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Page 1: Chapter 8 Infrared spectra of iodine complexes with five ...shodhganga.inflibnet.ac.in/bitstream/10603/34607/13... · Figure 2 (c) Analysis of Gaussian background profile in mid-IR

Ashvin B. Padhiyar / Ph. D. Thesis (Physics) / Sardar Patel University - 2010

246

Chapter 8

Infrared spectra of iodine complexes with

five amino acids

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1 INTRODUCTION

Organic charge transfer complexes have been known from a

long time many of which are found to be organic semiconductors

[1-3]. Some of them are found to be low-dimensional conductors

according to anisotropic electrical and optical properties [4,5].

Charge transfer interactions among biomolecules do exist [6] but

mostly homomolecular biomolecules have been studied [7].

However, initial studies of charge transfer interactions are also

carried out among amino acids and acceptors, purine and

riboflavin, purine and pyrimidine complexes with various

acceptors and some protein complexes [8-18]. Recently we have

established two-dimensional conducting systems which are

charge transfer complexes of four amino acids namely

asparagine, arginine, glutamine and histidine on the basis of

solid state infrared spectroscopy [19]. Some other charge transfer

complexes of macromolecular biomolecules have shown hopping

of small polarons induced by charge transfer interactions at the

chain-ends of macromolecules [20].

In the present study, we report iodine complexes of five

amino acids namely asparagine, arginine, glutamine, histidine

and tryptophan as studied with IR spectroscopy. The first four

are classified as having positively charged side chains and

tryptophan is the one having non-polar group as side-chain.

2 EXPERIMENTAL PROCEDURE

The five amino acids were obtained as analytical reagent

grades from sigma chemical company, USA. Resublimed iodine

used was also pure and in crystalline form. The amino acids were

mixed one by one with iodine in an agate mortar in 1:1

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molecular weight proportions and ground with a pastle till the

characteristic colours of the charge transfer complexes were

obtained. These CT complexes were reground with dry

spectrograde KBr powder (95%) to form homogeneous and finely

dispersd powders. These mixtures were manually compressed in a

die to form circular pellets which were weakly aborbing, mainly

transmitting and negligibly reflecting.

The spectra in the full IR range (400-4000cm-1) were recorded

using a GXFTIR single beam spectrometers manufactured by

Perkin-Elmer Company, USA. It is having a resolution of

0.15cm-1, a scan range of 15,000-30-1, a scan time 20 scan per

second, and OPD velocity of 0.20cm/sec and MIRTGS and

FIRTGS detectors. A beam splitter of opt KBr type was used

having a range of 7800-371cm-1.The spectra were recorded in

purge mode.

3 RESULTS AND DISCUSSION

The FTIR (Fourier-transformed infrared spectra) spectra of

the five amino acids namely asparagine, arginine, glutamine,

histidine and tryptophan were obtained and are shown below

(Figure 1).

Thus there are three broad envelopes in absorption (100-T)

formed by constitutive independent vibrational or rotational

levels. The envelopes are due to either symmetric groups and

side chains of amino acids. The formation of envelopes is

equivalent to the formation of wave-packets of longitudinal and

transverse optical phonons associated with intramolecular

vibrations.

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Wavenumbers (cm- 1)

Figure 1 Infrared spectra of

(a) Asparagines (b) Arginine (c) Glutamine

(d) Histidine and (e) Tryptophan

Figure 2(a)

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When charge transfer complexes are formed with iodine, the

charge densities of the amino acids are altered due to charge

transfer and even the intramolecular vibrations also show

changes in amplitudes and widths of the levels due to charge

transfer interactions at various sites. There is a multi-centered

charge transfer contact. The FTIR spectrum of aspargagine-

iodine complex is shown (Figure 2a) and nature of transition is

analyzed (Figure 2b).

Ahν vs hν is found to

be a straight line showing a

forbidden direct or an

allowed indirect transition

in a two-dimensional

system. The absorption

functions as a function of

photon energy for different

dimensionality are known

(Table I) Thus asparagines-

iodine is a layered

conductor.

Figure 2 (a) IR Spectrum of asparagine - iodine (1:1) complex

(b) Ahν vs hν showing interband transition

The set of levels in the region 2000-4000 cm-1 are blue

shifted due to anharmonic interactions in asparagine-iodine.

However, the oscillator strength which is proportional to area

under the band remains almost invariant. The anharmonic

interaction is related to stretching of bonds beyond elastic limit

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and this leads to an increase in the amplitude of vibration.

Because area under the band remains invariant, the width of each

level decreases leading to the reduction of width of the envelope

formed by large number of such levels. The increase in amplitude

beyond elastic limit menifests itself as blue shift in frequency

because in a quantum mechanical picture amplitude and

frequency are proportional due to quantization of lattice

vibrations [21].

The set of levels in the region 1000-2000 cm-1 form a

Gaussian envelope which has been fitted (Figure 2c).

Figure 2 (c) Analysis of Gaussian background profile in mid-IR range

There is reduction in frequencies of levels (red-shifts) due

to scattering of deformation vibrations because of the coupling

with electronic motions in this range. Red-shifts are associated

with increase in widths of vibrational levels to keep oscillator

strength invariant. Large polarons are formed and band motion of

such polarons lead to the formation of Gaussian envelope.

Gaussian distribution reveals free charge carriers and here charge

carriers are large polarons. Polarons are electrons or holes

surrounded by virtual phonon cloud. Polaron formation is a

consequence of electron-phonon interaction usually dominant in

ionic materials rather than covalently bonded materials. The

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electron-phonon coupling constant is given by ,

where is the longitudinal optical phonon frequency. As

decreases, signifying number of virtual phonons surrounding

a charge carrier or electron-phonon coupling constant increases.

Thus width of the envelope increases as the width of the

envelope is a measure of electron-phonon coupling potential.

Gaussian background reveals band motion of polarons in two

dimensions. In one-dimension, polarons hop being constrained to

one dimension and lead to beta density as envelope as found in α-

keratin [22] and in two dimensions, polarons show band motion

as found in a planar protein called elastin [23]. Gaussian

distributions are found in charge transfer complexes of

tetramenthylbenzidine,[24], triethylamine (DDQ)2, DPPD(DDQ)2

where DPPD=N,N’-diphenyl-p-phenylene diamine and

DDQ=2,3,-dichloro-5-6-dicyano-p-benzo-quinone[25] and charge

transfer complexes of indole [26]. These are the cases where

donor molecule is highly polarizable. Thus Gaussian envelope is

directly related with the high polarizability of asparagine

molecule due to highly polar aliphatic side chain. Red-shift of

entire set of vibrational levels are due to high polarizability

determined by strongly polar bonds.

The set of levels in the region 400-1000 cm-1 show blue

shift due to anharmonic interactions as dominant mechanism.

There is reduction in width of each level contributing to

reduction in width of entire envelope associated with reduction in

coupling constant. Increase in the frequency of longitudinal

optical phonon leads to decrease in electron-phonon coupling

constant. When there is blue shift, the phonons are not easily

excited and when there is red-shift phonons are easily excited.

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Excited phonons couple with charge carriers to from polarons.

Next is the spectrum of histidine-iodine (1:1) complex

which is shown here (Figure 3a).

Figure 3 (a) IR spectrum of histidine-iodine (1:1) complex

The region 2000-4000cm-1 contains a broad envelope with

reduction in width as compared to envelope in the spectrum of

hisitdine alone. Then there is flattening in transmission between

1800 cm-1 and 3200 cm-1 corresponding to constant absorption

coefficient. This is associated with allowed direct transition in

two dimensions in a polycrystalline material. Thus histidine-

iodine is also a layered semiconductor. In the second region

between 1100 cm-1 and 1800 cm-1, there is a Gaussian envelope

with increase in width. This is the region in which the set of

vibrational levels is red-shifted. The vibrational levels are

mainly consisted of ring stretching vibrations of histidine ring.

The ring stretching vibrations in aromatic or heterocyclic side

chains lie at lower frequency as compared with bond stretching

vibrations in aliphatic side chains. Thus the ring stretching

vibrations lie in the range of deformation vibrations along

aliphatic side chains. This is due to the coupling of strongly

delocalized π-electron cloud with ring vibrations. The last region

400-1000cm-1 contains again a Gaussian envelope which is weak

and very broad again revealing band motion of polarons. This

Gaussian has reduced width and set of constituent levels is

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blue shifted. Thus histidine-I2 contain two Gaussian background

envelops which are fitted by plotting ln A vs (K-K0)2 (Figures 3b

and 3c).

Figure 3 (b) Analysis of gaussian back ground profile in high frequency range

Figure 3 (c) Analysis of gaussian background profile in low frequency range Tryptophan-iodine (1:1) complex also reveals three

background envelop in the entire range (Figure 4a).

Figure 4 (a) IR Spectrum of tryptophan-iodine (1:1) complex The stretching vibrations are blue-shifted due to anharmonic

interactions dominating and the background profile shows

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reduction in width. Nature of transition when analyzed reveals

that Ahν vs hν as rectilinear plot (Figure 4b).

Figure 4(b) Ahν vs hν showing interband transition

Figure 4 (c) Analysis of Gaussian background profi le in the

mid IR- range

This is associated with forbidden direct or allowed indirect

transition in a disordered material in two dimensions. The

analysis reveals a layered nature of tryptophan-iodine complex.

The ring vibrations reveal red-shifting associated with softening

of these optical phonons and this is associated with an increase

in width of a Gaussian background which has been fitted (Figure

4c). The Gaussian lies between 900 cm-1 and 1800 cm-1. The

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tryptophan is an amino acid with non-polar side chain containing

a heterocyclic group. The non-polar side chain leads to neutral

character of tryptophan molecule. Tryptophan is a donor

molecule due to tryptophan ring system in side chain. There are

random orientations of tryptophan molecules in solid state due to

inert nature. This leads to isotropic disorder in tryptophan and its

iodine complex. The isotropic disorder related with equal

probability for all orientations reveals itself as a semicircular

distribution in absorption vs wavenumber graphs, i.e. in

absorption spectrum. A semicircular distribution is indeed

observed in the range 500-900 cm-1. An anisotropic disorder

leads to U-shaped distribution in biocytin complexes [27]. The

last region between 400 and 600 cm-1 contains a small envelope

which has reduced width with blue shifted levels of rocking,

wagging and group vibrations.

Next is the spectrum of arginine-Iodine complex

(Figure 5a).

Figure 5 (a) IR Spectrum of arginine-iodine (1:1) complex This spectrum consists of three regions of stretching

vibrations and lowest wavenumber region of rocking, wagging

and group vibrations contain envelopes which are red-shifted due

to softening of these modes. The intermediate frequency range

contains blue-shifted envelope with reduction in electron-

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phonon coupling constant. The envelope is a Gaussian in the

intermediate range corresponding to the band motion of large

polarons in two-dimensions (Figure 5b).

Figure 5 (b) Analysis of Gaussian background profile in the mid IR-range

Again there is a flattening of transmission with constant

absorption coefficient associated with direct allowed transition in

two-dimensions. The inverted behavior of blue and red shifts as

compared to earlier three cases is related with the symmetry of

the potential related with charge transfer from I2 molecules. Thus

there are two types of iodine complexes of amino acids: Two

extreme bands blue-shifted and two extreme bands red-shifted.

Also there is observation of noise in tryptophan-iodine and

arginine-iodine complexes both in highest wavenumber region

above 3400cm-1 and in the band gap region around 1700 cm-1.

This may be related with photoconducting nature of these

complexes and localization near the band edges.

The last is the spectrum of glutamine-iodine complex

(Figure 6a).

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Figure 6 (a) IR Spectrum of glutamine-iodine (1:1) complex This also belongs to second type and spectrum is similar to

arginine-iodine complex. The two extreme envelopes are red-

shifted and the central envelope is blue-shifted. The central

background envelope and the lowest wavenumber envelope are

both small Gaussian bands associated with band motion of

polarons in two-dimensions. (Figure 6b and 6c).

Figure 6 (b) Analysis of gaussian background profile in mid-IR range

Figure 6 (c) Analysis of gaussian background profile in low-frequency range

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The two dimensional nature is again verified with analysis of

nature of transitions. The absorption Ahν vs hν is found to have

rectilinear behavior indicating allowed direct transition in two

dimensions for a polycrystalline material (Figure 6d).

Figure 6 (d) Ahν vs hν showing inter band transition The nature of transitions are summarized for all five iodine

complexes of amino acids (Table II). All the Gaussian

distributions contain three parameters namely, central frequency,

maximum absorption and full-width at half-maximum (FWHM).

These parameters are tabulated (Table-III).

The electron-phonon coupling constant is also given by

Where is the average matrix element of

electron-phonon coupling, is the density of

states at the Fermi level and is average phonon frequency. The

density of states at the Fermi level is given by N( )=

where n=d-1 and d is the dimensionality of the electron system.

For layered materials d=2 and N( ) varies linearly with . Near

the Fermi level, N( ) is very small and this reduces electron-

phonon coupling constant . In a Peierls transition or Kohn

anomaly, a band gap or pseudo-gap opens at the Fermi level

which drastically lowers the density of states at Fermi level. This

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reduces electron-phonon coupling constant ( and since the mass

of a polarons is given by 1

(1 )6polm m . The effective mass of

polaron reduces. Since the electrical conductivity is given

by2

pol

ne

m , the conductivity increases. The large polarons move

fast showing a band motion. This is related with increase in

conductivity due to softening of phonons just above the Peierls

transition temperature observed in many one-dimensional

conductors. The velocity of a polarons is given by

where is the unperturbed velocity.

In the present study two iodine complexes namely histidine-

iodine and arginine-iodine show constant absorption just above

the band gap. This can be understood using a scaling hypothesis

leading to lnlim 2

lng

d gd

d l , where g is the dimensionless

conductance, is a characteristic length and d is the

dimensionality of the electron system. This relation can be easily

extended toln

lim 2ln

dd

d

, 1 1;

4

cn n

is real part of refractive

index, where is normalized absorption coefficient, λ is

wavelength of radiation and d is the dimensionality. For two-

dimensional system (d-2), lnα vs lnλ is horizontal line with slope

zero in the large absorption region. The above relation between

electrical conductivity and absorption coefficient. ( 1 1;4

cn n

is

being real part of refractive index). is the wavelength in the

medium depending on dispersion governed by refractive index as

a function of incident wavelength. Thus scaling hypothesis

explains constant absorption in two dimensional (layered)

materials.

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The electronic conductivity vs temperature also shows a

flattening at low temperature which can be also explained with

either correlated hopping or excitonic reduction of band gap

apart from scaling hypothesis. Thus the constant absorption near

the band gap can also be explained with correlated hopping or

excitonic reduction of the band gap down to zero.

4 CONCLUSIONS

All five iodine complexes of amino acids are found to be

two-dimensional (layered) conductors. Most of them reveal band

motion of large polarons in two dimensions consistent with ionic

nature of these charge transfer complexes. Anharmonic

interactions lead to blue shift of vibrations and softening of

phonon modes leads to red shift of vibrations. Two types of

iodine complexes are identified according to the nature of charge

transfer interaction potential.

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Table I

Dependence of absorption function on dimensionality. Value of r in = 0

(h - Eg)r / h or = 0 (h - Eg)r

Type of transition Direct Indirect

1 - d 2 - d 3 - d 1 - d 2 - d 3 - d

Allowed

Forbidden

1/2

3/2

0

1

1/2

3/2

2

3

1

2

2

3

Table II

Absorption functions and type of transition in amino acid-iodine

complexes

Name of the

complex

Absorption

function

Type of

transition

Plot

Asparagine - I2 h = A (h - Eg) Forbidden direct

or

Allowed indirect

h vs h

Histidine - I2 = Constant Allowed direct -

Tryptohan - I2 h = A (h - Eg) Forbidden direct

or

Allowed indirect

h vs h

Arginine - I2 = Constant Allowed direct -

Glutamine - I2 = Constant / h Allowed direct vs 1/h

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Table III

Gaussian parameters of the Gaussian profiles in amino acid-iodine

complexes.

Name of the Complex Gaussian Parameters

Central wave-

number

(cm- 1)

Absorption

maximum

(%)

FWHM

(cm- 1)

Asparagine - I2

(1)

1900

16.0

1400

Histidine - I2

(1)

(2)

1380

900

22.5

12.5

450

500

Tryptohan - I2

(1)

1350

27.5

1000

Arginine - I2

(1)

1670

15.0

320

Glutamine - I2

(1)

(2)

1650

780

23.0

12.5

200

400

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REFERENCES [1] F Gutman and L E Lyons, Organic semiconductors,

John Wiley and Sons, New York, 1967.

[2] Roy Foster, Organic charge transfer complexes, Academic

Press, New York, 1968

[3] S Kanda and K A Pohl in organic semi conducting

polymers, Ed. By J E Katon Marcel Dekker Inc. New York,

1968,p 67

[4] A F Garito and A J Heeger in One-dimensional conducting,

Ed. By H G Schuster,

[5] J J Andre, A-Bieber and F Gautier, Ann de Phy, I, 145,

1967

[6] M A Slifkin, Charge Transfer Interactions in Biomolecules,

Academic Press, London, 1971

[7] D D Eley in organic semiconducting polymers, Ed. By J E

Katon, Marcel Dekker Inc. New York 1968 p.

[8] A Szent – Gyorayi, bioenergetics, Academic Press, New

York, 1957

[9] M A Slifkin, Nature, 193, 464, 1962

[10] J B Birks and M A Slifkin, Nature, 197, 42, 1963

[11] M A Slifkin, Spectrochim, Acta, 20, 1543, 1964 p

[12] M A Slifkin, Nature, 197, 275, 1963

[13] G Weber, Biochem. J, 47,114, 1950

[14] I Isenberg and A Szent – Gyorgyi, Proc Nat. Acad – Sci

(USA), 44,857, 1958

[15] R Beuker and A Szent – Gyorgyi, Rec – Trav. Chim, 81,

541, 1962

[16] E M C Davis, D D Elay and R S Snart, Nature,186, 724,

1960

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Ashvin B. Padhiyar / Ph. D. Thesis (Physics) / Sardar Patel University - 2010

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[17] D D Elay and R S Snart, Biochem, Biophys, Acta 102, 379,

1965

[18] M A Slifkin, Biochim, Biophys – Acta, 103, 365,1965

[19] Ashvin Padhiyar, A J Patel and A T Oza, J. Phys.

Condenced Matter, 19, 486214, 2007

[20] G K Solanki, Anand Amin, Ashvin Padhiyar, A K Ray and

A T Oza, Ind. J. Biochem. Biophys, 45, 421, 2008

[21] C Kittel, Introduction to solid state physics, 7th Edition,

John Wiley & Sons Singapore, New York, Chichester

Brisbane, Toronto, 2004 (p. 108,297)

[22] Vishal Patel, M.Phil. Dissertation, Sardar Patel Univerity,

2010

[23] Pravinsinh Rathod, M. Phil, Dissertation, Sardar Patel

University, 2009

[24] Mukesh B Patel, S G Patel, Rajiv Vaidya and A T Oza, Ind.

J. Phys. B 77, 199,2003

[25] R G Patel, G K Solanki, S M Prajapati and A T Oza, Ind. J.

Phys. A, 78, 471, 2004

[26] G K Solanki, Mukesh Patel and A T Oza, Pragna – Sardar

Patel University, Research Jour.,16, 150, 2008

[27] Ashvin Padhiyar, A J Patel and A T Oza (To be published)