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CORRELATION BETWEEN SPONTANEOUS RAMAN INTENSITY AND
SECOND-ORDER NONLINEAR RESPONSE

By

Sandjaja Tjahajadiputra

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Chemistry

1 995

ABSTRACT

CORRELATION BETWEEN SPONTANEOUS RAMAN INTENSITY AND
SECOND-ORDER NONLINEAR RESPONSE

By

Sandjaja Tjahajadiputra

A theory that relates the density B(r,r',r";—(n,(o,0) of a second-order
nonlinear response to the derivatives of the molecular polarizability with respect
to normal mode coordinates has been established by Hunt et al. This suggests a
possible correlation between vibrational Raman intensities and the nonlinear
susceptibility B(-2co;m,m)responsible for frequency doubling. In this work, Raman
scattering experiments have been used to test for a correlation between the
spontaneous Raman scattering intensity and the second-order nonlinear
susceptibility p( - 2m;(o.m) .

The values of the derivatives of the isotropically averaged
polarizability, (EM) and the polarizability anisotropy, ( y'), taken with respect to
normal coordinates for mono-substituted benzene molecules (chlorobenzene,
bromobenzene, iodobenzene, aniline, toluene, and N-N-dimethylaniline) have
been evaluated in this work and plotted versus the molecular hyperpolarizability,
[3. Correlations are found between (5') and (7’) and the [3 values from the
literature sources. The extend of the correlation depends on the vibrational

mode involved.

To my parents, Mediarto and Sitanirawasih
My sisters, Evy and Yenita
My brother, Lucky
My fiancee, Herly

ACKNOWLEDGMENTS

I would like to take this opportunity to thank Prof. Katharine Hunt and
Prof. Gary Blanchard for their unending support and encouragement throughout
the course of my graduate studies and during the work that led to this thesis. I
am indeed grateful to them both as mentors and friends.

I would like to express my utmost appreciation to my family, especially my
mom and dad, for being so supportive and understanding all'this while. ‘

Last but not least, I would also like to thank the following people, all of
whom have made direct contributions, one way or another, to the enhancement
of my research: Dr. Tom Carter, who made the CW laser operate perfectly, and
also taught me how to use different peak fitting methods; the Hunt research
group members (Ed, Xiaoping, Pao-Hua and Glenda) for their inputs toward this
thesis; and the Blanchard research group members (Ying, Selezion, Patty,
Jennifer, Vince, Dave and Jeff) for letting me use the laboratory equipment and
chemicals needed to do this project, and for the incredible support I received

from them throughout the years.

iv

TABLE OF CONTENTS

. Page
List of Tables ...................................................................................................... vii
List of Figures .................................................................................................... viii
Chapter 1: Spontaneous Vibrational Raman Scattering Theory ............... I ............ 1
1.1. Introduction to Classical Raman Scattering Theory .................... 1
1.2. Raman Scattering Tensor ............................................................ 2
1.3. The Placzek Polarizability Theory ............................................... 3
1 .4. Conclusion ................................................................................... 7
1.5. References........................ .......................................................... 9
Chapter 2: Relationship Between the Spontaneous Raman Intensity and
the Second-order Nonlinear Response ............................................. 10
2.1. Introduction to Nonlinear Optical Susceptibility Theory ............ 10
2.2. Static Nonlocal Polarizability Density Theory ........................... 14
2.3. Frequency Dependent Nonlocal Polarizability Density
Theory ....................................................................................... 1 9
2.4. Relationship Between Raman Intensity and the
Hyperpolarizability Density ........................................................ 25
2.5. Conclusion ................................................................................. 27
2.6. References ................................................................................ 28
Chapter 3: Experimental Correlation Between Spontaneous Raman
Scattering and the Second-order Nonlinear Response ..................... 29
3.1. Overview of the Theoretical Parameters Used .......................... 29
3.2. Experimental Results ................................................................. 32
3.3. Conclusion ................................................................................. 62
3.4. References ................................................................................ 74

Chapter 4: Future Work75

4.1. Extension of These Experiments ............................................... 75
4.2. Computational Calculation on the Hyperpolarizability Density

in a Non-uniform Field Environment .......................................... 76
4.3. References ................ ........................................................... 77

vi

Table

Table 1.

Table 2.

Table 3.

Table 4.

Table 5.

Table 6.

Table 7.

LIST OF TABLES
Page

The Raman Intensity Obtained, Calculated Cross-section and
Concentration of the Desired Molecule ................................................ 37

The Experimentally Obtained Intensity, Cross-sectional Area,
Calculated Depolarization Ratio, (21"), and (y'), of Chlorobenzene

for Different Modes ............................................................................... 44

The Experimentally Obtained Intensity, Cross-sectional-Area,
Calculated Depolarization Ratio, (3’), and (7'), of Bromobenzene

for Different Modes ............................................................................... 47

The Experimentally Obtained Intensity, Cross-sectional Area,
Calculated Depolarization Ratio, (‘3'), and (7'), of lodobenzene

for Different Modes ............................................................................... 50

The Experimentally Obtained Intensity, Cross-sectional Area,
Calculated Depolarization Ratio, (6’), and (7'), of Toluene for

Different Modes .................................................................................... 53

The Experimentally Obtained Intensity, Cross-sectional Area,
Calculated Depolarization Ratio, (6'), and (7'), of Aniline for

Different Modes .................................................................................... 56

The Experimentally Obtained Intensity, Cross-sectional Area,
Calculated Depolarization Ratio, (F), and (y'), of
N-N-dimethylaniline for Different Modes ............................................. 59

vii

Figure Page
Figure 1. Schematic Diagram of the CW Argon-ion Laser............... .................. 36 I
Figure 2. Raman Spectra Taken for Different Polarizations of Chlorobenzene

(a) lIl , (b) lI“, (c) 'I,, and (d) 'I,I ................................................... 38
Figure 3. Raman Spectra Taken for Different Polarizations of Bromobenzene

(a) lIl , (b) iI", (c) '11, and (d) "I,I ................................................... 39
Figure 4. Raman Spectra Taken for Different Polarizations of lodobenzene

(a) lIl , (b) lI", (c) 'I,, and (d) 'Ill ................................................... 40
Figure 5. Raman Spectra Taken for Different Polarizations of Toluene

(a) iIl , (b) lI“, (c) '11, and (d) '1" ................................................... 41
Figure 6. Raman Spectra Taken for Different Polarizations of Aniline

(a) lIl ,(b) 1I", (c) 'I,, and (d) '1" ................................................. 2.42
Figure 7. Raman Spectra Taken for Different Polarizations of

N-N-dimethylaniline. (a) lIl , (b) lI", (c) '11, and (d)'I,, .................. 43
Figure 8. Graph of (3"), vs. B .................................. . ......................................... 64
Figure 9. Graph of (2?): vs. B ........................................................................... 64
Figure 10. Graph of (E?)3 vs. [3 ....................... . .................................................. 65
Figure 11. Graph of (Ex-'), vs. [3 ......................................................................... 65
Figure 12. Graph of (72")5 vs. B ......................................................................... 66
Figure 13. Graph of (y)1 vs. [3 ......................................................................... 66

LIST OF FIGURES

viii

Figure 14.
Figure 15.
Figure 16.
Figure 17.
Figure 18.
Figure 19.
Figure 20.
Figure 21.
Figure 22.
Figure 23.
Figure 24.
Figure 25.
Figure 26.

Figure 27.

Graph of (y')2 vs. B ......................................................................... 67
Graph of (y’)3 vs B ......................................................................... 67
Graph of (7'), vs. 13 ......................................................................... ea
Graph of (7')5 vs. B ......................................................................... 68
Graph of (3')1 vs B ......................................................................... 69
Graph of (FL vs B ......................................................................... 69
Graph of (z?)3 vs [3 ......................................................................... 7o
Graph of (3')“ vs B ......................................................................... 7O
Graph of (E?)s vs B ......................................................................... 71
Graph of (y’)1 vs B ......................................................................... 71
Graph of (y')2 vs B ......................................................................... 72
Graph of (7')3 vs B ......................................................................... 72
Graph of (7'), vs a ......................................................................... 73
Graph of (y')5 vs. B ......................................................................... 73

Chapter 1

Spontaneous Vibrational Raman Scattering Theory.
1.1. Introduction to Classical Raman Scattering Theory

Vibrational Raman scattering is essentially a vibronic process which
involves the initial, intermediate, and final vibronic states. Under special
circumstances, however, it can be viewed as a purely vibrational process similar
to infrared absorption. This possibility was first exploited by Placzek‘.

Placzek proved that if the initial electronic state is nondegenerate-and the
excitation is off-resonant, the vibrational Raman intensities are given
approximately by the vibrational matrix elements of the electronic polarizability.
Both of these conditions are satisfied for off-resonant vibrational Raman
scattering from molecules in their nondegenerate ground electronic states. The
Placzek polarizability also complements the existing classical theory of
vibrational Raman scattering, in which the oscillating dipole moment induced by
the incident electric field light is affected by the vibrational motions, resulting in

scattering with shifted frequencies.

1.2. Raman Scattering Tensor

The differential Raman cross-section at...“ is defined by? the ratio of the
number of scattered photons NM (per unit solid angle around the direction of

observation k.) linearly polarized in the I. direction, to the number of incident

photons F M (per unit area perpendicular to the direction of the incident light

beam k.) polarized in the I. direction. The unit vectors k, and I, are perpendicular
to each other and so are k. and I,

N MI. 20th,.“ F k1,, (1.1)

The cross-section for any combination of k.l, and k.l. can be expressed in

terms of the nine components of a Cartesian tensor of the second rankz. This is .

the ‘Raman scattering tensor’,

2

16K‘
ak,l,|t,-l, = 0‘ V0( V0 T Vm T Va)

3 yams-w...
. (mIRale)<e|Rp|n> (mlRa|e><e|Rp|n>
8”” (n<—m)=; h(v -vm—vo)-il'e +h(V‘-Vn -i~vo)-il',3

 

 

(1.2)

 

where a p, (ns—m) is the pa component of the Raman tensor for the transition
involving the initial (ml, intermediate (el, and final (n| vibronic states; p and o
are unit vectors parallel to the p and o axes; hvm, hve and by, represent the
energies of Im), Ie) and In) and hvo is that of the exciting radiation; 11“,, is the

damping term introduced to avoid the divergence of Eq. (1.2) under resonant

3
condition. The notation 2' means that (ml and (n| are excluded from the
summation. The first classical derivation of ap,(n<—m) was done by Kramers

and Heisenberg3 and later, quantum mechanically, by Dirac‘. [ Note that is the

Raman scattering tensor component, whereas a,” is a polarizability tensor

component].
1.3. The Placzek Polarizability Theory

The mean square of the Raman tensor components are correlated to the
Raman intensities for a randomly oriented molecular system. To perform an

averaging over all orientations, it is necessary to resolve the Raman tensor {am}
into three partsz,
{am}: {ag,}+{a;,}+{a;,,} (1.3)

where {ago}, {ago}, and {age} are the trace, symmetric and antisymmetric

parts of the Raman tensor the components of which are defined by’

{a;,,}= (ape +aap)—(a§,) (1.4)

4

The three Placzek constants 0°, 0’ and G“ are the square moduli of

{3° }, {3:0}, and {afw}, and are given by1

pa

G° =2 a3,

rm

G. =2 a;,, (1.5)

pp

6" =2 32,,

P!

 

 

which do not change in their values under rotation of the coordinates. The three
Placzek invariants determine the Raman intensities from randomly oriented
systems. .

By use of adiabatic approximations2 to the initial, intermediate, and final
states, Eq. (1.2) can be expressed in a more tractable form related to molecular

energy levels,

|m>=lg>li)
In) =lg>l1) (1.6)
|e) =le>lv)

assuming only transitions between the vibrational substates [i) and [ j) of the

ground electronic state |g). [v) indicates the vibrational substate of the excited

electronic state |e) acting as the intermediate state. The | ) and ( ] denote ket

vectors in the eleCtronic and vibrational spaces, respectively.

5
By combining Eqs. (1.2) and (1.6), an adiabatic expression for the

dispersion can be obtained’,

apa(I+—i)=.§g §{(’I<9 ISZI6)[V_)I:I<EE:IQ>[I)

oval

,(illg lRplellV)(V](e|R«|9>[f)}

(1.7) -
h( V”, + v,)— iI‘,,,

where hv , and hv 1 are the transition energies for |e)[v)(—|g)[i) and

ev.g

|e>[V)+-|g)[1')-

The Placzek polarizability theory assumes the following two conditions‘:

ev,g

First, the ground electronic state is non-degenerate and second, the energy of

the exciting radiation hvo is so far from the resonance energy hvow, that the
energy difference h( v“, — Va) is much larger than the vibrational energies.
The secondcondition leads to the following approximate relations‘: '
h( v”, + v,)— (rug-(14,8, + v0) (1.8)
and

h(VCt'.gt T VO)—ircv zh(Vao,go - V0) (1-9)

where hvwm is the pure electronic transition energy for (eh—(9|, and the

damping constant 1“,, is usually of the order of the vibrational energies and

6

hence is negligible compared with h(V.o,go - v0). Using the completeness
theorem of [v) in the vibrational spaces, I
:[v)(v]=1 ‘ (1.10)
the sum over [v) in Eq. (1.7) can be left out. The new equation is given by2
ap,(j+—i) 5(ilap,|j)

_ (QIR0Ie><e|Rp|9>+<g|Rp|e>(e|Ra|g) (1.11)

pa exg h(v,ogo—vo) h(v,o.go+vo)

 

where a” is the po component of the electronic polarizability tensor“. In Eq.

(1.11), the Raman tensor component am, is given approximately by the ij

vibrational matrix element of a which is expressed by the adiabatic kets with

W
the vibrational coordinates as parameters. Therefore, the Raman process
involving the vibronic transitions | g>[j)(—|e)[v)(—| g)[i) can be viewed as a
purely vibrational transition [j)+—[i).

For a free molecule with no external fields, there are two typical kinds of
electronic degeneracy: first, the degeneracy due to the spatial symmetry of the
electronic Hamiltonian, and second, the degeneracy due to time reversal
symmetry’. However, degenerate states are excluded by the first assumption in

Placzek theory. According to Kramers’ theorem“, all electronic states of a system

having an odd number of electrons must be at least be doubly degenerate

.7

because of time reversal symmetry. Therefore, the ground electronic state lg) in

Eq. (1.11) must be an orbitally non-degenerate singlet state or a non-degenerate

spin-orbit state of an even electron system. Time reversal symmetry implies that

(1) if la) is non-degenerate, (g|R,,|e) must be real, and (2) if |e) is degenerate,
<g|R°|e) can be made real by taking the appropriate linear combinations of the

components of la). From Eq. (1.11), the sum is taken over all degenerate

components of |e), which have exactly the same energy denominators, from the

earlier discussion, by taking proper linear combinations of (g|R°|e) and
<e|Rp|g> that are real, (g|RO|e><e|Rp|g> can be made real. Thus, the
polarizability tensor {aw} is real and symmetric if the first assumption in

Placzek theory is implied.
1.4. Conclusion

Eq. (1.11) gives the formal expression in the Placzek polarizability theory.
Under off-resonant conditions, the Raman tensor {am (j<—i)} is approximated
by the vibrational matrix element of the electronic polarizability tensor {aw},

which is real and symmetric given that |g) is non-degenerate. Consequently, the

Raman tensor itself is real and symmetric within the framework of the

8
polarizability theory. This leads to the conventional polarization rule of
vibrational Raman scattering, in which the values of the depolarization ratio p
are limited to OSpSO.75. I
The extension of the polarizability theory to degenerate ground electronic

states has been discussed by various authors"12 and is not treated here.

1.5.

9.

10.
11.
12.

References

G. Placzek, Handbuch der Radiologie, edited by E. Marx (Akademische
Verlagsgesellschaft, Leipzig, 1934), Vol. 6, Chap. 2, p. 205.

H. Hamaguchi, Advances in Infrared and Raman Spectroscopy, edited by
R. J. H. Clark and R. E. Hester, Vol. 12, Chap. 6, p. 273.

H. A. Kramers and W. Heisenberg, Z. Phys. 31, 681 (1925).

P. M. Dirac, Proc. Roy. Soc. (London) A114, 710 (1927).

P. M. Dirac, The Principles of Quantum Mechanics, 4th ed, Oxford
University Press, 1958.

H. Eyring, J. Walter, and G. E. Kimball, Quantum Chemistry, John Wiley
and Sons, New York, 1944.

E. P. Wigner, Group Theory, Academic Press, 1959.

H. A. Kramers, Koninkl. Ned. Akad. Wetenschap., Proc. 33, 959 (1930).
M. S. Child and H. C. Longuet-Higgins, Phil. Trans. Roy. Soc. (London)
A254, 259 (1961 ).

M. S. Child, Phil. Trans. Roy. Soc. (London) A255, 31 (1962).

J. A. Koningstein, J. Mol. Spectrosc. 58, 274 (1975).

J. A. Koningstein and T. Parameswaran, Mol. Phys. 32, 1311 (1976).

Chapter 2

Relationship Between the Spontaneous Raman Intensity and the
Second-order Nonlinear Response

2.1. introduction to Nonlinear Optical Susceptibility Theory

Nonlinear optics covers a wide range of applications - this field deals
with the nonlinear interaction of light with matter. All nonlinear optical processes
involve light-induced changes of the complex dielectric response. of a medium. In
each nonlinear optical process, an intense electric field induces a nonlinear
response in a medium, which reacts modifying the optical fields nonlinearly.

Electromagnetic phenomena are governed at the electronic level by the

Maxwell’s equations for the electric and magnetic fields E(r, t) and B(r, t)‘,

v =———
XE lam

16E 41!
V =—— —
x8 0 6t+ c J (2.1)
V-E=41rp

V-B=O .
where J(r, t) and p(r, t) are the current and charge densities, respectively.

charge conservation implies the equation of continuity',

69
v. ——= . .
J+m o (23

1O

11

One can expand J and p into series of multipolesz,

5P 0"
J: — V M — V-
Jo+ at +c x +at( Q)+ _ (2.3)
p=po -V-P—V(V-Q)+...
Here P, M, and Q, are the electric polarization, the magnetization, and the
electric quadrupole polarization, respectively. In many cases, it is more useful to

use J and p directly as the source terms in the Maxwell’s equations, or to use a

generalized electric polarization, P, defined by’,

6P
J=J — 2.4

where J, is the dc current density. “The generalized P reduces to the electric-
dipole polarization P, when the magnetic dipole and higher order multipoles are
neglected. The difference between P and P is that P is a nonlocal function of the
field and P is local“.

With Eqs. (2.2) and (2.4), Maxwell’s equations appear in the formz,

V E 16B
x =—-—
c at

V B-l-a—(E 4 P) 341.1
X -00"! + II + C a (2.5)

V-(E+47rP)=O
v-B=o

“where P is now the time-varying source term. In general, P is a function of E that

12
describes fully the response of the medium to the field“.
'The polarization P is usually a complicated nonlinear function of E. In. the

linear case P takes a simple linearized form given by“,
P(r,t)=_I: x‘” ( r—r.',t—t') -E( r',t')dr'dt' (2.6)

(I)

where x is the linear susceptibility. The medium is assumed to be invariant, in

obtaining Eq. (2.6), and 'if E is a monochromatic plane wave with

E(r,t)=E(k,m)=9(k,to)exp(Ik-r-itot), the Fourier transformation of (2.6)

yields’,
P r,t —)P k,a)
( )=1“(’(k,:i)-E(k,w) (2.7)
with”,
x‘”(k,m)=_I:x“’(r,t)exp(-Ik-r+icot)drdt (2.8)
The linear dielectric constant 3( k,a)) is related to 1‘” ( k,a)) byz,
e(k,a))=1+41txm(k,co). (2.9)

in the linear dipole approximation, 1‘”( k,w) is independent of r, and hence

both 1”’(k,w) and £(k.w) are independent of k. This applies for

homogeneous medium, treated at the macroscopic level, but not on the

microscopic level.

13
In the nonlinear case, when E is sufficiently weak, the polarization P as a

function of E can be expanded as power series in E given byz,

P(r,t)=I:x"’(r-r',t—t')-E(r,t')dr'dt
+_I:x‘” ( r-r,,t—t,; r—r2,t—t2):E( r,,t,)
xE(f2,I2)df1dtdfzdt

I” x“) (r-r,,t-t,; r-r,,t—t,;
+

“°° r-r3,t—t3)EE(r,,t,)
xdgmdgmmmmmmm

+...

(2.10)

where X” is the nm-order nonlinear susceptibility. If E can be expressed as group

of monochromatic plane waves E( r,t) = ZE( k,,co, ), then Fourier transformation

of (2.10) yieldsz,
P( k,w)=P"’( k,w)+1>“~”( k,w)+P‘3’( k,w)+... (2.11)
with

Pm(k,w)=z“’(k,w)-E( k,a))

Pm(k,w)=zm(k=k,+k,,w=a),:twj):E(k,,w,)E(k,,a)1)
(2.12)
P") ( k,a))-fzm( kzk, +15 +k,,w=a), in), m)

:E(k,,w, )£(k,,w,)1=:(k,,w,)

14
and
‘x("’(k=k,+k2 +...+k,,,oa=o),+co2 +...+co,,)
=J:x"”(r-r1,t-t,;...;r-r,,,t—t,,) (2.13)
xe"l"*‘ "“‘i‘ "'i ’*"°*"~‘ 1-..,1 "'~ ’1 dr, dt, ...dr,, dt,,

Similarly, in the electric dipole approximation, XI") (r,t) is independent of r,
or x‘"’(k,w) is independent of k. The linear and nonlinear susceptibilities
characterize the optical properties of a medium. Physically, 1‘") is related to the

microscopic structure of the medium via the nonlocal polarizability density.
2.2. Static Nonlocal Polarizability Density Theory

Nonlocal polarizability density theory characterizes the molecular"

response to a local field, on a microscopic level. The nonlocal polarizability
density a(r,r') is a linear-response tensor that determines the electronic
polarization induced at point r in a molecule, by an external field F‘, acting at
point r'. The electronic polarization satisfies‘

p(r)=-V-P(r) (2.14)
exactly; within a molecule, there is no ‘free’ charge, and P accounts for the

higher multipole charge densities, as well as the dipole density. Then P

.15
corresponds (on the microscopic level) to the generalized polarization of the
previous section. This relation also holds for the polarization and charge density
operators, P( r) and [2( r).
Hunt3 has shown that, for a molecule perturbed by a static external field

PM the total polarization of the electronic charge distribution is related to the

nonlocal polarizability density a(r,r') and the hyperpolarizability density
fl(r,r',r') by

P(r)=P‘°’ (r)+Idr'a(r,r' )- F’ (r) .
+y2jdr'dr"/i(r,r',r")l='(r')F(r")+... (2.15)
=P‘°’(r)+P‘"“(r)

where P‘°’ ( r) is the static polarization at r with no external perturbation.
As shown by Hunt”, Hunt et al.‘, Maaskant et al.7, Hafkensheid et al.”,
and Keyes et al.”, the nonlocal polarizability density a( r,r') determines the

linear response to the field E, and the expression for the ground—state
polarizability density in terms of the sum-over-states formulation is3

Pa(’r)lk><klf’fi(r')|0>

(Elk-£0)

 

 

. o
a,,,(r,r')=g,,,,2,< (2.16)

where Cap symmetrizes the expression with respect to the indices of the

16

operators Pa(r) and P,( r'). The prime on the summation indicates that, in

summing over the states k, the ground state is omitted.

Similarly, the expression for the nonlocal hyperpolarizability density
,6(r,r',r') is3

flaflr I r,r',r')
I <0

Wis... “alrllflljlmlr'llwhen)

(Ej—Eo )( E,-E,)

fi.(rl|0><0|f1(r')lkllklilr')Oil

(Ek_E0)2 I '

Hunt has shown that the derivatives of molecular properties with respect

 

 

O> (2.17)

 

 

 

 

-2“

to the nuclear coordinates depends on nonlocal polarizability densities. When a
nucleus changes its coordinate via an infinitesimal vector 6R’, there are two
contributions to the change in molecular dipole moment; the first is due to the
nuclear displacement and the other is due to the electronic response. This

change in the nuclear coordinates also changes the electric field 1" at the point r

(r-R‘)

l3
lr-Rl

(r-R‘).

|r-R' [3 +2l Tafi(r—R‘)5R'fl+...

=f”°) +§fl +...

a a

due to the nucleus I from Z‘ to11

r; =z'
(2.18)

17

where fa“) is the field due to nucleus I in its original coordinate. The electronic
charge distribution responds to the change 5f; via the nonlocal polarizability
density a( r,r'). At lowest order, the change in the electronic polarization 5P(r)
due to the shift 6R’ is‘1

5P, (r)=jdr'a,,(r,r')6f; (r') ' (2.19)
Using Eq. (2.19), one can find an expression for the change in electronic charge
density6p( r) induced by the shift 6R’ ; with Eqs. (2.15), (2.16) and (2.18):11

5p(r)=J' dr'zI V}, V; |r'—R’|'1 5R;

xz,[<0|fi(r)|k><k|133(r')l0>+<0|Pfl(r')lk><klfy(r)lo>] (2.20)

k . (E, 7 Eu) 1
where V; denotes the derivative with respect to ref.
Equations (2.18) and (2.19) imply11
5P(r)=jdr'a(r,r')-z‘7(r',R‘)-5R‘ (2.21)
to the lowest order in 6R‘. 1

The electronic contribution to the dipole moment is the integral of P(r)

over all space. Using equation (2.21) and adding the nuclear contribution gives11

18

an, _ 6,11; + 3p;
5R; ’ 512; 512; (2.22)

=Z16¢ +ZI Idrdr'aafi (r,r') Tm (r,R')

 

 

It also should be noted that the nonlocal polarizability’density has the
Born symmetry‘°' 1‘

aw(r,r')=a,,a(r',r). - (2.23)

_ - a
It is also possible to establish a relationship between Elia—I and 6f‘.

Suppose that a perturbing field PM is applied to a molecule; then the effective

nonlocal polarizability density changes from the unperturbed value a( r,r') to"

an", ( r, r')=aafl ( r,r’)+Idr" 1640,“: r’,r') F;( r")
+%Idr-dr- yw( r,r’,r',r')F;(r') F;(r~) (2.24)
+...

where ym6(r,r',r',r') is the second hyperpolarizability density. An infinitesimal

shift of nucleus 1 induces a response of the electrons to the change in the field
51” via the nonlocal hyperpolarizability densities11 - that is, the effect due to the
internal perturbation bf‘ cannot be distinguished from the effect of an external

perturbation F' of the same spatial variation. Therefore,

61:“, ( r, r')=aafi( r,r')+Idr' flaw“, r',r")Z‘ 7'r,,(r",RI )flt;

+...

(2.25)

19
The effective electronic polarizability is the integral of a;fi(r,r') with
respect to rand r' over all space. As a result, when a nuclearposition in a
molecule shifts infinitesimally, the change in a( r,r') is connected to the same
hyperpolarizability flafl,(r,r’,r') that describes the electronic charge

distribution’s response to external fields by11

56! p,
are;

 

=Idrdr'dr' flw(r,r’,r')Zl T&(r',RI) (225)

These results show that, when the nonlocal polarizability densities are
known, one can determine the dipole moment and polarizability derivatives with
respect to the nuclear coordinates. A change in position, however small, of the

nucleus will cause a change in the field on the electrons due to that nucleus.

aafir
an}, '

 

Using Eq. (2.26), one can perform a direct electrostatic calculation of

where all of the quantum mechanical effects are embodied in the functional
forms of the polarizabilities densities.
2.3. Frequency Dependent Nonlocal Polarizability Density Theory

Hunt at al.12 were able generalize on the static nonlocal polarizability

density theory to the frequency-dependent case.

20

The induced electronic polarization P“(r,cu), caused by a frequency-
dependent external field F(r,m) depends on the polarizability density a(r;r',w),

the hyperpolarizability density )6 (r;r',w',r',w'), and the other higher-order
nonlinear response tensors12
P“(r,o)=_Idr'a(r;r',cu)-F(r',m)
+-;-I:dm’Idr'dr'B(nr',co-tu',r',u)'):F(r;r',w—u)') P(r',m') (2.27)
+...
Just as in the static case, the induced polarization, P“(r,co), is related to p‘""(r,oo)
by”
v. P“ (r,w)=— p“(r,w). (2.28)
The frequencypependent polarizability density for a molecule in the

ground state is given by”

a¢(r;r',a))

=[1+C(w—)—a))]<0 (2.29)

 

Pa(r)G—(w)Pfi(r’)|O)
The equation is valid when the frequency co is off-resonance with

molecular transition frequencies, C(co—)—a)) is the operator for complex

conjugation and replacement of (l) by - (t). (7( w) is given by12

G’lw)=l 1- nllH-Eo—hw)"(1— n). (2.30)

21
where (do is the ground-state projection operator |0)(0|. The nonlocal
polarizability density fully determines the electronic charge redistribution linear

in the perturbing field F(r,co). Integrating a(r;r',0)) over all space with respect to

r and r' gives the dipole polarizability a(w), but moments of a(r;r',a)) also
yields all of the higher multipole, linear response tensors’.

Similarly, the hyperpolarizability density fl(r;r',w',r',a)') gives the
polarization induced at r by the lowest-order nonlinear response to a field of
frequency to’ acting at point r' and a field of frequency to" acting at r

Integrating flap, (r;r',a)',r',a)') with respect to r, r' and r" over all space yields

flap, (w',a)'), while moment integrals of flap, (r;r',a)',r',a)') yield all of the third-

order higher multipole susceptibilities. When u" is zero, the expression for the
hyperpolarizability density is given by”,
flap, (r,r', m', r" ,0) =
[1+C( as WM 0.1"(file)[i,(r")-P3°(r")]5(w)f’i(r')l0)
+(0‘W)()[ pV-Pl) 3°r()]5(0)f’,r"0()|)
+(0lMr")5(w)[i.(r)-P2°(r)]5(wP )r .r( 10>}-

This equation is derived by analogy with Eq. (43b) in Ref. 13. Also,

P§°(r)=(o P

 

 

 

 

r)|o) , and similarly for pg°(r') and Pf°(r~).

22
Hunt at al.“ proved that the change in polarizability density due to a

change 8f' in the internal field from nucleus I is determined by the same
hyperpolarizability density flaw, (r; r',(o',r', (0') that fixes the response to external

fields.

When Eq. (2.29) is differentiated with respect to Rfi, the result is1

60¢(r;r',a))

 

 

 

 

 

 

 

i 30 . _. . ,
51%, =[l+Cjw—>-w)][<a—&—, Pa(r)G(w)Pp(r ) O> .l (2.32)
+<O f’a(r)é:—§:,)—)f’fl(r') 0>+ <0 f’a(r)G-(w)lsfl(rv)laé’1;>’l

The derivative of the ground state with respect to any arbitrary parameter

n in the Hamiltonian is"

50 __ 5H
E>=-G(o)% o) (2.33)

and the derivative of the operator 5(a)) is given by”

 

56(0)) =—§(0)) dH‘Eo) 5(0))
an an
+afielelelohelomlefl’ie. (234)
an an

To obtain the derivatives needed in Eq. (2.32), one uses Eqs. (2.33) and

(2.34) with an;. The change in the Hamiltonian due to the shift 6R; is given

by

23

.61

(mi =Ier’Vilr-R’l i5(r) (2.35)

where Vi represents 31%" As in the static case, 35—1ng can be written in terms of

the polarization operator f’( r). Using Eq. (2.15), integrating by parts with respect

 

to r, and recalling that Valr — R‘lq = — Vf, |r — R’l4, Eq. (2.35) becomes12
3%: — Idr" zI f’5(r")Tfl(r",RI) (2.36)
Combining Eqs. (2.31), (2.32) - (2.34), and (2.36) the resultant equation12
aafl'gézr"w) =Idr",6fi,5(r;r',a),r",0)ZI T&(r",RI) (2.37)

where Zl is the charge on nucleus 1 and T&(r',R‘) is the dipole propagator.
When there is a shift 6R‘ in the position of nucleus 1 there is also a change in

the nuclear Coulomb field acting on the electrons; this equation proves that the
resulting change in polarizability density is determined by the same

hyperpolarizability density that fixes the response to external fields.
The derivative of the polarizability aw(w).with respect to the normal-
mode coordinate qv is given by a linear combination of the derivatives in Eq.

(2.37)”,

gawk") = 50¢(a’) 5R: . (2.38)

 

 

The Raman intensities are dependent to the matrix element (i]ap, [j )
Expanding a p, as a function of the normal mode coordinates, about the

equilibrium position (denoted by the superscript °),

5a

apa=a-p,({ql})+; axle (anal)

2
$2 23 0:, :1; lo(q.-q°.)(q..—q:.) (2.39)

v v’

 

 

+...

Then the matrix element becomes

 

 

(i]a,,.,[j) =a...({a:}) (i][j) +22" lo (i](q.—q:)[1)
+%Z Z :x’wlo (i](c7.—q°.)(q.. ~q‘L)[1) (2-40)

+...
The vibrational states are orthonormal, so for is: j , the first term on the right
hand side vanishes. The third and higher terms correspond to vibrational

overtones, which are neglected here. Then,

66: M

617v

 

(ila...[1) a Z In (i](q.—q:)[1). (2.41)

So the electronic property that determines the intensity of vibrational Raman

25
scattering is the derivative of the polarizability with respect to the normal mode

coordinate, within the approximation made here.

2.4 Relationship Between Raman Intensity and the Hyperpolarizability
Density

, aafir(r;r',a))
Equation 2.37 relates 0R,

 

to ,6(r,r',co,r",0). Integrating over all

5%,

 

space with respect to r and r' yields an equation that relates to

I
a

,6(r,r',a),r",0). It requires comparatively few assumptions; the chief requirement

aa
is that the Bom-Oppenheimer approximation be valid. Connecting j?— to the

Raman intensity requires assumptions of Placzeks’s Theory. Subject to these

conditions, the connection between Raman intensities and fl(r,r',a),r",0)is

quantum mechanically rigorous.
This suggests the possibility of a correlation between Raman intensities

and the [3 hyperpolarizability tensor that gives rise to frequency doubling (as a
nonlinear phenomenon). The frequency-doubling intensity depends on ,B(w,w) ,

which can be obtained by integrating the hyperpolarizability density

26
,B(r,r', w',r", (a) :
p(m,m)=j fi(r,r’,w,r",w)drdr'dr". (2.42)
There are two differences between the integral expressions for ,6(co,a))

5a,,
az‘ '

 

and for

1. The frequency dependence of the hyperpolarizability density

aaflr

differs; for fl(a),co) both frequencies are optical, but for 5R,

one frequency in the hyperpolarizability density is optical while
the other is zero.

2. The spatial integration has a dipoleopropagator weighting factor

flap,

 

for while there is no weighting factor in the integral for

I
,6(w,co). A molecule may have a large hyperpolarizability

5a
density and hence a large values of OR? , but a vanishing B

a

 

due to symmetry.

For these reasons, the theory does not yield a precise relation between

Raman intensities and ,6(w,w); however, it does suggest that a correlation may

27
exist. Experimental results and a literature survey to test for correlation are

discussed in Chapter 3.
2.5. Conclusion

Equation (2.37) gives a new physical interpretation for integrated
intensities of vibrational Raman bands, by showing that the band intensity
depends on the response of the molecule to the change in the Coulomb fields of

the nuclei via the B hyperpolarizability density. In Refs. 3 and 6, methods of

finding required components of a(r;r',0) are illustrated. With sufficient

information on ,6 (r; r', (0’, r',0), it should be possible to distinguish the regions of
the electronic charge distribution that contribute the most to the vibrational
Raman band intensities of isolated molecules. The dipole propagator tensors

5005(0))

—a_R‘_ weight the regions nearest to nucleus 1”. This behavior
7

appeanngin

supports additive approximations if ,6 (r;r',a)’,r',0) is largest for small |r—r'| and

|r—r")

28

2.6. References

pp)

V. D. Barger and M. G. Olsson, Classical Electricity and Magnetism, Allyn
and Bacon, Massachusetts, 1987.

Y. R. Shen, The Principles of Nonlinear Optics, John Wiley & Sons, New
York, 1984.

K. L. C. Hunt, J. Chem. Phys. 80, 393 (1984).

J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York,
1975.

K. L. C. Hunt, J. Chem. Phys. 78, 6149 (1983).

K L. C. Hunt and J. E. Bohr, J. Chem. Phys. 84, 6141 (1986).

W. J. A. Maaskant and L. J. Oosterhoff, Mol. Phys. 8, 319 (2964).

L. M. Hafkensheid and J. Vlieger, Physica 75, 57 (1974).

T. Keyes and B. M Landanyi, Mol. Phys. 33, 1271 (1977).

M. Born, Optik(Springer, Berlin, 1933), p. 406.

K L. C. Hunt, , J. Chem Phys. 90, 4909 (1989).

K L. C. Hunt, Y. Q. Liang, R. Nimalakirthi, and R. A. Harris, J. Chem.
Phys. 91, 5251 (1989).

B. J. Orr and J. F. Ward, Mol. Phys. 20, 513 (1971).

A. D. Buckingham, Proc. R. Soc. London, Ser. A 267, 271 (1962).

D. M. Bishop, Moi. Phys. 42, 1219 (1981); J. Chem. Phys. 88, 5613
(1987)

Chapter 3

Experlmental Correlatlon Between Spontaneous Raman Scattering
and the Second-order Nonlinear Response

3.1. Overview of the Theoretical Parameters Used

Theories of Raman scattering with changes in the molecular vibrational
state have been proposed by Behringer‘, Shoryginz, Van Vleck3, Placzek‘, and
Albrechts. However, the work of Peticolas et al.6 will be used in our discussion. In
spontaneous Raman scattering, an incident photon of frequency (01 is annihilated
and the photon of frequency (02 and the phonon of frequency cov are created.

wvzwpwz (31)
where the transition probability of such a process can be found's'7 by third-order
perturbation theory.

The interaction Hamiltonian between the molecular electrons and the

radiation field is given by“9 -u- E, where p is the dipole moment operator and E

is the electric field strength operator. The interaction between the electrons and

a molecular vibration is represented by (— ”'1 Q, where H is the Hamiltonian of

5Q

the electrons and Q is the normal coordinate of the molecular vibration. The

29

30
subscript 0 means that the derivative with respect to Q is taken at the
equilibrium position of the nuclei.
The differential Raman scattering cross section per molecule per

steradian in a liquid is given by‘°,

(21.7345 (0):) / a (01W2 (%j . (3.2)

. x(h(i7+1)/2w,)R(-w,,m,,w,)2L

 

 

where e(co1) and e(co2) are the dielectric constants of the liquid at m, and (02,
respectively, c is the velocity of light, 17=[exp(ha)/kT)-l]-l is the average
quantum number of the thermally excited vibrations of normal mode Q, L is the
local field correction factor, and R(—cal,a),,a)v) is a matrix element which is given
by".

R(— (01,602, 0),.)

a'Xa' la. 11 M

 

 

(g'le. I: II? X3 (33-)
= §{ (5,90 —ha)2)(Eago 4m.)
($3)
(5,0 .I...)(E:,o +....)

+ fourotherterms}

 

(3.3)

(9191 1‘ W X5 “X“. '92 '9 IQ. >

 

 

 

+

where e. and e; are the polarization vectors of the incident and the scattered
light, 3', a', and B' are the electronic wave functions of the ground and excited

electronic states, and EJ and Eog° are the energy differences between the

31
excited and the ground electronic states without coupling to the molecular
vibration.

When Q is a totally symmetric vibration, Kato et al.11 assumed that the

diagonal terms of (2.11] should dominate over the off-diagonal terms. Since

Q o
[aHI]
Q o

the last four terms in (3.3) become zero.

(g .)=(Z%|Qflsflj = o (3.4)

 

 

Thus, R(—q,m,,w,) is given by“

R("wl ’wz’wv)

= , 2[(E”0)2 +7160, 602] (3.5)

“' [(E..°)2-(ha1)2][(E..°)2-(ha5)2]
(3%].

where the wavefunctions are assumed to be real.

 

 

 

x (g'lezfl '0' X0 a'Xd lei/1 '8' >

The electrons localized on a molecule in a liquid interact with the local
field which differs from the macroscopic field due to the polarization of the other
molecules in the liquid. Using the results of Armstrong et al.12 and Eckhardt et
al.“ and treating the radiation field classically gives the local field correction

factor11

L: {[£(w() +2]/3}2 {[£(a)2) +2]/3}2 (3.6)

32
Furthermore, when the incident and scattered light have the same
polarization, R(-w,,a)2,co,) is equal to the squared polarizability derivative

(E')2+(;§)(y')2, where (67') and (7') are the average isotropy and the

anisotropy of the derived polarizability tensor with respect to the normal

coordinate at the equilibrium position.
3.2. Experimental Results

For our experimental study, we require a group of molecules that exhibit
good Raman scattering intensities. For our purposes, we chose to use a set of
mono-substituted benzenes. Besides being readily available, this particular,
group is known to possess a strong Raman scattering character. All chemicals
were purchased from Malinkrodt Chemical Company.

For a Raman scattering phenomenon, we can express the depolarization
ratio in terms of the derivation of the polarizability tensor associated with the k'“

normal mode (where k is arbitrary). The relation is“,

 

67:2
.. z . 3.7
p 45(5’)2 +77'2 ( )
The matrix element2 is
_ 4
|R|2 = ( a ' )2 +(-4—5-)(y’)2. (3.8)

Combining Eqs. (3.7) and (3.8), we obtain

IRIZ =——— a (3.9)

By use of Eq. (3.9), tedious mathematical expressions otherwise needed
to evaluate the matrix element can be avoided. Also, by combining equations
(3.2), (3.6), and (3.9), assuming that the dielectric constants are approximately
equal (81 a: $2) and also directly proportional to the square of the refractive index

of the molecule, we obtain

/ \

(WI-(64 ),[(nzi12)2]lil

 

pa

xlzf-éfi-olléé)

 

 

(3.10)

Similarly,

 

(r')2=[ 6—4§p,)[(nzi12)2][50:]4
x [h::;1)](g%)

To obtain the values of (c?)2 and (y')2, we need to find the values of the

(3.11)

 

depolarization ratio and the scattering cross-section.

The values of the depolarization ratio can be obtained experimentally,

- 4
ma1

34

p = -1,(%)+11,(%) (3,2)
" ”472% *I-(Vz)

where .L is an abbreviation for perpendicular and I is an abbreviation for parallel.

'I.(%) denotes the radiant intensity of scattered radiation plane-polarized

parallel to the scattering plane and propagating along a direction in the

scattering plane making an angle (7%) to the direction of the incident radiation

plane-polarized parallel to the scattering plane.

In our experimental study, we used a CW Argon ion laser, with a 488 nm
excitation wavelength and the schematic layout shown in Figure 1. A 1-cm
pathlength cuvette was used as the sample holder.

The first experiments that were carried out yielded the average cross-
sectional area for the mono-substituted benzene molecules (chlorobenzene,
bromobenzene, iodobenzene, toluene, aniline and N-N-dimethylaniline) using
benzene as the standard. Table 1 summarizes the values obtained and
calculated, for the respective normal mode, from these experiments.

Figures 2 to 7 show the various bands intensities as functions of the
polarization of the radiation field. In order to calculate the intensities, a peak
fitting module program, called Origin, was used. The Origin peak fitting module
is primarily designed to analyze data with many peaks. The kernel of the module

is the Levenberg-Marquardt non-linear least-squares curve fitter, the Lorentzian

fitting function had been used‘s,

35

__2_A, w
y 7’ 4(x-xc)2+ca2

 

(3.13)

where xc is the center of the peak, A is the area and a) is the full width at half

maximum.

After determining the values of the intensities, the rest of the calculations

were doneusing the equations given above to obtain the values of (a?)2 and
(7')2, and thus (6') and (7') . These values are tabulated in Tables 2 to 7.

There are differences in the values of the intensities with different polarizations
because laser power used is different from day to day.

In the case of N-N-dimethylaniline, there are only four possible modes
that can be observed because N-N-dimethylaniline fluoresces after sometime
during the experiment; and hence, peaks that are located in the lower Raman
shift frequency (less than 400 cm") are harder to determine due to the

fluorescence effects.

36

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o ..

1400
2000 (b)

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1500 In

 

 

V I V V

I I I
1G” 8d) 6“)

V

51

I I
1200 400

 

 

 

 

 

I r l

I l I l U I I
1400 1200 1000 8(1) 6CD 400 2(1)

1500 (c) llIi

1000
500
O.

Intensity
C

 

 

 

 

 

V V j t T V I

I I I I I I
1400 1200 1Q!) 8(1) 6(1) 400 200

(6) III"

2G1)
1500
1000

5°3,‘ Ii

 

 

 

 

 

I T I ' 1 W
1400 1200 1000 800 600 400 200
Raman Shift (cm'1)
Figure 2 Raman spectra taken for different polarizations of Chlorobenzene.

(a) 11., (b) *1... (c) '1,, (d) "I...

39

 

 

 

 

 

 

 

 

I v I v I I I
1400 1200 1000 8(1) 6th 400 200

 

 

 

 

 

 

I I
1400 1200 1(XD 8m 60) 4(1) 2(1)

 

 

 

 

 

 

I 1'I—“*I'I
12001000 800 600 400 200

 

 

 

 

 

Y I fly I V I 1
1000 800 600
Raman shift(cm°1)

l'l I'T‘
“(0120) 400200

Figure 3. Raman spectra taken for different polarizations of bromobenzene.
(a) *1), (b) 11... (c) '1‘, (d) "1...

40

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

: (3) .LI
20000- i
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1 ‘ I. L
0..
T fl I I f I ' I I r i
1200 1000 800 6a) 400 200
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1000-
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1200 1000 800 Gm 400 200
3000
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2000 i
1(XD
0
r I r I ' I j I ' 1 ' I
1200 1m 8% 6(1) 400 2(1)
2000.. (‘9 ll
.. Ill
1W-
0.

 

 

Raman shift (cm'1)

Figure 4. Raman spectra taken for different polarizations of iodobenzene.
(a) l1., (b) iI... (c) "1,, (d) "I...

Intensity

Figure 5.

2000
1000

 

 

 

 

 

41

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

- (a) I
J II
J
.1 I t I v I r I i I 1 I 1
1600 1400 1200 1000 800 6m 400
6» i1
ll
1 l v I v T r ' , r f I I
1600 1400 1200 1000 800 600 400
(C) II
I.L
' I ' I ' I T I T I T I I
1600 1400 1200 1000 8m 6(1) 400
(d) "1
ll
' I 1 I + I j I ' .1 f I I
1600 1400 1200 1000 8(1) 600 400

Raman shift (cm‘1)

Raman spectra taken for different polarizations of toluene.
(a) *II. (b) iIn. (C) "I I. (d) "In

40000

 

42

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) .LI
30000 I
20030
10000 f‘ .
0 ' I M I T I I T f T
1400 1200 1000 8C!) 600 400
1000 ~
g .
U)
5 0 I T l 1 1 I ' T T l ‘
E 1400 1200 1000 8(1) 600 400
3000: (c) "I
. I
2000 -
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o I ' l l T l ‘ I ' l
1400 1200 1000 8(1) 600 400
3000 a
, (d) "1,I
2000 -
1000 -
0 I l l ‘ l l ‘ 1 fl
1400 1200 1000 8C!) 600 400
Raman shift (cm'1)
Figure 6. Raman spectra taken for different polarizations of aniline.

(a) 11,, (b) *1", (c) “1,, (d) "I...

43

 

 

 

 

 

 

 

 

 

 

 

 

 

1400 1200 1000 800 6C!)
20000 -
‘ (b)

10000 - .LI

Q ll

‘0 q

5

E I I T I T I r

1400 1200 1000 800 600

 

 

 

 

 

 

 

 

 

 

 

 

T I T I f I T I
1400 1200 1000 800

Raman shift (cm'1)
Figure 7.

Raman spectra taken for different polarizations of N-N-
dimethylaniline. (a) *1.. (b) *1., (c) "1., (d) '1...

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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46

 

 

 

 

 

 

 

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61

 

 

 

 

 

 

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62

Figures 8 to 27 show the graphs of (61") and (y) plotted with respect to

the hyperpolarizability, B. The values of the hyperpolarizability, B, in figures 8 to

17 are taken from ref. 16 whereas in figures 18 to 27 the values are taken from
ref. 17. The one main difference between these two references is that in ref. 16

the value of aniline is tabulated and in ref. 17 the value of N-N-dimethylaniline is

tabulated.

3.3. Conclusion

Data in their current form show a definite correlation between the Raman
intensities and the 8 hyperpolarizabilities of the species and vibrations studied.

There are strong correlations between B and the derivative of the
isotropically averaged polarizability with respect to vibrational mode #2 in this
work, based on either set of data for the B hyperpolarizabilities. R values for the

straight line fits are ~ 0.97 in one case and ~ 0.96 in the other. A relatively high

level of correlation between (5'), and B is observed for [3 values from the first

set of literature data, and vibrational modes i = 1 to 4 (R ranges from ~ 0.87 to ~

0.97), and moderate correlations are found for (7'), and B, i = 1 to 5 (R ranges

from ~ 0.68 to ~ 0.88). Generally, correlations are weaker based on the second
set of [3 values from the literature, although even in this case, for particular
modes and particular choice of isotropic vs. depolarized Raman scattering, high

R values can be found (R ~ 0.96 and 0.92 in two cases).

63-

To determine the validity of the correlations and to determine whether
differences in R values between modes and between a and y derivatives are
chemically meaningful, it will be necessary to obtain highly reliable data on the

Raman intensities and to discriminate among literature values of B.

64

Figure 8. Graph of (601' vs. B.

 

 

 

 

 

 

 

 

 

 

6 a
CoHsNHz
r: i
b /
E 4 1 c H c1
“2 s 5
o
F 1
X
V.- coHsCHa Panama-nu
a 2 -1
'5. C H Br A01.0117900.79675
_ 5 5 30121143700423
<
‘ / +11%“; 11 mm
/ so-1.1543.~-5
/ ”0.05557
0 7 I J i I Y j V I ' I ' if ' T
-‘l 0 1 2 3 4 5
Beta ()1 1040 m4 V-1)
Figure 9. Graph of (502 vs. [3.
6 - ,
/
‘ /
5 a
Ce 5 H2
::~ 4
b: /
E ‘ ‘
"P . /,
O /
Q- //
X 3 d ,
v
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3. - 801M0018183
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< o s /Csl"lt.f3fl3 50:04:93. 11:5
/ Paoooam
/ .
‘ ' A T * T ' I Y r ' I Y I r
-1 - O 1 2 3 4 5

Beta ()1 1040 m4 V4)

65

Figure 10. Graph of (62")3 vs. 8.

 

 

 

 

 

 

 

 

 

 

 

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5‘ I
b
E s-
q n
o
P 1
X
v
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<
a -o.m
80-0.67026.N-5
p-onouz
2 I ' I I I
-1 4 5 6
Beta (x 1040 m4 V")
Figure 11. Graph of (EFL vs. [3.
5 - l/
< CGHSNH
4 -
p _
'c» /
'5 3- /
,/
“I, //”
o * /‘
P __ .
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/" '
. / R so.911a1
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o / p-omm
I I I 7 I I I I ' I I I
.1 O 1 2 3 4 5

Beta (1110.40 m4 V4)

66

Figure 12. Graph of (515 vs. B.

 

 

 

 

     

 

 

2-

: CBHSCIlE—l
b 1
19
O
‘-
x
V

in 1"
«I
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< %%

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C‘H5B | ° 5 13:04:10:
‘ T v T V I ' fl
1

2 3 . 4
Beta ()1 104° m4V-1)

' T
-1 0

Figure 13. Graph of ( 7')l vs. B.

ugJ

 

 

 

 

 

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C 6HE’NH2
~’.-‘ 4 iii
a
E CSHSCI ,
0
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o 7/"
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X
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P - 0.15005
0 v ‘ v T v I V I V 1 f
'1 O ‘ 1 2 3 4

Beta ()1 10.40 m4 V-1)

Figure 14.

Figure 15.

67

Graph of ( y')2 vs. B.

A
l

U
J J

Gamma 2 (X 10-5 cm4 g-l)

 

  

 

   

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so - 0.04404. N - 5
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1 2 3 4 5
Beta (x 1040 m4 V“)

Graph of ( y')3 vs. B.

 

 

 

 

 

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A CGHSNH
b
"3 6 - /
O //
C H Br .
n ' 5 tlic H ' PannOVoluoDod
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P - 0.00195
2 f I I I v I I I I I I I
-‘| O 1 2 3 4 5

Beta (x 1040 m‘ V“)

68

Figure 16. Graph of ( M4 vs. B.

  

 

   

 

 

 

 

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v? 4 -
a
’9
o 3..
P
X
v
V. . °5H5°H3
0 WW
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0 e . +51
1 R =000240
so=003200~=5
9:00:751
1 I T I I I If I I I I I l
-1 0 1 2 3 4 5

Beta (x 1040 m4 V-‘)

Figure 17. Graph of ( 7')5 vs. B.

 

 

 

 

 

/
4 /

:3 //:I§15NH2

a 1 /

0 CGHsCL

5 +14 /
I9 3 ‘

o

P

X

V

ID

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/ 9:000701
1 f fl ' I ' T f T ' I j 1
-1 O 1 2 3 4 5

Beta ( x 1040 m4 V“)

69.

Figure 18. Graph of (6')1 vs. B.

 

ParamOVatuoDad

 

002277220150312
000150110030914 , C$H5NH2

 

. R IO.28225
80‘220062.N-5
PIO.64645

CGH5CI

 
   

Cal-15W. 2

Alpha 1 (X 1050014911)
[E

 

 

Beta (x 1040 m4 V-‘)

Figure 19. Graph of (77')2 vs. 0.

 

 

 

 

 

 

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q
I? 6 ~
0
up
a
"' 4 -
is ’/ ParamUValuand
N
a 401 140920001039
5 C H I 80004244001441:
0. g6 5
< CGH53' “ R - 0.95070
2" Eli , . so-0.00240,~-5
. C H C' P-0.00990
, 0 5
A," V t I ' 1 ' 7 V
0 2 4 6

Beta ( x 10°4°m4 V-1)

70

Figure 20. Graph of (E’)3 vs. B.

 

 

 

    

 

 

 

 

 

 

 

 

Panmovmou
8- 00320015013023 u
000402420030701 ceHsNH
:3 R -0.05540
'3 . so-1.0405.H-5
E P3022981
I: 3 ‘
O
1-
X
v
n
a
‘5 4-
2 E
‘ CGH5I
I I n I I T f I
0 2 4 6 8
Beta (x 104°m4 V-‘)
Figure 21. Graph of (5) vs. 0.
ParamDValand
401277730002013 El
4 4 000203030019300 csHsNHz
:2 R -0.04505
'5; so - 11500. N - 5
E P-O.23929
o
"P
o
1-
X
V
I 2 ‘ c H or /'
a 5 /’f
.1:
3 Etc H l
< alCGHsar 5 5
I I I I I I I i
0 2 4 6 8

Beta ()1 104°m4 V")

71

Figure 22. Graph of (01")5 vs. B.

 

 

 

 

 

 

 

 

 

 

 

2 - Paramovuuooad
40100020043790
000.027410014013
R -01315
c 50-057109-4
b c H Cl 59-03685
E 6 5
up
so- CSH5NH2
x
v
ID
2
C H l
% CGHsBr $16 5
r 1 ' I ' I f I fl I ' I I I
.1 0 1 2 3 4 5 6
Beta ( x 10-40m‘ V-‘)
Figure 23. Graph of (y')l vs. B.
0 E
'1 C 0H5NH2
:5
E»
up
D 4 d
..
x
v
Q‘- c H BI PII‘IMUVOMUN
E 8 5
E 403.2410001.50050
00 2 000222090035025
(9 ‘ -
C0H5' R -0.33973
so - 213153.19 - 5
E p-0.57591
I ' I r I I I ' I
O 2 4 6 8

Beta ( x 104°m4 V-1)

72

Figure 24. Graph of (y’)2 vs. B.

 

 

 

 

 

 

 

 

 

 

Par-110mm m
8 ' 40170590111153 O‘HSNMoz
800722400020249
A 4
'7 R =004000
a so-1.57054,~s5
g 6a 92007001
19
$3 I c H NH
0 5 2
X m
V ‘-
N
O mcoHsar
E C H I
g m6 5
0 2 . mceHscl
I ' I ' I I I I
O 2 4 6
Beta (x 104°m4 V")
Figure 25. Graph of ( y')3 vs. B.
8-1 ParamUVatL-Dad m
AD3322640106225 chsNHz
000 437100024049
A
'7 R -071259
5’ so- 140070.19-5
E 9-017077 g
9 6 - C0H5NM°2
o
P
X C0H58' /,/
V" 7,
g 1’ E93 C0H5'
e 1‘
G C H Cl
0 E 6 5
1
I ' r i I I I
O 2 4 6

Beta (x 1040 m4 V-‘)

73

Figure 26. Graph of ( y')‘ vs. B.

 

 

 

 

 

 

 

 

 

 

 

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74

3.4. References

10.
11.
12.

13.
14.
15.
16.
17.

J. Behringer, Raman Spectroscopy, H. A. Szymanski, Ed. (Plenum, New
York, 1967), p. 168.

P. P. Shorygin and T. M. lvanova, Opt. Spektrosk. 25, 200 (1968) [Opt
Spectrosc. 25, 107 (1 968)].

J. H. Van Vleck, Proc. Natl. Acad. Sci. (US) 15, 754, (1929).

G. Placzek, Handbuch der Radiologie, edited by E. Marx (Akademische
Verlagsgesellschaft, Leipzig, 1934), Vol. 6, Chap. 2, p. 205.

A. C. Albrecht, J. Chem. Phys. 34, 1476 ( 1961 ).

L. Peticolas, L. Nafie, P. Stein, and B. Fanconi, J. Chem. Phys. 52, 1576
(1970)

F. A. Savin and l. I. Sobel’man, Opt. Spektrosk. 7, 733 (1959) [Opt.
Spectrosc. 7, 435 (1959)].

E. A. Power and S. Zienau, Phil. Trans. Roy. Soc. (London) A251, 427
(1959)

J. Fiutak, Can. J. Phys. 41, 12 (1963).

Y. Kato and H. Takuma, J. Opt. Soc. Am. 61, 347 (1971).

Y. Kato and H. Takuma, J. Chem. Phys. 54, 5398 (1971 ).

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys.
Rev. 127, 1918 (1962).

G. Eckhardt and W. G. Wagner, J. Mol. Spectry. 19, 407 (1966).

D. A. Long, Raman Spectroscopy (McGraw—Hill, Great Britain, 1977) p. 59.
Microcal Software, Inc, The Peak Fitting Module Manual, p. 41.

B. F. Levine and C. G. Bethea, J. Chem. Phys. 63, 2666 (1975).

J. L. Oudar and H. Le Person, Opt. Commun. 15, 258 (1975).

Chapter 4

Future Work

4.1. Extension of These Experiments

In our earlier experiments, we have obtained (2?) and (7') for 5 modes

of six monosubstituted benzene molecules. However, to test more adequately for
a correlation between Raman intensities and hyperpolarizabilities, additional
data are required.

We have propose to continue this experiment using other species with
known B values that can also be easily handled in the lab.

Calculations on two particular molecules (bromobenzene and N-N-
dimethylaniline) from the earlier experiments need to be redone. The data from
bromobenzene did not give a satisfactory result; and N-N-dimethylaniline
fluoresced during the experiment making it difficult to obtain a ‘clean’ spectra.
What we have proposed is to use the Ti-Sapphire laser to obtain a better

spectra in the case of N-N-dimethylaniline.

75

76

4.2. Computational Calculation on the Hyperpolarizability Density in a
Non-uniform Field Environment

In the 1960s, Lipscomb et al."5 proposed a set of computational
calculation on molecular properties based upon a perturbed Hartree-Fock,
calculations. Lipscomb et al. solved the limited basis set Hartree-Fock problem
in the presence of a perturbation term in the Hamiltonian to obtain the first-order
perturbed wavefunction, in a uniform field. They then applied the formulation to
the calculation of electric polarizability, magnetic susceptibility, and magnetic
shielding all in an invariant and uniform electric field environment.

What we propose to do is to compute the exact kind of calculation but in a

non-uniform electric field environment.

77
4.3. References

1. R. M. Stevens, R. M. Pitzer, and W. N. Lipscomb, J. Chem. Phys. ‘38, 550
(1963).

R. M. Stevens and W. N. Lipscomb, J. Chem. Phys. 40, 2238 (1964).

R. M. Stevens and W. N. Lipscomb, J. Chem. Phys. 41, 184 (1964).

R. M. Stevens and W. N. Lipscomb, J. Chem. Phys. 41, 3710 (1964).

R. M. Stevens and W. N. Lipscomb, J. Chem. Phys. 42, 3666 (1965).

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