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

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, stretching modes of Metalloporphyrins and its substituted 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) algebraic mode Hamiltonian. The higher excited energy levels of Cm-H, Cm-C, Cm-D and Cb-Cb vibrational modes of Porphyrin and its substituted 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.


INTRODUCTION
Recently measurement of highly-excited overtone-combination spectra of molecules have renewed in a theoretical 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 obtained either by appropriately modifying ab-initio calculations or by more phenomenological methods.In this article, we begin a systematic analysis of overtonecombination spectra of molecules in terms of novel ap-proach: 3) Vibron model [1][2][3][4].This model is a formulation of the molecular spectral problem in terms of elements 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 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 Porphyrins and its substitute form molecule results are discussed.Finally, the conclusions are presented in Section 4.
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 different words, the second-quantization picture suited to describe anharmonic vibrational modes, is specialized through an extended use of Lie group theory and dynamical symmetries.By means of this formalism, one can attain algebraic expressions for eigenvalues and eigenvectors of even complex Hamiltonian operators, including 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 calculation 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 comparison to harmonic series, especially when facing medium-or large-size molecules.However, it should also be noticed that, as a possible drawback of purely local Hamiltonian formulations (either algebraic or not) compared with traditional perturbative approaches, the actual eigenvectors of the physical system.Yet, for very local situations, the aforementioned disadvantage is not a serious 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 enhanced 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.

Hamiltonian Operators
We address here the explicit problem of the construction of the vibrational Hamiltonian operator for the Metalloporphyrin molecules.According to the general algebraic description for one-dimensional degrees of freedom, a dynamically-symmetric Hamiltonian operator for n-interacting (not necessarily equivalent) oscillators can be written as In this expresssion, one finds three different classes of effective contributions.The first one, 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, ] ] We note, in particular, th the expressions above depe ral Hamiltonian operator (1) can be adapted to at nd 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 4 & Table 5 the values of the Vibron numbers used in the present 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-dimensional anharmonic oscillators to each atom.This procedure eventually leads to a redundant picture of the whole molecule, as it will include spurious (i.e translational/rotational) degrees of freedom.However, it is possible to remove easily such spurious modes through m -H/C m -D/C m -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 basis (3).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 algebraic parameters A i , A ij , λ ij of Eq.1 need to be calibrated to reproduce the observed spectrum.Let us clarify the actual meaning of these two steps by considering explicitly the C m -H/C m -D/C m -N stretches manifold of the Nickel Metalloporphyrin molecule.
We limit ourselves to in-plane C retching motions i.e., without including possible coupling terms with ring deformation.So, we can write for these remaining four degrees of freedom the Hamiltonian operator, 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 r 1 , r 2 , r 3 ,…….quantized through the algebra (2) With m = N , N -2,…..,1 or 0 (N -odd or even).The M ) where C is the invar (5) For non-interac orse Hamiltonian (2) can be written, in the algebraic approach, simply as

Interaction of the taken into account in i i i i
The interaction potential can be written as reduces to the usual harmonic force field when the displacements are small V(s i , s j ) ≈ type Eq.7 can be the algebraic approach by introducing two terms [26].One of these terms is the Casimir operator, C ij , of the combined (2) (2) algebra.The matrix elements of this operator in s Eq.3 are given by The operator C tu total Hamiltonian for n stretching vibrations is ij is diagonal and the vibrational quanm 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 considering an operator C ij ´ whose matrix elements are ). second term is the Majorana operator, M ij .This erator has both diagonal and off-diagonal matrix elements  (11) If λ ij =0 the vibrations have local behavior.As the λ ij s in ymmetry-Adapted Operators int group n i j   crease, one goes more and more into normal vibrations.

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 operators M ij , states are degenerate.The introduction of the Majorana operators has two effects: 1) it splits the degeneracies 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 geometric symmetry of the molecule.The total Majorana operator is divided into subsets reflecting the symmetry of the 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 (D 4h ) discussed.

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 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.2.The Poschl-Teller Hamiltonian have absorbed the λ(λ -1) part into D, can be written, in the algebraic approach, as is identical to that of stretching vibion (Eq.3).The only difference is that the coefficients 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 eigenvalues of uncoupled Poschl-Teller oscillators are, however, still given by

Th
The construction of the symmetry-adapted ope of the Hamiltonian operator of polyatomic molecule illustrated 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 D 4h symmetry here, the operators (on the basis of e considerations mentioned above) are lization of S produces states that carry reprentations transform according to the representations A 1g , B 1g , A 2g , B 2g , and E 1u of D 4h .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).
where  e and  e x e are the spectroscopic constants of diatomic molecules [27].This numerical value must be seen as initial guess; depending on the specific molecular 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 rting guess for the pa (N -1). ( 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 singleoscillator fundamental mode as E (ν = 1) = -4 A sent case we have three different energies, rresponding to symmetric and antisymmetric combinations 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) 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 Finally a numerical fitting procedure is to be carried to  are calcula study the hi

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 parameters A and λ starting from values Eq.20 and Eq.21, and A΄ (whose initial guess can be zero).
Using the Eqs.20, 21and 22, A,  and ted [4,[5][6][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  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 useful to the experimentalist to analyze the predicted vibrational frequencies with the observed data.The model pre sented here describes the splitting of local stretching/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][29][30][31][32][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 assignments of known levels.The study of vibrational excitations of these bio-molecules (proteins) has numerous importance not only in human life but also in scientific research.

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 author is very much grateful to the anonymous referee of this paper for his valuable suggestions and comments, which greatly helped to improve the quality of the paper.

Comp een th tu ed and Cal Ni(OEP)-d uencies of C ations of N ctaeth
n species E Obs (cm -1 )  (a) All values in cm -1 except N, which is dimensionless.Openly accessible at  Openly accessible at

3 W
spectra of the Porphyrin and its substituted form molecules.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

3 , 4 , 5 , 6 , 7 , 8
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)-d 4 .Eight bands are studied, which can be labeled the C m -H, C b -C b and only for Ni (OEP)-d 4 the bands labeled are C m -D, C b -C b 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 corresponding to the C m -H, C b -C b of Ni (TPP).Figures 4 and 5 gives the levels corresponding to the C m -H, C b -C b of Cu (TPP).Figures 6 and 7 gives the levels corresponding to the C m -H, C b -C b of Cu (TMP).Figures 8 and 9 gives the levels corresponding to the C m -D, C b -C b of Ni (OEP)-d 4 .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, 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 C b -C and Pyrrol breathing up to Second overtone ( = 2), the combinational bands of Nickel Octaethyl Porphyrin [Ni(OEP)] and its substituted form Ni(OEP)-d 4 molecules are given in Table

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

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

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

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

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

Figure 9 .
Figure 9. C b -C b band vibrational energy level of Ni(OEP)-d 4 .
leads to cross-anhar- monicities between pairs of distinct local oscillators in terms of the operators C ij .The third one,

Table 1 .
Comparison b substitution forms.