Vibrational spectra of distorted structure Macro & Nano molecules: An algebraic approach

doi:10.4236/jbpc.2012.33031

Vol.3, No.3, 259-268 (2012) Journal of Biophysical Chemistry

http://dx.doi.org/10.4236/jbpc.2012.33031

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: [email protected]

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

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

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

2δδ

11

δδ

11

δ

iijj

i

iiii

ijiiijij

ijiijjij

ijiiijjj

ijjjjiii

CN

CNN

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

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+1exp

ss

H

pspDs













(3)

For each oscillator i, states are characterized by repre-

sentations of



22

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

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

,;,;,

4

4.

iijjijiijj

ijijij

ijiiii

ijijjj

NνNνCN,νNν

NN

NNNNvv





























(10)

The second term is the Majorana operator, M

ij

. This

operator has both diagonal and off-diagonal matrix ele-

ments









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

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

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 Nν–ν



(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

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

Ca

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

1

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

263

For these molecules, the locality parameters are



1

2tan8,,1,2,3.

iijiij

AAij











(20)

Corresponding to the two bonds. A global locality pa-

rameter for XYZ molecules can be defined as the geo-

metric mean [20]



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

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

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

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

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

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.

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