by
JOHN EDWitRD RitUOl
A TIESIS
submitted to
in partial fulfillment of
the requirements for the
degree of
MS1ER OF SCIEN
June 1960
in charge of
?jor
QiáÌrman of Department of Chemistry
Date thesis is presented
Typed by Mrs. V.
L
1iite, Jr.
This thesis is
dI«t.d to Profossai
J. C. Decius
whose efforts and suggestions uade this thesis possible.
chapter
Page
..s..ess$.ee*ø
i
DfrROOUCUON.
Ii
cLASSIcAL TLEORY OF
:îii
QOANTUMTJEoRYO'TlERAEFFEcr
Iv
ØYPERPGIARIZABILITY
V
VE
VI
TiE
eh
Sh
vn
TIE
ah
VI II
NUJSION
B ZBUOGRAPRY
wPEM:Ix
i
s
2
RAMAN EFFEC)
,
,
.
.
. . .
.
,.,..,,
. ...
.
,
,
.
$1JNTRY POINT GROUP
,
a
$1TRT
.
.
a a
a
ea
a
General
a
a
O1$T GROUP
.
a
s.seU
S
a
Forlas
a
e
a
*
*
a
a
6
9
..
*
C
25
a
a
32
s
s e
,s**s
37
ci Tx*asZoritks
1FvFhs1o*ssssss0s
Tables of characters
(nz2, 6,o°).
s,
$urpi
tk
e
for an Operation
APPEMIX
3
Gharacter for
R
.
APPENDIX
4
EulerIan Angles and the Averages of
Needed Products of Direction Co.insø
.
APPEMflX
3
.........
f orthe
APPErmu
TE
I
a
41
.
0
s
51
,
5 = Sunuary of the Forni and Range of the
Various
n of
Point Groups
the Species of the
.
.
.
s
gq'ENDIx
6
The iverage Intensity From
Oscillating Dipole
APPEMIX
7
The Dependence of the Intensity of a
Hyper-Ranun Une Upon Ion Concentration
s
a Classical
.
, .
.
s
5?
HYPERPO[ARJZABILITY AND FORBIDDEN
LISES
Ri'MAN
QL4PThR I
INTRODUCTION
In the
weak
Raman
spectrum of certain molecules and ions
tzansitions occur which are forbidden to take place
when one considers the symmetry of the free molecule.
article by Gray and Waddinyton
(
9O1O8
6, p.
)
An
presents a
sunmiry of the articles which have been written on the azide
ion which is usually
a linear
ny investigators
syustric triatomic ion.
ion have observed its
symmetry, i. e.,
assund to have
2'
and
3)
bas
of this
in the Raman spectrum.
In lead azide a partial covalent bond has been postulated to
exist between the lead and the azide which destroys the
symnietry for the azide ion.
the alkali
stals
Ïiowever, for the azide salts of
this could not
be
postulated. It
ts
stated
that the normal selection rules are relaxed under the influence
of the perturbing
It
electric fields
of neighboring ions.
has also been noted by Welsh,
that carbon dioxide at high density
the P and V
sg.,
( 11,
p. 99-110 )
the existence of both
shows
bands in the Raman spectrum.
They suggested that
the molecule was distorted under an influence of the field of
neighboring molecules in such a way that
type
symitry, i. e., the molecule
the molecule took
on
remained linear but
one carbon oxygen bond became longer than the other.
conclusion was drawn from the fact the
fainter
nzin*zrn while the
J
band had a double
14.
band was narrow,
of low
strong Q branch with rotational wings
This
indicating a
intensity.
They
nade no depolarization measurennts.
Evans and Bernstein
(
3, p.
11274133
)
found that in
concentrated solution of carbon disuluide using cyclopentane
as a solvent that the existing 7
polarizsd
(
and
0.63 for both bands
1.
)
and that the
of carbon disulfide showed no indication
triatomic molecule with
be two polarized
Iace
( J'
V,3 )
C-
bands were both
J)
bands
of a double peak.
A
syraietry requires that there
bands and one depolarized band
(
it was concluded that distoxtion in a carbon disulfide
molecule under the influence of a perturbing neighboring field
is not only along
this axis.
the ScS axis but also at a right angle to
They gave the distorted carbon disulfids molecule
a symmetry
ofe
of the
band to the
2-3
.
They also found the ratio of the intensity
3'
band of carbon disulfide to be
proportional to the volume-fraction of carbon disulftde
cyclopentane to the four-thirds power.
This paper will be an attempt to discuss the appearance
of such forbidden bands, retaining the symmetry of the free
molecule.
).
3
EFFECT
tIASSICAL flEORY OF ThE
The introduction of an atom or molecule Into an electric
field E induces an electric dipole moment P in the system.
a component ai the
electric dipole is
If
expanded by a Taylor's
series expansion in terms of the components of the electric
field,
(1)
-
+
F"
Thus, the induced dipole, neglecting quadradic and higher
terms of the components of the
(2)
PF
the components of the polarizability.
Let
It
electric field, is
can be shown ( 12, p. 44 )
that a set of axes in a molecule
exists such that
(3)
r'
if light
=
c(,
E
p
of frequency
falls
P}
c
on an
ion or molecule, the
varying electric field produced by the light
(4)
ThIs
=-
field produces
t. E}
ny
be represented
a0cos(ri-)
a varying dipole moment which
itself
causes
4
emission of light of the same frequency as the incident light.
For a classical oscillating dipole the intensity of the emitted
radiation,
is given by
1,
(
______
±
5)
see appendix 6)
a
This accounts for the Rayleigh scattering
which is responsible for the phenomena of dispersion and the
Tyndall effect.
The polarizability must change if internuclear
distances change
(
vibrations
)
or il the molecule undergoes
rotation, since the polarizability depends on the orientation
of the molecule.
ability,
o(
(6)
By expanding the components of the pclariz
in terms of the
c
-
cKy
nornl coordinates
-
/i)0Qk*
.:
is a normal coordinate of ths
of the molecule
.
.
aolcule and changos with
a frequency characteristic of the molecule.
(7)
Now,
=
Qfr
coiuing
(8)
Q; Ccos(zi
equations 3, 4. 6, and 7
7-.I
(ck)
Cos(t1t)
(9)
-F
(L)0
¼
Q;
Cos2(idi)t
CûS?Yj
Thus, Whfl one considers stnall changas in
o(
with vibration,
the induced dipole moment changes not only with the frequency
z
of the incident light but also with
with frequencies
Z2zfor
rotations ).
frequenc1esZ(
The Rariìn
or
lines dis-
placed toward the longer wave lengths are called Stokes lines
(Z-
)
and those displaced
)
are
tord
calld anti-Stokes
the shorter wave lengths
lines.
Thus, one should
expect displacements on either side of the excitation line
equal toz4
4
for vibrations and2i4 for
is fixed while
can take on any
rotations.
Cls*sicafly
value. Hence, one should
expect a continuous spectrum about the exciting Rayleigh line
due to
rotation
and
that
displacent should have
length
the long wave length cononent of the
the sane intensity as the short wave
displacement. These two expectations are not observed.
Thus, although the classical treatnnt gives a qualitative
picture of the
picture.
C
Rann
effect it does not give a quantitative
12, p. 43-48
)
[i
QUAPIflIM
TIEORY OF TIE RAMI4N EFFF.(
molecules in the kth state which are
If there are
of B louer energy and
in the
asrgy, the average intensity
tr*nsitiin
k-'n is
th state which is cf a higher
of the line arisIng
proportional t, Nk
frm
)"
while that
arising Erom the transition n-.k is proportional to N
IA
(-i)V
V0-V) N
i
since Nfl/Nk Is the
8eltzn
the
(V0k).
-(v0v4 factor betmean the two states,
This simple derivation explains why the intensity of the Stokes
lines are generaIly mere intense than those of the antl-'Stokes
lines.
The mean rate of total radiation of an induced dipole is
givan classically by equation (5).
(
Dirac radiation theory
)
The hypothesis is nade
that the classical forzadas can be
converted into quantum formulas by replacing P0
in which
[F°]
(11)
=
5O
by 4
d
thus,
(12)
1
d]
pOJ
nn
I
Now,
(13)
-y*
6fo(><
LPTTh
+ Ef&xy
VCcfl* Ef°<2
V'
ations hold for
Similar
and
and
ire the
components
light
ve,
the integrals
and
04)
xxJ
[Ponm.
the amplitudes of
of
\4ÇT
/-
Ex°
the
Ey0
incident
f°(xx
are the matrix elements of the six components of the polar-
izability tensor. In
effect the electric vector of
the Raman
the incident illumination, acting on the charges of the molecule,
perturbs its wave function.
using perturbed
is
found
as
the
t'ave
functions
that for n=rn there
itegra1s
If the
C
are calculated
insisdiug time factors J, it
are terms o! the san* frequency
incident light, but for n'm
frequency4J±zj.
P0m
there are terms
Z
with the
terms with unshifted frequency are
responsible for Rayleigh scattering while those of frequency
z4/give rise
to
the Rauan
effect.
allowed if at least one of the six
the approximation that the
product
immn
transition is
quantitiesßF,/JmO.
ve function can be written as a
of a rotational function,
and a vibrational wave function,
with quantum numbers R
\/f,
the integrai
or
f,jr
LJ
(f5)
since
mo1ecuIarfid axu
in
coordinates of rotation
(
)
12, p. 52 ).
f'dî
does not involve the
The orthoonality of the
rotational wave £unctions, therefore, leads to the result
I
-SR/f]V
E'F
(16)
ConsequtiJflm
same.
zero unless the rotational states a
i
These integrals can be treated exactly the
sar
7
thà
U
the integrals of the electric moment; that Is to say, they vanish
i
unless the synuitry of the set of functions
levels *iìd.r consideration
species
the
f
norl
%
etc.
,
e1lf
y'ìfor the two
some species In common with the
For fundamentals this requires that
vibrations for a given frequency fall into one of the
species associated with the
in the
t,-
'rb,
Rann
effect.
(
's,
if this frequency is to occur
12, p. 48uu53
J
cnwTER IV
HWERPOMRIZILITY
in the case of the p.1ui*eb*Ift, a component of the
As
electric dipole
of the
tnon*nt ney be expanded
in terus of the components
ctric field utin9 upon the molecule
in question.
Hence,
(D
(17)
EF'
(ÇfJ)O
-.
EE
The components of the poiarizabflity are
-
(-kF
(p3)
EF'I°
that for
Suppose
a given
vibration of a molecule all of the
partial derivations of the components of the polarizability
with respect to the norail coordinates of this particular
vibration vanish, i. e., the line is forbidden by polarizability
theory
(
for a free molecule or ion the line would also be
forbidden for the hyper-Rainan effect because of the
dependence
tipon
concentration ).
C
intensity
See appendix 7 ).
In this
case the intensity of the emitted light will be dependent
the partial derivatives of the
conents
upon
of the hyperpolariz-
ability with respect to the normal coordinates of this particular vibration provided they do not vanish.
The components of
the hyperpolarizability are given by
(19)
/8Fr'f"
=
______
L
In the hyper-.Rann effect the induced dipole mon*nt of
a molecule
iy
be represented as
pIJ
(20)
EF')
It"
f'
In turn can be expanded in terms of the normal co
The fi,
,, 9
ordinates of the molecule,
=
/,i#i
(21)
TheFfcan
+(:'1)
be shown to be all zero for molecules with a
center of symmetry.
r be non-vanishing for
The
molecules with no center of syn*netry which will give rise to
a pure rotational spectrum.
If one separates EF into the field,
due to the light source, EF? and the field due to the neighboring
ions and molecules,ZEF
,
EF
(22)
Ei becomes:
¿
-1-
y be represented by
EF
f23)
where
o5
Th
is thè frequency of the incident
light.
¿t.E,
may
be represented as
cosc')
(24)
where
,
is a normal frequency of a neighbor and
time independent part of the surrounding field.
EFDis the
The normal
Fil
co. diute,
y siso b. written
as)
sor1
ii ths
re
C.ining
()
Q°fr
=
PF
fr.queney of the
Z
j:"
I
(")
*E° COS
C OS
Qk°
i.c.i. i.
sti.n.
i(t
Cus
E(
t
r7,t)(E°'COS2117-
Cosirt)J
jp
(27)
Cos(1T7)
these eqstiosst
,
E*nding
u
this beccíss
=II2
F
(è
o
t'
) Qk
"
CÜ5 2Cos,?i1U'
E(cos)(Cos71Zz')
-i-E;
¿1&,
E;iì(cos)(ccs)(co5't)
*-dEi0
£i, (c.5 g
l:ff'oilØ
CF»
-E;E1
ir
t)(Ccs
1p-ir
')
Cos1TZ1
(cos'rrvt)(cos-ir i,t)
(Cos 2ir7t)(cosii1t)(co
iri-t)(cos
'rrVt)(cos
f7/)
ir)
ar4t)]
12
Thus, the
oscillating
dipole due to the hyper-Rarn effect
will have frequencies of the following types:
A
Rann apparatus is usually only designed to observe slight
shifts
from the exciting frequency.
involve
and
Hence, only the terms which
are of importance.
?-
Terms of
the type
which involve frequencies 1tZ.1 are dependent upon
Terris of frequencies
,r
seems logical to consider the
than
(28)
terms.
LJC,
terms as being much smaller
This being the case,
2:
f:
are dependent upon E,L1E,', It
O
f"
r'
+
)7
for observable lines in the hyperRaman effect.
Depolarization Ratios:
The two
(
quantities
12, p. 43-47
)
most conmionly measured
are the relative
intensities of the Raman shifts and their depolarization ratios.
The depolarization ratios will now be discussed.
For the laboratory fixed axes, the total radiation emitted
per unit solid angle in the
¡
direction is given by
77,ì
(29)
c
-
t see appendix
Consider
6 )
that the direction of propagation of the incident
13
incident light
coincides with the
Incident T
Polarized
Light
4I
Incident 2
Polarized
Light
-+f7
y
-_-//7
(cBS
'
1
axis.
)
'
A
1(OBS.1.)
(CBS.
J-)
g:::
;
lIght is plane polarized in the
1f the incident
and the
total scattered intensity is
axis, this total scattered
I1OBS. li).
j direction
observed along the T
intensity is
If the incident light is
represented by
Zpolarized and the
total scattered intensity is observed along the T axis, this
total scattered latensity is represented by
IT(OBS.
in this second case the
scattered light
J.
).
If
is polarized parallel
to the Z axis, the scattered intensty is represented by
If we now consider only lines which are Lcr-
111
bidden in the f irst-order-Ran effect, but allowed in
tk
byper-Rann effect, the intensities according to equation
(17) and (29) and the above
(30)
'T
ii)
-1-
=
definitions are
YFF'(C'XF)(
L, PZFFfx&Fx +&F»'X&pK
F)FttF]
-t-
14
2
TT (oBs.i) =
r'(E2z
(F
(os,L
c3
EF)(S4'Z
[_III
FP'
where
yFF'EE'
FF'
r-F'
is the electric vector of the external light
(E, ¿
)
source and
(EEy)
is the electric vector acting
the molecule in question due to the molecules
These equations apply only
which are
on
surround
which
to a single radiator; for
it.
N
free to assume al]. orientations with respect to the
observer's axes with equal probability, the equations are to be
multiplied by
found as the
It
N
and the average of each expression
radiator is allowed to
can be shown
C
see appendix i )
of transformation of Pp
i'
=
Nnv
consider,
j
be
all orientations.
that the general f orrt1as
assume
in laboratory fixed axes to
in molecular f 1d axes are given by
(31)
is to
-íJ
15
(32)
_-4
=
=
E,1
z
(E
4Ef
F F"
If the incident light from the source is such that
then,
pp'"
,
If we now consider
'X&'"x A
p''X +
On1y(EF'
type terms to be those cb1ch
produce the hyper-Raman effect
(
see equation 28 ),
be reduced for our purposes to
(p'f"f'X
(33)
Upon expansion this gives,
=
13pkX
(s-
i-
)
P-
* /'FXZK4&z + 13FyXeXe,Y
However,
(L\
ElàEf/)
(34)
¡3FF'F"
...
-
=
Then
(35)
-,
p,
Hence,
(36)
(see next page)
-I-
i,y
may
-4
-
[2
(
'
/3
'
E
-i-
z
it is
equation (36)
From
FXF
A'c'
FLX' GL]
apparent that the intensity of the
exciting light is proportional to
can be
The termL
.
computed by introducing sorne assumptions regarding the intermolecular forces
F
F
is needed.
(
=¿
see appendix 7
"L"
Since
.
FG
But
)
Also only the
F'F'
"FLLL'GL"
p<
O, A4
-
2.
FX'
(37)
)('j
"JLL'L"
cI
and
average value of
reduces to
+2
I
The depolarization ratio is
'J
"defined as the ratio of the
scattered intensity which is polarized perpendicular
toi,
that is, in the direction of propagation of the incident light,
to the intensity paralled
tol."
incident light is natural
(
ratio
ny
(
12, p. 47
)
"
If the
unpolarized ) , the dep.iarization
be computed by considering the scattersd light to
represent the sum of the intensities of the observations oflde
parallel
and perpendicular
polarized bern.
Th*
t part
to the incident electric vector of a
of the
light
from the
parallel
observation, being unpolaxi*i, outributes one-half Its
17
intensity to the scattered light polarized, respectively, parallel
and perpendicular to
I(o&_I,,(oBS.J)
(38)
i11(os.i)--
±SIT0U)
IT(oES.)/)
t
12, p. 4?
)
Nowlet
Then
(39)p
21 f3y
-1--
I
-f-Z &
I.Mf3
Tf3'"
2
A°r
"1
This may be written more concisely as
(40)
¡on=
2
F
1
¡XF
f±
zF
F
Y2'yi
í'
18
zh S1MIflY
TIE
POINT GROUP
First it will be necessary to determine which of the
ten distinct s exist as linear couthinations which transform
as ce*tain of the synmetry species of the group. One ny cbsørve
the following
transformations under the operations of the group:
Xty'Ztl:
(41)
ZU; VY'Z'
iii7)c*)..rm:
X 'Y
'Z'
(iz) (y) (-*)= XU
X'Y'Z' 2ì(z)(.y)(_s)m;
(
See appendix 2
fnce, this
shows
/)Q(2
4YZ
lW (.y) (s) -ZU
Z'Y'Z'°4s)(y)(Z)'ZYZ
Z 'VZ'
for the character table of
that
In a like manner
360).
XY'Z'(X)(7)cÍI)-=
it
transforms as
( 12,
easily
that
may be
and I!Yyz transform as
3
,
y,
shown
p. 359-
and izzY
transform as
The
E
character per unshifted atom ( see appendix 3) for
equals the nuther of coordinates. This is also the total
number of
linear conJ.inations of the coordinates
transform as the various species
( 12,
shows
will
p. 106). Hence, in the
infrared there are three cothinations; in
six combinations. This
which
the Ranun there are
that the ten byperpolarizability
components listed are all of the ones for this point group.
Zn o*der to illustrate that there are molecules which
allow vibrations which are inactive in the ordinary
Raman
effect
to becoue active in the hyper-Raman effect, consider the
ethylene molecule.
By
(12,
standard methods
p. 106-113
), it is
found that
the vibrational normal coordinates are of species:
The
f ellowing table
summarizes the nuthar of frequencies
respectively in infrared absorption, the ordinary
Raman, and
the hyper-Raman effects.
n(()
1!
ï
n(O
iL
n(()
byoer-R
3
Rl
o
02
1
g
B
z
A
i
u
2
2
a2
2
2
B
I
i
B
'u
n (Y)
IR
n (Y)
n)
R
is the
number of
is the nuther of
infrared modes.
Raman modes.
hyper-R is the nunther of hyper-Raman modes.
active
Hence, by observing the
frequencies
and
are
k character table the infrared
in the
hyper-Raman effect
alled
an additional fundanental frequency is
is
hyperRain effect which
kainan effect.
or
allowed
seen to be
Next the
of the
B1
lt
(
p.
observable
group
h point
Let
species.
in the
not al1ovd In either the infrared
106409 )
quantities
for the various species
f
will be tabulated.
First consider the
Q represent a normal coordinate
for this
u
species.
2,
Then from the character table for
ah
in
appendix
it follows that:
(42)
-o
but all other
For simplicity of notation let
(43)
-
5
nc., since the order of
the
4=
cartesian subscripts is irrelevant,
non-vanishing terms for these species are
1
/ß33)
3/3') A3/')
4W ,
f3i13)
From equation
(
23)
and
F
/32) /32}
pzF
evaluated.
(44)
(45)
(3
f'ii3
.: /XF
+
3333
need to be
y')3
(45) continued -+f33I
:iff#3
*
=
The
3/lY3X
+
+
and other
averages
/k
similarly needed averages
are calculated in appendix 4.
(46)
z,
/3
t
lcdi
/3'"X!
f333
w
f /3333 ¿3223 /3(3
+1313,/33,1
=
Tb,
-
freni
:
(47)
V!X3Y3X/ +
(//3 p333
esth*s
_
Z3 p333 +
13
th
$Et,
the value
Y!
XJ-Yxa
f2fl3ZZ)<3
ìì
*
3
* e:z
(44) , (45) , and (46),
.
,
i/ /dZ 3
-1-
°tfFi
*
?i /3A:23
7'
3 f3333 * //3 f223)
needed which is given by
-t
(46):;
The
first
term on the right-4iand side of equation (48)
be expanded.
Thus,
may
)
22
(49)
+
+
fS233
f3ß
zì3)
/3))3
I%I!
±?.Z3)
+Az3(1z
From the average values of the direction cosines,
($0)
i:
%i
f3/3I
3ff3
-+
33
+
The second term on the right-hand side of equation (48)
ny
be expanded to gIve
7!.Ç3il3
f3333
Il)
+
* 7j31)3Z?-3
¡73Z:5(3333
-J-
f313
+j32Z3
Thus, troni equations (48), (50), and (51),
(52)
(3)3
+
+f9113333
--
:
-
F
7
(3.Êg
-1
7
33
:
+
;:(R3/:3333
/3ìì3
(23
Us ing equation
+4
1.4/3)3
(53)
&
-
Ci1f33
The values of
f3)3=
X
+
íì3
33 33
7
values ofÇ3.
L
and
I3333
.
For
siUcity,
øence,
zC4x?+ 4Y-,.Zz
(54)
C
il
X
Il
yz
4Zz + XY+YZ ± Xz]
will experience a relative extreimim value when
p
(55)
-e--
3J3333
iit. restricted as shown by determIning
misim
fS22.3:
,c333
4f!33+
are
the nuxinim and
/3
-=-
W
-
-
let
Lflx2
A
(56)
Hyz
+
+
indicated
Now, performing the
4z
differentiation
partial
L22X-y+ZJ
2_1e.X-Y-ZJ
(57)
xy+xz-i-yzJ
-F
AZ
-zCy-,>zJ 2C2Z-yXJ
32
A
Three sets of solutions
(56)
,4z.
-
obi ;ained
from these equations:
A{x1v, Z3yJ, B{x=--4y1z31] Cy-4x23XJ
'4/l3.
Solution A gives
R,
are
o
=
6/7. 4/13 ,s
Solutions B and C both gii
found to be the absolute tninir*nn and
6/7 was found to be the absoluta nxiitm of
Similar solutions for
are obtained for the B
maining species
yz
species
the ten ,g '
non-vanishing
to the one for the 81u
and B
species
ifl2h for which'
O;
Of&2h.
species
The re
's appear is &.
For this
while the derivatives of the other nine of
are zero.
's
f.
is before,
become (ß1123,
i'
let
IZ3
32, *2/,3,
In this case the required terms for the determination of
(59)
LI f3
ççi,
f3'
/
-
The
3
i°k
are
=
4
_3_
¡3I/,
Efl
2
Iz3
'iL'
-j ß/t3
-I(
AppI7tnØ e*ation (40) to the eqisation given in (59)
(60)
A
sury
gronp and the
appendix 5.
of the
,
Ti
vedous species et this
andO&Qb point gxonps will be found in
iii,
As In the case of the
gr°u,
LIii
let us deterTdfle which
cf the ten distinct linear cothinations of the
atad with the various species of the group.
similar to thea
transforn as
,
as
F'm -'xyy
(
,zx'
+
( ¶'
*7*
J3
'
are associ»
In a process
of equations (41) it can be shown that
g,75
and
's
transforms as
1:;'rn
m-
transforms as E
3
transforms
f-'
P')
(
transform as
)
P'zyy
i
8,
-
lu
J'
and
'
(see appendix 3).
Bausana is a representativa molecule of this group and a
procedure alMiar to that which
of
£2h
ts
ehylene
carried out
will be carried out for this molecule.
,'
-
The z.'.axis is a six..
fold symmetric axis.
7-
I,
The normal vibrations of C
lg
*2g+
Elg±
4Ef
Ij,,
+
are:
Ø+
2u
*1U
Then, since the infrared activa species are
ordinary
the
AZu,
Ra*
*CtIVe species are
hyper-*as aøtive species
B,
Ein, sad
E,
the
and
and
1%.
2u
Eit
the
while
are, as indicated above,
nueers
of activa frequencies are
as given in the following tablee
n (Y)
(Y)
nCi)
hyjer-R
IR
2
B
E
1g
A
lu
'2u
Blu
B
2u
E
lu
SI
E
Ja
Hence, the infrared frequncies are allowed in the hyper-Raman
effect and six
add1t11
fuds*ental frequencies are allowed
also which are not in either the infrared or Raman spectrum.
xt the observable quantities,
species of the
consider the B1
6h
point group will be calculated.
species.
for this species.
i2, for the various
Let Q represent a normai coordinate
Hence:
0
(61)
and ali other of the lisiar
group are zero.
First
coinat1esi
of the
From this it is found that
(62) see next page
for this
(62)
q-
=
and ali othez
--7';1
's
vanish,
O
Lat
=-
Hence the non-vanlshiig te
for this species
y be
112
represented
Ç
(63)
1i2t/1zi1
(222'
'I!
thiS case,
/21.
/5
(Z2
O
JL"" ,L''&!:/f
L
0r
o
L
ipp1ying equation RO), to the equations of (63),
(C)//7
=
The species B2
to that of
B.
has linear combination of the
This yields that
for B
/
's
similar
is given
by
(65)
1lso
¡Ç
the species
of the degenerate pair
E
acts identically as the
)
¡
of the species
of the point group
=
(66)
for the E2
species.
Next, consider the
's
of the A2
of
species.
U
Q
represents a nornd coordinate
for
this species, then
Thus, it is found that
IL
O)
(68)
and
o
-
(67)
all
of the other
(69)
o
are zero ter this
's
-
species. [at
t3
Hence, the non-vanisbing terms
for this species
may be
re-
presented
P131' I'311' I223' ß232'
ll3'
P"s are exactly those obtained in the Bi
These
the
2bPOiflt
/,Y/33 _/8
-A
,L.
/3
A
f'223
7?1l3.
,c?'
to
,G'
&3
_
1'1i3
and ,c:333
PN
As before
-
,G'J
,
J;'37
7(-
the maxinuzm and mininnim values of
restriction on the range of
//333
But, for this species
46h
This reduces
E
z:
YO)
sain
species of
of
This leads directly to
group.
Ei/- 4ß3
for the
species
322h
p
put a useful
For simplicity, let
,,9.
Hence,
y::&Y
'4
731Z
the relative extreniiru
values are found by setting
7
__
(71)
=
This leads to the following set of
2S.-
(7e)
x
z(7+y2_ay)(4+2y)
4y
Z3x4y+Z
(Z3x2+4yZy)2
Two sets of solutions are
(Z3x24yz+Zxy)
obtained
Af3x/]
Solution A gives
4/13
/2x)
Y
23xa#4yZ+Zxy
(73)
equations:
from these equations:
5f2v->]
_;
Solution B
4/13.
6/7.
gives
is found to be the absolute mininnim
and 6/7
is found
for species A
to be the absolute maxiurnm of
of the group
196h'
The remaining species ofL96h which
of (s's associated with it is
E1.
has llaerar
If Q be a
combinations
norn1 coordinate
for this species, then
o
0
(74)
while
(75)
(yy(xxy)
o=13zz
__Q
and so forth.
Thus, it is
(76) see next page
found
that
(76)
è'
/9X
and al].
's are zero for this species
et the other
particular
normal coordinate.
Hence,
te non'-vanishing
and
Let
= /&/J3 )
(77)
C),'
LÌLY
K
j6?
= Pi
terms for
18X/,V:::
this species ny
be re
bJ111. P122
2zl'
I-2l2'
ì33' 1'3l3' ß$$
set is equivalent to the set of non-vanishing 1$ ef
presented
This
the
iu species of the
A?
cg:
['ßiz-ii ?3
for the
E'
(78)
point group. This leads dt*ectly to
2//-3J/_fi//Gz7
But, for this species
6h
This reduces
fL = I /
p
/
2
7
Li
to
/'/i
Again the relative extreimam values of
by setting
- /&
Piz j
11/
species of [)
lll3'?l22*
2h
where
/
f
x9111
P/il
i3J
will be determined
and
Hence
(79)
(80)
f17
Z
,
bXz
-'úox81
?L.
O%299y2/2x/
_______________
.
31
(80) continued
Two sets of solutions are obtaIned from these equations:
SolutIon A
0.363
gins
f
=
0.363.
Solution B
gins
)r
0.732.
s found to b. the absolute ainiz*a and 0.732 was found
to be the absolute waxioum of
A suwnery of the
for the species
E.
for the various species of this
group will be found in appendix 5.
Lc4
¿y
14'Ò
y
/4-O
i
rr
LP1ER VII
TIE
0,h SYMETRY
in the case of the
2h
POINT GROUP
group, let us determine
which of the ten distinct linear coInations of the
are associated with the various species of the group.
In a
process similar to that shown by equations (14), it can be
shown that
(
,
/2?
1'
and
-t
and
(
f
transform as the species
3
3
f33,77
transform as the species 7I
fByyz
transforms as the species L.u, and
-
xxx
,c,cy' (
transforms as the specIes
xyy
(3xyz
(,(3,- 3(3,yy(3yyy
(see appendix 2).
fl
rbon dioxide is a representative molecule of this
group and a procedure similar ta that for ethylene of
2h
will be carried out for this molecule.
-
y.,
The normal vibrations of
2
-z
the z axis is the
molecular syietric axis
are of the
Thus, since dipole monnt is of species
polarizabllity is of species
are as given in the table.
*7Tg*Ag
(see next page)
and ordinary
the active modes
n«)
nI)
hpru4
I
- i
z_
I
i
-*
For a linear triatomic
sjitric
molecule, such as
carbon dioxide, there are four fundamental
molecular
with
vibration.
The
frequencies of
totally synetric
vibration associated
is active only in the Raman effect, comparing only the
Raman and infrared spectra, while on the same basis the anti
symmetric
stretch of the
molecule associated with the
species is active only In the infrared spectrum, and the de-
generate
bending
motion, associated
with the
also active only in the infrared spectrum.
1TU species, is
Thus, it is readily
seen that the existence of linear coninatIons of the
associated with both
active
and
1's
insure that the Infrared
fundamentals will also be active
in the hyper-Raman
effect.
Next, consider the species
flu.
consideration of
either Bi
or B2
of the point group
6b
will show
that
6/7 for this species also.
II
If the
the
fl's
of the species
of the species A
Qn --
(70)
z
f
for the
(
01
S6h
+
7
23f3//3
hare compared with
of
it is apparent that
,
33
-fri
-t-
//3j43
species and that the range of
is given by
± L
(81)
13
The last species of
the
fs
Ehin
which linear cothinations of
are associated is j7.
A comparison of the
this species with those of the species Ei
0e
's
of
of the point group
ShOWS that
(78)
fn -
for the species '7T'
4(s1ßì
-1-
5O/3,
1
/333 -3ßìiììj
-1--
The range of
n
again being given
by
o.3
(82)
A
sury
of the
:
f
O's
4 o73
for the various species of this
group will be found in appendix 5.
HAP1ER VIII
.ji, ('JI
retical expectation for the
hyper-Raman
to be d. The theo
colsis utratlon dep**. of the
this subject is y$
work on
Much
lines should be determined for cases of ion-ion,
ion-dipole, dipole-dipole, etc., interactions in
detail
much more
than in appendix 7. The quantum mechanical aspect of this work
should be worked out to show the
approach.
flowever, the experinental side of the theory can not
be overlooked.
Although the values
ratios for the Z
been
nasured
and 7T species
fall within
and
is offered
funda*etal
for carbon disuluide have
A
mode
of vibration is
allod
effect. This
predicted range,
very interesting prediction
by the hyperpolarizability theory.
effect which is not
Raman
for the depolarization
the theoretically
cases should be studied.
ny more
usual
validity of the classical
For ethylene a
alld in the byper-Raman
in either the infrared or the
mode
corresponds to the
species.
Theoretically this line should have a depolarization ratio
of 6/7.
In the case of benzene even more apparent should be
the confirmation of the theory. Allowed fundamental modes of
vibra tion in the hyper-Rainan effect correspond to the species
B1«, B2«, and
E2.
All
six of these allowed lines should
have
depolarization ratlos of 6/7.
This theory should not be
considered as a
contradiction
or a rival theory of those previously presented in papers
by Evans and Bernstein, and Welsh,
However, it should
be considered as a complementary theory describing the same
phenomena only considered from a slightly different point of
visw.
37
BiBLIOGRAPHY
1.
Buckingham, A. D. and J. A. Popie. Theotetical studies
of the Keri effect. I. Deviations from a linear polarization law. The Proceedings
Section A 68: 9O69O9. 1955.
of
the Physical Society
2. Goulson, C. A., Allan Maccoil and L. E. Sutton. The
polarizability of the molecules in strong electric fields.
Transactions of the Faraday Society 48: 106-113. 1952.
3. Evans, J. C. and
effect.
the
H.
J. Bernstein. intensity in the Rain
interaction on
V. The effect of intermolecular
Raman spectrum of carbon disulfide.
of chemistry 34: 11274133.
Canadian Journal
1956.
4. Glasstone, Samuel. Textbook of physical chemistry.
Second ed. Princeton, New Jersey, D. Van Nostrand, 1958.
1320 p.
Irbert. Classical machanics. Reading,
Massachusetts, Addison-Wesely, 1969. 399 p.
5. Goldstein,
6. Gray, Peter and T. C. Waddington. Fundamental vibration
frequencies and force constants in the aside ion,
Transactions of the Faraday Society 53: 901-Q08. 1967.
7. Herzberg, Gerhard. Infrared and
polyatomic molecules. New York,
632 p.
Raman spectra of
D. Van Nostrand, 1945.
8. Lindsay, Robert Bruce. Concepts and mathods of theoretical
physics. New York, D. Van Nostrand, 1951. 515 p.
9. Sewell, Geoffrey L. Stark effect for a hydrogen atom in
its ground state. Proceedings of the cambridge Philosophical
Society 45: 678-679. 1949.
10. Sherbini, M. A. El. 11. Third-order terms in the theory
of the Stark effect. The London, Edinburgh, and Dublin
Philosophical MagazIne and Journal of Science 13: 24-23.
1932.
il.
L,
P. E. Pashler and B. P, Stoicheff. Density
Raman spectrum of carbon dioxide. Canadian
Journal of Physics 30: 99.110. 1952.
Weisch, H.
effects in the
12.
Wilson, E. Bright Jr., John Courtney Decius and Paul C.
Cross.
Molecular vibrations. New York, MGraw-Hul1,
1955.
388 p.
s,
I
GEFERAL FOR11IAS OF TRANSFORMItTION FOR TiE
1*
the hyper-.Ratwi effect
(i)
is given
,,U
PF
F'F'1
E1
¡3
..
by
EF»
also
(2)
E) Is
exe lE1, E2,
light along the
tbe electric vector of the Incident
tbe asscules
ss
and (E,,
E, IQ is
the electric vector et the Incident light along the 1abor
are the direction cosines between
atory fixed sass.
the two
fIeld
syst,
1*
is guss
(3
Now,
L product of two
th «, Y
cononents of the electric
Z system In terne of the (1, 2, 3) system
b7
F
E' - ¿T
conthining equations (1) and (2)
(4)
,.4/p
F'F"
1ut,,pi* the U.:
(S)
,X1F
Y.
Z system
-
in the (1, 2, 3) system.
1A(
Let
is
gi'n by
(6)
,«f
-Z (,flFF'F»
F
F''F'
Then, solving for
-:2X
,41i
since
;y
Ay
(7)
3X
are the elennts of an orthogonal matrix.
thßgF
ExpS*sion of the determinant gives
a)
_/u
-
[
(
(A)(FF1
E
+ [2: (.
E'J(z3y
3ZZ)
F1'](-3
:zxy)
FF'
'a'
--
/SZFF
E
'F'P) E
Hence,
(9)
ß/oaod
3z -:L3YZZ)
4
S2x)
t;:l
fPF'1F
= F" /FF
'
''(zx3y
F Ill/F
P
X>i)
'
Thus, in generai,
(10)
,s
F3"1
1FF'F" -=
2:
F
'1ci1
zi&
E
Ag
I
i
I
i
81g
6(zx)
6(yz)
i
6(117)
i
I
i
1
1
"i
I
i
4
4
B29
t
4
i
-1
1
4
14
B3g
I
4
4
1
1
1
.i1
Au
I
i
I
I
iii
4
I
I
i
I
.1
1
4
1
4
1
1
B«
Ay
ig
4
1
1
4
1
4
1
.1
*
c(xx,efy,c1zz
Rz
OIxy
R7
IZX
'z
Au
Blu
BZu
3u
441
.1
Ç
4
6b
*19
E
2C
2C3
C2
34 3G2"
i
2S
ZGb*
-*1r.J--JuF
1
1
Aagli
Big
i 4
1
1
1
1
11
1
1.14
.1
1
4
1
4
i
1*1
1
4-1
1
0
0
2
*1
4
0
0
0
0
24 4
2
0
0
1
¡
1
1
1i-14
1
4
1
i14
020
14
EIg
2
1%
2isl
*i
i
i
I
1
1
1
1
1
444
4 2
i
j
Ji
-4
4 4
1
1
1
1
1
444 144
4 4 4 4
1
1
14 141.1 114 14
14 1.14
414 114
21.1.2 00 «241 20 0
1
B
Elu
E
1
2442
f)
0
.21
140
0
oh
;±
1L4
g
$...
J.
I
...
I
i
I
...
i
i
I..
1
I
i
...
D&rj
2,
i ç
O
O 2
2COS?4S
.. .. .,.
e. Se
e.
j
...
iø'I
4
...
4
tel
,,
i
2eos4.
O
2
i..
i
...
«4.4
j.(
2
2ccs.
04
2
2Cû2.
O
2
4cos24.
o
2Cob.
O
2
2.
O
e
e.
e.
S.
ecc
e.
-
ce.
(T.T)
e.
T
nrt& 1:IT:ur
ruM
M
_
L
1LId1
n
t-
j-
i
1 t
1J
.
.
r
o
T
-
,
1±i
,
-
I
;1Y7*
z;
-:TA
(J3yzz,
1JL_,_L
J
r:
u
ßrn)
-
-----
(
f
Lt
yJ3 *72
fyyy3ßxxy)
(
yz,
0ZX)
(xiry'xy)
i.
I
ilr.
(Rx,Ry)
O
::L:
t:
oXtc%Y)cZ
I
O 2
2
- 1
¿__1(
coCz
i
1,
1W ølmuutr
Tc átaia
tsaasf*t1e.X*,
of a
nøods only to nata hat dsftnition, i, a.,
(12, p.92)
(I)
In ii1ch R
1$
d1
s
a
NsIt*$*Iat
1*1W
$t$iat
=
I
**
1.
1k
then add up
jmønl
1« torna of the tnansfcat1on.
for u
t1*s
4*
a.s lbs Z *ds.
fosnatlan Is
&Ç;
1
(3)
siw
y'
N&
X
-
I
X
+
cas
Y
y
2'Z
mats,
x
,
k
-=
i+ec
ton a general sotary reflection, S
2# X -
X'
-
cas
y'
-
s1N4x
2'-Z
XS
-H
+
--
HIM.,
(5)
,;
-
±1-t-
zco'
S 'N
4
y
Thun.
the txans».
positive sign is used for proper rotations, III$S
where the
rotations,
for itçroper
and the negative sIgn Is used
rot*tInes,
rotary reflections.
This concept can be extended for the transformatIon of
tk kind which
would be
Involve the product of two coordinates
This
the character of a transformation for polarizability
components.
Thus,
fe
general for a rotation
the trans
formation is
(6)
-
(X
(
yi)2
(
(z')2
csd
x - 51A/d
v)
SIW&
=-
X-
X/)(y))(C0
(x')(')
(c
(y(Z')
¿N
1y)(5wdX
cos\/)
MY)(Z)
X + COSd y)(Z)
where
hence,
(7)
Nci
co
(
+ i)
as
consider a general rotary reflection,
fies that
S.
The only changes
of the rotation
(8)
x')(2')
- - (coSo
=.
hence, (3e*
*ezt page)
-
(3/iVO
;x
X
-1-
sìivd
co&d
y)C7)
)(.J
47
(9)
;2
COSO
-))
Thus,
(10)
cOsc
-
caonents
The
±ì)
(,2CoS
hpnp*krizabi1ity transform as
of the
the product of three coordinates. llenes, the character of a
transformstion for the hyperpolarlzabiUty components ny be
found in the sams vy
in
as In the previous two cases.
gsza1 for a rotation,
11-»
U9-:?
((s)
X
(smoL)
')3
((sINC()
X
!- (CßSO)
(z')
(X)a <
U')
')
is
y) 3
)
3
z3
¿:(so
)2= («$
(X')2(z')
(X
c, ue transformation
Thus,
(Z ')k::
(T92(z')
(SINd) )'(srncx+sc
X
X
(SThc
((SoL)
X
(SINs)
( (Cß$
X
(SI$
r) z
X --
«os
)2 z
X+
coos:'6
= (sx
(V) «.)2=
)
((SINcô
(X') (Y') (Z')= ((sb
x
r)((szro
x-4cosô
'r)
)2 z
i)
(SmO
r)
sir,c*.
z-t.
«xso
bnce,
(12)
xíf))z
Y,)
2CO.cd- -I)
2 cosc (4co,s--
r) z
Now consider a general rotary reflection,
changes from that of the rotation
(13)
(Z')3-
are
-z3
iZ,= - (wosd z -
sm
(V)2 Z9
cx'
(j
The only
4.
'z'
((sc X
z
(SINd
+ scb
(SThc
r)
2
z
r) 2 z
r)(siNc
x-toeso
Hence,
(14)
Thus,
X
=
co
4CoSo ±2c&4-I)
r)z
The Eulerian ang'es are
fIasd as th
$kr.. successive
engles of rotation b7 asans of which a gives cartesian coor
y be transforn*d to another (S, p. 107.109
dinate system
Here the system as shown in Goldstein
will be used,
's
Ç)sica1
).
chanics
Hence, the Eulerian angles are detlned by the
following three rotations:
C'
(1)
where
,
5
and i
,
represent column matrices ;
sents the coordinates of the
repre
nttia1 cartesian coordinate
system and Z' represents the coordinates of the final system,
D, C, and
(2)
B are the ntrices
D =
Jcos4
sxr
o
(-S1N4
cos4
e
o
i
s
0
ko
/1
c!o
o
0
.sin&
fcosq'
o
The product natrix
AB
cos&
si/'
o
o
o
i
Bf.4ZNV'
t.
sme
will furnish the direction cosines
which connect the initial and final cartesian coordinate
systems.
a)
/
sV-'
s.ws
sx
sxr
-SlN Y'szNçía.00se S
NbSC/)SIN&SZN
A
$lNS
SINeSIN
cos&
There are only four distinct types of products ci direction
cosines which need to be evaluated.
y;)
fxi
These are
X/y2)
X!YI
For cu,terian angles the average of a function, f (
X2YZ
'4'
),
given by
(4)
(S)&)
8ffff(e)iìded
Hence,
(5)
'/5
z
/I5
-
is
L1
')
TIE P3U
TiE POINT
SUMMARY OF
OF
.,!4
_-:--
81u
iip7+1I;7
GROUPS
OF TIE SPECIES
2h'D6h'
!__._-__
-
f?zzz±xz
v-
-1*_
-.--
WT
-,L-r*
Z')YY
Bi
ç3
rpi-Ii
Bi m)f?:?z
6dI
4I1$rt
6/7
.
?u7,yzz
4/13F1,
i
i:
;Y;m
z$
Y11
YYYY
-
p*vy
pu1
!(r
.
T-37z
--
T
11
p
yzz
413;771/3?77
7
2L4*41.P
f
-
u?xy*
_-__
4;Yz
g44ii
B2«
B2
ii
a
f f;x
.
TIE VJRIOUS
RANGE OF
A?
'4
:
6/7
6/7
+1L
n
___zzç
zzz+zzzç/pzxxg
tìrJr
Ta
flZ0
¿/9
ç)
XXZ
¿/9
__
(zL/jxxEy*stz)
rn
¿19
sa
73
6k
----
_J
_uz, íz
yz
---
-.--
-
23Z49
¡;
1
;
)
L'
1
7cxy
(,Lz Iry*, 1'zy*
'-
ß
ßíz
LITI
L.M
L
6/1
--
;)
P'x2
4(5-i8).. 3/z
P*í
s- ---
2
J-
ZZZ
-i) *TEI
"Consider a linear current
a1on
.i.nt
the Z axis with ita center at the
r-
QOL
d
length L
zi1a." (8,
p.
416)
/1'
Let the current flowing through the element be given by
(1)
I
=
10SIHwt
The magnetic induction1 and the electric field strength
it
may be written in terms of a vector potentialTand a scalar
potential
Thus,
-3
B7XA 1E+-=
-.3
(2)
A solution of
and4 may be given
as a retarded potential:
(3)
-;
Ä;
where
-
,
ç()j
C
1
Hence,
b
neu
ny
that the etectzic field Intensity
cttai*d by evaluating
A
F() and
(i).
for cur problem
A=A2Z1
(4)
Az
j1,rN
cren
and S is the
w(t,R/Cl
setii
Tkapese
is
z
: it sui easdietor.
"R Is tt* distance Eroe P to tke
(8,p. 416).
rn
Z axis
is directed along the
since
(5)
Is
Jint
of length da.'
túeaastreespac.sothit
udijC, Th.$.lstionofAsft.ri.$.nhf*
bec
A
(6)
z
tas tis
$r
or
w(t.iI.)
sisathe øs$ r»/Z.
Ii epatica! coordinates
sise
(7)
-
ø=VXA
Thus, sisee
(Q)
fl
r
=0
;
e=
Cr
srneL oes
(uts.kr)
SIN
(utii4tr)j
The scaler potential is given by the Lorsata relation
(9)
t
iba sølution of
Ç
trom the
puavias*$yiMaed***
s
OES & cOS(wt.kr)
(10)
+
we
LQ
k
S9 SXN(wt-kr)
wer
Now the components of the electric field intensity may be solved
Thus,
for.
______
(11)
F
SIN(wt-kr)
j2GDS(wt'u.kr)]
SIN(wt-kr) -
L151114k2oe8(wtwkr)
wr
S(wt-kr)
LSJ
for the oscillating dipole becomes
The Poynting vector
(12)
The time average
o
-CF
(13)
-
e)
,,.
,L,l
-;:ZJoAIc
FThe total radiation may be obtained by integrating the
Poynting vector over a surface of a sphere of radius r.
rsinødd4.
element of area is given by
(14)
.11
pZ
o
.
rx ;i:7).ra
î
I1'
and
e d& d4
J
_
The
Mter performing
integration, the resultis
vhere
oeS(wtu,kr
becs*
the Poynting vect*r
-
(
-
J
3C
this
APPENDIX 7
TIE C*W1DEN
OF TIE DTXENSITY OF A UYPER-.RAMAN
UPON ION ODNOERTION
UT'E
In the Debye-Hückel theory the electrostatic potential,
cL
at a given point in the vicinity of a positive
,
)
(1)
çb:=
zet
D
JC
for a
±
-for a
ri
or
ion is given by ( 4, p. 956-.958 )
a.ths
mere
(
is the distance from the ion,
constant,
is the charge
D
positive ion
negative ion
is the
dielectric
Z
on one electron,
is the nunther of
charges on the ion, and K is given by
K2 =
(2)
YìZf'
4tn constant
where k is t
and n
is the
nuner
of each
kind of ion per cc.
The electric field at a distance r
finding
(3)
e
E11
ny
be obtained by
Hence,
____
= -
However, the electric field
at this point
+ kì)
due
to
the ion
itself is given by
(4)
cj'
-
____
Thus, the electric field on the ion i$*SIf due to the sur-.
rounding atmosphere is
U
J\
these
is sn*11
(6)
=
=
'and
-4(J'
.4(J\
e
y be expanded as
+
SSD
r
Using these e*pansions,
zCe
(7)
Thus,!
j
[-e]=
2eAL
way be given by
(6)
This reduces to
(9)
Equation
tA
()
of the text gives
(10)
=
snd equation (29)
(il)
:-
d
L
the text gives
2---
T
- =
#
This waans that the intensity in the hyper-Rawan effect is
proportional to
(
)2
a constant
_______
(12)
qDg-
+2
nzzJ
L
The above argunent ignores
in contrast to
indeterminate
asj-+o
the con1icìtion that
is apparently
(ne resolution of this difficulty
ney involve the assumption of
finite radius for the Ions.