### Paper Menu >>

### Journal Menu >>

physical problem at issue (thus permitting a direct calcu-

lation of eigenvectors in this same local basis) and 2)

algebraic expansions are intrinsically anharmonic at their

zero-order approximation. This fact allows one to reduce

drastically the number of arbitrary parameters in com-

parison to harmonic series, especially when facing me-

dium- or large-size molecules. It should be however also

noticed that, as a possible drawback of purely local Ha-

miltonian formulations (either algebraic or not) com-

pared with traditional perturbative approaches, the actual

eigenvectors of the physical system. Yet, for very local

situations, the aforementioned disadvantage is not a se-

rious one. A further point of import here is found in the

ease of accounting for proper symmetry adaptation of

vibrational wave functions. This can be of great help in

the systematic study of highly excited overtones of

not-so-small molecules, such as the present one. Last but

not least, the local mode picture of a molecule is en-

hanced from the very beginning within the algebraic

framework. This is an aspect perfectly lined up with the

current tendencies of privileging local over normal mode

pictures in the description of most topical situations.

We address here the explicit problem of the construc-

tion of the vibrational Hamiltonian operator for the po-

lyatomic molecule. According to the general algebraic

description for one-dimensional degrees of freedom, a

dynamically-symmetric Hamiltonian operator for n in-

teracting (not necessarily equivalent) oscillators cab

written as

0

.

iiijijijij

H

= E+AC+AC+M

(1)

In this expresssion, one finds three different classes of

effective contributions. The first one,

1

n

i

A

i

C

i

is devoted

to the description of n independent, anharmonic sequences

of vibrational levels (associted wih n independent, local

oscillator) in terms of the operators C

i

. The second

one,

1

n

ijij

i

A

C

leads to cross-anharmonicities between

pairs of distinct local oscillators in terms of the operators

C

ij

. The third one,

1

n

ijij

i

M

, describes anharmonic,

non-diagonal interactions involving pairs of local oscil-

lators in terms of the operators M

ij

. The C

i

, C

ij

operators

are invariant (Casimir) operators of certain Lie algebras,

whilst the M

ij

are invariant (Majorana) operators as-

sociated with coupling schemes involving algebras na-

turally arising from a systematic study of the algebraic

formulation of the one-dimensional model for n inter-

acting oscillators. We work in the local (uuncoupled os-

cillaators) vibrational basis written as

123

........

n

In which the aforementioned operators have the

following matrix elements

!!

!!

!

!

12

!

11

12

!

1

4

4

2δδ

11

δδ

11

δ

iijj

iijj

i

iiii

ijiiijij

ijiijjij

ijiiijjj

ijjjjiii

CN

CNN

MNN

MNN

MNN

!

1

δ.

ijj

(2)

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

pend on the numbers N

i

(Vibron numbers). Such numbers

have to be seen as predetermined parameters of well-

defined physical meaning, as they relate to the intrinsic

anharmonicity of a single, uncoupled oscillator through

the simple relation. We report in Table the values of the

Vibron numbers used in the present study.

The general Hamiltonian operator 1) can be adapted to

describe he internal, vibrational degrees of freedom of

any polyatomic molecule in two distinct steps. First, we

associate three mutually perpendicular one-dimensional

anharmonic oscillators to each atom. This procedure even-

tually leads to a redundant picture of the whole molecule,

as it will include spurious (i.e. translational/rotational)

degrees of freedom. It is however possible to remove

easily such spurious modes through a technique de-

S. R. Karumuri et al. / Journal of Biophysical Chemistry 3 (2012) 259-268

Copyright © 2012 SciRes. OPEN ACCESS

261

scribed elsewhere [19-21]. One is thus left with a Ham-

iltonian operator dealing only with true vibrations. Such

modes are given in terms of coupled oscillators in the

local basis; 2) The coupling is induced by the Majorana

operators. A sensible use of these operators is such that

the correct symmetries of vibrational wave functions are

properly taken into account. As a second step, the alge-

braic parameters A

i

, A

ij

, λ

ij

of Eq.1 need to be calibrated

to reproduce the observed spectrum.

The algebraic theory of polyatomic molecules consists

in the separate quantization of rotations and vibrations in

terms of vector coordinates r

1

, r

2

, r

3

, . quantized

through the algebra

123

222GUUU

For the stretching vibrations of polyatomic molecules

correspond to the quantization of anharmonic Morse os-

cillators, with classical Hamiltonian

2

2

,= 2+1exp

ss

H

pspDs

(3)

For each oscillator i, states are characterized by repre-

sentations of

22

ii

ii

UO

Nm

(4)

with m

i

= N

i

, N

i

– 2, , 1 or 0 (N

i

—odd or even). The

Morse Hamiltonian (3) can be written, in the algebraic

approach, simply as

0iiii

H=ε+AC, (5)

where C

i

is the invariant operator of O

i

(2), with eigen

values

22

0

–.

iiiii

ε= ε+AmN

Introducing the vibrational quantum number

–2

iii

Nm

, [20] one has

2

0

4

iiiiii

εεANνν (6)

For non-interacting oscillators the total Hamiltonian is

i

i

H

H

with eigen-values

2

0

4

iiiii

ii

E=EANvv

(7)

2.1. Hamiltonian for Stretching

Vibrations

The interaction potential can be written as

,1exp1exp,

ijijiijj

Vsskαsαs

(8)

which reduces to the usual harmonic force field when the

displacements are small

,

ijijij

Vss kss

.

Interaction of the type Eq.8 can be taken into account

in the algebraic approach by introducing two terms. One

of these terms is the Casimir operator, C

ij

, of the com-

bined

22

ij

OO algebra. The matrix elements of this

operator in the basis Eq.2 are given by

2

?

4 .

ii ;jjijii ;j j

ijijij

N,νN,νCN,νN,ν

NN

(9)

The operator C

ij

is diagonal and the vibrational quan-

tum numbers ν

i

have been used instead of m

i

. In practical

calculations, it is sometime convenient to substract from

C

ij

a contribution that can be absorbed in the Casimir

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

ing an operator

'

ij

C whose matrix elements are

2

2

2

,;,;,

4

4

4.

iijjijiijj

ijijij

ijiiii

ijijjj

NνNνCN,νNν

NN

NNNNvv

NNNNvv

(10)

The second term is the Majorana operator, M

ij

. This

operator has both diagonal and off-diagonal matrix ele-

ments

12

12

,;,;,

2

,+1; ,1,; ,

11

,1;,1,;,

11.

iijjijiijj

ijjiij

ii jjijiijj

jiiijj

ii jjijiijj

ijjjji

NνNνMN,νNν

NvNv

NνNνMNνNν

νν+NνNν+

NνNνMNνNν

νν+NνNν+

(11)

The Majorana operators M

ij

annihilâtes one quantum

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

versa.

2.2. Symmetry-Adapted Operators

In polyatomic molecules, the geometric point group

symmetry of the molecule plays an important role. States

must transform according to representations of the point

symmetry group. In the absence of the Majorana opera-

tors M

ij

, states are degenerate. The introduction of the

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

S. R. Karumuri et al. / Journal of Biophysical Chemistry 3 (2012) 259-268

Copyright © 2012 SciRes. OPEN ACCESS

262

generacies of figure and 2) in addition it generates states

with the appropriate transformation properties under the

point group. In order to achieve this result the λ

ij

must

be chosen in an appropriate way that reflects the geo-

metric symmetry of the molecule. The total Majorana

operator

n

ij

ij

SM

(12)

is divided into subsets reflecting the symmetry of the

molecule

=++S SS

(13)

The operators =++S SS

are the symmetryadapt-

ed operators. The construction of the symmetryadapted

operators of any molecule will become clear in the fol-

lowing sections where the cases of Metalloporphyrins

(D

4h

) will be discussed.

2.3. Hamiltonian for Bending Vibrations

We emphasize once more that the quantization scheme

of bending vibrations in U(2) is rather different from U(4)

and implies a complete separation between rotations and

vibrations. If this separation applies, one can quantize

each bending oscillator i by means of an algebra U

i

(2) as

in

Eq.4. The Poschl-Teller Hamiltonian

2

,=2cosh

s

H

ps Ds

(14)

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

written, in the algebraic approach, as

0

,

iiii

H=ε+AC (15)

This Hamiltonian is identical to that of stretching vi-

bration (

Eq.5). The only difference is that the coeffi-

cients A

i

in front of C

i

are related to the parameters of the

potential, D and α, in a way that is different for Morse

and Poschl-Teller potentials. The energy eigen-values of

uncoupled Poschl-Teller oscillators are, however, still

given by

2

0

4.

iiiii

ii

E=ε=EA 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

Y

C

a

C

m

C

m

N

Ca

C

b

C

b

C

a

N

C

m

C

a

C

b

C

b

M

C

a

C

b

C

b

C

a

N

C

m

Y

Y

Y

Y

Y

Y

Ca

X

X

X

X

N

Figure 1. Structure of Metalloporphyrins.

By inspection of the figure, one can see that two types

of interactions in Metalloporphyrins:

1) First-neighbor couplings (Adjacent interactions)

2) Second-neighbor couplings (Opposite interactions)

The symmetry-adapted operators of Metalloporphyrins

with symmetry D

4h

are those corresponding to these two

couplings, that is,

,,

nn

ijijijij

i<ji<j

S=cMS=cM

(17)

with

12233445

13243546

12233445

13243546

1

0

0

1

c=c=c=c==

c=c=c=c==

c=c=c=c==

c=c=c=c==

The total Majorana operator S is the sum

111

SSS (18)

Diagonalization of S produces states that carry repre-

sentations of S, the group of permutations of objects,

while Diagonalization of the other operators produces

states that transform according to the representations A

1g

,

A

2g

, B

1g

, B

2g

and E

1u

of D

4h

.

2.5. Local to Normal Transition: The Locality

Parameter (

)

The local-to-normal transition is governed by the dimen-

sionless locality parameter (

). The local-to-normal tran-

sition can be studied [19,20] for polyatomic molecules,

for which the Hamiltonian is

local

1212

iiijijijij

H

HMACACM

(19)

S. R. Karumuri et al. / Journal of Biophysical Chemistry 3 (2012) 259-268

Copyright © 2012 SciRes. OPEN ACCESS

263

For these molecules, the locality parameters are

1

2tan8,,1,2,3.

iijiij

AAij