Molecular Spectroscopy chemistry homework

This is homework for the Molecular Spectroscopy chemistry course. Any tutor could do it? 

1.Use the ladder operator formalism for harmonic oscillator to derive the selection rule on

⟨𝑣 ′|(𝑅 − 𝑅𝑒 )
𝑛 |𝑣”⟩ for arbitrary n.

2. For a heteronuclear diatomic molecule AB, the dipole moment function in the neighborhood of

R=Re is given by

𝜇(𝑅) = 𝑎 + 𝑏(𝑅 − 𝑅𝑒 ) + 𝑐(𝑅 − 𝑅𝑒 )
2 + 𝑑(𝑅 − 𝑅𝑒 )

3

In which a, b, c and d are constants. Treating this molecule as a harmonic oscillator (using ladder

operator), expand dipole moment in Taylor series around R2 and then calculate the relative intensity

of v=0->1, v=0->2 and v=0->3 transitions in terms of these constant and harmonic oscillator

constants μ and ω.

3. (McHale chapter10. Problem7) A general harmonic potential function for water is

𝑉 =
1

2
𝑘𝑟 (∆𝑟1)

2 +
1

2
𝑘𝑟 (∆𝑟2)

2 +
1

2
𝑘𝜃 (𝑟∆𝜃)

2 + 𝑘𝑟𝑟 ∆𝑟1∆𝑟2 + 𝑘𝑟𝜃 𝑟∆𝑟1∆𝜃 + 𝑘𝑟𝜃 𝑟∆𝑟2∆𝜃

The last three terms contain off-diagonal force constants, while the first three are diagonal. In

matrix form, this can be expressed as 2V=RTFR, where R=(∆𝑟1 ∆𝑟2 ∆𝜃) is the vector whose

elements are the internal coordinates. Find the symmetry coordinates S1, S2 and S3 for water,

and the diagonal force constant f which permits the potential energy in form written STfS

4. For raman spectroscopy, show that the following equation leads to a symmetric tensor, 𝛼𝜌𝜎 =

𝛼𝜎𝜌, in the limit 𝜔0 ≪ 𝜔𝑒𝑔 .

(𝛼𝜌𝜎 )𝑖𝑓 =
1

ℏ
∑[

⟨𝑖|𝜇𝜌|𝑛⟩⟨𝑛|𝜇𝜎 |𝑓⟩

𝜔0 + 𝜔𝑛𝑓 + 𝑖Γ

𝑛

−

⟨𝑖|𝜇𝜎 |𝑛⟩⟨𝑛|𝜇𝜌|𝑓⟩

𝜔0 − 𝜔𝑛𝑖 − 𝑖Γ𝑛
]

𝑛

r In>=/§n”Xn”u ” > far. excited electronic
states

^ 11<7--180%0>¥¥T÷.↳=**.>
=
cntereyenerkfasteleetrennohcm

=cXn”u ” / Mn.io/Xoo>
T
electronic transition dipole

.

wkri-won.tw?-nni.– Won “Tlarge in
Wi=Wz=W

ciisc-i-sok-z.E-i-U.in/n..xEu-..d–E—
Won

”
-1W

+¥Éñ÷¥¥É¥¥)
=É

✗ polarizability ,
=ÉÉ

selection rule

a-xot-ZI-ieai-IE.la?&-q.)QQj
-1

–
–

. .

Karla/xoi-xosoo-iz.BE?I
☒ Halation>

00=-1-1
042¥. transition

pelyeriahab.it#y.&-.y=&a-aisatensn.-l&!:&:&:&:*:)true
Oxy* Unity

→g
§
SO → @

Pg
size changes

.

←o-o-o→¥a=o 1¥40
8-8-5%21=0 ¥-0- + –

Czuib – rotation
math in>

Tn
#e’IET.IE?m1k3-l/0eS1XusslXR”>

II.u.ie?esmtEdlks=4oekXuskxrlutEXr7IXosSl&eS–so/ekxu’
tulku”> Ide>ftp.IUE/XpeD–vib,Transi-uendephe

. riot oJ=I/

I i¥¥:÷¥÷÷
”

sina.eu/#rah7iIL.?aat-r

solekxvtutr.sn/Xviil0/eS–Sfdrdrxqr,4Ecr.rDucr.rl4dr.M
Xu (R)

permanent =)drdetcr.NUCR.NO/eCr.R)=GeSelee-shekR)
d .-pete .

home one
IUECRJIXU,>

nuderuelr)
taylor expansion .

dipole UelR7=UelRe)t¥4R-Re) -1 hemoment It *
O

@ – –
‘

lxoluecrstxv.is

=UlReixt Fekete
④ < Xu' IR-Relaisortlenerl

Edrlre to transition dipole .

x
=

my
Txu , HR-Re)lXw,>

c. 3.3 I -Eau)’s (atat)
Attn>=#Intl> raising op
Aln> =Tnln – I > Lang op.
xsxuslatatln>
⇒

xodrnlnty-sxplntlntD-rncxutn-lstrntiexvln.is
orthenaml =Tn Su!n- I thtfv ‘s htt

00=11

selection rule .
① Terran dip FLIR to
the permanent n’booster d.pete
does net matter

0–0=0
asym

Sym← → →←→
0€20 0=5-0

UCR)=O NIKKO

Q Owo LIFO
1 OR
R X active

P I also homeunder diatonic maleate
ill does not have ZR abs .

t l R ② ou =I l
J– o Q at the sometimeofIt

,
OM-0,11

– l P c don’t need permanent rib dipole>

MH . C3 electronic
UB

na.>etq.io/exxvsixrs=imsaediffvenoE=lO,adown”PES .
motorhomes#de” >this>her”> Tks

g
”
I’d
” “>

smiuyk , uteteirtudr,
isoekxuluelrslxwilrfeidxrtuek
→ Colette.>sxutcxriluncrllxr”>Hes←
sole.la/udUecr7txuxl/0e7=sxwl#erl/elXuD

transitions,

£-4
dipole Ueglkktfecrsmluelrlrfekr.RS
troneg-oe-jdrofgcr.sk/BUdbfeIbR)

Ugelr)=UgeCRe)t2Lqe_frfr-Re)t – – –
lXvlUgeCR3lXu
=?*”¥a”÷:÷÷”me¥m

Frank -4XVIXv”> detains which 0
”

→U
‘

Condon
tragic-en has the Largest probability .Footer
ngelke) # O

‘¥i÷i¥¥¥
.

D. pdgaeenic vibrations .
µ . H . Zi
10.2 – F- Naomichi 3Ncard ,”Y ‘

3 card terrains is¥¥¥!Iy, 3. enad for nott.EE?mzeondforuetlinen
3N- 615) vibration for nonlinear

( linen)notate
.

27

.

ftm

–

A-x-k-m.r.mn 0-¥☐m
, ¥-p, -d -¥Em§

10 -¥a
. ¥ -a

/ →
/
Qana

,

b. *H=lÉ¥÷¥EÉ = .
Cc ↳ Cos

21m¥ -d) ¥*-m•)
‘

-1¥ -a)4¥-4–0
X.im/-aXz–mk-a-#sd5-0

T

Translation
calculate eigenstates et de Conant nodes)

ditmas
0

1÷÷÷÷¥÷¥
./1¥:¥Y¥|-O =¥m• 0

–

–
–

-¥m
,

12=0 ⇒ Lz=o

_tÉm☐4
‘

”

– ¥am↳”=o

= – L’s”

normalization I -_I
iii. iii.”714¥;) -4
Lili -11%2=1
E-RE c÷=F£l”=EÉ)
“Y÷÷÷÷¥)÷µ⇒”l÷¥:)
M=2MA +MB

Q =L’E

a”=oE o -E)(¥;)
=¥1 , – ¥93

Q%lmIm)k£
,

– 2(M¥m)”I,-111m¥:
9-i
–

-Mikki

Q5=(m⇒%1 , -11M¥)k9→
+ (Tmt)

”
29-3

Q”
‘ E-CI

sym

a
” F-c¥

asym

AM 0%-0-7 translational
.

why no bending ?
selection rule on=-4

IR
sepeetnue peak.
N-t ~ 3200cm

”

-2=-2- ICE loaoemt
-⇐c-

– •

sooocent

w=F¥m -I-I- 1200am -1-0
–

Topic 5 . Raman speetnayy
Mchale highor cooler perturbationheavy .
12 – ~ IT. D , S – E

① ¥cmtD=¥_zcn(⇒e→%m+1k>

.
–

Cnet> = Snk

ciiitu-z-h-fjle-iwkmtkmliot.lk>de
now substance Éii into

–

① ag-a.in ?

3-ecintb-fg-zcn%De-iwn-mtmlwc.is/ng–#*Y-zne-iwnmt.cm/wT-Dlns.fIe-iwkntTn1niktDlk
dtz

– a –

–
–
– –

.

Ciii a) = 1¥)%n|jéi%ñ↳mlÑHDIn>dt,
→

go.tte-iwkn-zcnlw.tk/k7dtz
–

1k>¥1 in>¥1m>

TPA
f-

Im> THE -1k>
Twa
–
– – . -2in>

h?¥
.
-Ein>

tart –
-1k> ¥1m>

Raman stokes anti stakes .

tinny
–

F-
– Em> –

g-
– – Ein>

vh-wz-ii.im> “±Éi
,

1k>

ÑHD=-ñ£É(eiwi-k-iEF-p-e-iwik.fi#*yWHz)—UJ-E-(eiwa-4-
.it#+o+e–iwa-4-tiE’aE-o)sn1WtW1k

> = -€n1ñE¥¥⇐’e±””
-¥kÉ÷!!¥É’e÷wik

= -±e-ik-ireius-4-EEEEeikie-ia.at
.

Umn

CIFHT-i-h-Y-z.nfje-iwnm-hle-mneiwzti-un.netӴ,
☒Jj ‘

e-iwkm-zfu-nkewi-z-llnke-iwikd-ktlmntlnk-TP.lt
ÑummnnTank –

T.P.E.IT#nnUnic—Raman
t.eu?::.::i..ciiiEts–i’-+)InUmnTenkJIe–iWnm-4e-iodide

,

e-
“”
de

,

✗J.jae-iwkn-ke-iwikdt.se
→=÷÷7

[m⇒= – i-qp-zumnu-m.fi#e-ilWkm+WztWkm-wDtidy,
✗Ina
–

+⇒ =¥Eumnñ÷+÷÷÷÷÷⇒
t
,
→

sw=¥¥÷É¥ÑEumnñnk

Raman : #-)In(ETIE.tn?I!u-Yk*-iE-imE.:::#E-E)
✗slw÷III

L

–

b
H=Tt

V

T — III.milVip — LIE

,
milking?-1¥?)

define mass- weighted coerel
Gi = (Imi)” ki – Xie)

T — IEEE
inn M mechanics

. . I is a column Veeder
cheek -7 ad T= Ip

i

‘

. I

a-

KE

EFE.EE#iEitEEiEIEet.i.?.io

.

at bottenTV(Ii-01=0 of thepotential
teen

vaI÷⇒¥⇒¥¥o
E.F. kij LiEj

U=IE
‘

KE

Lagrangian mechanics to solve theme E-AM .

”

define ⇐ TCI;) – Hei)
Lagrange’s equated

date)a→. – Rated o
single particle case L=’zmx’2- thx)
Ex — mi 3¥ = -2¥
dafcmx

.

) +31×-0
ma = m = -Ex = F

–

for polyatomic vibrations
↳ T- V — t.EE?-I?.?gkijEiGj
2L

⇐i )si? Ii

IIT. )%= – ftp.Ekijkj his gonej due to double
E.AM→ Ii 1- 3- kijqj =o

counting .

–

general solution is
Ei= Ai Ces Hkt to,

sub Gj into E – O – M

O = -NAicos#¥)t.fkijajceslxkt.io#O=-dAitZjkijAj
”

E⇒y÷÷?
‘

iii.Ii:
–

: :
– –

ti
:÷÷f.\ , I

ksn.tl
”

”

KSN
,3N
-X

det (E -RE)-o ⇒ detlktxdee
do diagonalize k .
aset efs.ee# solute en d
3.N solutions ←

eigenvalue .

3. N- 516) often are allowed crib freq .
di = 4172022
Thi i. angular freq
Vi is frequency

find some diagonalize k . →Normalmeet.es
WIFEY.fr

‘
‘

Qk= ↳ Ej linear asnperpoaiten.
a-= .co/umveeee-.E–EEE– EE
I i

i

.
I’ is he

transposeof
E

Ei — Et

aye . . . .
EtkE=o⇒E + EEE-0

–
–
row EEE EE + EEEE-0
✓eeer

.

Ekmatrix IE
=L: t E-o

V

E-Koi .
.

) e
find Etta can mark F as
a diagonal matrix .

eigenvalue .

*=f÷÷:
Et EE-0

E.am + ai -0 Ii onlydepend
on Qi

all meteors are decoupled . Cnormal modes?
2T=E”EYE’ ‘(EAT

=

‘EEE — a”ai

zu = I’EIETTFE.IE#aT=ETEEE’a
Risk.IE?Ialiud–eTEE—zxiQEnooffdigtijmso

Example : a linear molecule .
O C O

0=0-0
MA MB me
I

. I, T’s
←
see
a
Sz

2V= kisitkzsfsi-xz-XT.SE#-xI
= k, (Xi -Xz)

–

t kzlxz -XD
–

= k
,
Xftk

,
XE- 2K , Xixztkzxztkzxz

‘

-2kHz
Ii = MikeXz

3V=¥aEtmCktkdEEt¥eE5
–

–
214
¥3 Ii Iz

– 2kg

calculate
¥

–

– – M
1213

i
e 2Ii2Gj

Ii:÷÷÷
‘

:÷÷÷: * IEV2479,21
,

Ehs III -X

=

” ‘¥7:& ‘
imam
, mis

-d -¥

• ¥*. ÷::!
consider 0=0–0 MaeMe

KEKEk