddτ
isth
esp
ecifi
cin
tegr
atio
nvo
lum
eel
emen
t;f∗
isth
eco
mpl
ex-c
onju
gate
dfo
rmoff.
Sum
mar
y:
Inth
eei
genf
unct
ion
basis,
the
solu
tion
ofEq.
(1.1
)is
am
atrix
equa
tion
:
c(t)=U(t,t
0)c(t
0)
(2.1
4)
whe
re
Unm(t,t
0)=
{0,n6=m
exp(−
iωn(t−t 0),n=m
(2.1
5)
and
c m(t
0)=
〈χm|Ψ(t
0)〉
(2.1
6)
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t21
2.1.
3N
atur
alin
tegr
atio
n
The
met
hod
pres
ente
din
the
prev
ious
sect
ion
goes
back
toE.S
chrö
ding
er.
Itfo
rms
the
basis
ofth
eso
lution
ofth
eSc
hröd
inge
req
uation
from
tim
ein
depe
nden
tsp
ectr
osco
py.
Tim
ede
pend
ent
dyna
mic
sfrom
tim
ein
depe
nden
tsp
ectr
osco
py
E.Sc
hröd
inge
r,N
atur
wisse
nsch
afte
n14
(192
6)
Letψ(t,x)=c 1(t)χ1(x)+c 2(t)χ2(x)+...
andHχi=Eiχi.
spec
tros
copi
cst
ates
The
n
c i(t)=c i(0)exp(
−i2πEi
ht)
.
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t22
2.1.
4Pra
gmat
ical
inte
grat
ion
inan
yba
sis
repr
esen
tation
:so
lution
bydi
agon
aliz
atio
n
Nor
mal
ly,
and
unfo
rtun
atel
y,ei
genf
unct
ions
are
unkn
own
expe
rimen
tally
.In
orde
rto
cal-
cula
teth
emnu
mer
ical
ly,
we
repr
esen
tth
emin
aba
sis,
say,
ofsq
uare
inte
grab
lefu
nction
s:{φ
n(r)|n
=1,...,N}.
The
num
berN
isa
finite
num
berin
alln
umer
ical
trea
tmen
ts.
We
assu
me
thatN
can
bech
osen
such
the
repr
esen
tation
issu
ffici
ently
good
.It
can
beex
pect
edth
at,th
ela
rgerN
,th
ebe
tter
isth
ere
pres
enta
tion
.H
ence
let
χ(r)=
N∑
n
Znφn(r)
(2.1
7)
Inse
rtin
gth
isfo
rmin
toEq.
(2.7
)le
ads
first
to
Hχ(r)=
N∑
n
ZnHφn(r)
(linearity)
=E
N∑
n
Znφn(r)
(2.1
8)
Mole
cula
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mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t23
We
now
assu
me
that
the
basis
isor
thon
orm
al.
Mul
tipl
icat
ion
from
the
left
ofEq.
(2.1
8)by
asp
ecifi
cba
sis
func
tion
φm
and
subs
eque
ntin
tegr
atio
nov
erth
ede
finitio
nsp
ace
ofth
efu
nction
s(p
roje
ctio
nonφm
),le
adth
ento
〈φm|Hχ(r)〉
=
N∑
n
Zn〈φ
m|H
φn〉=
E
N∑
n
Zm
〈φm|φn〉
︸︷︷︸
δ nm
=EZm
(2.1
9)
Eq.
(2.1
9)is
am
atrix
equa
tion
:
Hz=E
z.
(2.2
0)
with
mat
rixel
emen
ts
Hmn=〈φ
m|H
φn〉=
<φm|H
|φn>=
∫
dτφ∗ m(r)Hφn(r).
(2.2
1)
Mole
cula
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mD
ynam
ics
2019
Pro
f.R
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arq
uard
t24
The
solu
tion
sof
this
equa
tion
are
the
eige
nval
ueE
and
the
eige
nvec
torz,w
hose
elem
ents
are
the
repr
esen
tation
coeffi
cien
tsZn
ofth
eei
genf
unct
ionχ
inth
eba
sis{φ
n}.
The
rank
ofth
em
atrix
HisN
.AsH
isse
lf-ad
join
t,H
isH
erm
itia
n.So
we
may
ex-
pect
tofin
dN
real
solu
tion
sE
1,...,E
Nw
ith,
for
each
solu
tion
eige
nval
ue,a
own
solu
tion
eige
nvec
torz1,...,z
N.
Eig
enve
ctor
sar
epr
agm
atic
ally
arra
nged
asco
lum
nve
ctor
sof
am
atrix
Z.
LetZnm
beth
en-t
hel
emen
tof
eige
nvec
torzm;it
ispo
sition
edat
the
cros
sing
ofro
wn
and
colu
mnm
.
From
linea
ral
gebr
ait
iskn
own
that
,ifH
isH
erm
itia
n,Z
isun
itar
y,i.e
.Z
−1=
Z† ,
whe
reZ
†is
the
adjo
int
mat
rixto
Z:Z
† nm=Z
∗ mn.
SoZ
−1
nm=Z
∗ mn.
The
mat
rixZ
isth
etr
ansf
orm
atio
nm
atrix
from
the
repr
esen
tation
basis{φ
n}
toth
eaf
ore-
men
tion
edsu
bset
ofei
genf
unct
ions
ofth
eha
milt
onia
n:
χm(r)=
N∑
n
Znmφn(r)
(2.2
2)
Shor
tly,
itis
used
tore
pres
ent
anei
genf
unct
ion
ina
set
ofba
sis
func
tion
s.N
otic
eth
eor
der
Mole
cula
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mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t25
ofin
dice
s:Eq.
(2.2
2)is
not
asim
ple
mat
rixve
ctor
mul
tipl
icat
ion.
Let
now
Ψ(t,r)=
N∑
n
b n(t)φn(r)
(2.2
3)
beth
ere
pres
enta
tion
ofth
eso
lution
ofth
eSc
hröd
inge
req
uation
inth
atba
sis.
The
vect
orb(t)=(b
1(t),...,b N
(t))T
isth
est
ate
vect
orin
the
basis
unde
rco
nsid
erat
ion.
Inse
rtio
nof
Eq.
(2.2
2)in
toEq.
(2.1
0),an
dso
me
rear
rang
emen
t,th
enle
ads
to:
Ψ(t,r)=
N∑
n
(N∑
m
c m(t
0)exp(−
iωm(t−t 0))Znm
)
φn(r)
(2.2
4)
Mole
cula
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mD
ynam
ics
2019
Pro
f.R
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arq
uard
t26
and
inse
rtio
nin
toEq.
(2.1
6)le
ads
to:
c m(t
0)=
N∑
n
Z∗ nm〈φ
n|Ψ(t
0)〉
(2.2
5)
But
the
brac
ket
isth
ein
itia
lcon
dition
repr
esen
tation
:
b n(t
0)=
〈φn|Ψ(t
0)〉
(2.2
6)
One
dedu
ces,
afte
rin
sert
ion
and
som
ead
dition
alre
arra
ngem
ents
:
Ψ(t,r)=
N∑
n
(N∑
n′
N∑
m
Z∗ n′ mZnmb n
′ (t 0)exp(−
iωm(t−t 0))
︸︷︷
︸
≡b n(t)
)
φn(r)
(2.2
7)
But
the
term
unde
rth
epa
rent
hese
sis
tobe
inte
rpre
ted
asb n(t).
Mole
cula
rQ
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mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t27
Sum
mar
y:
Inan
yba
sis,
the
solu
tion
ofEq.
(1.1
)is
am
atrix
equa
tion
:
b(t)=U(t,t
0)b(t
0)
(2.2
8)
whe
re
Unm(t,t
0)=
N∑
k
ZmkZ
∗ nkexp(−
iωk(t−t 0))
(2.2
9)
=N∑
k
Zmkexp(−
iωk(t−t 0))Z
−1
kn
(2.3
0)
=δ m
nfo
rt=t 0.
(2.3
1)
isth
em
atrix
repr
esen
tation
ofth
epr
opag
ator
inth
eba
sis{φ
n},
and
b n(t
0)=
〈φn|Ψ(t
0)〉
the
initia
lcon
dition
.
Mole
cula
rQ
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mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t28
Itis
wor
thco
nsid
erin
gth
ean
alog
uetr
ansf
orm
atio
nru
les.
IfH
isth
em
atrix
repr
esen
tation
ofth
eha
milt
onia
nin
the
basis{φ
n},
then
Z†H
Z=
E1
00···
00E
20···
0. . .
. . .. . .
. . .. . .
00
0···EN
(2
.32)
isits
repr
esen
tation
inth
eba
sis
ofei
gens
tate
s,w
hich
may
beca
lledd(E
).T
his
isa
diag
onal
mat
rix.
Rec
ipro
cally
,
U(t,t
0)=Z
exp(−
iω1(t−t 0))
00···
00
exp(−
iω2(t−t 0))
0···
0. . .
. . .. . .
. . .. . .
00
0···exp(−
iωN(t−t 0))
Z
†
(2.3
3)
isth
ere
pres
enta
tion
ofth
epr
opag
ator
inth
eba
sis{φ
n},
asits
repr
esen
tation
inth
eba
sis
ofei
gens
tate
sis
diag
onal
.
Mole
cula
rQ
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mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t29
Rem
ark:
Bec
ause
ofm
ultidi
men
sion
ality
,th
est
ate
vect
orin
dexn
may
desc
ribe
aw
hole
set
ofin
dice
san
dqu
antu
mnu
mbe
rs.
For
ak-d
imen
sion
alsy
stem
,n
stan
dsfo
rak-t
uple
t(n
1,...,n
k).
Exa
mpl
es:
-2
coup
led
osci
llato
rs,1
0st
ates
each
:n=(v
1,v
2).
n1
23
···
1011
12···
100
v 10
12
···
01
2···
9v 2
00
0···
11
1···
9
-T
hehy
drog
enat
om.
i1
23
45
67
8···
n1
12
22
22
2···
ℓ0
00
01
11
1···
m0
00
0-1
-10
0···
s-1
1-1
1-1
1-1
1···
Mole
cula
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mD
ynam
ics
2019
Pro
f.R
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arq
uard
t30
2.1.
5D
irec
tso
lution
byiter
atio
n(1
stan
d2n
dor
der)
Effi
cien
tdi
agon
aliz
atio
nal
gorit
hms
offu
llm
atric
esar
eav
aila
ble
toda
yfo
rm
atric
era
nks
onth
eor
der
of10
4.
Alre
ady
“sm
all”
syst
ems
such
asa
tetr
a-at
omic
mol
ecul
ew
ith
6de
gree
sof
free
dom
and
10D
VR
prim
itiv
efu
nction
sfo
rea
chde
gree
offree
dom
yiel
dsa
mat
rixw
ith
rank
106.
Subs
ets
ofei
genv
alue
san
dei
genv
ecto
rsm
aybe
calc
ulat
edfo
rsu
chm
atric
esby
iter
ativ
eal
gorit
hms,
such
asth
eLa
nczo
sal
gorit
hm.
How
ever
,in
orde
rto
use
Eq.
(2.3
1),th
efu
llse
tof
eige
nvec
tors
and
eige
nval
ues
isne
eded
.
The
refo
re,
the
mos
tco
mm
only
used
algo
rithm
for
prop
agat
ion
isth
esh
ort
tim
eiter
ativ
em
etho
d.T
hetim
eax
isis
disc
retize
d,no
rmal
lyw
ith
equa
lste
ps,
t∈
{t0,t
1,...,tNt}
t n=n∆t+t 0,n=0,...,Nt
(2.3
4)
and
the
tim
epr
opag
ator
mat
rixis
eval
uate
dap
prox
imat
ivel
yat
ever
ydi
scre
tetim
est
epby
trun
cation
afte
rth
elin
ear
term
:
U(tn+1,tn)≈
Ulin(∆t)=1−i2π h
H∆t
(2.3
5)
Mole
cula
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ynam
ics
2019
Pro
f.R
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arq
uard
t31
The
stat
eve
ctor
isth
enpr
opag
ated
step
wise:
b(tn+1)≈
Ulin(∆t)·b
(tn)
(2.3
6)
Sinc
eU
lin
isan
appr
oxim
atio
nto
Uat
ever
ytim
est
ep,
the
erro
ris
prop
agat
edan
dm
aycu
mul
ate
quite
rapi
dly.
Cle
arly,in
orde
rto
min
imiz
eth
est
eper
ror,
∆t||H
||<<h.
(2.3
7)
Wha
tis
typi
cally
asu
ffici
ent
smal
l∆t?
One
mig
htna
ivel
yth
ink
that
the
proc
essw
illbe
over
befo
reer
rorh
asac
cum
ulat
edsign
ifica
ntly.
Mole
cula
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ynam
ics
2019
Pro
f.R
.M
arq
uard
t32
As
anex
ampl
e,co
nsid
era
sim
ple
two
leve
ldy
nam
ics,
whe
rea
quan
tum
stat
eis
coup
led
reso
nant
lyvi
aa
coup
ling
cons
tantV
toa
seco
nd,iso-
ener
getic
quan
tum
stat
e-
this
isth
esim
ples
tm
odel
ofa
quan
tum
mec
hani
calt
unne
ling
mot
ion.
The
exac
tev
olut
ionP(t)=cos(2πt/τ)
ofth
epo
pula
tion
ofth
ein
itia
lsta
teis
show
nin
the
figur
ebe
low
asa
cont
inuo
uslin
e,w
hereτ=h/2V
isth
ety
pica
levo
lution
tim
e.
0
0.5 1
1.5
0 1
2
P(t)
t / τ
linea
r-1p
linea
r-2p
exac
t
Mole
cula
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mD
ynam
ics
2019
Pro
f.R
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arq
uard
t33
The
dott
edlin
esh
ows
the
sam
eev
olut
ion
calc
ulat
edvi
aEq.
(2.3
5)(“
linea
r-1p
”la
bel).
The
accr
ued
prop
agat
ion
erro
rbe
com
esas
larg
eas
10%
afte
rju
ston
epe
riod
ofth
eev
olut
ion,
if∆t
ison
ly1%
ofth
ety
pica
levo
lution
tim
eτ.
Nor
mal
ly,
ther
eis
also
erro
rpr
opag
atio
nin
term
sof
the
phas
esof
the
tim
eev
olvi
ngst
ate
vect
or.
The
erro
rpr
opag
atio
nle
adsto
ase
vere
non
cons
erva
tion
ofth
eno
rm.
The
erro
rca
nbe
muc
hre
duce
d,in
this
case
,vi
ath
eus
eof
a2n
dor
der
form
ula
(“lin
ear-
2p”
labe
l)w
hich
relie
son
the
eval
uation
ofth
ewav
efu
nction
attw
oea
rlier
tim
est
epst k
andt k
−1
[10]
,th
eso
-cal
led
Cra
nk-N
icho
lson
met
hod
[11]
.
The
sem
etho
dsw
illno
tbe
disc
usse
dfu
rthe
rin
this
lect
ure,
and
the
read
eris
dire
cted
toth
eex
istig
liter
atur
e[2
,5,1
2]fo
rad
dition
alin
form
atio
n.
Mole
cula
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ynam
ics
2019
Pro
f.R
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arq
uard
t34
2.2
Ope
nsy
stem
s
Forop
enm
olec
ular
syst
ems,H
=H(t),
and
ener
gyis
not
cons
erve
d.
Inpr
actice
,th
etim
epr
opag
ator
isre
pres
ente
das
am
atrix
.
The
sim
ples
tin
tegr
atio
nsc
hem
eco
nsists
oftr
unca
ting
the
Mag
nus
expa
nsio
naf
terth
elin
ear
term
.Add
itio
nnal
ly,t
hetim
eva
riation
ofth
eha
milt
onia
nis
assu
med
tobe
smal
lin
each
tim
est
ep,su
chth
at
U(tn+1,tn)≈
Ulin(∆t,t n)≈
1−i2π h
H(tn)∆t.
(2.3
8)
For
open
syst
ems,
the
tim
est
eper
ror
isth
ustw
ofol
d!O
nepo
ssib
ility
tore
duce
the
erro
ris
disc
usse
dat
the
exam
ple
ofth
etr
eatm
ent
ofm
olec
ule-
radi
atio
nin
tera
ctio
nin
the
follo
win
g.
Mole
cula
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mD
ynam
ics
2019
Pro
f.R
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arq
uard
t35
2.3
Mol
ecul
e-ra
diat
ion
inte
ract
ion
Am
olec
ule
exci
ted
byel
ectr
omag
netic
radi
atio
nm
aybe
cons
ider
edto
bean
open
syst
emw
ith
the
tim
ede
pend
ent
ham
ilton
ian H
(t)=H
0+H
1(t)
(2.3
9)
whe
reH
0is
the
tim
ein
depe
nden
tm
olec
ular
ham
ilton
ian,
andH
1(t)
isth
eco
uplin
gop
erat
orto
the
elec
trom
agne
tic
radi
atio
n.In
the
elec
tric
dipo
leap
prox
imat
ion,
this
oper
ator
has
the
form
H1(t)=−µ(r)E(t)
(2.4
0)
whe
reµ(r)
isth
eth
ree
dim
ension
alve
ctor
oper
ator
ofth
eel
ectr
icdi
pole
mom
ent,
usua
llya
loca
lfu
nction
ofth
enu
clea
rco
ordi
nate
son
ly,an
dE(t)
isth
etim
ede
pend
ent
elec
tros
tatic
field
vect
orof
the
radi
atio
n.Vec
tors
are
defin
edin
the
labo
rato
ryfix
edfram
e.
Mole
cula
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ynam
ics
2019
Pro
f.R
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arq
uard
t36
For
cohe
rent
,m
onoc
hrom
atic
radi
atio
nat
freq
uenc
yν,
the
elec
tros
tatic
field
stre
ngth
may
begi
ven
byE(t)=E
0cos(ωt+η)
(2.4
1)
whe
reω=2π
νis
the
circ
ular
freq
uenc
yan
dη
isa
phas
efa
ctor
.U
sual
ly,it
isas
sum
edth
atth
eel
ectr
icfie
ldha
san
insign
ifica
ntva
riation
inth
ein
tera
ctio
nre
gion
.T
heph
ase
fact
orth
enty
pica
llyes
tabl
ishe
sth
eph
ase
ofth
eel
ectr
omag
netic
field
atth
ece
nter
-of-m
ass
position
ofth
em
olec
ule
inth
ela
bora
tory
fixed
axes
syst
em.
Inth
ese
mi-cl
assica
lap
prox
imat
ion
ofth
em
atte
r-ra
diat
ion
inte
ract
ion,
the
elec
trom
agne
tic
radi
atio
nis
trea
ted
clas
sica
ly,w
here
asth
em
olec
ular
syst
emis
fully
quan
tize
d.T
his
appr
oxi-
mat
ion
isas
sum
edto
beva
lidfo
rsu
ffici
ently
high
inte
nsitie
s.
The
inte
nsity
ofm
odek,λ
ofth
eel
ectr
omag
netic
field
isde
fined
asth
em
odul
eof
the
Poy
ntin
gve
ctor
,
I k,λ=|S
k,λ|=
c 4πEk,λH
(irr)
k,λ
=c 4πE
2 k,λ
(Gau
ss)
Ek,λHk,λ
=ǫ 0cE
2 k,λ
(SI)
,(2
.42)
whe
reǫ 0
isth
eel
ectr
icco
nsta
ntan
dc
isth
eva
cuum
spee
dof
light
.T
hela
steq
uation
sar
eva
lidin
char
gefree
spac
es.
Mole
cula
rQ
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mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t37
The
seex
pres
sion
sm
aybe
cast
into
the
prac
tica
l,di
men
sion
less
rela
tion
s
|E0|
Vcm
−1≈
27.45
√
<I>
Wcm
−2
(2.4
3)
H1
hccm
−1≈
−0.4609
µ D
√
<I>
MW
cm−2.
(2.4
4)
whe
re<
I>
isth
etim
eav
erag
edin
tens
ity,
whi
cheq
uals
half
the
peak
inte
nsity
for
am
onoc
hrom
atic
radi
atio
n.
For
very
inte
nse
field
s,ho
wev
er,
the
elec
tric
dipo
leap
prox
imat
ion
may
beco
me
inad
equa
te,
and
high
eror
derco
uplin
gte
rmsm
ustbe
cons
ider
edin
addi
tion
toth
eel
ectr
icdi
pole
coup
ling,
such
asth
epo
lariz
abili
tyco
uplin
g−(α
E(t))
E(t),
the
elec
tric
quad
rupo
leor
even
mag
-ne
tic
dipo
leco
uplin
gs.
Forve
ryin
tens
era
diat
ion,
mol
ecul
ario
niza
tion
proc
esse
sm
ayeq
ually
beco
me
non-
negl
igib
le,w
hich
infe
rth
eus
eof
mul
tipl
eel
ectr
onic
surfac
esin
the
calc
ulat
ion
ofth
eex
cita
tion
proc
ess.
Mole
cula
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mD
ynam
ics
2019
Pro
f.R
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uard
t38
Tim
ede
pend
ent
lase
rpu
lses
are
stric
tly
mul
tich
ro-
mat
ic.
Figu
re2.
1sh
ows
aty
pica
ltim
eev
olut
ion
ofsu
cha
pulse,
whi
chm
aybe
mod
elle
dby
afu
nction
ofth
ety
peE(t)=E
0(t)cos(ωt+η)
(2.4
5)
The
enve
lope
ofth
efie
ldam
plitud
e|E
0(t)
|is
norm
ally
aslow
lyva
ryin
gfu
nction
oftim
e,co
mpa
red
toth
era
pid
osci
llation
sof
the
phas
e.
The
perio
dof
phas
eos
cilla
tion
sis
half
the
perio
dof
the
cent
ralf
requ
ency
τ=
2π ω.
(2.4
6)
0
0.2
0.4
0.6
0.8 1
0 0
.5 1
1.5
2 2
.5 3
I(t)/Imax
t/ps
Fig
ure
2.1
:Typ
ical
form
ofa
lase
rpu
lse
(qua
litat
ive)
Mole
cula
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ynam
ics
2019
Pro
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t39
2.3.
1T
heFlo
quet
-Lia
puno
ffm
etho
d
Forst
rictly
perio
dic
tim
ede
pend
ent
ham
ilton
ian,
H(t)=H(t+τ)
(2.4
7)
e.g.
forst
rictly
mon
ochr
omat
icra
diat
ion,
withτ=2π/ω
,th
etim
epr
opag
ator
may
begi
ven
byth
eex
pres
sion
[13,
14] U
(t,t
0)=F(t)exp(A
(t−t 0))F
−1(t
0)
(2.4
8)
whe
reF(t)=F(t+τ)
(2.4
9)
andA
isa
tim
ein
depe
nden
top
erat
or.
Inpr
actice
solu
tion
sar
eob
tain
edby
disc
retiza
tion
ofth
etim
eax
ist n,m
=t 0+n·τ
+m
·∆τ
(m·∆τ<τ;n,m
≥0)
and
num
eric
alin
tegr
atio
n.T
here
sultin
geq
uation
forth
epr
opag
ator
mat
rixm
aybe
give
nas
U(tn,m,t
0)=U(m
∆τ+t 0,t
0)U
n(τ
+t 0,t
0).
(2.5
0)
U(m
∆τ+t 0,t
0)is
calc
ulat
edw
ithi
nth
epr
ogra
mURIMIR5a
[15]
form
=1,...,nτ
bydi
rect
num
eric
alin
tegr
atio
n(w
ithnτ∆τ=τ);
the
initia
lco
effici
ent
vect
orb(t
0)
ispr
opag
ated
tob(tn,m)
withURIMIR5b
[15]
bydi
agon
aliz
ing
the
com
plex
tim
eev
olut
ion
oper
ator
mat
rixU(τ
+t 0,t
0)
toob
tain
thenth
power
acco
rdin
gto
Eq.
(2.5
0).
Mole
cula
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ynam
ics
2019
Pro
f.R
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t40
2.3.
2T
hequ
asi-re
sona
ntap
prox
imat
ion
for
period
icpr
oble
ms
Inso
me
case
s,th
equ
asi-r
eson
ant
appr
oxim
a-tion
(QRA)
isal
sous
ed[1
6–18
],in
stea
dof
Eq.
(2.5
0).
The
appr
oxim
atio
nis
expe
cted
tobe
good
,if
the
coup
ling
stre
ngth
|Vij|=∣ ∣ ∣ ∣
2πH
1ij
h
∣ ∣ ∣ ∣(2
.51)
(in
units
ofci
rcul
arfreq
uenc
ies)
and
the
reso
-na
nce
defe
ct
Xk≡ωk−nkωL,
(2.5
2)
are
both
muc
hsm
alle
rth
anth
eca
rrie
rex
ci-
tation
freq
uenc
yωL.
The
inte
gernk
used
tode
fine
the
quas
i-re
sona
ntle
vels
isan
appr
o-pr
iate
inte
ger,
such
that
|Xk|<
ωL/2
(see
Figu
re2.
2).
ω 1
ω 2ω 3
ω 4
ω 5ω 6
ω 7ω 8
...ω 1
2
ω Lx12
Fig
ure
2.2
:Typ
ical
,qu
alitat
ive
leve
l
sche
me
for
mul
tiph
oton
exci
tation
pro-
cess
esin
pol
yato
mic
mol
ecul
es
Mole
cula
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mD
ynam
ics
2019
Pro
f.R
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arq
uard
t41
Thi
sap
prox
imat
ion
cons
ists
inso
lvin
gth
eeq
uation
id dta
(t)={X
+1 2V
QRA}a
(t)=
2π hH
QRAa(t)
(2.5
3)
whe
reX
=Dia
g(...,X
k,...),
VQRA
kj
=
{Vkj
if|n
k−nj|=
1
0if|n
k−nj|6=
1(2
.54)
HQRA
isa
cons
tant
mat
rix,
the
tim
eev
olut
ion
oper
ator
mat
rixm
aybe
calc
ulat
edas
inEq.
(2.3
1).
The
exac
tso
lution
isth
enap
prox
imat
edby
b(t)≈
bQRA(t)=a(t)Dia
g(exp(−
i·nkωLt)).
(2.5
5)
Mole
cula
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ynam
ics
2019
Pro
f.R
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arq
uard
t45
3Pot
ential
ener
gysu
rfac
es
3.1
Brie
fhi
stor
ical
rem
arks
The
conc
ept
ofpo
tent
iale
nerg
ysu
rfac
es(P
ES)
was
evoq
ued
inth
eda
ysof
the
old
quan
tum
theo
ry,
for
inst
ance
ina
pape
rby
Bje
rrum
[19]
onth
ein
terp
reta
tion
ofsp
ectr
alba
nds
ofga
spha
sem
olec
ules
1 .Fo
llow
ing
Bje
rrum
,th
enu
clei
mov
edu
eto
“val
ence
forc
es”
whi
chsh
ould
bede
term
inab
lefrom
reco
rded
spec
tra
2.
With
this
idea
inm
ind,
and
the
addi
tion
alwor
king
hypo
thes
isth
atth
em
otio
nof
the
nucl
eico
rres
pond
to(h
arm
onic
)vi
brat
ions
,Bje
rrum
deriv
eda
quad
ratic
forc
efie
ldfrom
the
term
valu
esof
obse
rved
fund
amen
talvi
brat
iona
ltr
ansition
sin
CO2,
prop
osin
gth
uspe
rhap
sth
efir
stfo
rmul
atio
nof
anan
alyt
ical
repr
esen
tation
ofan
effec
tive
mol
ecul
arpo
tent
ial
ener
gyhy
pers
urfa
ce.
1 The
sent
ence
in[1
9]”N
ach
neue
ren
Unt
ersu
chun
gen
ents
tehe
ndi
em
eist
enul
trar
oten
Spek
tral
band
endu
rch
Bew
egun
gen
von...Ato
men
oder
Ato
mgr
uppen
,w
ähre
nddi
eLin
ien
imsich
tbar
enun
dul
trav
iole
tten
Spek
trum
aufEle
ktro
nens
chw
ingu
ngen
ber
uhen
”[1
9],is
likel
yre
ferr
ing
toD
rude
[20]
,Ein
stei
n[2
1],an
dN
erns
t[2
2]2 ”
Das
Stu
dium
derul
trar
oten
Spek
tren
mus
s...fü
run
sere
Ken
ntni
sse
zude
nAto
mbew
egun
gen
von
gros
sem
Nut
zen
sein
könn
en”
[19]
Mole
cula
rQ
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mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t46
Late
ron
anha
rmon
icfo
rce
field
swer
ede
rived
ona
sim
ilar
way
from
wel
lre
solv
edin
frar
edsp
ectr
aof
halo
gen
hydr
ides
[23]
.T
hese
forc
efie
lds
wer
ede
velo
ped
astr
unca
ted
Tay
lor
expa
nsio
nsof
nucl
ear
dist
ance
disp
lace
men
tsfrom
equi
libriu
m.
The
spec
tra
also
allo
wed
tore
cogn
ize
clea
rlyth
ere
lation
betw
een
rota
tion
san
dvi
brat
ions
[24]
.Ana
logo
uswor
kon
the
harm
onic
forc
efie
ldof
amm
onia
[25]
and
met
hane
[26]
follo
wed
soon
afte
r.
The
orig
inof
the
effec
tive
pote
ntia
lsfo
rth
enu
clea
rm
otio
nis
expl
aine
din
the
theo
ret-
ical
fram
ewor
kse
tup
byBor
nan
dH
eise
nber
g(in
the
old
form
ulat
ion
ofqu
antu
mth
e-or
y)[2
7],Con
don
[28,
29]3
,Sl
ater
[30]
4 ,H
eitler
and
Lond
on[3
1]5
aswel
las
Bor
nan
dO
p-pe
nhei
mer
[32]
6 .In
thes
eth
eorie
s,th
eas
sum
ptio
nis
mad
eth
atth
em
otio
nof
elec
tron
san
dnu
clei
are
adia
batica
llyse
para
ble:
the
nucl
eim
ove
rela
tive
lyslow
lyw
ith
resp
ect
toth
eel
ectr
ons
whi
chad
apt
them
selv
esra
pidl
yto
any
disp
lace
men
tof
the
nucl
ei.
3 rec
eive
dM
arch
19,19
27,su
bmitte
dfirs
tin
Göt
ting
en,an
dth
enin
Mun
ich
onA
pril
18,19
274 r
ecei
ved
onA
pril
26,19
275 r
ecei
ved
onJu
ne30
,19
276 r
ecei
ved
onA
ugus
t25
,19
27
Mole
cula
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mD
ynam
ics
2019
Pro
f.R
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arq
uard
t47
Late
r,th
ehy
poth
esis
was
form
ulat
edth
atth
ead
iaba
tic
sepa
ration
ofel
ectr
onic
and
nucl
ear
mot
ion
follo
win
gBor
nan
dO
ppen
heim
erm
aybe
used
toal
sode
scrib
epr
edisso
ciat
ion
and
unim
olec
ular
deca
y[3
3,an
dre
fere
nces
ther
ein]
7 .
Larg
eam
plitud
em
otio
nof
atom
sin
boun
dm
olec
ular
syst
ems
may
lead
tobo
nddi
ssoc
iation
,fo
rmat
ion
ofne
wbo
nds
orne
wm
olec
ular
conf
orm
atio
ns(e
.g.
stru
ctur
alisom
erisat
ions
).La
ter
form
ulat
ions
[34]
ofth
eor
igin
alBor
n-O
ppen
heim
erth
eory
allo
wfo
rno
n-pe
rtur
bative
trea
tmen
tof
larg
eam
plitud
em
olec
ular
dyna
mic
sev
enin
case
sw
here
the
adia
batic
sepa
ration
turn
sou
tto
bea
poor
appr
oxim
atio
n,su
chas
atco
nica
lint
erse
ctio
ns[6
].
7 ”It
will
gene
rally
be
conc
eded
that
the
abov
eun
per
turb
edei
genv
alue
san
dei
genf
unct
ions
will
give
ina
fairly
corr
ect
man
ner
the
ener
gyle
vels
and
diss
ocia
tion
limits
ofa
diat
omic
mol
ecul
e,an
dhe
nce
the
corr
espon
ding
per
turb
atio
nsw
illgi
veco
rrec
tly
such
thin
gsas
the
rate
atw
hich
am
olec
ule
goes
from
adi
scre
test
ate
toa
cont
inuu
m.”
[33,
page
1459
]
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t48
3.2
PES
from
“exp
erim
ent”
PES
are
not
dire
ctly
obse
rvab
lequ
antities
.H
owev
er,
they
may
bere
pres
ente
dan
alyt
ical
lyan
dpa
ram
eter
sin
trod
uced
inth
atway
may
bede
term
ined
from
expe
rimen
talda
tasu
chas
spec
tros
copi
ctr
ansition
s.
An
exam
ple
isth
eha
rmon
icfo
rce
field
ofa
diat
omic
mol
ecul
e,th
ean
alyt
ical
form
ofw
hich
is
V(r)=
1 2f(r
−r e
q)2
(3.1
)
whe
rer
isth
ein
tera
tom
icdi
stan
ce;he
re,f
(Hoo
ke’s
forc
eco
nsta
nt)
andr e
qar
epa
ram
eter
sof
the
pote
ntia
lene
rgy
func
tionV(r).
Whi
ler e
qm
aybe
dete
rmin
ed,
typi
cally
,from
mic
rowav
esp
ectr
osco
pyvi
ade
term
inat
ion
ofth
ero
tation
alco
nsta
nt“B
e”(e
.g.
from
aju
dici
ous
extr
apol
atio
ntov=0
ofth
eob
serv
able
quan
titiesBv):
r eq=
h
8π2cB
eµ
(3.2
)
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t49
the
forc
eco
nsta
ntis
dete
rmin
edfrom
the
line
position
sof
the
infrar
edsp
ectr
um,w
hich
yiel
dth
efreq
uenc
yte
rmν:
f=µν2.
(3.3
)
Inbo
theq
uation
s,µ
isth
ere
duce
dm
ass
ofth
edi
atom
icm
olec
ule.
Mor
ege
nera
lly,q
uant
itie
ssu
chasr e
qan
df
are
para
met
ersth
atm
aybe
obta
ined
from
spec
tro-
scop
icda
tavi
alin
earo
rnon
-line
arad
just
men
ts(fi
ts)of
theo
retica
lter
mva
luesν(the)(r
eq,f,...)
toex
perim
enta
llyde
term
ined
term
valu
esν(exp) .
Adj
ustm
ent
may
beac
hiev
edby
min
imiz
atio
nof
the
root
-mea
n-sq
uare
(rm
s)de
viat
ion,
e.g.
∆νrm
s=
√ √ √ √ √
ndata
∑ i=1
(ν(the)
i(p
1,p
2,...)−ν(exp)
i)2
ndata
.(3
.4)
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t50
Forin
stan
ce,t
heor
etic
alte
rmva
lues
rela
ted
toth
equ
adra
tic
forc
efie
ldof
adi
atom
icm
olec
ule
are
give
nby
ν(the)
i(f)=(i−1)
1 c
√
f µ(i=1,2,...)
(3.5
)
Cle
arly,it
ises
sent
ialt
oha
vean
alyt
ical
repr
esen
tation
sof
PES
for
this
purp
ose.
Mor
ere
alistic
anal
ytic
alre
pres
enta
tion
sin
volv
eth
ead
just
men
tof
addi
tion
alpa
ram
eter
s,su
chas
the
anha
rmon
icity
para
met
era
inth
eM
orse
pote
ntia
l[35
]
V(r)=D
e(exp(−a(r
−r e
q))−1)
2(3
.6)
whe
reD
eis
the
“disso
ciat
ion
ener
gyw
ith
resp
ectto
the
bott
omof
the
pote
ntia
l”;th
isqu
antity
isre
late
dto
the
quad
ratic
forc
efie
ldco
nsta
ntf
byf=2D
ea2.
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t51
Forth
eM
orse
osci
llato
r,th
eore
tica
lter
mva
lues
are
give
nan
alyt
ical
lyby
ν(the)
i(D
e,a)=(ω
e−ωex
e)(i−1)
−ωex
e(i−1)
2(i=1,2,...)
(3.7
)
whe
re
ωe=
1 c
√
2Dea
2
µ(3
.8)
ωex
e=ha2
2cµ=hcωe2
4De
(3.9
)
Mor
eco
mpl
icat
edan
alyt
ical
pote
ntia
lsm
aybe
deriv
ed.
How
ever
,te
rmva
lues
cann
otge
n-er
ally
beca
stin
tosim
ple
anal
ytic
alfo
rms
and
mus
tbe
trea
ted
num
eric
ally
.Fo
rpo
lyat
omic
mol
ecul
es,th
eore
tica
lter
mva
lues
are
alm
ost
excl
usiv
ely
give
nnu
mer
ical
ly,of
ten
afte
rdi
ago-
naliz
atio
nof
aneff
ective
ham
ilton
ian
mat
rix.
Ani
ceex
ampl
eof
the
deriv
atio
nof
anan
alyt
ical
PES
from
spec
tros
copi
cda
tais
give
nin
[36]
.
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t52
3.3
PES
from
“the
ory”
Pot
ential
ener
gysu
rfac
esm
aybe
asse
ssed
from
abin
itio
calc
ulat
ions
ofth
eel
ectr
onic
stru
c-tu
re;
mod
ern
com
puta
tion
alm
etho
dsan
dla
rge
com
pute
rsal
low
toac
hiev
eun
prec
eden
ted
high
accu
racy
.
Star
ting
poin
tis
the
mol
ecul
arha
milt
onia
n(e
xpre
ssed
here
inth
eno
n-re
lativi
stic
form
;qu
an-
tities
are
assu
med
tobe
give
nin
atom
icun
its)
:
H=
−1 2
(∑
a
∇2 a+∑
A
1 mA∇
2 A
)
+∑ a<b
1 r ab
−∑ a,A
ZA
r aA
+∑ A<B
ZAZB
r AB.
(3.1
0)
The
cent
ralwor
king
hypo
thes
isle
adin
gto
the
theo
retica
lde
finitio
nof
PES
isth
ead
iaba
tic
appr
oxim
atio
n.
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t53
3.3.
1Adi
abat
icap
prox
imat
ion
Inth
efo
llow
ing,
the
sym
bolQ
isus
edfo
rnu
clea
rco
ordi
nate
sx(n) ,
andq
for
the
elec
tron
icco
ordi
nate
sx(e) .
The
non-
rela
tivi
stic
ham
ilton
ian
isth
enof
the
form
H=T
(n) (Q,∂
Q)+T
(e) (q,Q
,∂q)+V(q,Q
)(3
.11)
The
goal
isto
find
solu
tion
sof
the
tim
ein
depe
nden
tSc
hröd
inge
req
uation
8
HΨ(q,Q
)=EΨ(q,Q
)(3
.12)
The
adia
batic
ansa
tzco
nsists
ofse
ttin
g
Ψ(q,Q
)=ψ(n) (Q)·ψ
(e) (q;Q
)(3
.13)
with
adia
batica
llyse
para
ted
wav
efu
nction
sψ(n) (Q)
andψ(e) (q;Q
)th
atar
eno
rmal
ized
,re
spec
tive
ly,
inth
eir
defin
itio
nsp
aces
;th
esy
mbo
l“;Q
”inψ(e) (q;Q
)m
eans
that
nucl
ear
coor
dina
tes
are
cons
ider
edas
para
met
ers,
rath
erth
anfu
nction
varia
bles
.
Inse
rtio
nof
this
ansa
tzin
Eq.
(3.1
2)yi
elds
8 We
refrai
nhe
refrom
indi
cating
the
depen
denc
eon
spin
san
dth
eap
prop
riat
ebos
onor
ferm
ion
sym
met
ry
ofnu
clei
orel
ectr
ons.
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t54
HΨ
=ψ(e) (q;Q
)(T
(n) (Q,∂
Q)ψ(n) (Q))
+ψ(n) (Q)(T
(n) (Q,∂
Q)ψ(e) (q;Q
))
+ψ(n) (Q)(T
(e) (q,Q
,∂q)ψ(e) (q;Q
))+V(q,Q
)ψ(n) (Q)ψ(e) (q;Q
)(3
.14)
Bec
ause
ofth
egr
eat
mas
sdi
ffer
ence
sfo
und
inm
olec
ular
syst
ems,mA>>
1(in
atom
icun
its)
,th
ehy
poth
esis
ism
ade
that
∣ ∣ ∣ ∣
∫
dτ qψ(e)∗(q;Q
)T
(n) (Q,∂
Q)ψ
(e) (q;Q
)∣ ∣ ∣ ∣<<
∣ ∣ ∣ ∣
∫
dτ qψ(e)∗(q;Q
)T
(e) (q,Q
,∂q)ψ
(e) (q;Q
)∣ ∣ ∣ ∣
(3.1
5)
The
refo
re,in
Eq.
(3.1
2),th
ete
rmT
(n) (Q,∂
Q)ψ(e) (q;Q
)is
negl
ecte
dw
ith
resp
ect
toth
ete
rmT
(e) (q,Q
,∂q)ψ(e) (q;Q
)(B
orn-
Opp
enhe
imer
adia
batic
appr
oxim
atio
n).
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t55
The
tim
ein
depe
nden
tSc
hröd
inge
req
uation
then
beco
mes
ψ(e) (q;Q
)(T
(n) (Q,∂
Q)ψ(n) (Q))
+ψ(n) (Q)(T
(e) (q,Q
,∂q)ψ(e) (q;Q
))
+V(q,Q
)ψ(n) (Q)ψ(e) (q;Q
)≈Eψ(n) (Q)ψ(e) (q;Q
)(3
.16)
Solu
tion
she
reof
are
obta
ined
intw
ost
eps.
3.3.
2Ste
p1:
“ele
ctro
nic
stru
ctur
e”,ad
iaba
tic
pote
ntia
len
ergy
surfac
es
LetE
(e)
k(Q
)be
thek-t
hei
genv
alue
ofth
eel
ectr
onic
Schr
ödin
ger
equa
tion
T(e) (q,Q
,∂q)ψ(e)
k(q;Q
)+V(q,Q
)ψ(e)
k(q;Q
)=E
(e)
k(Q
)ψ(e)
k(q;Q
)(3
.17)
atfix
edva
lues
ofth
enu
clea
rco
ordi
nate
sQ
.Pra
gmat
ical
ly,in
elec
tron
icst
ruct
ure
calc
ula-
tion
s,Q
andq
are
cart
esia
nco
ordi
nate
san
dT
(e) (q,Q
,∂q)≡T
(e) (∂q).
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t56
Let
Qbe
a(o
ne-d
imen
sion
al)
nucl
ear
coor
dina
te(e
.g.
the
inte
rato
mic
dist
ance
ofa
diat
omic
mol
ecul
e).
One
firs
tde
fine
s
agr
idof
nucl
ear
coor
dina
tesQ
1,Q
2,...
that
will
be
cons
ider
edas
fixe
dpa
ram
e-
ters
for
the
calc
ulat
ion
ofth
eel
ectr
onic
stru
ctur
e.
The
non
eca
lcul
ates
differ
ent
elec
tron
ic
ener
giesE
(e)
k(Q
)at
each
nucl
ear
pos
itio
n
Q=Qlin
divi
dual
ly.
The
quan
tum
num
-
berk
just
coun
tsso
lution
sst
arting
from
the
lowes
ton
e.
The
figu
reon
the
righ
tha
ndside
in-
dica
tes
sche
mat
ical
lyth
epos
itio
nof
eige
nval
uesE
(e)
k(Q
)w
ith
resp
ect
toa
(one
-dim
ension
al)
variat
ion
ofth
enu
clea
r
pos
itio
npa
ram
eterQ
.
E1(
1)
E2(
1)
E3(
1)
��� ����� ���� ��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
��
E1(
4)
E2(
4)
E3(
4)
��� ������ ����� ��
E3(
3)
E2(
3)
E1(
3)E
1(2)
E2(
2)
E3(
2)
Q(1
)Q
(2)
Q(3
)Q
(4)
Eel 3
Eel 2
Eel 1
Q
Vn(Q
)
Fig
ure
3.1
:Sch
emat
icvi
ewof
the
abin
itio
calc
ulat
ion
of
pot
ential
ener
gysu
rfac
es(t
osim
plify
the
nota
tion
,Ek(j)=
E(e)
k(Q
)fo
rQ
=Qj,he
re).
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t57
Bec
ause
ofth
eco
ntin
uity
ofpa
ram
eterQ
inEq.
(3.1
7),
one
may
link
ener
gypo
ints
corr
e-sp
ondi
ngto
the
sam
equ
antu
mnu
mbe
rk
tode
fine
ahy
per-
surfac
eVk(Q
)=E
(e)
k(Q
)th
ein
mul
ti-d
imen
sion
alsp
ace.
The
sear
eth
ead
iaba
tic
pote
ntia
lene
rgy
surfac
es.
Cor
resp
ondi
ngly,th
eso
lution
sψ(e)
k(q;Q
)of
Eq.
(3.1
7)ar
eca
lled
adia
batic
elec
tron
icst
ates
.
The
rear
etw
oim
port
antdi
stin
ctca
sesof
adia
batic
pote
ntia
lene
rgy
surfac
es(s
eeFi
gure
3.2
):ad
iaba
tic
pote
ntia
lsth
at“c
ross
”an
dth
ose
that
“avo
id”
each
othe
r.In
Figu
re3.
2a
,po
tent
ial
cros
sing
sar
eav
oide
d.It
mig
htbe
that
elec
tron
icwav
efu
nction
sha
vece
rtai
npr
oper
ties
that
dono
tfo
llow
adia
batica
llyth
em
otio
nof
nucl
ei.
For
inst
ance
,le
tQ
bea
diss
ocia
tion
coor
dina
tean
dle
tth
eso
lution
sEk(1),Ek(2)
andEk(4)
atQ
1,Q
2
andQ
4be
the
solu
tion
sof
wav
efu
nction
sth
atha
veio
nic
char
acte
r(s
uch
asin
HF)
.T
he“c
orre
lation
”of
this
prop
erty
isin
dica
ted
byth
eda
shed
line
inFi
gure
3.2
a.
How
ever
,at
poin
tQ
3,
the
wav
efu
nction
sha
ppen
tobe
long
toirr
educ
tibl
ere
pres
enta
tion
sof
the
sym
met
rygr
oup
pert
aini
ngto
the
mol
ecul
e,th
epr
oduc
tof
whi
chco
ntai
nsth
eto
tally
sym
met
ricre
pres
enta
tion
9Con
sequ
ently,
ther
eis
aco
uplin
gbe
twee
nth
e“c
orre
late
d”wav
efu
nction
satQ
3an
don
eob
tain
s,in
deed
,tw
odi
ffer
ent
ener
gyso
lution
sEk(3)6=Ek(3).
9 One
ofte
nsa
ys“t
hest
ates
have
the
sam
esy
mm
etry
.”
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t58
a
Ek(
4)
El(
4)
Ek(
2)
El(
2)
El(
1)
Ek(
1)E
l(3)
Ek(
3)
Q(1
)Q
(2)
Q(3
)Q
(4)
Q
V n(Q
)b
Ek(
3) =
Ek(
4)
El(
4)
Ek(
2)
El(
2)
El(
1)
Ek(
1)
El(
3)
Q(1
)Q
(2)
Q(3
)Q
(4)
Q
V n(Q
)
Fig
ure
3.2
:Sch
emat
icvi
ews
ofad
iabat
icpot
ential
ener
gysu
rfac
esVk(Q
)an
dVl(Q)
.
InFi
gure
3.2
b,
wav
efu
nction
sha
ppen
tobe
long
todi
ffer
ent
irred
uctibl
ere
pres
enta
tion
sth
atdo
not
cont
ain
the
tota
llysy
mm
etric
spec
ies.
Inth
atca
se,
pote
ntia
lsm
aycr
oss
(e.g
.Ek(3)=Ek(3))
.In
mul
ti-d
imen
sion
alsy
stem
scr
ossing
sm
ight
occu
rw
hen
asin-
gle
disjoi
ntco
ordi
nate
has
asp
ecia
lval
ue,at
whi
chth
em
olec
ular
stru
ctur
ega
ins
sym
met
ry.
Inth
atca
seon
eha
sth
eoc
cure
nce
ofa
coni
calin
ters
ection
.Adi
abat
icpo
tent
ials
may
thus
beco
me
sing
ular
atce
rtai
npo
ints
ofco
nfigu
ration
spac
e!
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t59
3.3.
3Ste
p2:
“vib
ration
alst
ruct
ure”
The
seco
ndst
epof
the
solu
tion
ofth
etim
ein
depe
nden
tSc
hröd
inge
req
uation
usin
gth
ead
iaba
tic
ansa
tzco
nsists
ofso
lvin
gfo
rth
enu
clea
rpr
oble
m.
We
repl
ace,
inEq.
(3.1
6),
ψ(e)
k(q;Q
)(T
(n) (Q,∂
Q)ψ(n) (Q))
+
ψ(n) (Q)(
T(e) (q,Q
,∂q)ψ(e)
k(q;Q
)+V(q,Q
)ψ(e)
k(q;Q
))
︸︷︷
︸
=E
(e)
k(Q
)ψ(e)
k(q;Q
)
≈Eψ(n) (Q)ψ(e)
k(q;Q
),
mul
tipl
ybyψ(e)∗
l(q;Q
)an
din
tegr
ate
over
the
elec
tron
icco
ordi
nate
s;be
caus
eth
eψ(e)
k(q;Q
)ar
eor
thon
orm
al(a
tev
ery
poin
tQ
),on
eob
tain
s
T(n) (Q,∂
Q)ψ(n)
j,k(Q
)+E
(e)
k(Q
)ψ(n)
j,k(Q
)=Ej,kψ(n)
j,k(Q
).(3
.18)
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t60
InEq.
(3.1
8),E
(e)
k(Q
)=Vk(Q
)is
the
pote
ntia
len
ergy
surfac
eof
the
adia
batic
elec
tron
icst
atek,a
ndEj,kar
ead
iaba
tic
vibr
onic
eige
nval
ues,
whi
chm
aybe
give
nasEj,k=Ej(k)+E
el k,
whe
reE
el kis
defin
edas
the
min
umum
ofth
epo
tent
ials
urfa
ceVk(Q
)(s
eeal
soFi
gure
3.1
).So
met
imesE
el kis
calle
dth
e“e
lect
roni
c”en
ergy
,w
hich
isan
over
-sim
plifi
cation
.
3.3.
4Bor
n-O
ppen
heim
erex
pans
ion,
adia
batic
basis
Stric
tly,
the
solu
tion
sψ(e)
k(q;Q
)of
Eq.
(3.1
7)ar
eno
tph
ysic
ally
obse
rvab
lequ
antities
10,
beca
use
they
are
notto
talw
ave
func
tion
sof
the
mol
ecul
arsy
stem
.H
owev
er,sinc
eth
eyfo
rma
com
plet
eba
sisse
tof
the
elec
tron
iclin
earsp
ace,
the
tota
lwav
efu
nction
may
bede
com
pose
das
Ψ(q,Q
)=∑
k
ψ(n)
k(Q
)ψ(e)
k(q;Q
)(3
.19)
Eq.
(3.1
9)is
som
etim
esca
lled
the
Bor
n-O
ppen
heim
erex
pans
ion.
The
expa
nsio
nco
effici
ents
ψ(n)
k(Q
)fo
rmth
usco
ntra
varia
ntve
ctor
s,w
here
asth
ead
iaba
tic
wav
efu
nction
sψ(e)
k(q;Q
)fo
rmco
varia
ntve
ctor
sin
afo
rmal
lyin
finite
vect
orsp
ace.
10Eve
nm
odul
oa
phas
efa
ctor
.
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t61
The
set{ψ
(e)
k(q;Q
)|k=
1,2,...}
,w
hereψ(e)
k(q;Q
)is
aso
lution
ofEq.
(3.1
7),de
fines
the
adia
batic
basis
ofth
eel
ectr
onic
prob
lem
.
Cle
arly,e
achψ(n)
k(Q
)m
ayon
itstu
rnbe
deco
mpo
sed
inei
genf
unct
ionsψ(n)
j,k(Q
)of
the
nucl
ear
prob
lem
:
ψ(n)
k(Q
)=∑
j
c j,kψ(n)
j,k(Q
).(3
.20)
For
the
sam
ere
ason
asgi
ven
abov
efo
rth
eel
ectr
onic
wav
efu
nction
s,th
eψ(n)
j,k(Q
)ar
eno
tph
ysic
ally
obse
rvab
le.
How
ever
,a
tota
lad
iaba
tic
wav
efu
nction
Ψ(q,Q
)=ψ(n) (Q)·ψ
(e) (q;Q
)be
com
esan
ob-
serv
able
quan
tity
,of
ten
toa
very
good
appr
oxim
atio
n,w
hen
the
non-
adia
batic
coup
lings
can
bene
glec
ted.
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t62
3.4
Non
-adi
abat
iceff
ects
,di
abat
icpo
tent
iale
nerg
ysu
rfac
es
Thi
sse
ctio
nfo
llow
scl
osel
y[3
7].
3.4.
1N
on-a
diab
atic
coup
ling
mat
rix
Let
Λjk
=δ jkT
(n) (Q,∂
Q)
−∫
dτ qψ(e)∗
j(q;Q
)T
(n) (Q,∂
Q)ψ(e)
k(q;Q
).(3
.21)
The
sequ
antities
are
herm
itia
nop
erat
ors
onth
enu
clei
.T
hey
defin
ea
mat
rixΛ
whi
chis
the
repr
esen
tation
inth
ead
iaba
tic
basis
ofth
atpa
rtin
the
non-
rela
tivi
stic
ham
ilton
ian
that
coup
les
vibr
atio
nsto
elec
tron
icm
otio
nno
n-ad
iaba
tica
lly.
Off-d
iago
nalm
atrix
elem
ents
Λjk
(i6=k)
are
iden
tica
lto
the
quan
titiesH
(n)
jkm
ention
edon
page
10.
Inth
ead
iaba
tic
appr
oxim
atio
n,al
lm
atrix
elem
ents
Λij=
0.T
his
isso
met
imes
also
calle
dth
eBor
n-O
ppen
heim
erad
iaba
tic
appr
oxim
atio
nan
d,m
ore
com
mon
ly,al
thou
ghle
sspr
ecisel
y,al
soth
eBor
n-O
ppen
heim
erap
prox
imat
ion.
Mole
cula
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ynam
ics
2019
Pro
f.R
.M
arq
uard
t63
Aslig
htly
less
strin
gent
appr
oxim
atio
n,so
met
imes
calle
dth
eBor
n-H
uang
adia
batic
appr
oxi-
mat
ion,
requ
ires
that
only
the
offdi
agon
alel
emen
tsof
Λbe
negl
ecte
d.
Inor
der
todi
scus
sth
efo
rmof
non-
adia
batic
coup
ling
mat
rix
elem
ents
inm
ore
deta
il,le
tQ
beth
eto
tals
etof
mas
swei
ghte
dca
rtes
ian
coor
dina
tes
ofth
enu
clei
11.
The
n,th
eex
pres
sion
forT
(n) (Q,∂
Q)from
Eq.
(3.1
0)be
com
es
T(n) (Q,∂
Q)=−
1
2M
∑
n
∂2 Qn,
(3.2
2)
whe
reM
isth
eav
erag
ednu
clea
rm
ass
ofth
em
olec
ule.
One
may
then
derive
the
follo
win
gex
pres
sion
forno
n-ad
iaba
tic
coup
ling
mat
rixel
emen
ts:
Λjk
=1
2M
(
2∑
n
(Fjk
;n(Q
)·∂
Qn)+Gjk(Q
))
=1
2M(2Fjk(Q
)·∂
Q+Gjk(Q
))(3
.23)
11T
his
isno
tth
ebes
tch
oice
for
prag
mat
icca
lcul
atio
nsof
non-
adia
batic
coup
ling
mat
rix
elem
ents
!
Mole
cula
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ynam
ics
2019
Pro
f.R
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arq
uard
t64
For
the
rem
aind
erof
this
disc
ussion
,we
adop
tth
eno
tation
that
bold
quan
tities
such
asQ
orFjk
are
vect
ors
orm
atric
esde
fined
inth
e(fi
nite
dim
ension
al)
coor
dina
tesp
ace
ofnu
clei
,w
here
asun
derli
ned
quan
tities
such
asΛ
are
mat
rices
orve
ctor
sde
fined
inth
e(f
orm
ally
infin
ite)
vect
orsp
ace
defin
edby
the
Bor
n-O
ppen
heim
erex
pans
ion
Eq.
(3.1
9).
For
the
part
icul
arch
oice
ofm
ass
wei
ghte
dnu
clea
rco
ordi
nate
sm
ade
abov
e,th
equ
antities
Fjk
are
give
nby
Fjk(Q
)=
∫
dτ qψ(e)∗
j(q;Q
)∂
Qψ(e)
k(q;Q
);(3
.24)
quan
tities
ofth
isty
pear
eal
soca
lled
non-
adia
batic
deriva
tive
coup
lings
.It
can
besh
own
that
the
mat
rixF
isan
ti-h
erm
itia
n.Fu
rthe
rmor
e,if
the
adia
batic
basis
isco
mpo
sed
ofre
alfu
nction
s,th
edi
agon
alco
mpo
nent
sof
this
mat
rixva
nish
iden
tica
lly:Fjj(Q
)≡
0.
The
quan
titiesGjk
are
give
nby
Gjk(Q
)=
∫
dτ qψ(e)∗
j(q;Q
)(∂
Q·∂
Q)ψ(e)
k(q;Q
);(3
.25)
they
are
calle
dno
n-ad
iaba
tic
scal
arco
uplin
gs.
Mole
cula
rQ
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mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t65
Whi
leth
esc
alar
coup
lings
are
norm
ally
smal
lan
dof
ten
negi
glib
le,
the
deriv
ativ
eco
uplin
gsm
ayin
deed
beco
me
quite
larg
e.It
isst
raig
htfo
rwar
dto
show
12th
at
Fjk(Q
)=
∫
dτ qψ(e)∗
j(q;Q
)(∂QH
(e))ψ(e)
k(q;Q
)
Vk(Q
)−Vj(Q)
(3.2
6)
whe
reH
(e)=T
(e) (q,Q
,∂q)+V(q,Q
).Pra
gmat
ical
ly,in
elec
tron
icst
ruct
ure
calc
ulat
ions
,∂QH
(e)=∂QV(q,Q
),w
hereV(q,Q
)is
the
tota
lmol
ecul
arCou
lom
bpo
tent
ial.
Thu
s,at
thos
epo
sition
sin
confi
gura
tion
spac
ew
here
two
differ
ent
PESVj
andVk
cros
s,th
eno
n-ad
iaba
tic
deriv
ativ
eco
uplin
gsm
aydi
verg
e.So
met
imes
dive
rgen
ceis
avoi
ded
ifth
enu
mer
ator
inEq.
(3.2
6)va
nish
es,
i.e.
bysy
mm
etry
.D
iver
genc
eof
deriv
ativ
eco
uplin
gsar
epa
rtic
ular
lyim
port
antw
hen
adia
batic
pote
ntia
lsbe
com
esing
ular
,e.g
.at
coni
cali
nter
sect
ions
.
Inge
nera
l,ho
wev
er,
whe
n“a
diab
atic
elec
tron
icst
ates
”ar
ewel
lse
para
ted
inen
ergy
(i.e
.|Vj(Q)−Vk(Q
)|>>
0in
the
rele
vant
regi
onof
confi
gura
tion
spac
e),
the
adia
batic
ap-
prox
imat
ion
isgo
od.
Qui
teof
ten
the
elec
tron
icgr
ound
stat
eof
clos
edsh
ellsy
stem
sis
wel
lse
para
ted
from
high
erly
ing
stat
es,an
dth
ead
iaba
tic
appr
oxim
atio
nis
good
.
12by
appl
ying
the
grad
ient
∂Q
tobot
hside
sof
Eq.
(3.1
7)
Mole
cula
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ynam
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2019
Pro
f.R
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Whe
nth
eap
prox
imat
ion
brea
ksdo
wn,
inpa
rtic
ular
atdi
verg
ence
sof
the
deriv
ativ
eco
uplin
gs,
adia
batic
stat
esbe
com
em
eani
ngle
ss.
The
prag
mat
icca
lcul
atio
nof
non-
adia
batic
deriv
ativ
ean
dsc
alar
coup
ling
elem
ents
,an
dth
eso
lution
ofth
efu
lltim
ein
depe
nden
tSc
hröd
inge
req
uation
are
com
plic
ated
.In
case
sw
here
the
adia
batic
appr
oxim
atio
nis
likel
yto
brea
kdo
wn,
othe
rpo
ssib
ilities
forth
eso
lution
ofth
eco
uple
dnu
clea
ran
del
ectr
onic
prob
lem
shou
ldbe
cons
ider
ed.
One
such
poss
ibili
tyis
the
tran
sfor
mat
ion
ofth
ead
iaba
tic
basis
desc
ription
toth
atof
the
diab
atic
basis.
3.4.
2D
iaba
tic
base
s
The
Bor
n-O
ppen
heim
erex
pans
ion
Eq.
(3.1
9)m
akes
itev
iden
tth
atth
ead
iaba
tic
basis
isju
ston
lyon
epo
ssib
leba
sisfo
rth
ere
pres
enta
tion
ofth
eto
taln
ucle
aran
del
ectr
onic
wav
efu
nction
.O
nem
ight
then
ask,
whe
ther
ther
eco
uld
not
beba
sesψ(e)
k(q;Q
)ot
her
than
the
adia
batic
basis,
inw
hich
the
num
erat
orin
Eq.
(3.2
6)va
nish
esin
the
entire
nucl
earco
nfigu
ration
spac
e,av
oidi
ngth
ussing
ular
itie
s.
Mole
cula
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ynam
ics
2019
Pro
f.R
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arq
uard
t67
Letψ(e) (q;Q
)be
the
(cov
aria
nt)
vect
orin
the
(for
mal
lyin
finite)
vect
orsp
ace
defin
edby
the
Bor
n-O
ppen
heim
erex
pans
ion,
that
isco
mpo
sed
ofth
ead
iaba
tic
elec
tron
icwav
efu
nction
sψ(e)
k(q;Q
)(i.e
.th
eso
lution
sof
Eq.
(3.1
7)).
Acc
ordi
ngly,le
tψ(n) (q;Q
)be
the
(con
trav
ari-
ant)
vect
or,in
the
dual
vect
orsp
ace,
com
pose
dof
the
adia
batica
llyse
para
ted
nucl
ear
wav
efu
nction
sψ(n)
k(Q
).T
hen,
form
ally,
the
Bor
n-O
ppen
heim
erex
pans
ion
may
begi
ven
byth
esc
alar
prod
uct
Ψ(q,Q
)=ψ(n) (q;Q
)·ψ
(e) (q;Q
)(3
.27)
Let
furt
herm
ore
ψ(e) (q;Q
)=U(Q
)ψ(e) (q;Q
),(3
.28)
whe
reψ(e) (q;Q
)is
ane
w(c
ovar
iant
)ve
ctor
inth
eve
ctor
spac
ede
fined
byth
eBor
n-O
ppen
heim
erex
pans
ion
andU
isa
unitar
ym
atrix
:U·U
†=1 .
Cle
arly,U
=U(Q
),an
dun
itar
itym
ust
hold
inth
een
tire
confi
gura
tion
spac
e.
Mole
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ynam
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2019
Pro
f.R
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arq
uard
t68
Aco
mpo
nent
ofψ(e) (q;Q
)is
inde
eda
new
elec
tron
icwav
efu
nction
ψ(e)
k(q;Q
),w
hich
isde
fined
asa
linea
rco
mbi
nation
ofad
iaba
tic
elec
tron
icwav
efu
nction
s:
ψ(e)
k(q;Q
)=∑
j
Ukj(Q)ψ(e)
j(q;Q
)(3
.29)
The
coeffi
cien
tsUkj(Q)
ofth
isex
pans
ion
need
yet
tobe
dete
rmin
ed.
Eq.
(3.2
8)de
fines
aba
sis
tran
sfor
mat
ion.
We
note
that
,in
orde
rto
keep
the
desc
ription
ofth
eto
talw
ave
func
tion
inva
riant
unde
rth
isba
sis
tran
sfor
mat
ion,
the
adia
batica
llyse
para
ted
nucl
ear
wav
efu
nction
sne
edto
betr
ansf
orm
edac
cord
ingl
y:
ψ(n) (q;Q
)=U
† (Q)ψ(n) (q;Q
),(3
.30)
thus
ψ(n)
j(Q
)=∑
k
U∗ kj(Q)ψ(n)
k(Q
)(3
.31)
Mole
cula
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ynam
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2019
Pro
f.R
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arq
uard
t69
Itca
nbe
show
nth
at,
due
toth
eba
sis
tran
sfor
mat
ion
ofEq.
(3.2
8),
the
mat
rixF
ofno
n-ad
iaba
tic
deriv
ativ
eco
uplin
gsis
tran
sfor
med
as
F=U
† FU+U
† ∂QU
(3.3
2)
foral
lval
ues
ofQ
.
LetU(Q
)be
the
mat
rixsa
tisf
ying
the
follo
win
geq
uation
s:
0=F(Q
)U(Q
)+∂
QU(Q
)(3
.33)
Not
eth
atth
isis
afo
rmal
lyin
finite
set
ofpa
rtia
ldi
ffer
ential
equa
tion
sfo
rth
efu
nction
sUjk(Q
).
The
elec
tron
icwav
efu
nction
sψ(e)
k(q;Q
)ob
tain
edw
ith
the
solu
tion
sof
thes
eeq
uation
sar
eca
lled
diab
atic
elec
tron
icwav
efu
nction
san
dth
eco
rres
pond
ing
basis
diab
atic
basis.
Dia
-ba
tic
stat
esar
eth
usde
fined
such
that
the
non-
adia
batic
deriva
tive
coup
ling
mat
rixva
nish
esid
entica
lly.
The
reis
afo
rmal
proo
fth
atso
lution
sof
Eq.
(3.3
3),
and
diab
atic
stat
es,
exist,
inth
eid
eal
case
ofa
com
plet
eba
sisse
t.In
prax
is,h
owev
er,o
nem
usttr
unca
teth
elin
earel
ectr
onic
spac
e,an
din
that
case
itca
nbe
show
(see
[37,
and
refe
renc
esci
ted
ther
ein]
),th
atst
rict
diab
atic
stat
esdo
not
exist,
inge
nera
l,ex
cept
for
diat
omic
mol
ecul
es.
Mole
cula
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ynam
ics
2019
Pro
f.R
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arq
uard
t70
How
ever
,it
ispo
ssib
leto
defin
e-
and
calc
ulat
epr
agm
atic
ally
-qu
asi-di
abat
icst
ates
,w
hich
are
alm
ostdi
abat
ic,a
ndw
hich
allo
wto
rem
ove
alls
ingu
larities
inth
eno
n-ad
iaba
tic
coup
lings
.
The
reex
ist
toda
ym
any
way
sof
calc
ulat
ing
(qua
si-)
diab
atic
stat
es,
whi
chw
illal
lde
fine
differ
ent
(qua
si-)
diab
atic
base
s.O
nepo
ssib
ility
invo
lves
the
bloc
kdi
agon
aliz
atio
nof
the
elec
tron
icha
milt
onia
n(s
ee[ 3
8],w
hich
also
give
san
over
view
ofot
her
met
hods
used
inth
efie
ld).
3.4.
3D
iaba
tic
pote
ntia
len
ergy
surfac
es
Exp
ress
edin
atr
unca
ted,
quas
i-dia
batic
basis,
the
tota
lham
ilton
ian
mat
rixH
can
beof
the
form
H=
H(n)
11H
(n)
12···
H(n)
21H
(n)
22···
. . .. . .
. ..
(3
.34)
Mole
cula
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ynam
ics
2019
Pro
f.R
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arq
uard
t71
whe
re
H(n)
ik=H
(n)
ik(Q
,∂Q)=
{
T(n) (Q,∂
Q)+
Vk(Q
)(i=k)
Wik(Q
)(i6=k)
(3.3
5)
Thi
sfo
rmis
som
etim
esca
lled
diab
atic
ham
ilton
ian.
The
diag
onal
elem
entsVk(Q
)ar
eca
lled
diab
atic
pote
ntia
len
ergy
surfac
es.
The
off-d
iago
nalel
emen
tsW
ik(Q
)=W
ki(Q)
(ass
um-
ing
real
elec
tron
icwav
efu
nction
s)ar
eso
met
imes
calle
dno
n-ad
iaba
tic
coup
ling
pote
ntia
ls.
Figu
re3.
3de
pict
ssc
hem
atic
ally
adia
batic
and
diab
atic
pote
ntia
lfu
nction
sfo
ra
diat
omic
mol
ecul
e.
Dia
batic
pote
ntia
lsar
eof
ten
smoo
ther
than
adia
batic
pote
ntia
ls,a
ndso
met
imes
they
corr
elat
ece
rtai
nm
olec
ular
prop
erties
and
char
gedi
strib
utio
ns.
Not
e,ho
wev
er,th
atth
ese
tof
diab
atic
pote
ntia
lsVkQ)
isin
suffi
cien
tto
desc
ribe
the
nucl
ear
quan
tum
dyna
mic
s;fo
rthi
spu
rpos
eit
ism
anda
tory
tokn
owth
eto
tals
etof
coup
ling
pote
ntia
lsgl
obal
ly,i.e
.in
the
entire
confi
gura
tion
spac
e.
Mole
cula
rQ
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ynam
ics
2019
Pro
f.R
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arq
uard
t72
Fig
ure
3.3
:Sch
emat
icvi
ews
ofad
iabat
ic(lef
tha
ndside
)an
ddia
bat
icpot
ential
ener
gyfu
nction
s(r
ight
hand
side
)in
adi
atom
icm
olec
ule.
The
dott
edlin
esin
dica
teth
ere
lative
pos
itio
nof
the
corr
espon
ding
diab
atic
(lef
t)
and
adia
batic
(rig
ht)
pot
ential
s.N
ote
that
abin
itio
poi
nts
are
inex
iste
nton
the
diab
atic
pot
ential
s.
a
Ek(
4)
El(
4)
Ek(
2)
El(
2)
El(
1)
Ek(
1)E
l(3)
Ek(
3)
r(2
)r(
3)
r(4
)r(
1)
r
Vj(r
)b
r(2
)r(
3)
r(4
)r(
1)
r
Vj(r
)
Mole
cula
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ynam
ics
2019
Pro
f.R
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uard
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3.5
Exa
mpl
es
3.5.
1Am
mon
iadi
ssoc
iation
Fig
ure
3.4
:(8
,7)
CA
S-S
CF
stud
yof
plan
ar
amm
onia
diss
ocia
tion
[39]
;r
ison
eN
Hbon
d
leng
th.
⋄A1
stat
ein
C2v
∗B1
stat
ein
C2v
Mole
cula
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ynam
ics
2019
Pro
f.R
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uard
t74
−1
~ 4
400
hc
cm
−1
hc
cm
~ 5
9000
Fig
ure
3.5
:Pot
ential
ener
gyda
tafo
rth
e
NH3→
NH2+H
reac
tion
.
⋄CCSD
(T)
data
∗M
RCIda
taA1
stat
ein
C2v
B1
stat
ein
C2v
Mole
cula
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ynam
ics
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Pro
f.R
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uard
t75
3.5.
2M
etha
nest
ereo
mut
atio
npo
tent
ial
Fig
ure
3.6
:Ste
reom
utat
ion
pot
ential
sfo
r
met
hane
(fro
m[4
0]).
The
sepo
tent
ials
wer
eca
lcul
ated
from
agl
obal
,an
alyt
ical
repr
esen
tation
ofth
ePES
ofm
etha
nein
the
grou
ndel
ectr
onic
stat
e[4
1].
Tha
tre
pres
en-
tation
was
intu
rnob
tain
edfrom
ano
n-lin
ear
adju
stm
ent
toM
RD
-CIda
taan
dex
perim
enta
llyav
aila
ble
over
tone
tran
sition
sof
the
CH
chro
mop
hore
inCHD3.
The
yco
rres
pond
tost
eepe
stde
scen
tpa
ths
from
the
resp
ective
sadd
lepo
int
stru
ctur
es.
Mole
cula
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ynam
ics
2019
Pro
f.R
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t76
3.5.
3Vib
ration
alte
rmva
lues
inhy
drog
enflu
orid
e
The
follo
win
gta
ble,
adap
ted
from
[42]
,allo
wsfo
ran
over
view
of“s
tate
ofth
ear
t”co
mpu
tation
ofvi
brat
iona
ltra
nsitio
nwav
enu
mbe
rs.
νexp
n/c
m−1
(νth n−νexp
n)/
cm−1
nCCSD
[T]
CCSD
[T]-R12
CCSD
TCCSD
T-R
12CCSD
T-R
12+
rel
1032
311.
79[4
3]-5
3.68
4.55
-18.
0529
.53
5.83
929
781.
33[4
3]-2
1.66
17.0
5-1
.02
29.8
68.
42
827
097.
87[4
3]-0
.41
23.6
310
.51
28.7
49.
48
724
262.
18[4
3]12
.26
25.7
217
.16
26.2
69.
18
621
273.
69[4
3]18
.43
24.6
619
.81
22.7
57.
90
518
130.
97[4
3]19
.05
21.5
819
.44
18.6
16.
04
414
831.
63[4
4]18
.39
17.4
917
.04
14.3
34.
10
311
372.
78[4
4]14
.85
12.8
913
.29
9.50
2.25
277
50.7
9[4
4]10
.22
8.22
8.84
6.05
0.76
139
61.4
2[4
4]5.
153.
844.
292.
62-0
.07
Tab
le3.
1:
Vib
ration
alte
rmva
lues
inH
F:ex
per
imen
tan
dth
eory
.
Mole
cula
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ynam
ics
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Pro
f.R
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uard
t77
App
endi
xto
the
lect
ure
note
s
AD
iago
naliz
atio
nof
a2×
2sy
mm
etric
(her
mitia
n)m
atrix
Let
A=
(S−D
CC
S+D
)
(A.1
)
whe
reD
≥0.
A.1
Det
erm
inat
ion
ofth
eei
genv
alue
s(λ
1an
dλ2)
Eig
enva
lues
are
zero
sof
the
dete
rmin
ant
ofth
ese
cula
rm
atrix:
D(
Aλ
)
=
∣ ∣ ∣ ∣
S−D
−λ
CC
S+D
−λ
∣ ∣ ∣ ∣=
(S−λ)2−
(D2+C
2)
︸︷︷
︸
≡W
2
! =0
(A.2
)
⇒λ
=S±W
(W=√D
2+C
2)
(A.3
)
Mole
cula
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ynam
ics
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Pro
f.R
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uard
t78
Bec
auseD
2+C
2≥
0ei
genv
alue
sw
illal
way
sbe
real
.It
can
besh
own
that
allhe
rmitia
nm
atic
es,of
whi
chsy
mm
etric
mat
rices
are
spec
ialc
ases
,ha
vere
alei
genv
alue
s.
Let
λ1=S−W
(A.4
)
λ2=S+W
(A.5
)
such
thatλ2−λ1=2W
≥0.
Mole
cula
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ynam
ics
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Pro
f.R
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t79
A.2
Det
erm
inat
ion
ofth
eei
genv
ecto
rbe
long
ing
toλ1
The
vect
orz1=
(Z11
Z21
)
isso
lution
of
Aλ1z1=0
⇔{(S
−D
−(S
−W
)Z11+CZ21=0
CZ11+(S
+D
−(S
−W
)Z21=0
(A.6
)
Bec
auseD(A
λ1)=
0,th
etw
oeq
uation
sar
elin
early
depe
nden
t,an
don
lyon
eeq
uation
is
effec
tive
lyto
beus
ed,e.
g.th
efir
ston
e:∗
Z11=−
C
W−DZ21
(A.7
)
∗O
nem
aych
eck
that
the
solu
tion
ofth
efirs
teq
uation
isau
tom
atic
ally
aso
lution
ofth
ese
cond
one:
CZ11+(S
+D−(S
−W
)Z21=(−C
2/(W
−D)+D+W
)Z21=(−C
2−D
2+(D
2+C
2))Z21/(W
−D)=0.
Mole
cula
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ynam
ics
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Pro
f.R
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t80
The
com
plet
eso
lution
isob
tain
edup
onco
nsid
erat
ion
ofth
eno
rmal
isat
ion
cond
itio
n:
Z2 11+Z
2 21=1⇒(
C2
(W−D)2+1)
Z2 21=1
⇒Z
2 21=
(W−D)2
C2+(W
−D)2
=(W
−D)2
C2+(W
)2−2D
W+D
2
=(W
−D)2
2(D
2+C
2−DW
)
=(W
−D)2
2(W
−D)W
=W
−D
2W
(A.8
)
Mole
cula
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ynam
ics
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Pro
f.R
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t81
Con
sequ
ently:
Z21=±√
1 2
W−D
W.
(A.9
)
Itis
not
poss
ible
tode
term
ine
the
abso
lute
sign
ofth
eso
lution
!
The
abso
lute
phas
eof
the
eign
evec
torca
nnot
bede
term
ined
.
The
expr
ession
forZ11
can
befu
rthe
rsim
plifi
ed.
Let
C
W−D
=sig(C
)
√D
2+C
2−D
2
W−D
=sig(C
)
√
W+D
W−D
(A.1
0)
whe
reth
esign
umfu
nction
(sig(x))
isde
fined
by
sig(x)=
{1,
six≥
0−1,
six<
0(A
.11)
Mole
cula
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ynam
ics
2019
Pro
f.R
.M
arq
uard
t82
One
obta
ins
forZ11
Z11
=−
C
W−DZ21
=∓sig(C
)√
W+D
W−D
√
1 2
W−D
W
=∓sig(C
)√
1 2
W+D
W(A
.12)
Mole
cula
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ynam
ics
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Pro
f.R
.M
arq
uard
t83
Itis
usef
ulto
cons
ider
the
ratio
R=D W
(A.1
3)
whe
re0≤R
≤1.
The
eige
nvec
torpe
rtai
ning
toλ1=S−W
isth
engi
ven
as
Z11=∓sig(C
)√
1 2(1
+R)
Z21=±√
1 2(1
−R)
whi
chyi
elds
,as
afu
nction
ofth
esign
ofC
,tw
opo
ssib
lech
oice
s:
C≥
0C<
0
Z11=−√
1 2(1
+R)
Z11=√
1 2(1
+R)
Z21=√
1 2(1
−R)
Z21=√
1 2(1
−R)
ou
C≥
0C<
0
Z11=√
1 2(1
+R)
Z11=−√
1 2(1
+R)
Z21=−√
1 2(1
−R)
Z21=−√
1 2(1
−R)
(A.1
4)
Mole
cula
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ynam
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Pro
f.R
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uard
t84
A.3
Det
erm
inat
ion
ofth
eei
genv
ecto
rbe
long
ing
toλ2
The
vect
orz2=
(Z12
Z22
)
isso
lution
of
Aλ2z2=0
⇔{(S
−D
−(S
+W
)Z12+CZ22=0
CZ12+(S
+D
−(S
+W
)Z22=0
(A.1
5)
Sim
ilarly
asfo
rz1,on
em
aylim
itth
eca
lcul
atio
nto
one
ofth
etw
oeq
uation
s:
Z12=
C
W+DZ22
(A.1
6)
Mole
cula
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ynam
ics
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Pro
f.R
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uard
t85
Using
the
norm
alisat
ion
cond
itio
n
Z2 12+Z
2 22=1⇒(
C2
(W+D)2+1)
Z2 22=1
⇒Z
2 22=
(W+D)2
C2+(W
+D)2
=(W
+D)2
C2+(W
)2+2D
W+D
2
=(W
+D)2
2(D
2+C
2+DW
)
=(W
+D)2
2(W
+D)W
=W
+D
2W
(A.1
7)
Mole
cula
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ynam
ics
2019
Pro
f.R
.M
arq
uard
t86
one
obta
ins,
cons
eque
ntly,
Z22
=±√
1 2
W+D
W(A
.18)
=
√
1 2(1
+R)
(A.1
9)
and
the
sim
plifi
edex
pres
sion
forZ12
byse
ttin
g
C
W+D
=sig(C
)
√D
2+C
2−D
2
W+D
=sig(C
)
√
W−D
W+D
(A.2
0)
and
Z12
=C
W+DZ22
=±sig(C
)√
W−D
W+D
√
1 2
W+D
W
=±sig(C
)√
W−D
2W
(A.2
1)
Mole
cula
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ynam
ics
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Pro
f.R
.M
arq
uard
t87
The
eige
nvec
torpe
rtai
ning
toλ2=S+W
is,th
en:
Z12=±sig(C
)√
1 2(1
−R)
Z22=±√
1 2(1
+R)
whi
chyi
elds
two
poss
ibili
ties
,de
pend
ing
onth
esign
ofC
:
C≥
0C<
0
Z12=√
1 2(1
−R)
Z12=−√
1 2(1
−R)
Z22=√
1 2(1
+R)
Z22=√
1 2(1
+R)
ou
C≥
0C<
0
Z12=−√
1 2(1
−R)
Z12=√
1 2(1
−R)
Z22=
−√
1 2(1
+R)
Z22=
−√
1 2(1
+R)
(A.2
2)
Mole
cula
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ynam
ics
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Pro
f.R
.M
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uard
t88
A.4
Bas
istr
ansf
orm
atio
nm
atrix
Bec
ause
the
abso
lute
sign
ofan
eige
nvec
toris
unkn
own,
four
poss
ible
vers
ions
ofth
etr
ans-
form
atio
nm
atrix
Zar
epo
ssib
le.
The
sem
atric
esde
fine
the
eige
nvec
tors
inth
e2×
2ba
sis
orig
inal
lyus
edto
set
upth
em
atrix
repr
esen
tation
ofth
eha
milt
onia
n.
Z=
(Z11
Z12
Z21
Z22
)
=
sig(C
)
√1+R
√2
sig(C
)
√1−R
√2
−√1−R
√2
√1+R
√2
or
sig(C
)
√1+R
√2
−sig(C
)
√1−R
√2
−√1−R
√2
−√1+R
√2
or
−sig(C
)
√1+R
√2
sig(C
)
√1−R
√2
√1−R
√2
√1+R
√2
or
−sig(C
)
√1+R
√2
−sig(C
)
√1−R
√2
√1−R
√2
−√1+R
√2
Mole
cula
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ynam
ics
2019
Pro
f.R
.M
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uard
t89
BT
ight
bind
ing
ham
ilton
ian
The
tigh
t-bi
ndin
gha
milt
onia
nis
the
repr
esen
tation
ofth
een
ergy
oper
ator
ina
basis
ofN
dege
nera
tequ
antu
mst
ates
with
anen
ergyE
and
neig
hbor
-to-
neig
hbor
coup
lingV
:
H=
EV
0···
00
VEV
···
00
0V
E···
00
. . .. . .
. . .. .
.. . .
. . .0
00···EV
00
0···V
E
(B.2
3)
The
eige
nsta
tesre
sultin
gfrom
the
diag
onal
izat
ion
ofth
eha
milt
onia
nsp
read
ina
band
betw
een
E−2V
andE
+2V
acco
rdin
gto
the
follo
win
gfo
rmul
a:
En=E
+2V
cos(nαN)
(n=1,...,N)
(B.2
4)
whe
reαN=π/(N
+1)
.
Mole
cula
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ynam
ics
2019
Pro
f.R
.M
arq
uard
t90
The
eige
nsta
tenu
mbe
rn
isgi
ven
by |n〉=
N∑ m=1
Zmn|m
〉0(B
.25)
whe
re
Zmn=
√
2
N+1sin(m
nαN)
(n,m
=1,...,N)
(B.2
6)
and|m
〉0is
aba
sis
stat
e.
Inth
ene
xttw
ose
ctio
nsth
epr
oofof
thes
est
atem
ents
isgi
ven
follo
win
gre
f.45
.T
heei
gen-
valu
esof
the
tigh
t-bi
ndin
gm
atrix
are
also
know
nfrom
Hüc
kel
theo
ryth
atgo
esba
ckto
1932
[46]
(see
also
ref.
47fo
ra
very
peda
gogi
cald
eriv
atio
n).
Inth
isco
ntex
t,K
utze
lnig
gha
sal
sopr
opos
eda
nice
over
view
ofth
eth
eory
[48]
.
Not
eth
atei
genv
alue
san
dei
genv
ecto
rsca
nbe
form
ulat
edal
tern
ativ
ely
as
En=E
−2V
cos(nαN)
(n=1,...,N)
(B.2
7)
Zmn=
√
2
N+1(−
1)m
sin(m
nαN)
(n,m
=1,...,N)
(B.2
8)
Mole
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Pro
f.R
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uard
t91
B.1
Pro
of:
eige
nval
ues
Let
|HEn|=
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
E−En
V0···
00
VE
−En
V···
00
0VE
−En···
00
. . .. . .
. . .. .
.. . .
. . .0
00···E
−En
V0
00···
VE
−En
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
(B.2
9)
beth
ede
term
inan
tof
the
secu
lar
mat
rix.
Itca
nbe
show
nby
indu
ctio
n(s
eeal
soW
alto
n,20
07[4
9]),
that
|HEn|=
VUN(−xn)
(B.3
0)
whe
rexn=
(En−E)/2V
andUN(x)
isth
eChe
bysh
evpo
lyno
mia
lof
the
seco
ndki
ndof
orde
rN
.
Hen
ce,th
eei
genv
alue
sar
eth
eze
ros
ofUN(−x).
Rec
allt
hatUN(x)=(−
1)NUN(−x)
[50]
.Fo
r|x|≤
1[5
0]
UN(x)=
sin((N
+1)arccos(x))
sin(arccos(x))
(B.3
1)
Mole
cula
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ynam
ics
2019
Pro
f.R
.M
arq
uard
t92
and
poss
ible
zero
sar
e
xn=cos
(nπ
N+1
)
(n=1,...,N)
(B.3
2)
from
whe
refo
llow
sEq.
(B.2
4).
B.2
Pro
of:
eige
nvec
tors
The
com
pone
ntsZmn
ofth
eei
genv
ecto
rre
pres
enta
tion
pert
aini
ngto
eige
nval
ueEn
mus
tso
lve
the
set
ofeq
uation
s
−2x
nZ1n+Z2n
=0
. . .Zm−1,n−2x
nZmn+Zm+1,n=
0. . .
ZN−1,n−2x
nZNn=
0
(B.3
3)
whe
reth
ela
steq
uation
isre
dund
ant.
Inad
dition
,th
eyar
eno
rmal
ized
:|Z
1n|2+...+
|ZNn|2=1.
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t93
The
Che
bysh
evpo
lyno
mia
lsof
the
seco
ndki
ndha
veth
ere
curr
ence
rela
tion
ship
[50]
:
U0(x)=
1(B
.34)
U1(x)=
2x(B
.35)
Um−1(x)−
2xUm(x)+Um+1(x)=
0(B
.36)
Let
Zmn≡c nUm−1(x
n)
(B.3
7)
whe
rec n
isa
norm
aliz
atio
nco
nsta
nt.
The
n
Z1n
=c n
(B.3
8)
Z2n
=c n
2xn
(B.3
9)
Zm+1,n=
2xnZmn−Zm−1,n
(B.4
0)
Bec
auseZN+1,n∝UN(x
n)=0,
the
last
setof
equa
tion
ssh
owth
atZmn
defin
edby
Eq.
(B.3
7)fu
lfills
Eq.
(B.3
3).
Hen
ce
Zmn=c nUm−1(x
n)
=c n
sin(m
nαN)
sin(nαN)
(B.4
1)
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t94
The
norm
aliz
atio
nco
nsta
ntis
give
nby
1=c2 n
N∑ m=1
U2 m−1(x
n)
(B.4
2)
Not
ing
that
13
N∑ m=1
sin2(m
nαN)=
N∑ m=1
(
2−ei2mnαN−e−
i2mnαN
4
)
=N 2
−1 4
N∑ m=1
ei2mnαN−
1 4
N∑ m=1
e−i2mnαN
=N 2
−1−
ei2(N
+1)nαN
4(
1−ei2nαN
)+1 4−
1−
e−i2
(N+1)nαN
4(
1−e−
i2nαN
)+1 4
=N 2
+1 2
(B.4
3)
13n ∑ k=0
xk=(1
−xn+1)/(1
−x)
Mole
cula
rQ
uantu
mD
ynam
ics
2019
Pro
f.R
.M
arq
uard
t95
beca
useei2(N
+1)nαN=ei2πn=1
for
alln
.T
here
fore
N∑ m=1
U2 m−1(x
n)=
N+1
2sin2(nαN)
(B.4
4)
and
c2 n=
2sin2(x
n)
N+1
(B.4
5)
One
then
obta
ins
Zmn=
sin(m
nαN)
√
2
N+1
(B.4
6)
whi
chpr
oves
Eq.
(B.2
6).
