ass="t m0 x1 hf y89 ff4 fs7 fc0 sc0 ls53 ws82">modes lead to a (local) Hamiltonian formulation of the
physical problem at issue (thus permitting a direct calcu-
lation of eigenvectors in this same local basis) and 2)
algebraic expansions are intrinsically anharmonic at their
zero-order approximation. This fact allows one to reduce
drastically the number of arbitrary parameters in com-
parison to harmonic series, especially when facing me-
dium- or large-size molecules. It should be however also
noticed that, as a possible drawback of purely local Ha-
miltonian formulations (either algebraic or not) com-
pared with traditional perturbative approaches, the actual
eigenvectors of the physical system. Yet, for very local
situations, the aforementioned disadvantage is not a se-
rious one. A further point of import here is found in the
ease of accounting for proper symmetry adaptation of
vibrational wave functions. This can be of great help in
the systematic study of highly excited overtones of
not-so-small molecules, such as the present one. Last but
not least, the local mode picture of a molecule is en-
hanced from the very beginning within the algebraic
framework. This is an aspect perfectly lined up with the
current tendencies of privileging local over normal mode
pictures in the description of most topical situations.
We address here the explicit problem of the construc-
tion of the vibrational Hamiltonian operator for the po-
lyatomic molecule. According to the general algebraic
description for one-dimensional degrees of freedom, a
dynamically-symmetric Hamiltonian operator for n in-
teracting (not necessarily equivalent) oscillators cab
written as
0
.
iiijijijij
H
= E+AC+AC+M
(1)
In this expresssion, one finds three different classes of
effective contributions. The first one,
1
n
i
A
i
C
i
is devoted
to the description of n independent, anharmonic sequences
of vibrational levels (associted wih n independent, local
oscillator) in terms of the operators C
i
. The second
one,
1
n
ijij
i
A
C
leads to cross-anharmonicities between
pairs of distinct local oscillators in terms of the operators
C
ij
. The third one,
1
n
ijij
i
M
, describes anharmonic,
non-diagonal interactions involving pairs of local oscil-
lators in terms of the operators M
ij
. The C
i
, C
ij
operators
are invariant (Casimir) operators of certain Lie algebras,
whilst the M
ij
are invariant (Majorana) operators as-
sociated with coupling schemes involving algebras na-
turally arising from a systematic study of the algebraic
formulation of the one-dimensional model for n inter-
acting oscillators. We work in the local (uuncoupled os-
cillaators) vibrational basis written as
123
........
n

In which the aforementioned operators have the
following matrix elements







!!
!!
!
!
12
!
11
12
!
1
4
4
2δδ
11
δδ
11
δ
iijj
iijj
i
iiii
ijiiijij
ijiijjij
ijiiijjj
ijjjjiii
CN
CNN
MNN
MNN
MNN













!
1
δ.
ijj

(2)
We note, in particular, that the expressions above de-
pend on the numbers N
i
(Vibron numbers). Such numbers
have to be seen as predetermined parameters of well-
defined physical meaning, as they relate to the intrinsic
anharmonicity of a single, uncoupled oscillator through
the simple relation. We report in Table the values of the
Vibron numbers used in the present study.
The general Hamiltonian operator 1) can be adapted to
describe he internal, vibrational degrees of freedom of
any polyatomic molecule in two distinct steps. First, we
associate three mutually perpendicular one-dimensional
anharmonic oscillators to each atom. This procedure even-
tually leads to a redundant picture of the whole molecule,
as it will include spurious (i.e. translational/rotational)
degrees of freedom. It is however possible to remove
easily such spurious modes through a technique de-
S. R. Karumuri et al. / Journal of Biophysical Chemistry 3 (2012) 259-268
Copyright © 2012 SciRes. OPEN ACCESS
261
scribed elsewhere [19-21]. One is thus left with a Ham-
iltonian operator dealing only with true vibrations. Such
modes are given in terms of coupled oscillators in the
local basis; 2) The coupling is induced by the Majorana
operators. A sensible use of these operators is such that
the correct symmetries of vibrational wave functions are
properly taken into account. As a second step, the alge-
braic parameters A
i
, A
ij
, λ
ij
of Eq.1 need to be calibrated
to reproduce the observed spectrum.
The algebraic theory of polyatomic molecules consists
in the separate quantization of rotations and vibrations in
terms of vector coordinates r
1
, r
2
, r
3
, . quantized
through the algebra


123
222GUUU
For the stretching vibrations of polyatomic molecules
correspond to the quantization of anharmonic Morse os-
cillators, with classical Hamiltonian


2
2
,= 2+1exp
ss
pspDs


(3)
For each oscillator i, states are characterized by repre-
sentations of

22
ii
ii
UO
Nm

(4)
with m
i
= N
i
, N
i
– 2, , 1 or 0 (N
i
—odd or even). The
Morse Hamiltonian (3) can be written, in the algebraic
approach, simply as
0iiii
H=ε+AC, (5)
where C
i
is the invariant operator of O
i
(2), with eigen
values
22
0
–.
iiiii
ε= ε+AmN
Introducing the vibrational quantum number

–2
iii
Nm
, [20] one has

2
0
4
iiiiii
εεANνν (6)
For non-interacting oscillators the total Hamiltonian is
i
i
H
H
with eigen-values

2
0
4
iiiii
ii
E=EANvv


(7)
2.1. Hamiltonian for Stretching
Vibrations
The interaction potential can be written as



,1exp1exp,
ijijiijj
Vsskαsαs





(8)
which reduces to the usual harmonic force field when the
displacements are small
,
ijijij
Vss kss
.
Interaction of the type Eq.8 can be taken into account
in the algebraic approach by introducing two terms. One
of these terms is the Casimir operator, C
ij
, of the com-
bined
22
ij
OO algebra. The matrix elements of this
operator in the basis Eq.2 are given by

2
?
4 .
ii ;jjijii ;j j
ijijij
N,νN,νCN,νN,ν
NN


(9)
The operator C
ij
is diagonal and the vibrational quan-
tum numbers ν
i
have been used instead of m
i
. In practical
calculations, it is sometime convenient to substract from
C
ij
a contribution that can be absorbed in the Casimir
operators of the individual modes i and j, thus consider-
ing an operator
'
ij
C whose matrix elements are





2
2
2
,;,;,
4
4
4.
iijjijiijj
ijijij
ijiiii
ijijjj
NνNνCN,νNν
NN
NNNNvv
NNNNvv








(10)
The second term is the Majorana operator, M
ij
. This
operator has both diagonal and off-diagonal matrix ele-
ments




12
12
,;,;,
2
,+1; ,1,; ,
11
,1;,1,;,
11.
iijjijiijj
ijjiij
ii jjijiijj
jiiijj
ii jjijiijj
ijjjji
NνNνMN,νNν
NvNv
NνNνMNνNν
νν+NνNν+
NνNνMNνNν
νν+NνNν+





(11)
The Majorana operators M
ij
annihilâtes one quantum
of vibration in bond i and create one in bond j, or vice
versa.
2.2. Symmetry-Adapted Operators
In polyatomic molecules, the geometric point group
symmetry of the molecule plays an important role. States
must transform according to representations of the point
symmetry group. In the absence of the Majorana opera-
tors M
ij
, states are degenerate. The introduction of the
Majorana operators has two effects: 1) It splits the de-
S. R. Karumuri et al. / Journal of Biophysical Chemistry 3 (2012) 259-268
Copyright © 2012 SciRes. OPEN ACCESS
262
generacies of figure and 2) in addition it generates states
with the appropriate transformation properties under the
point group. In order to achieve this result the λ
ij
must
be chosen in an appropriate way that reflects the geo-
metric symmetry of the molecule. The total Majorana
operator
n
ij
ij
SM
(12)
is divided into subsets reflecting the symmetry of the
molecule
=++S SS

(13)
The operators =++S SS

are the symmetryadapt-
ed operators. The construction of the symmetryadapted
operators of any molecule will become clear in the fol-
lowing sections where the cases of Metalloporphyrins
(D
4h
) will be discussed.
2.3. Hamiltonian for Bending Vibrations
We emphasize once more that the quantization scheme
of bending vibrations in U(2) is rather different from U(4)
and implies a complete separation between rotations and
vibrations. If this separation applies, one can quantize
each bending oscillator i by means of an algebra U
i
(2) as
in
Eq.4. The Poschl-Teller Hamiltonian


2
,=2cosh
s
H
ps Ds
(14)
where we have absorbed the λ(λ – 1) part into D, can be
written, in the algebraic approach, as
0
,
iiii
H=ε+AC (15)
This Hamiltonian is identical to that of stretching vi-
bration (
Eq.5). The only difference is that the coeffi-
cients A
i
in front of C
i
are related to the parameters of the
potential, D and α, in a way that is different for Morse
and Poschl-Teller potentials. The energy eigen-values of
uncoupled Poschl-Teller oscillators are, however, still
given by

2
0
4.
iiiii
ii
E=ε=EA Nνν

(16)
One can then proceed to couple the oscillators as done
previously and repeat the same treatment.
2.4. The Metalloporphyrins Molecule
The construction of the symmetry-adapted operators
and of the Hamiltonian operator of polyatomic molecules
will be illustrated using the example of Metallopor-
phyrins. In order to do the construction, draw a figure
corresponding to the geometric structure of the molecule
(
Figure 1). Number of degree of freedom we wish to
describe.
C
b
C
b
C
a
Y
Y
C
a
C
m
C
m
N
Ca
C
b
C
b
C
a
N
C
m
C
a
C
b
C
b
M
C
a
C
b
C
b
C
a
N
C
m
Y
Y
Y
Y
Y
Y
Ca
X
X
X
X
N
Figure 1. Structure of Metalloporphyrins.
By inspection of the figure, one can see that two types
of interactions in Metalloporphyrins:
1) First-neighbor couplings (Adjacent interactions)
2) Second-neighbor couplings (Opposite interactions)
The symmetry-adapted operators of Metalloporphyrins
with symmetry D
4h
are those corresponding to these two
couplings, that is,
,,
nn
ijijijij
i<ji<j
S=cMS=cM


(17)
with
12233445
13243546
12233445
13243546
1
0
0
1
c=c=c=c==
c=c=c=c==
c=c=c=c==
c=c=c=c==




The total Majorana operator S is the sum
111
SSS (18)
Diagonalization of S produces states that carry repre-
sentations of S, the group of permutations of objects,
while Diagonalization of the other operators produces
states that transform according to the representations A
1g
,
A
2g
, B
1g
, B
2g
and E
1u
of D
4h
.
2.5. Local to Normal Transition: The Locality
Parameter (
)
The local-to-normal transition is governed by the dimen-
sionless locality parameter (
). The local-to-normal tran-
sition can be studied [19,20] for polyatomic molecules,
for which the Hamiltonian is
local
1212
iiijijijij
H
HMACACM

 (19)
S. R. Karumuri et al. / Journal of Biophysical Chemistry 3 (2012) 259-268
Copyright © 2012 SciRes. OPEN ACCESS
263
For these molecules, the locality parameters are


1
2tan8,,1,2,3.
iijiij
AAij



Vibrational spectra of distorted structure Macro & Nano molecules: An algebraic approach
Vol.3, No.3, 259-268 (2012) Journal of Biophysical Chemistry
http://dx.doi.org/10.4236/jbpc.2012.33031
Copyright © 2012 SciRes. OPEN ACCESS
Vibrational spectra of distorted structure macro &
nano molecules: An algebraic approach
Srinivasa Rao Karumuri
1*
, J. Vijayasekhar
2
, Velagapudi Uma Maheswara Rao
3
,
Ganganagunta Srinivas
4
, Aappikatla Hanumaiah
5
1
Department of Electronics & Instrumentation, Lakireddy Bali Reddy College of Engineering, Mylavaram, India;
*
Corresponding Author: srinivasakarumuri@gmail.com
2
Department of Mathematics, GITAM University, Hyderabad, India
3
Department of Applied Mathematics, Andhra University, Vishakhapatnam, India
4
Department of Physics, KL University, Guntur, India
5
Department of Sciences & Humanities, Lara Vigyan Institute of Science & Technology, Vadlamudi, India
Received 18 April 2012; revised 20 June 2012; accepted 10 July 2012
ABSTRACT
Using the Lie algebraic method the vibrational
frequencies of 97 resonances Raman lines (A
1g
+
B
1g
+ A
2g
+ B
2g
) and 38 infrared bands (E
u
) of oc-
taethylporphyrinato-Ni (II) and its mesodeuter-
ated and
15
N-substituted derivates and Fullere-
nes C
60
and C
70
of 7 vibrational bands are cal-
culated using U(2) algebraic Hamiltonian with
four fitting algebraic parameters. The results ob-
tained by the algebraic technique have been com-
pared with experimental data; and they show
great accuracy.
Keywords: Lie Algebra; Vibrational Spectra; Ni
(OEP); Ni (OEP)-d
4
& Ni (OEP)-N
4
; Fullerenes
1. INTRODUCTION
Nanoscience is an interdisciplinary field that seeks to
bring about nature nanotechnology. Focusing on the na-
noscale intersection of fields such as Physics, Biology,
Engineering, Chemistry, Computer Sciences and more,
Nanoscience is rapidly expanding [1]. A comprehensive
treatment and understanding of spectroscopic features
of nano-size molecules is by far one of the most chal-
lenging aspects of current studies in molecular spec-
troscopy. On one side, experimental techniques are
producing a rapidly increasing amount of data and clear
evidence for intriguing mechanisms characterizing sev-
eral aspects of molecular dynamics in nano-bio mole-
cules [2]. On the other side, theoretical approaches are
heavily pushed towards their intrinsic limits; in the at-
tempt to provide reliable answers to hitherto unresolved
questions concerning very complex situations of nano-
bio molecules. The appearance of new experimental
techniques to produce higher vibrational excitations in
nano-bio polyatomic molecule requires reliable theo-
retical methods for their interpretation. Two approa-
ches have mostly been used so far in an analysis of
experimental data: 1) the familiar Dunham like expan-
sion of energy levels in terms of rotations-vibrations
quantum numbers and 2) the solution of Schrodinger
equation with potentials obtained either by appropri-
ately modifying ab-initio calculations or by more phe-
nomenological methods. In this article, we begin a sys-
tematic analysis of vibrational spectra of bio-nano
molecules in terms of novel approach; 3) Vibron model
[3-6].
Recently Lie algebraic model introduction [7-18] could
proved itself to be a successful model in the study of
vibrational spectra of small, medium size and polya-
tomic molecules [19,20]. The algebraic model is fully
based on the dynamical symmetry and through the lan-
guage of Lie algebra. For the triatomic, tetratomic, Tet-
rahedral and poly-atomic Bio-molecules (i.e. metal-
loporphyrins, Ni (OEP), Ni (TTP), Ni Porphyrin) we
studied earlier [21-25] using algebraic model. Using the
algebraic model in this study we have calculated the
vibrational frequencies of octaethylporphyrinato-Ni(II)
and its meso-deurated and N substituted derivatives for
97 vibrational bands each using U(2) algebraic model
Hamiltonian. In our study we used four fitting parame-
ters which provide better comparisons between the ex-
perimental and theoretical calculations throughout the
study.
In this paper, we have considered only the In-Plane
Vibrations of Nickel Octaethylporphyrin and its meso
substituent and
15
N derivatives for 97 vibrational bands
and fullerenes C
60
and C
70
for 7 vibrational bands (both
stretching and bending) are calculated by using U(2)
algebraic mode Hamiltonian.
S. R. Karumuri et al. / Journal of Biophysical Chemistry 3 (2012) 259-268
Copyright © 2012 SciRes. OPEN ACCESS
260
2. ALGEBRAIC FRAMEWORK
A complete description of the theoretical foundations
needed to formulate the algebraic model for a vibrating
molecule. We apply the one-dimensional algebraic model,
consisting of a formal replacement of the interatomic,
bond coordinates with unitary algebras. To say it in dif-
ferent words, the second-quantization picture suited to
describe anharmonic vibrational modes, is specialized
through an extended use of Lie group theory and dy-
namical symmetries. By means of this formalism, one
can attain algebraic expressions for eigenvalues and ei-
genvectors of even complex Hamiltonian operators, in-
cluding intermode coupling terms as well expectation
values of any operator of interest (such as electric dipole
and quadrupole interactions). Algebraic model are not
ab-initio methods, as the Hamiltonian operator depends
on a certain number of a priori undetermined parameters.
As a consequence, algebraic techniques can be more con-
vincingly compared with semi-empirical approaches mak-
ing use of expansions over power and products of vi-
brational quantum numbers, such as a Dunham-like se-
ries. However, two noticeable advantages of algebraic
expansions over conventional ones are that 1) algebraic

(20)
Corresponding to the two bonds. A global locality pa-
rameter for XYZ molecules can be defined as the geo-
metric mean [20]

12
12
.

(21)
Locality parameters of this metalloporphyrins is given
in the results and discussions With this definition, due to
Child and Halonen [21], local-mode molecules are near
to the
= 0 limit, normal mode molecules have
1.
3. RESULTS AND DISCUSSIONS
The number N [total number of bosons, label of the
ire-ducible representation of U(4)] is related to the total
number of bound states supported by the potential
well.
Equivalently it can be put in a one-to-one correspond-
dence with the anharmonicity parameters x
e
by means of
1
.
2
e
x
N
(22)
We can rewrite the
Eq.22 as

11,2.
e
i
ee
Ni
x
 (23)
Now, for a blood cell molecule, we can have the val-
ues of ω
e
and ω
e
x
e
for the distinct bonds (say CH, CC,
CD, CN etc.) from the study of K. Nakamoto [22] and
that of K.P. Huber and G. Herzberg [23]. Using the val-
ues of ωe and ωexe for the bond CH/CC we can have the
initial guess for the value of the vibron number N.
Depending on the specific molecular structure N
i
can
vary between ±20% of the original value. The vibron
number N between the diatomic molecules C-H and C-C
are 44 and 140 respectively. Since the bonds are equiva-
lent, the value of N is kept fixed. This is equivalent to
change the single-bond anharmonicity according to the
specific molecular environment, in which it can be
slightly different.
Again the energy expression for the single-oscillator in
fundamental mode is

141EvAN
(24)
In the present case we have three and six different en-
ergies corresponding to symmetric and antisymmetric
combinations of the different local mode.

41
A
EN (25)
where E = Average energy, The initial guess for λ can be
obtained by
12
2EEN
 (26)
A numerical fitting procedure is adopted to adjust the
parameters A and λ starting from the values above and A’
whose initial guess can be zero.
The complete Calculation data in stretching and bend-
ing modes of different Bio & Nano molecules are pre-
sented in
Tables 1-5 and the corresponding algebraic
parameters are presented in
Tables 6 and 7.
4. CONCLUSIONS
We have presented here a vibrational analysis of the
stretching/bending modes of Bio molecules (i.e. Nickel
Porphyrins) and Nano molecules (Fullerenes C
60
, C
70
) in
terms of one-dimensional Vibron model i.e. U(2) alge-
braic model.
From the view of group theory, the molecule of
Ni(OEP), Ni(OEP)-d
4
& Ni(OEP)-
15
N
4
takes a square
planar structure with the D
4h
symmetry point group.
Molecular vibrations of metalloporphyrins are classi-
fied into the in-plane and out of plane modes. For Oc-
taethyl dimmers of D
4h
structure assuming the periph-
eral ethyl group is point mass the in-plane vibrations of
Octaethyl dimmers are factorized into 35 gerade and 18
ungerade. Out of planes are factorized into 8 gerade and
18 ungerade modes. The A
2u
and E
u
modes are IR active
where the A
1g
, B
1g
, A
2g
, B
2g
& E
g
modes are Raman
active in an ordinary sense. The Nano-molecules C
60
and C
70
are I
h
and D
5h
point group symmetry respec-
tively.
In this study the resonance Raman spectra of Ni(OEP),
Ni(OEP)-d
4
and Ni(OEP)
15
N
4
for 97 vibrational bands,
we obtain the RMS deviation i.e. (r.m.s) = 40.92 cm
–1
,
33.03 cm
–1
, 4.04 cm
–1
and the locality parameters are
1
= 0.0765,
2
= 0.0468,
3
= 0.0685 respectively.
In this study the vibrational frequencies of Nano
molecules C
60
and C
70
for 7 vibrational bands, we obtain
the RMS deviation i.e. (r.m.s) = 6.439 cm
–1
, 3.2029
cm
–1
, and the locality parameters are
1
= 0.0384,
2
=
0.0493,
3
= 0.0590 respectively.
Using improved set of algebraic parameters, the RMS
deviation we reported in this study for Bio and Nano
molecule is lying near about the experimental accuracy.
Using only four algebraic parameters, the RMS deviation
we reported in this study for Bio-Nano molecule are bet-
ter fit.
The above two points confirm that in four parameters
fit, the set of algebraic parameters we reported in this
study of local to normal transition provide the best fit to
the spectra of Bio-Nano molecules.
We hope that this work will be stimulate further re-
search in analysis of vibrational spectra of other Nano
molecules like fullerenes and protein molecules where the
algebraic approach has not been applied so far.
S. R. Karumuri et al. / Journal of Biophysical Chemistry 3 (2012) 259-268
Copyright © 2012 SciRes. OPEN ACCESS
264
Table 1. Comparison between the experimental and Calculated frequencies of the resonance Raman active fundamental modes
of Ni(OEP) (cm
–1
).
Symmetry Mode Description Exp
a
Cal
(Exp-Calc)
1
(Cm - H)
- 3041.94 -
2
(Cb - Cb)
1602 1602.04 –0.04
3
(Ca-Cm)sym
1519 1525.06 –6.06
4
(Pyrhalf-ring)sym
1383 1383.45 –0.45
5
(Cb - C)sym
1025 1010.52 14.48
6
(Pyr breathing)
806 803.76 2.24
7
δ (Pyr def)sym 674 685.99 –11.99
8
(Ni - N)
344 344.36 –0.36
A
1g
9
δ (Cb - C)sym 226 225.66 0.34
10
’(Ca - Cm)sym
1655 1639.26 15.74
11
(Cb - Cb)
1576 1577.96 –1.96
12
(Pyr half-ring)sym
- 1293.43 -
13
δ (Cm - H) 1220 1219.69 0.31
14
(Cb - C)sym
- 1065.55 -
15
(Pyr breathing)
- 750.51 -
16
δ (Pyr def)sym 751 752.43 –1.43
17
δ (Cb - C)sym - 304.08 -
B
1g
18
(Ni-N)
- 423.15 -
19
’(Ca - Cm)sym
1603 1589.26 13.74
20
(Pyr quarter-ring)
1397 1396.89 0.11
21
δ (Cm - H) 1308 1327.40 –19.40
22
’(Pyr half-ring)sym
1121 1121.87 –0.87
23
’ (Cb - C)sym
- 1104.44 -
24
δ’(Pyr def)sym 739 732.81 6.19
25
δ (Pyr rot) - 523.70 -
A
2g
26
δ’(Cb - C)sym - 382.51 -
27
(Cm - H)
- 3040.95 -
28
’ (Ca - Cm)sym
- 1507.32 -
29
(Pyr quarter-ring)
1409 1408.53 0.47
30
’ (Pyr half-ring)sym
1159 1142.34 16.66
31
’ (Cb - C)sym
- 1159.46 -
32
δ’ (Pyr def)sym 785 773.06 11.94
33
δ (Pyr rot) - 528.26 -
34
δ’ (Cb - C)sym - 437.96 -
B
2g
35
δ (Pyr transl) - 178.96 -
36
(Cm - H)
- 3040.95 -
37
’ (Ca - Cm)sym
1604 1642.79 –38.79
38
(Cb - Cb)
1557 1604.28 –47.28
39
(Ca - Cm)sym
1487 1474.88 12.12
40
(Pyr quarter-ring)
1443 1442.36 0.64
41
’ (Pyr half-ring)sym
1389 1392.45 –3.45
42
δ (Cm-H) 1268 1266.81 1.19
43
’ (Cb-C)sym
1148 1143.34 4.66
44
’ (Pyr half-ring)sym
1113 1113.39 –0.39
45
’ (Cb-C)sym
993 994.40 –1.60
46
δ’ (Pyr)sym 924 925.56 –1.56
47
(Pyr breathing)
726 727.94 –1.94
48
δ (Pyr)sym 605 604.92 0.08
49
δ (Pyr rot) 550 552.48 –2.48
50
(Ni - N)
- 501.94 -
51
δ’ (Cb - C)sym - 460.94 -
52
δ (Cb - C)sym 287 288.26 –1.26
Eu
53
δ (Pyr transl) - 183.77 -
a
Experimental data has taken from Reference [24]. (r.m.s) = 40.92 cm
–1
S. R. Karumuri et al. / Journal of Biophysical Chemistry 3 (2012) 259-268
Copyright © 2012 SciRes. OPEN ACCESS
265
Table 2.
Comparison between the experimental and Calculated frequencies of the resonance Raman active fundamental modes
of Ni(OEP)-d
4
(cm
–1
).
Symmetry Mode Description Exp
a
Cal
(Exp-Calc)
1
(Cm - D)
- 2265.10 -
2
(Cb - Cb)
1602 1612.99 –10.99
3
(Ca - Cm)sym
1512 1513.86 –1.86
4
(Pyr half-ring)sym
1382 1384.58 –2.58
5
(Cb - C)sym
1026 1026.81 –0.81
6
(Pyr breathing)
802 802.86 –0.86
7
δ (Pyr def)sym 667 668.34 –1.34
8
(Ni - N)
342 340.76 1.24
A
1g
9
δ (Cb - C)sym 226 227.40 –1.40
10
’ (Ca - Cm)sym
1645 1650.13 –5.13
11
(Cb - Cb)
1576 1578.24 –2.24
12
(Pyr half-ring)sym
- 1272.24 -
13
δ (Cm - D) 950 951.44 –1.44
14
(Cb - C)sym
1187 1188.94 –1.94
15
(Pyr breathing)
- 762.93 -
16
δ (Pyr def)sym 684 683.18 0.82
17
δ (Cb - C)sym - 296.58 -
B
1g
18
(Ni - N)
- 171.41 -
19
’ (Ca - Cm)sym
1582 1565.75 16.25
20
(Pyr quater-ring)
1397 1397.12 –0.12
21
δ (Cm - D) 890 891.24 –1.24
22
’ (Pyr half-ring)sym
1202 1203.64 –0.64
23
’ (Cb - C)sym
1029 1028.43 0.57
24
δ’ (Pyr def)sym 733 736.96 –3.96
25
δ (Pyr rot) - 524.78 -
A
2g
26
δ’ (Cb - C)sym - 277.35 -
27
(Cm - D)
- 2268.85 -
28
’ (Ca - Cm)sym
- 1712.19 -
29
(Pyr quater-ring)
1408 1408.90 –0.90
30
’ (Pyr half-ring)sym
1159 1159.91 –0.91
31
’ (Cb - C)sym
- 1165.43 -
32
δ’ (Pyr def)sym 785 805.59 –20.59
33
δ (Pyr rot) - 493.73 -
34
δ’ (Cb - C)sym - 252.37 -
B
2g
35
δ (Pyr transl) - 182.17 -
36
(Cm - D)
- 3040.95 -
37
’ (Ca - Cm)sym
1595 1592.94 2.06
38
(Cb - Cb)
1542 1543.50 –1.50
39
(Ca - Cm)sym
1480 1480.76 –0.76
40
(Pyr quarter-ring)
1440 1445.84 –5.84
41
(Pyr half-ring)sym
1383 1384.58 –1.58
42
δ (Cm - D) 1175 1175.14 –1.14
43
’ (Cb - C)sym
1114 1112.05 1.95
44
’ (Pyr half-ring)sym
1018 1002.44 15.56
45
’ (Cb - C)sym
943 944.41 –1.41
46
δ’ (Pyr)sym 843 840.20 2.80
47
(Pyr breathing)
722 723.01 –0.01
48
δ (Pyr)sym 597 598.28 –1.28
49
δ (Pyr rot) 537 536.54 –1.95
50
(Ni - N)
- 256.08 -
51
δ’ (Cb - C)sym - 302.32 -
52
δ (Cb - C)sym - 295.01 -
Eu
53
δ (Pyr transl) - 188.27 -
a
Experimental data has taken from Reference [24]. (r.m.s) = 33.03 cm
–1
S. R. Karumuri et al. / Journal of Biophysical Chemistry 3 (2012) 259-268
Copyright © 2012 SciRes. OPEN ACCESS
266
Table 3. Comparison between the experimental and Calculated frequencies of the resonance Raman active fundamental modes
of Ni(OEP)-
15
N
4
(cm
–1
).
Symmetry Mode Description Exp
a
Cal
(Exp-Calc)
1
(Cm - N)
- 2089.8 -
2
(Cb - Cb)
1602 1603.02 –1.02
3
(Ca - Cm)sym
1519 1525.06 –6.06
4
(Pyr half-ring)sym
1377 1371.10 5.90
5
(Cb - C)sym
1022 1021.12 0.88
6
(Pyr breathing)
801 803.76 2.76
7
δ (Pyr def)sym 673 685.99 –12.99
8
(Ni - N)
344 344.26 –0.26
A
1g
9
δ (Cb - C)sym 226 226.28 –0.28
10
’ (Ca - Cm)sym
1655 1639.26 15.74
11
(Cb - Cb)
1576 1575.39 0.61
12
(Pyr half-ring)sym
- 1337.81 -
13
δ (Cm - N) 1220 1220.05 –0.05
14
(Cb - C)sym
- 1273.68 -
15
(Pyr breathing)
- 750.51 -
16
δ (Pyr def)sym 749 752.43 –3.43
17
δ (Cb - C)sym - 330.59 -
B
1g
18
(Ni - N)
- 369.38 -
19
’ (Ca - Cm)sym
1603 1589.26 13.74
20
(Pyr quater-ring)
1396 1396.89 –0.89
21
δ (Cm - N) 1305 1309.28 –4.28
22
’ (Pyr half-ring)sym
1108 1113.39 –5.39
23
’ (Cb-C)sym
- 1065.55
24
δ’ (Pyr def)sym 732 732.81 –0.81
25
δ (Pyr rot) - 523.70 -
A
2g
26
δ’ (Cb - C)sym - 304.08 -
27
(Cm - N)
- 2105.12 -
28
’ (Ca - Cm)sym
- 1474.88 -
29
(Pyr quater-ring)
1408 1408.53 –0.53
30
’ (Pyr half-ring)sym
1150 1142.34 7.66
31
’ (Cb - C)sym
- 1010. 52 -
32
δ’ (Pyr def)sym 785 773.06 11.94
33
δ (Pyr rot) - 528.26 -
34
δ’ (Cb - C)sym - 460.94 -
B
2g
35
δ (Pyr transl) - 178.96 -
36
(Cm - N)
- 2120.38 -
37
’ (Ca - Cm)sym
1603 1603.02 –0.02
38
(Cb - Cb)
1555 1562.23 –7.23
39
(Ca - Cm)sym
1484 1483.30 0.70
40
(Pyr quarter-ring)
1442 1442.36 0.36
41
(Pyr half-ring)sym
1386 1383.45 2.55
42
δ (Cm - N) 1266 1265.11 0.89
43
’ (Cb - C)sym
1140 1147.40 –7.40
44
’ (Pyr half-ring)sym
1108 1113.39 –5.39
45
’ (Cb - C)sym
986 994.40 –8.40
46
δ’ (Pyr)sym 921 918.06 2.94
47
(Pyr breathing)
719 727.94 –8.94
48
δ (Pyr)sym 602 601.10 0.90
49
δ (Pyr rot) 550 552.48 –2.48
50
(Ni - N)
- 394.50 -
51
δ’ (Cb - C)sym - 374.90 -
52
δ (Cb - C)sym - 288.26 -
Eu
53
δ (Pyr transl) - 183.77 -
a
Experimental data has taken from Reference [24]. (r.m.s) = 4.04 cm
–1
.
S. R. Karumuri et al. / Journal of Biophysical Chemistry 3 (2012) 259-268
Copyright © 2012 SciRes. OPEN ACCESS
267
Table 4.
Comparisons between the experimental and calculated frequencies of the Raman active fundamental modes of C
60
(cm
–1
).
Vibrational mode Exp
b
Cal
(Exp-Calc)
1
273 275.9303 –2.9303
2
497 498.3048 –1.3048
3
528 530.2039 –2.2039
4
577 577.3049 –0.3049
5
1183 1182.2093 0.7907
6
1429 1431.9848 –2.9848
7
1469 1470.5968 –1.5968
(r.m.s) = 6.439 cm
–1
.
Table 5. Comparisons between the experimental and Calculated frequencies of the Raman active fundamental modes of C
70
(cm
–1
).
Vibrational mode Exp
b
Cal
(Exp-Calc)
1
260 259.3543 0.6457
2
571 573.0294 –2.0294
3
1062 1064.3029 –2.3029
4
1185 1186.0928 –1.0928
5
1232 1233.2930 –1.2930
6
1513 1513.2087 –2.9848
7
1568 1565.3392 2.6608
b
Experimental data has taken from Reference[25], (r.m.s) = 3.2029 cm
–1
.
Table 6. Fitting algebraic parameters of octaethylporphyrinato Ni(II) and its meso-deuterated and N-substituted derivatives.
Cm-H Cb-Cb Cb-C Ca-Cm Ni-N Pyr.half Pyr.quater Pyr.breath Pyr.rot Pyr.def
Ni(OEP) molecule
A –1.8972 –1.7829 –1.8293 –1.5403 –2.2832–1.0293 –2.3940 –1.2930 –1.2394 –1.2930
A’ –0.3094 –0.3049 –0.3833 –0.3209 –0.4954–0.4859 –0.4930 –0.4938 –0.2918 –0.3820
λ 0.0394 0.0238 0.0495 0.0594 0.0293 0.0433 0.0867 0.0594 0.0637 0.0322
λ 0.1029 0.0384 0.3902 0.0293 0.0390 0.0902 0.0293 0.0783 0.0394 0.9200
Ni(OEP)-d
4
molecule
A –1.9567 –1.7394 –1.7574 –1.4839 –2.4758–1.9438 –1.5783 –1.4839 –1.3489 –1.4938
A’ –0.4039 –0.5493 –0.4938 –0.2345 –0.5489–0.2390 –0.4465 –0.3493 –0.2930 –0.4930
λ 0.0840 0.0349 0.0657 0.0405 0.0349 0.0128 0.0928 0.0647 0.0493 0.0574
λ’ 0.2349 0.0504 0.0394 0.0192 0.0128 0.0495 0.0112 0.0349 0.0325 0.0932
Ni(OEP)-15N
4
molecule
A –1.7849 –1.7839 –1.8495 –1.3849 –2.3948–1.0490 –2.4930 –1.3049 –1.3829 –1.2389
A’ –0.4302 –0.3940 –0.3647 –0.2784 –0.4304–0.3920 –0.4289 –0.3940 –0.3920 –0.4673
λ 0.0333 0.0394 0.0432 0.0394 0.0239 0.0320 0.0788 0.0403 0.0433 0.0333
λ 0.0938 0.0574 0.2987 0.0293 0.0293 0.0843 0.0392 0.0563 0.0233 0.0945
Table 7. Fitting algebraic parameters of fullerenes C
60
and C
70.
Vibron number Algebraic parameters
N A A’ λ λ
C60 140 –1.4309 0.0384 0.0739 –0.4932
C70 140 0.9837 0.0456 0.0348 0.5903
All values in cm
–1
except N, which is dimensionless.
S. R. Karumuri et al. / Journal of Biophysical Chemistry 3 (2012) 259-268
Copyright © 2012 SciRes. OPEN ACCESS
268
5. ACKNOWLEDGEMENTS
The author Srinivasa Rao Karumuri would like to thank Prof. Thom-
son G. Spiro for providing necessary literature for this study.
The author Srinivasa Rao Karumuri also would like to thank Uni-
versity of Grant Commission (UGC), New Delhi, India, for providing
the financial assistance for this study.
REFERENCES
[1] Kroto, H.W., Heath, J.R., Brien, S.C.O., Curl, R.F. and
Smalley, R.E.
(1985) C
60
: Buckminsterfullerene. Nature,
318
, 162-163. doi:10.1038/318162a0
[2]
Treubig Jr., J.M. and Brown, P.R. (2002) Analysis of C
60
and C
70
fullerenes using high-performance liquid chro-
matography—Fourier transform infrared spectroscopy.
Jour-
nal of Chromatography A
, 960, 135-142.
[3]
Levine, R.D. (1982) Representation of one-dimensional
motion in a morse potential by a quadratic Hamiltonian.
Chemical Physics Letters, 95, 87-90.
doi:10.1016/0009-2614(83)85071-4
[4]
Iachello, F. and Levine, R.D. (1982) Algebraic approach
to molecular rotation
vibration spectra. I. Diatomic mo-
lecules.
Journal of Chemical Physics, 77, 3046-3055.
doi:10.1063/1.444228
[5]
van Roosmalen, O.S., Dieperink, A.E.L. and Iachello, F.
(1982) A unified algebraic model description for inter-
acting vibrational modes in ABA molecules.
Chemical
Physics Letters
, 85, 32-36.
doi:10.1016/0009-2614(82)83455-6
[6]
van Roosmalen, O.S., Iachello, F., Levine, R.D. and Die-
perink, A.E.L. (1983) The geometrical-classical limit of
algebraic Hamiltonians of molecular vibrational spectra.
Chemical Physics Letters, 79, 2515.
doi:10.1063/1.446164
[7]
Sarkar, N.K., Choudhury, J. and Bhattacharjee, R. (2006)
An algebraic approach to the study of the vibrational
spectra of HCN.
Molecular Physics, 104, 3051-3055.
doi:10.1080/00268970600954235
[8]
Sarkar, N.K., Choudhury, J., Karumuri, S.R. and Bhat-
tacharjee, R. (2008) An algebraic approach to the com-
parative study of the vibrational spectra of monofluo-
roacetylene (HCCF) and deuterated acetylene (HCCD).
Molecular Physics, 106, 693-702.
doi:10.1080/00268970801939019
[9]
Sarkar, N.K., Choudhury, J. and Bhattacharjee, R. (2008)
Study of vibrational spectra of some linear triatomic mole-
cules.
Indian Journal of Physics, 82, 767.
[10]
Choudhury, J., Karumuri, S.R., Sarkar, N.K. and Bhat-
tacharjee, R. (2008) Vibrational spectroscopy of CCl
4
and SnBr
4
using lie algebraic approach. Physics and As-
tronomy
, 71, 439-445. doi:10.1007/s12043-008-0123-z
[11]
Choudhury, J., Karumuri, S.R. and Bhattacharjee, R.,
(2008) Algebraic approach to analyze the vibrational spec-
tra of tetrahedral molecules.
Indian Journal of Physics,
82, 561-565.
[12]
Karumuri, S.R., Sarkar, N.K., Choudhury, J. and Bhat-
tacharjee, R.
(2008) Vibrational spectroscopy of C
m
-H,
C
-C
stretching vibrations of Nickel metalloporphyrins.
Molecular Physics, 106, 1733-1738.
doi:10.1080/00268970802248998
[13]
Karumuri, S.R., Choudhury, J., Sarkar, N.K. and Bhat-
tacharjee, R. (2008) Analysis of resonance raman spectra
of nickeloctaethyl porphyrin using lie algebra.
Journal of
Environmental Research and Development
, 3, 250-256.
[14]
Karumuri, S.R., Sarkar, N.K., Choudhury, J. and Bhat-
tacharjee, R. (2009) Study of vibrational spectra of Nickel
metalloporphyrins: An algebraic approach.
Pramana
Journal of Physics, 72, 517-525.
[15]
Karumuri, S.R., Sarkar, N.K., Choudhury, J. and Bhat-
tacharjee, R. (2009) Vibrational spectroscopy of stretch-
ing and bending modes of nickel tetraphenyl porphyrin:
An algebraic approach.
Chinese Physics Letters, 26, 093-
301. doi:10.1088/0256-307X/26/9/093301
[16]
Karumuri, S.R., Sarkar, N.K., Choudhury, J. and Bhat-
tacharjee, R. (2009
) U(2) algebraic model applied to vi-
brational spectra of Nickel Metalloporphyrins.
Journal of
Molecular Spectroscopy
, 255, 183-188.
doi:10.1016/j.jms.2009.03.014
[17]
Karumuri, S.R., Choudhury, J., Sarkar, N.K. and Bhat-
tacharjee, R. (2010) Vibrational Spectroscopy of C
m
-C/
C
b
-C
b
stretching vibrations of Copper Tetramesityl Por-
phyrin Cu (TMP): An algebraic approach.
PramanaJour-
nal of Physics
, 74, 57-66.
doi:10.1007/s12043-010-0007-x
[18]
Karumuri, S.R. (2010) Calculation of vibrational spectra
by an algebraic approach: Applications to Copper Tetra-
mesityl Porphyrins and its Cation radicals.
Journal of Mo-
lecular Spectroscopy
, 259, 86-92.
doi:10.1016/j.jms.2009.11.005
[19]
Iachello, F. and Levine, R.D. (1995) Algebraic theory of
molecules. Oxford University Press, Oxford.
[20]
Iachello, F. and Oss, S. (2002), Algebraic methods in quan-
tum mechanics: From molecules to polymers
. Physics
and Astronomy
, 19, 307-314.
doi:10.1140/epjd/e20020089
[21]
Child, M.S. and Halonen, L.O. (1984) Overtone frequen-
cies and intensities in the local mode picture.
Advances
in Chemical Physics
, 57, 1.
doi:10.1002/9780470142813.ch1
[22]
Wood, B.R., Stoddart, P.R. and McNaughton, D. (2007)
Molecular imaging of red blood cells by raman spec-
troscopy.
Australian Journal of Chemistry, 387, 1691.
[23]
Phuber, K. and Herzberg, G., (1979) Molecular spectra
and molecular structure IV: Constants of diatomic mole-
cules. Van Nostrand Reinhold Co., New York.
[24]
Kitagawa, T., Abe, M. and Ogoshi, H. (1978) Resonance
Raman spectra of octaethylporphyrinato
Ni(II) and meso
deuterated and
15
N substituted derivatives. II. A normal
coordinate analysis.
Journal of Chemical Physics, 69,
4526. doi:10.1063/1.436450
[25]
Schettino, V., Pagliai M. and Cardini, G. (2002) The in-
frared and raman spectra of fullerene C
70
. DFT calcula-
tions and correlation with C
60
. The Journal of Physical
Chemistry A
, 106, 1815-1823. doi:10.1021/jp012680d.