Overtone spectra of porphyrins and its substituted forms: an algebraic approach

doi:10.4236/jbpc.2010.12015

Vol.1, No.2, 119-133 (2010)

doi:10.4236/jbpc.2010.12015

Journal of Biophysical Chemistry

Overtone spectra of porphyrins and its substituted

forms: an algebraic approach

Srinivasa Rao Karumuri1*, Ambati Siva Rama Prasad2, Nirmal Kumar Sarkar3, Joydeep

Choudhury4, Ramendu Bhattacharjee4

1Faculty of Sciences & Humanities, Sri Viveka Institute of Technology (SVIT), Vijayawada, India; *Corresponding Author:

drsrkarumuri@rediffmail.com

2Faculty, Department of Mathematics, P.B. Siddhartha College of Arts & Sciences, Vijayawada, India

3Faculty, Department of Physics, Karimganj College, Karimganj, India

4Faculty, Department of Physics, Assam (A Central) University, Silchar, India

Received 8 June 2010; revised 13 July 2010; accepted 18 July 2010.

ABSTRACT

We introduce an algebraic model to vibrations

of polyatomic Bio-molecules and present, as an

example, the vibrational analysis of Cm-H, Cm-C,

Cm-D, Cb-Cb, pyrrol breathing and Cb-C, stretch-

ing modes of Metalloporphyrins and its substi-

tuted forms. The excited energy levels of Cb-C,

pyrrol breathing stretching modes of Ni(OEP)

and Ni(OEP)-d4 are calculated by using U(2) al-

gebraic mode Hamiltonian. The higher excited

energy levels of Cm-H, Cm-C, Cm-D and Cb-Cb

vibrational modes of Porphyrin and its substi-

tuted forms are predicted upto second overtone.

It shows that the energy levels are clustering at

the higher overtones. The results obtained by

this method are accuracy with experimental

data.

Keywords: Algebraic Model; Vibrational Spectra;

Energy Levels; Metalloporphyrins

1. INTRODUCTION

Recently measurement of highly-excited overtone-com-

bination spectra of molecules have renewed in a theo-

retical description and understanding of the observed

spectral properties. Two approaches have been mostly

used so far in an analysis of experimental data: 1) the

familiar Dunham like expansion of energy levels in

terms of rotations-vibrations quantum numbers and 2)

the solution of Schrodinger equation with potentials ob-

tained either by appropriately modifying ab-initio calcu-

lations or by more phenomenological methods. In this

article, we begin a systematic analysis of overtone-

combination spectra of molecules in terms of novel ap-

proach: 3) Vibron model [1-4]. This model is a formula-

tion of the molecular spectral problem in terms of ele-

ments of Lie algebra and it contains the same physical

information of the Dunham and potential approach.

However, by making use of the powerful methods of

group theory, one is able to obtain the desired results in a

much faster and straightforward way.

In recent years, these polyatomic bio-molecules (i.e

Metalloporphyrins) have numerous importances in the

field of Chemical Physics. In case of polyatomic bio-

molecules the parameters play major role in the Vibron

model. Of course, we have explicitly described with few

parameters, the vibrational bands of the triatomic linear

molecules HCN, OCS, HCP [5-7] and tetratomic mole-

cules HCCF, HCCD by using an algebraic approach [8].

We have also reported the vibrational bands of tetrahe-

dral molecules CCl4, SnBr4 and Propadiene [9-11] and

polyatomic bio-molecules like Nickel Octaethyl porphy-

rin, Nickel porphyrin molecules using U (2) Vibron

model respectively [12-18]. The advantage of the alge-

braic approach, as compared to that of Dunham or pho-

nological potential models, is that typically it requires

few parameters to obtain the same level of accuracy. It

also provides a simultaneous description of bending and

stretching modes [19-25].

In Section 2, review the theory of algebraic model to

polyatomic molecules is described. In Section 3, the

calculation procedure of Vibron number and the fitting

algebraic parameters corresponding to various Porphy-

rins and its substitute form molecule results are dis-

cussed. Finally, the conclusions are presented in Section 4.

2. THEORY: AN ALGEBRAIC

APPROACH

A complete description of the theoretical foundations

S. R. Karumuri et al. / Journal of Biophysical Chemistry 1 (2010) 119-133

120

Openly accessible at

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

convincingly compared with semi-empirical approaches

making use of expansions over power and products of

vibrational quantum numbers, such as a Dunham-like

series. 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. However, it should also

be noticed that, as a possible drawback of purely local

Hamiltonian 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 a 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.

2.1. Hamiltonian Operators

We address here the explicit problem of the construction

of the vibrational Hamiltonian operator for the Metal-

loporphyrin molecules. According to the general alge-

braic description for one-dimensional degrees of free-

dom, a dynamically-symmetric Hamiltonian operator for

n-interacting (not necessarily equivalent) oscillators can

be written as

H = E0 +

1

n

i



AiCi +

n

ij



Aij Cij +



ij Mij (1)

n

ij



In this expresssion, one finds three different classes of

effective contributions. The first one, AiCi

is devo-

ted to the description of n independent, anharmonic

sequences of vibrational levels (associted wih n inde-

pendent, local oscillator) in terms of the operators Ci.

The second one,

1

n

i



n

ij



Aij C

ij leads to cross-anhar-

monicities between pairs of distinct local oscillators in

terms of the operators Cij. The third one,

n

ij





ij M

ij,

describes anharmonic, non-diagonal interactions involving

pairs of local oscillators in terms of the operators Mij.

The Ci, Cij operators are invariant (Casimir) operators of

certain Lie algebras, whilst the Mij are invariant (Ma-

jorana) operators associated with coupling schemes

involving algebras naturally arising from a systematic

study of the algebraic formulation of the one-dimen-

sional model for n interacting oscillators. We work in the

local (uncoupled oscillaators) vibrational basis written as

123

....... n



 



In which the aforementioned operators have the

following matrix elements

4( )

iii

CN

i







4( )()

iji iij ij

CNN



 

 

!!

!(2)

iij j

ijiijji j

MNN





 



!!

!1

11

[(1)()(1)]

iij j

ijiii jjj

MNN

/2



 

 



 

!!

!1

11

[(1)() (1)]

iij j

ijjjj iii

MNN

/2



 

 





We note, in particular, th the expressions above de-

pe

ral Hamiltonian operator (1) can be adapted

to

at

nd on the numbers Ni (Vibron numbers). Such num-

bers have to be seen as predetermined parameters of

well-defined physical meaning, as they relate to the in-

trinsic anharmonicity of a single, uncoupled oscillator

through the simple relation. We report in Table 4 & Ta-

ble 5 the values of the Vibron numbers used in the pre-

sent study.

The gene

describe the internal, vibrational degrees of freedom

of any polyatomic molecule in two distinct steps. First,

we associate three mutually perpendicular one-dimen-

sional anharmonic oscillators to each atom. This proce-

dure eventually leads to a redundant picture of the whole

molecule, as it will include spurious (i.e transla-

tional/rotational) degrees of freedom. However, it is

possible to remove easily such spurious modes through

S. R. Karumuri et al. / Journal of Biophysical Chemistry 1 (2010) 119-133

121

m-H/Cm-D/Cm-N

st

Openly accessible at

different techniques. One is thus left with a Hamiltonian

operator dealing only with true vibrations. Such modes

are given in terms of coupled oscillators in the local ba-

sis (3). The coupling is induced by the Majorana opera-

tors. 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 Ai, Aij, λij of Eq.1 need to be calibrated

to reproduce the observed spectrum. Let us clarify the

actual meaning of these two steps by considering explic-

itly the Cm-H/Cm-D/Cm-N stretches manifold of the

Nickel Metalloporphyrin molecule.

We limit ourselves to in-plane C

retching motions i.e ., without including possible cou-

pling terms with ring deformation. So, we can write for

these remaining four degrees of freedom the Hamilto-

nian operator,

44 4

!

1

CHi iijijijij

iijij

H

ACA CM



 

 

 

The algebraic theory of polyatomic molecules consists

in

For the stretching vibrations of polyatomic olecules

co

p(–βs)] (2)

For

re

the separate quantization of rotations and vibrations in

terms of vector coordinates r1, r2, r3,……. quantized

through the algebra

1

( 2)GU23

(2)(2) ..........U U

m

rrespond to the quantization of anharmonic Morse

oscillators, with classical Hamiltonian

H(ps, s) = ps

2/2μ + D[1 – ex2

each oscillator i, states are characterized by rep-

sentations of

(2) (2)

ii

UO

Nm





(3)

With m = N, N – 2,…..,1 or 0 (N - odd or even). The

M

H = ε0i + A C, (4)

where C is the invar

ε =ε0i+Ai (m2 – N2).

Introducing th number ν =

(N

A (N ν – ν2 ). (5)

For non-interac

iiii

orse Hamiltonian (2) can be written, in the algebraic

approach, simply as

ii i

iant operator of Oi(2), with eigen

i

values

ii i

e vibrational quantumi

i-mi)/2, [26] one has

εi =ε0i – 4iiii

ting oscillators the total Hamiltonian is

H = i

i

H

,

With eigenvalues

E = i

i



= E0 –

i



4A (N ν –ν2). (6)

2.2. Hamiltonian for Stretching Vibrations

–αj sj)], (7)

which

kij si sj .

Interaction of thetaken into account

in

ii ii

The interaction potential can be written as

V(si, sj)=kij´[1 – exp(–αi si)][1 – exp(

reduces to the usual harmonic force field when

the displacements are small

V(si, sj) ≈

type Eq.7 can be

the algebraic approach by introducing two terms [26].

One of these terms is the Casimir operator, Cij, of the

combined (2) (2)

ij

OO



algebra. The matrix elements

of this operator ins Eq. 3 are given by

 Ni, νi; Nj, νjCij Ni, νi; Nj, νj  = 4[(νi + ν

the basi

j)2 –

(8)

The operator C

tu

ν)2

–

(9)

The

op

νi; Nj, νjMijNi, νi; Nj,. νj = (Niνj + Njνi – 2νiνj)

 

= –10)

Th

of

total Hamiltonian for n stretching vibrations is

H = E0 +

(νi + νj)(Ni + Nj)]

ij is diagonal and the vibrational quan-

m numbers νi have been used instead of mi. In practical

calculations, it is sometime convenient to substract from

Cij a contribution that can be absorbed in the Casimir

operators of the individual modes i and j, thus consider-

ing an operator Cij´ whose matrix elements are

 Ni, νi; Nj, νjCij Ni, νi; Nj, νj  = 4[(νi + j

(νi + νj)(Ni + Nj)] + [(Ni + Nj)/Ni]4 (Niνi – νi

2)

+ [(Ni + Nj)/Nj]4(Njνj – νj

2).

second term is the Majorana operator, Mij. This

erator has both diagonal and off-diagonal matrix ele-

ments

 Ni,

 Ni, νi +1 ; Nj, νj – 1MijNi, νi; Nj,. νj

= – [νj(νi + 1)(Ni – νi)(Nj – νj + 1)]1/2

Ni, νi – 1; Nj, νj + 1MijNi, νi; Nj,.νj

[νi(νj + 1)(Nj – νj)(Ni – νi + 1)]1/2 (

e Majorana operators Mij annihilâtes one quantum

vibration in bond i and create one in bond j, or vice

versa.

The

1i

n



n

ij





A C +

i iAij Cij +



ij Mij (11)

If λij =0 the vibrations have local behavior. As the λij s

in

ymmetry- Adapted Operators

int group

n

ij





crease, one goes more and more into normal vibra-

tions.

2.3. S

In polyatomic molecules, the geometric po

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 Mij, states are degenerate. The introduction of the

Majorana operators has two effects: 1) it splits the de-

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 λij must be

chosen in an appropriate way that reflects the geometric

symmetry of the molecule. The total Majorana operator

S. R. Karumuri et al. / Journal of Biophysical Chemistry 1 (2010) 119-133

122

Openly accessible at

S =

n

ij

M



 (12)

is divided into subsets reflecting the symmetry of the

S = S + S + -------------. (13)

The operators

op

e

(14)

Where we

(15)

This Hamiltonian

rat

E =

molecule

S, S, ------ are the symmetry-adapted

erators. The construction of the symmetry-adapted

operators of any molecule become clear in the following

sections where the cases of Porphyrins (D4h) discussed.

2.4. Hamiltonian for Bending Vibrations

We emphasize once more that the quantization schem

of bending vibrations in U(2) is rather different from

U(4) and implies a complete separation between rota-

tions and vibrations. If this separation applies, one can

quantize each bending oscillator i by means of an alge-

bra Ui(2) as in Eq.2. The Poschl-Teller Hamiltonian

H(ps, s) = ps

2/2µ - D/cosh2(αs)

have absorbed the λ(λ – 1) part into D, can be

written, in the algebraic approach, as

Hi = ε0i + Ai Ci,

is identical to that of stretching vib-

ion (Eq.3). The only difference is that the coeffici-

ents Ai in front of Ci 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 eigenvalues of

uncoupled Poschl-Teller oscillators are, however, still

given by

i





n pro

= E0 – 4Ai (Ni νi – νi

2). (16)

One ca

pr

e Metalloporphyrin Molecule

rators and

types

of

ent interactions)

s)

th

ij ijij Mij.

c12 = c23 = c34 = c45= - - - - - = 1, c13 = c24 = c35 = c46

i



coupn theceed to le the oscillators as done

eviously and repeat the same treatment of Eqs.2, 3,

and 4.

2.5. Th

The construction of the symmetry-adapted ope

of the Hamiltonian operator of polyatomic molecule illu-

strated using the example of Metalloporphyrin. 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.

By inspection of the figure, one can see that two

interactions in Metalloporphyrin:

1) First-neighbor couplings (Adjac

2) Second–neighbor couplings (Opposite interaction

With D4h symmetry here, the operators (on the basis of

e considerations mentioned above) are

n

S=

n M, S=

n

ij

SM



,

ij



c

ij



c

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

XX

X

N

12

3

4

Figure 1. The structure of Metalloporphyrin.

yl

- - - - - =0,

34 = c45 = - - - - - = 0, c13 = c24 = c35

7)

Diagona

se

. RESULTS AND DISCUSSIONS

e have used U (2) algebraic model to study vibrational

Ni (OEP) → X = H, Y = Ethyl,

Ni (OEP)-d4 → X = D, Y = Ethyl.

Ni (TPP) → X = Phen, Y = H,

Cu (TMP) → X = Mesityl, Y = H.

Ni Por → X = H, Y = H.

=

c12 = c23 = c

= c46 = - - - - - = 1, (1

lization of S produces states that carry repre-

ntations transform according to the representations A1g,

B1g, A2g, B2g, and E1u of D4h. The S operator is thus the

“symmetry adapter” operator. This result, which, at first

sight, appears to be surprising, can be easily verified by

computing the characters of the representations carried

by the eigenstates of S in the usual way. Here, in this

case the value of n is either 4 (j = 4, i = 3) or 8 (j = 8, i =

7).

3

W

spectra of the Porphyrin and its substituted form mole-

cules. The fitting algebraic parameters are A, A΄, λ, λ΄

and N (Vibron number). The values of Vibron number (N)

can be determined by the relation

1

e



i

ee

Nx





, (i = 1, 2,……) (18)

where



e and



exe are the spectroscopic constants of

diatomic molecules [27]. This numerical value must be

seen as initial guess; depending on the specific molecu-

lar structure, one can expect changes in such an estimate,

which, however, should not be larger than ± 20% of the

original value (Eq.18). The Vibron number N between

S. R. Karumuri et al. / Journal of Biophysical Chemistry 1 (2010) 119-133

123

rting guess for the

pa

(N – 1). (19)

In the pre

co

the diatomic molecule C-C, C-H and C-D are 140, 44

and 59 respectively. From the figure 1, it is noticed that

some of the bonds are equivalent. It may be noted that

during the calculation of the vibrational frequencies of

Porphyrins and substituted forms, the value of N is kept

fixed and not used as free parameter.

The second step is to obtain a sta

rameter A. As such, the expression for the single-

oscillator fundamental mode as

E (ν = 1) = –4 A

sent case we have three different energies,

rresponding to symmetric and antisymmetric combi-

nations of the different local modes. A possible strategy

is to use the center of gravity of these modes, so the

guess for

Openly accessible at

4(1 )

E

AN





(20)

The third step is to obtain an initial guess for λ. Its

role is to split the initially degenerate local modes,

placed here at the common value E used in Eq.19. Such

an estimate is obtained by considering the simple matrix

structure, we can find

1

2

EuA g

EE

N





 (21)

& 21

!

4

Bg Ag

EE

N





 (22)

Finally a numerical fitting procedure is to be carried

to



 are calcu-

la

study the

hi

4. CONCLUSIONS

nted a systematic analysis of

st

f the method is that it allows one to

do

MENTS

e to thank Prof. Thom-

adjust (in a least- square sense, for example) the para-

meters A and λ starting from values Eq.20 and Eq.21,

and A΄ (whose initial guess can be zero).

Using the Eqs.20, 21 and 22, A,



and

ted [4,5-7,27] using the available data points. We have

taken



 = 0 (In this case, the next nearest neighbor

couplings are omitted). As one can see from Table 1 &

Table 2, the agreement with experiment is good and thus

we think that the parameter set of Table 4 & Table 5 can

be used reliably to compute energies of highly excited

overtones. We note that in Table 2 & Table 3, there are

many predicted overtones that have not been studied

experimentally. We have explicit calculations up to the

second overtone (energy up to ≈ 10000 cm-1).

We have used the algebraic Hamiltonian to

ghly excited vibrational levels of the molecule Ni

(TPP), Cu (OEP), Mg (OEP), Cu (TPP), Cu (TMP), Ni

Porphyrin, Ni (OEP) and its substitution form Ni

(OEP)-d4. Eight bands are studied, which can be labeled

the Cm-H, Cb-Cb and only for Ni (OEP)-d4 the bands

labeled are Cm-D, Cb-Cb respectively. The highly excited

vibrational levels, calculated by using the algebraic

Hamiltonian Eq .11, are shown in Figures 2, 3, 4, and 5

(The detail calculated vibrational energy levels are listed

in Tables 3). Figures 2 and 3 gives the levels corre-

sponding to the Cm-H, Cb-Cb of Ni (TPP). Figures 4 and

5 gives the levels corresponding to the Cm-H, Cb-Cb of

Cu (TPP). Figures 6 and 7 gives the levels correspond-

ing to the Cm-H, Cb-Cb of Cu (TMP). Figures 8 and 9

gives the levels corresponding to the Cm-D, Cb-Cb of Ni

(OEP)-d4. When the quantum number ν increases in a

fixed band, the numbers of energy levels increase rapidly.

Usually, the degeneracy or quanti-degeneracy of energy

levels is called clustering. It may be seen from Figures 2,

3, 4, 5, 6, 7, 8 and 9 that the vibrational energy levels of

Porphyrin and its substituted form make up clusters.

In this paper, we have prese

vibrational spectra of Porphyrin and its substituted

forms in the algebraic framework making use of the

one-dimensional Vibron model i.e. U (2) Vibron model.

Using the U (2) algebraic model Hamiltonian, the

retching frequencies of Cb-C and Pyrrol breathing up to

Second overtone (



= 2), the combinational bands of

Nickel Octaethyl Porphyrin [Ni(OEP)] and its substitu-

ted form Ni(OEP)-d4 molecules are given in Table 2.

However, due to lack of sufficient data base, we could

not compare the calculated vibrational frequencies with

that of observed data of Nickel Metalloporphyrin and its

substituted forms at higher overtones. This study is use-

ful to the experimentalist to analyze the predicted vibra-

tional frequencies with the observed data. The model pre

sented here describes the splitting of local stretch-

ing/bending modes due to residual interbond interactions.

The splitting pattern determines the nature of interaction

(Parameter



׳). Once we get the parameter, we predict

the splitting pattern of overtones. It is worth to point out

that most applications of previous algebraic models

available in literature [28-33] are restricted to vibrations

of Bio-molecules.

The importance o

a global analysis of all molecular species in terms of

few algebraic parameters. In turn provides a way to

make assignments of unknown levels or to check as-

signments of known levels. The study of vibrational ex-

citations of these bio-molecules (proteins) has numerous

importance not only in human life but also in scientific

research.

5. ACKNOWKEDGE

The author Dr. Srinivasa Rao Karumuri would lik

son G Spiro for providing the necessary literature for this study. The

authors Dr.Srinivasa Rao Karumuri and Prof. Ramendu Bhattacharjee

are grateful to the DST, New Delhi for supporting this work. The au-

thor is very much grateful to the anonymous referee of this paper for

his valuable suggestions and comments, which greatly helped to im-

prove the quality of the paper.

S. R. Karumuri et al. / Journal of Biophysical Chemistry 1 (2010) 119-133

124

etween the Observed and Calculated frequencies of the fundamental vibrations of Porphyrin and its

Obs(cm-1) ECalc(cm-1) Δ (Obs-Calc) (cm-1) EObs(cm-1) ECalc(cm-1) Δ (Obs-Calc)(cm-1)

Table 1. Comparison b

substitution forms.

Sym E

Ni (OEP) Molecule Ni Porphyrin Molecule

Cm-H vibrationamode l

A1g(1) 3041 3041.9321 3042 3042.0322 -0.0322 -0.9321

B2g(27) 3041 3040.8903 +0.1103 3041 3041.0420 -0.0420

E1u(36) 3040 3040.0189 -0.0189 3041 3041.4001 -0.4001

Cb-Cb vibrate ional mod

B1g(2) 1602 1602.0445 1579 1579.0590 -0.0590 -0.0445

A1g(11) 1577 1577.9601 -0.9601 1509 1509.0589 -0.0589

E1u(38) 1604 1604.2822 -0.2822 1547 1548.9754 -1.9754

Cu (OEP) M Mg (Oolecolecule EP) Mule

Cm-H vibrational mode

A1g(1) 3041 3042.3001 3041 3039.3000 +1.7000 -1.3001

B2g(27) ........ 3052.3226 ......... ........ 3052.3207 .........

E1u(36) ........ 3062.2923 ......... ........ 3062.2910 .........

Cb-Cb vibrational mode

B1g(2) 1592 1594.6856 ....... 1596.8243 ........ -2.6856

A1g(11) 1568 1575.6450 -7.6450 1578 1578.0634 -

PP) Mole Cu Molecu

0.0634

E1u(38) ....... 1613.7239 ........ ....... 1615.58 ........

Ni (Tcule (TPP)le

Cm-C vibrational mode

A1g(1) 1235 1234.4200 1234 1235.6438 -1.6438 +0.5800

B2g(27) 1269 1270.6832 -1.6832 ......... 1245.0709 .........

E1u(36) ........ 1306.9467 ........ ......... 1271.1534 .........

Cb-Cb vibrational mode

B1g(2) 1572 1571.9300 1530 1532.5800 -2.5200 +0.0700

A1g(11) 1504 1504.6540 -0.6540 ......... 1524.7402 .........

E1u(38) ....... 1639.2243 ......... .

(OEP)-d4

........

1626.8234 .........

Ni Cu (TMP)

Cm-D vibrational mode Cm-C vibrational mode

A1g(1) 3041 1236.8843 -1.8843 3038.6800 +2.3200 1235

B2g(27) 3041 3040.4000 +0.6000 ........ 1246.3940 .......

E1u(36) 3041 3042.1219 -1.1219 1256 1256.7238 -

Cb-Cb vibrate

0.7238

ional mod

B1g(2) 1602 1602.0484 1567 1567.5587 -0.5587 -0.0484

A1g(11) 1576 1577.9629 -1.9629 1495 1495.2902 -0.2902

E1u(38) 1604 1604.2889 -0.2889 ------- 1639.7793 ........

S. R. Karumuri et al. / Journal of Biophysical Chemistry 1 (2010) 119-133

125

Table arison betwe Observculated freqb-C stretching vibrickel Oyl Por-

phyrin [Ni(OEP)] and its substited form i.e 4.

2. Compeen th

tu

ed and Cal

Ni(OEP)-d

uencies of Cations of Nctaeth

n species EObs(cm-1) ECalc(cm-1)

Δ

(Obs-C -1 -1 -1 -1

alc) (cm) EObs(cm) ECalc(cm) Δ(Obs-Calc)(cm )

Ni(OPculei(Eolecu E) Mole NOP)-d4 Mle

C-C vibrational

bmode

A1g (5) 1025 1729 1.8271 1043.3471 -18.3471 1026 1024.

B1g (14) 1151 1134.4480 1187 1168.2545 18.7455

tone

16.5520

A2g (23) 1022 1045.5548 -23.5548 1029 1026.7901 2.2099

B2g (31) 1019 1010.5280 8.4720 999 1006.2923 -7.2923

Eu (43) …… 1159.4680 ………. 1165.7804

Eu (45) ……. 994.4080 ………. 978.6798

First Over

A 1831 1828 4.3351

1g(5+6)1837.3438 -6.3438 1823.6649

Eu 1981.8380 1924.4935

A1g(23) 2135 -10.06 2216 -30.47

A1g(31) 2494 3.8610 2460 7.3031

Second Overtone

B1g 2039.4881 1948.6704

A2g 2040.7594 2019.2772

Eu 2089.0281 2088.5504

Eu 2126.0643 2189.3792

22+2145.5906 592246.3847 38

Eu 2190.3683 2289.3098

Eu 2261.3456 2293.3070

B1g 2289.6720 2309.3572

Eu 2324.6758 2365.8686

Eu 2356.7683 2371.8268

Eu 2454.6856 2398.7437

Eu 2488.9103 2404.7578

28+2490.1380 2452.6969

B1g 2497.4836 2479.8699

A1g 2502.3920 2529.2334

Eu 2519.2033 2530.8538

A2g 23) 2541 3.72539 -2.9693

A2g(29) 2568 -1.3040 2607 -2.5773

B1g(14) 2670 -19.94 2710 6.6453

E

Pyr breathing vibrational mode

(3+2537.2890 110 2541.9693

14+2569.3040 2609.5773

Eu 2582.2093 2601.3572

A1g(19+23) 2614 2606.4803 7.5197 2604 2599.9575 4.0425

3+2689.3094 302703.3547

Eu 2730.3802 2734.8643

A1g 2772.9689 2767.8346

u 2826.3283 2798.3468

A1

g

(

6

)

806 1.82418020.3622804.1759 801.6378

B1

g

(15) 761 751.2534 758759.7356 -1.73569.7466

E

u

(47) …… 765.7932 783.3495

First Overtone

A1

g

1496.0304 1398.7654

E

u

1524.7395 1404.8475

E

u

1548.6704 1439.8364

B1

g

(16) 1557 -2.277214814.1636

6+1559.2772 1476.8364

Eu 1668.5904 1558.8239

S. R. Karumuri et al. / Journal of Biophysical Chemistry 1 (2010) 119-133

126

Eu 1782.3804 1702.7529

A1g 6) 1831 -33.26 1828 2.5296

Second Overtone

(5+1864.0326 031825.4704

Eu 1893.4056 1870.7354

Eu 1899.0328 1892.7253

A2g(21) 2109 -22.44 214 10.6 6+2131.9844 9812103.7364 263

B1g(6+20) 2144 2148.4379 2148 2152.7856 -4.7856

0

24

1g

A2g(19) 2626 9.1962 2628 -1.7439

E

-4.4379

A1g(4+6) 2184 2231.9844 -47.9844 2189 2190.4356 -1.4356

Eu 2260.5412 2237.2455

Eu 2339.6766 2290.5472

A1g 2310.9441 2367.3864

Eu 2363.584 2443.9763

Eu 2417.2044 2489.8322

Eu 81.1734 2547.3453

B2542.5424 2601.3643

5+2616.8038 2629.7439

u 2683.3023 2668.5233

Os takeneferenc

Tcies of Cm-H stretching vibrations of Porphyrins & its substituted forms (Cm-1).

bserved value from the re [34-36]

able 3. Calculated excited vibrational frequen

Sym EObs E

Calc E

Obs E

Calc E

Obs E

CalC E

Obs E

Calc

Ni (OEP) Ni Porphyrin Cu (OEP) Mg (OEP)

12.30 A

1g(1) 3041 304.93 3042 3042.03 3041 3042.30 3041 304

n = 1 3052.32

1g 942.44 951.37 946.23 923.63

= 2

1u 708.52 731.73 709.53 676.95

= 3

lculated eited vibrationcies of C

B2g(27) 3041 3040.89 3041 3041.04 ........ 3052.32 ........

E

1u(36) 3040 3040.01 3041 3041.40 ........ 3062.29 ........ 3062.29

E

1u 5941.47 5951.92 5945.15 5922.78

B 5 5 5 5

A

2g 5944.38 5951.73 5947.14 5924.36

n E1u 5945.16 5952.00 5948.14 5924.89

E

1u 5946.84 5953.35 5986.70 5924.65

E

1u 5947.29 5953.72 5994.28 5925.15

E

1u 5948.07 5954.35 6007.48 5925.94

A

1g 8707.29 8730.98 8708.54 8676.16

E 8 8 8 8

E

1u 8708.97 8731.97 8709.90 8677.24

E

1u 8709.75 8732.33 8712.60 8679.38

B

1g 8712.30 8731.90 8713.53 8680.12

n E1u 8713.45 8732.96 8812.42 8681.44

E

1u 8715.91 8735.95 8853.97 8683.71

A

1g 8716.09 8735.02 8858.35 8683.79

E

1u 8703.35 8737.94 8874.75 8684.60

E

1u 8717.14 8737.94 8899.68 8685.87

B

1g 8712.13 8738.08 8904.67 8691.29

E

1u 8719.56 8738.93 8921.42 8695.86

Ca xcnal freque

b

- C

b

stretratioof Porphyrinbstitd forms ching vibns s & its suute

Sym E E

Calc Obs E

Calc E

Obs E

Ca E

Obs ECalc

Obs E lC

S. R. Karumuri et al. / Journal of Biophysical Chemistry 1 (2010) 119-133

127

P) Ni (OEP) Ni Por Cu (OEP) Mg (OE

1596.82 B

1g(2) 1602 1602.04 1579 1579.05 1592 1594.68

n = 1 578.06

1u 159.17 033.88 157.61 142.67

= 2

1u 720.62 559.73 707.83 695.26

= 3

Calculated excited vibrational s of -C & Cm-D vibrons of Porphs substituted forms

A1g(11) 1577 1577.96 1509 1509.05 1568 1575.64 1578 1

E

1u(38) 1604 1604.28 1547 1548.97 1613.72 1615.58

A

2g 3143.88 3021.25 3138.57 3137.07

E 3 3 3 3

B

2g 3170.90 3061.13 3164.55 3144.71

n E1u 3183.36 3161.24 3176.65 3148.27

E

1u 3192.04 3081.13 3190.09 3174.59

E

1u 3200.82 3231.24 3195.69 3181.43

E

1u 3216.12 3068.45 3215.85 3193.35

A

1g 4696.54 4539.77 4688.79 4689.66

E 4 4 4 4

E

1u 4729.40 4567.02 4714.77 4697.30

E

1u 4744.70 4579.69 4766.28 4712.45

B

1g 4794.54 4621.02 4783.99 4717.66

n E1u 4816.94 4639.59 4869.11 4820.98

E

1u 4865.10 4679.51 4878.12 4827.54

A

1g 4868.71 4682.52 4894.87 4839.74

E

1u 4817.61 5183.77 4925.89 4862.25

E

1u 4889.18 4699.48 4927.63 4867.88

B

1g 4762.34 4889.77 4933.51 4872.57

E

1u 4895.60 4939.95 4874.32

frequencie Cmstretching atiyrins & it

Sym EOb E

Calc E

Ob Calc E s E

CalC E s E

Calc

s s E Ob Ob

) Ni (TPP) Cu (TPP) Cu (TMP Ni (OEP)-d

4 (Cm - D

3038.68 A

1g(1) 1235 1234.42 1234 1235.64 1235 1236.88 3041

n = 1 2710.40

1g 495.39 427.57 474.51 953.77

= 2

1u 713.46 554.51 692.22 776.62

= 3

B2g(27) 1269 10.68 245.07 1246.39 3041 304

E

1u(36) 1306.94 1271.15 1256 1256.72 3041 3042.12

E

1u 2459.13 2417.34 2464.18 5952.05

B 2 2 2 5

A

2g 2508.62 2436.20 2475.39 5953.91

n E1u 2531.65 2452.85 2484.84 5955.49

E

1u 2777.49 2459.33 2524.26 6007.76

E

1u 2835.59 2470.61 2535.23 6017.93

E

1u 2936.67 2445.64 2554.31 6035.62

A

1g 3674.12 3545.08 3681.89 8774.90

E 3 3 3 8

E

1u 3710.38 3557.95 3695.99 8777.25

E

1u 3821.69 3583.47 3702.55 8778.34

B

1g 4321.98 3633.87 3723.94 8781.91

n E1u 3855.42 3669.38 3804.16 8888.28

E

1u 4470.02 3675.60 3832.11 8914.19

A

1g 4788.38 3687.14 3892.19 8969.91

S. R. Karumuri et al. / Journal of Biophysical Chemistry 1 (2010) 119-133

128

E

1u 4812.25 3704.90 3902.71 8979.66

E

1u 4947.56 3708.45 3922.24 8997.77

B

1g 5138.57 3617.35 3952.28 9025.63

E

1u 5187.30 3718.93 3958.29 9031.21

Calculated eited vibrationcies of Cb-Cb stretationof Porphyrinstitud forms xc nal freque ching vibrs s & its subte

Sym EObs Calc E

Obs E

Calc E s E

CalC E

Obs E

Calc E Ob

Ni (TPP) Cu (TPP) Cu (TMP) Ni (OEP)-d

4

(1602.04 A

1g 1) 1572 1571.93 1530 1532.58 1567 1567.55 1602

n = 1 B2g(27 11577.96

1g 062.66 027.25 229.30 159.17

1u 571.45 478.84 810.31 720.62

= 3

) 1504 504.65 1524.74 1495 1495.29 1576

E

1u(36) 1639.22 1626.82 1639.77 1604 1604.28

E

1u 3035.20 3018.75 3201.84 3143.88

B 3 3 3 3

A

2g 3065.00 3034.43 3231.64 3170.90

E

1u 3090.13 3042.27 3256.77 3183.36

E

1u 3169.76 3120.83 3346.37 3192.04

E

1u 3194.32 3139.46 3372.75 3200.82

E

1u 3237.05 3171.88 3418.64 3216.12

A

1g 4543.99 4471.00 4782.85 4696.54

E 4 4 4 4

E

1u 4581.48 4481.70 4820.34 4729.40

E

1u 4598.92 4502.90 4837.78 4744.70

B

1g 4655.78 4678.74 4894.64 4794.54

E

1u 4817.83 4726.22 5076.98 4816.94

n E1u 4880.41 4828.30 5144.19 4865.10

A

1g 5014.97 4846.17 5288.72 4868.71

E

1u 5038.52 4879.35 5314.01 4817.61

E

1u 5082.26 4930.39 5360.99 4889.18

B

1g 5149.54 4940.60 5433.26 4762.34

E

1u 5163.00 4945.56 5447.71 4895.60

TVa lueof Algebraic Parame in the calculatio, CStretchin of rphyrins ansti-

tuted forms.

(C

m -D)

able 4. s (a ) ters Usedn of Cm-H m-D g ModesPod its sub

Ni(OEP) Cu(OEP) Mg(OEP) Ni(TPP) Cu(TPP) Cu(TMP) Ni Por Ni(OEP)-d

4

N 44 44 44 44 44 44 44 59

A -12 -10 -14 -2 -70 -23 - -13

0

7.680 7.6827.61.213.174.21717.653.04

A΄ -0.24 -0.25 -0.28 -0.9985 -0.4302 -1.0182 -1.3108 -0.2782

λ 0.014 .011360.009 0.1295 0.1072 0.0369 0.0113 0.0146

λ΄ 0.011 0.012 0.012

0.5685 0.2018 0.1073 0.0021 0.2361

(a) All values in cm-1 except N, which is dimensionless.

S. R. Karumuri et al. / Journal of Biophysical Chemistry 1 (2010) 119-133

129

Table Va lu e s (a) ofbbtretching Modes of Porphyriforms.

Ni(OEP) Cu(OEP) Mg(OEP) Ni(TPP) Cu(TPP) Cu(TMP) Ni Por Ni(OEP)-d

4

5. Algebraic Parameters Used in the calculation of C-C Sns and its substituted

N 140 140 140 140 140 140 140 140

A -2.83 -2.825 -2.835 -2.7205 -2.7502 -2.8825 -2.691 -2.83

A΄ -1.223 -1.286 -0.452 -1.986 1.0921 -1.223 -3.216 -1.223

λ 0.086 0.068 0.067 0.2403 0.028 0.2581 0.0713 0.086

λ΄ 0.047 0.092 0.020 0.0981 0.1823 0.0981 0.25 0.047

(a) All values in cm-1 except N, which nsionles

is dimes.

Figure 2. Cm-H band vibrational energy level of Ni(TPP).

Figure 3. Cb-Cb band vibrational energy level of Ni(TPP).

Openly accessible at

S. R. Karumuri et al. / Journal of Biophysical Chemistry 1 (2010) 119-133

130

Figure 4. Cm-H band vibrational energy level of Cu(TPP).

Figure 5. Cb-Cb band vibrational energy level of Cu(TPP).

Figure 6. Cm-H band vibrational energy level of Cu(TMP).

Openly accessible at

S. R. Karumuri et al. / Journal of Biophysical Chemistry 1 (2010) 119-133

131

Figure 7. Cb-Cb band vibrational energy level of Cu(TMP).

F

igure 8. Cm-H band vibrational energy level of Ni(OEP)-d4.

Figure 9. Cb-Cb band vibrational energy level of Ni(OEP)-d4.

Openly accessible at

S. R. Karumuri et al. / Journal of Biophysical Chemistry 1 (2010) 119-133

132

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