theory of vibrations of tetra-atomic symmetric bent … · of this tetra-atomic molecule in the...

THEORY OF VIBRATIONS OF TETRA-ATOMIC SYMMETRIC

BENT MOLECULES

Alexander I. Melker1, 2*, Maria Krupina3, Vitaly Kotov2

1Department of Physics of Strength and Plasticity of Materials 2Department of Physics and Mathematical Modeling in Mechanics

3Department of Experimental Physics

St.Petersburg State Polytechnic University, Polytekhnicheskaya 29

195251, St. Petersburg, Russia

*e-mail: [email protected]

Abstract. In this contribution we submit a theory of vibrations of a tetra-atomic symmetric bent molecule ABBA in cis-conformation. Compared to the previous calculations, where some approximations have been made, now we gave a more rigorous version. For the molecule studied we have calculated frequencies of all five vibrations. Among them three vibrations, ax1, ay1, and sy1, are normal or nearly normal. The longitudinal vibrations are of two types: ( ) and ( ). 1. Introduction Structure of molecules and macromolecules manifest themselves in vibration spectra. Modern physics, chemistry, and biology widely use the vibration spectra for solving numerous and diverse problems which refer to the study of molecule and macromolecule structure and its changing in physical, chemical and biological processes. Molecule vibrations play a crucial role in relaxation phenomena, in kinetics of chemical reactions, in self-organization of polymers and biopolymers with the resulting formation of a particular structure [1]. For this reason the theory of molecule vibrations is of fundamental importance for condensed matter physics, molecular physics, biology, and even for nanotechnology.

Vibration frequencies of molecules and macromolecules, after the dissociation energy and the energy of chemical bonds, are the most important constants of these substances [2]. One of the most reliable and precise methods for the experimental determination of eigen frequencies of molecules and macromolecules are infrared spectroscopy and Raman spectroscopy. To analyze the vibration spectra of complex molecules, which are very complicated, we must be able to establish purely theoretically characteristic vibrations of atomic groups incorporated in one or another molecule. Such vibrations depend very little on the presence of other atomic groups.

By the purely theoretical approach to molecule vibrations we understand any analytical solution obtained in the framework of mechanics or another science without using semi-empirical potential functions with a lot of parameters, usually elastic constants, which are necessary for a numerical solution and which number exceeds, usually significantly, the number of atoms in a molecule. The appearance of excess parameters had led to endless discussion about the correctness of parameter choice instead of developing new approaches, e.g. electronic theory of molecule vibrations was created with the purpose to gain better

Materials Physics and Mechanics 13 (2012) 9-21 Received: March 13, 2012

© 2012, Institute of Problems of Mechanical Engineering

insight into the nature of those vibrations [3]. Careful analysis of this situation has been given in [4]. Unfortunately, the majority of researchers prefer calculating instead of thinking. Probably for this reason, the purely analytical theory was created, at this moment, only for triatomic molecules in the framework of mechanics [2, 5] and relatively recently by the combination of mechanics and the bond-charge theory which takes into account electronic degrees of freedom [3]. The mechanical theory has become a classical one and was incorporated into text-books long ago [2, 5]. The new theory enhances the possibilities for studying properties of molecules significantly. In particular, it gives a unified approach to vibrations and rotations of molecule and macromolecules [6]. Nevertheless it is not worth leaving the classical mechanical theory out of scientific reckoning. As before, it remains the basis for classification of vibrations.

We have developed the theory of molecule vibrations in the frame work of classical mechanical approach for tetra-atomic linear symmetric molecules [7] and for tetra-atomic bent symmetric molecules [8]. However, in the last case, due to some approximations one of vibrations was lost. In this contribution we give a revised and more complete version of the calculations done in [8] without the previous approximations. 2. Target setting Consider a tetra-atomic bent symmetric molecule ABBA. Let in an equilibrium position the molecule has the cis–form of a symmetric bent chain with the valence angles θ, the equilibrium distance between atoms A and B being equal to a1 and between atoms B, respectively a2 (Fig. 1).

Denote by MA , MB , MB , MA the masses of the atoms and by 4321 ,, uuuu their

displacements from equilibrium positions. Suppose, as before [7], that interatomic forces both stretching the bounds and bending the angle between bonds are of a short-range character. The forces acting between the nearest neighbors are central. Let k1 , k2 , k1 be their elastic constants. The forces changing the valence angles θ, whose vertices are occupied by the second and the third atom, are of noncentral character. Let kθ be their elastic constant. The valence angle and elastic constants are input parameters. Hence we will study the vibrations of this tetra-atomic molecule in the framework of a four–parameter model. 3. Geometric constraints and new coordinates In the same way as before [7], we exclude translation motion of the molecule along the axes x and y putting the total linear momentum of the molecule equal to zero. It means that the inertia center of the molecule is immobile so the following relation takes place

1u

O

y

x

A

B

A

1v

4u

4v

2v

3v

3u 2u B

Fig. 1. Displacements of atoms in a tetra-atomic symmetric bent molecule.

10 Alexander I. Melker, Maria Krupina, Vitaly Kotov

0M iiu .

In our case this leads to appearance of the geometric constraints of atomic displacements what look like

0)uu(M)uu(M 32B41A ,

0)vv(M)vv(M 32B41A .

Introduce the new coordinates

411ax uuq , 322ax uuq ,

141sx uuq , 232sx uuq ,

411ya vvq , 322ya vvq ,

141ys vvq , 232ys vvq .

At that the geometric constraints take the form

1axB

A2ax q

M

Mq , 1ya

B

A2ya q

M

Mq .

To exclude rotation of the molecule, one needs to assume that its total angular momentum is zero. For small vibrations this condition takes the form [7]

0,M i0ii ur ,

where 0ir is the radius vector of the immobile equilibrium location of i atom.

Let us project the displacement of atom 4 on the direction normal to the line OA (Fig.2).

This gives the displacement producing the rotation 4 of the molecule around the axis z

going through the origin of coordinates

Fig. 2. Angles in a bent tetra-atomic symmetric molecule used in calculation.

4

y

O x2

1

3

n

A

n 4v

4u

11Theory of vibrations of tetra-atomic symmetric bent molecules

cosusinv)(b 444 .

Here

cos)a2/a(

sintan

12 , ,

OAb , 2121

222 aaa

4

ab .

The displacement producing the rotation 1 of the molecule in opposite direction can be

obtained by the action of rotation )y(C2 on )(b 4 ,

cosusinv)(b)y(C)(b 11321 .

If the rotation does not take place, then

32

B4A22

B1A v2

aM)(bMv

2

aM)(bM ,

or

)vv(2

aMcos)uu(sin)vv(bM 23

2B4141A .

Therefore

cos)uu(sin)vv(a

b2

M

Mvv 4141

2B

A23 .

In the new coordinates

sinqcosqa

b2

M

Mq 1ys1ax

2B

A2ys .

Thus three coordinates, namely 2ys2ya2ax q,q,q , are superfluous.

4. Kinetic energy The kinetic energy of the molecule is

23

22

B24

21

A4321kin 2

M

2

M),,,(E uuuuuuuu

23

22

23

22

B24

21

24

21

A vvuu2

Mvvuu

2

M .


Write down the kinetic energy in the new coordinates. Since

)qq(2

1uu 2

1sx2

1xa24

21 , )qq(

2

1uu 2

2sx2

2xa23

22 ,

)qq(2

1vv 2

1ys2

1ya24

21 , )qq(

2

1vv 2

2ys2

2ya23

22 ,

we have

22sy

22ay

22sx

22ax

B21sy

21ay

21sx

21ax

Akin qqqq

4

Mqqqq

4

ME .

Eliminate the superfluous coordinates

1axB

A2ax q

M

Mq , 1ya

B

A2ya q

M

Mq

Because of

21ax

2

B

A22ax q

M

Mq

, 2

1ay

2

B

A22ay q

M

Mq

,

the second term takes the form

22sy

22sx

B21ay

21ax

B

2A qq

4

Mqq

M4

M .

As a result, we have

22sy

22sx

B21sy

21sx

21ay

21ax

B

AAkin qq

4

Mqqqq

M

M1

4

ME

.

Now eliminate the superfluous coordinate

sinqcosqa

b2

M

Mq 1ys1ax

2B

A2ys .

Because of

sincosqq2sinqcosqa

b2

M

Mq 1sy1ax

221sy

221ax

2

2

2

B

A22sy

,

the second term takes the form


2sinqqsinqcosqa

b

M

Mq

4

M1sy1ax

221sy

221ax

2

2B

2A2

2sxB

Finally we obtain

2sinqqsinqcosqa

b

M

Mq

4

M

qqqqM

M1

4

ME

1sy1ax22

1sy22

1ax

2

2B

2A2

2sxB

21sy

21sx

21ay

21ax

B

AAkin

.

Or in another form

)qq(2sina

b

M

M

qsina

b2

M

M1

4

Mqcos

a

b2

M

M

M

M1

4

M

q4

Mq

4

Mq

M

M1

4

ME

1sy1ax

2

2B

2A

21sy

2

2

2B

AA21ax

2

2

2B

A

B

AA

22sx

B21sx

A21ay

B

AAkin

5. Potential energy The potential energy of the molecule consists of three parts

3214321 UUU),,(U uuuu ,

where

23

21

11 )l()l(

2

kU , 2

22

2 )l(2

kU , 2

32

22

3 )()(a2

kU .

Here 1l , 2l and 3l are the length changes of the interatomic bonds AB, BB, and BA,

respectively, is the deviation of the valence angle from its equilibrium value θ during

deformation vibrations, 21 aaa2 .

We can find the value 3l projecting the vector 34 uu on the direction BA,

cos)vv(sin)uu(l 34343 ,

where 2/ . From this it follows that

cossin)vv()uu(2cos)vv(sin)uu()l( 343422

3422

342

3 .

Acting by the rotation )y(C2 on 2

3 )l( , we find 21 )l(



2221

2221

21 .

The sum of these two last expressions is

2234

221

2234

221

23

21 cos)vv()vv(sin)uu()uu()l()l(

cossin)vv()uu()vv()uu(2 21213434 .

Since

)uuuu(2uuuu)uu()uu( 432123

22

24

21

234

221 ,

)vvvv(2vvvv)vv()vv( 4321

23

22

24

21

234

221 ,

)qq(u 1sx1xa2

11 , )qq(u 2sx2xa2

12 ,

)qq(u 2sx2xa2

13 , )qq(u 1sx1xa2

14 ,

)qq()qq(uu 2sx2xa1sx1xa4

121 , )qq()qq(uu 1sx1xa2sx2xa4

143 ,

)qqqq(uuuu 2s1s2a1a2

14321 , )qq(uu 2

2sx2

2xa41

32 ,

and analogous formulas are valid for the components iv , we have

22ys1ys2ya1ya

22ys

22ya

21ys

21ya

1

22sx1sx2xa1xa

22sx

22xa

21sx

21xa

11

cosqq2qq2qqqq4

k

sinqq2qq2qqqq4

kU

.

Let us eliminate the superfluous coordinates

1axB

A2ax q

M

Mq , 1ya

B

A2ya q

M

Mq

Because of

21ax

2

B

A22ax q

M

Mq

, 2

1ay

2

B

A22ay q

MM

q

,

we have

2

2 2 2 2 2 211 1 1 1 2 1 1 22 2 sin

4A A

a x sx ax sx ax sx sxB B

k M MU q q q q q q q

M M


2

2 2 2 2 2 211 1 1 2 1 1 22 2 cos

4A A

a y s y ay s y ay s y s yB B

k M Mq q q q q q q

M M

.

Now eliminate the superfluous coordinate

sinqcosqa

b2

M

Mq 1ys1ax

2B

A2ys .

Because of

sincosqq2sinqcosqa

b2

M

Mq 1sy1ax

221sy

221ax

2

2

2

B

A22sy

,

we obtain

1sy1ax2B

A2

2B

A1

21ys

2

2B

A2121ya

2

B

A212sx1sx

21

22sx

2121sx

2121xa

2222

2

2B

2A2

2

B

A11

qqa

b2

M

Msin1coscos2

a

b2

M

M

4

k

qa

b2

M

Msin1cos

4

kq

M

M1cos

4

kqq2sin

4

k

qsin4

kqsin

4

kqcoscos

a

b4

M

Msin

M

M1

4

kU

The next term is the potential energy 2U . Here

22sy

22sx

223

223

22 qq )vv()uu()l(

and we have

)(2

)(2

22

22

222

22 sysx qq

kl

kU

Again let us eliminate the superfluous coordinate

sinqcosqa

b2

M

Mq 1ys1ax

2B

A2ys .

Because of

sincosqq2sinqcosqa

b2

M

Mq 1sy1ax

221sy

221ax

2

2

2

B

A22sy

,


we obtain

)sincosqq2sinqcosqa

b2

M

Mq(

2

kU 1sy1ax

221sy

221ax

2

2

2

B

A22sx

22

Now find the change of the angle BBA for the bent molecule. For this purpose, project

the vector 34 uu on the direction normal to BA. Then we have

cos)uu(sin)vv(a

134343 .

The angle change 1 can be found by acting the rotation )y(C2 on 3

cos)uu(sin)vv(a

1)y(C 2121321 .

Therefore

cossin)vv()uu(2cos)uu(sin)vv(a 212122

2122

2121

2 ,

cossin)vv()uu(2cos)uu(sin)vv(a 343422

3422

3423

2 .

Comparing these expressions with


2122

212

1 ,


3422

342

3 ,

and the potential energies

23

22

23 )()(a

2

kU 2

32

11

1 )l()l(2

kU ,

with each other, we can write right away

1sy1ax2B

A2

2B

A

21ys

2

2B

A221ya

2

B

A22sx1sx

2

22sx

221sx

221xa

2222

2

2B

2A2

2

B

A3

qqa

b2

M

Msin1sincos2

a

b2

M

M

4

k

qa

b2

M

Msin1sin

4

kq

M

M1sin

4

kqq2cos

4

k

qcos4

kqcos

4

kqsincos

a

b4

M

Mcos

M

M1

4

kU


6. Lagrange function and normal coordinates Combining all the previous results obtained, we obtain the Lagrange function

2 2 21 1 21

4 4 4A A A B

ay sx sxB

M M M ML q q q

M

2 2

2 2 2 21 1

2 2

2 22 2 22 2 21

1 1 2 22 2

2 21 cos 1 sin

4 4

4sin 2 ( ) 1 sin cos cos

4

A A A A Aax sy

B B B

A A Aax sy a x

B B B

M M M M Mb bq q

M M a M a

M k M Mb bq q q

M a M M a

21

2

2 2 2 2 2 2 21 1 1 11 2 1 2 1

2

2 2 21 11 1 1

2 22

2

222

2

sin sin sin 2 cos 14 4 4 4

2 22cos 1 sin 2cos cos 1 sin4 4

2(

2

Asx sx sx sx a y

B

A AAs y ax sy

B BB

Asx

B

k k k k Mq q q q q

M

M Mb bk k M bq q qM a M aM a

k M bq

M a

2

2 2 2 21 1 1 1

2 2 22 2 2 2 2 2 2 2

1 1 22 22

2

2 2 2 21 2 1

cos sin 2 cos sin )

41 cos cos sin cos cos

4 4 4

cos 2 sin 1 sin 1 sin4 4 4

ax sy ax sy

A Aa x sx sx

B B

Asx sx a y

B

q q q q

k k kM M bq q q

M M a

k k kMq q q

M

2

21

2

21 1

2 2

2

2 22cos sin 1 sin

4

As y

B

A Aax sy

B B

M bq

M a

k M Mb bq q

M a M a

Usually MA<<MB, as in hydrogen peroxide H2O2 . The Lagrange function contains the terms of different order of smallness: of order MA and of order MA

2/MB. Eliminating small quantities of order MA

2/MB, one is able to separate out three functions which relate to normal and nearly normal vibrations

22 2

122

2 2 22 2 2 21

12 22

22 22 22 2 2 2 2 22

1 12 2 2 22 2

4( 1) 1 cos

4

41 sin cos cos

4

4 4cos 1 cos cos sin

2 4

A A Aax

B B

A Aa x

B B

A A Aa x a x

B B B

M M M bL a x q

M M a

k M M bq

M M a

kk M M Mb bq q

M a M M a


21ya

221

2

B

A21ya

B

AA qsinkcoskM

M1

4

1q

M

M1

4

M)1ya(L

21sy

222

2

2B

2A2

21sy

221

2

2B

A21sy

222

2

B

AA

qsina

b4

M

M

2

k

qsinkcoska

b2

M

Msin1

4

1qsin

a

b4

M

M1

4

M)1sy(L

The frequencies of these vibrations are equal to

222

2

B

A

B

AA

2222

2

2B

2A2

2

B

A

222

2

B

A

B

AA

222

2

2B

2A

222

22

2

2B

2A2

2

B

A1

21xa

cosa

b4

M

M

M

M1M

sincosa

b4

M

Mcos

M

M1k

cosa

b4

M

M

M

M1M

cosa

b4

M

Mk2coscos

a

b4

M

Msin

M

M1k

22

1

2

B

A

ABA

B21ya sinkcosk

M

M1

MMM

M

222

2

B

AA

222

2

2B

2A

222

1

2

2B

A

21sy

sina

b4

M

M1M

sina

b4

M

Mk2sinkcosk

a

b2

M

Msin1

7. Lagrange function and longitudinal vibrations Consider the remainder of the Lagrange function

22sx

B21sx

A q4

Mq

4

M2xs,1xsL 2

2sx2 q

2

k

2sx1sx2

2sx2

1sx22

1 qq2qqcosksink4

1 .

According to the general formula

0q

U

q

E

dt

d

nn

kin

,


find the equations of motion

2 2 2 21 1 1 1 2

2 2 2 22 2 1 2 1 1

sin cos sin cos 0

2 sin cos sin cos 0

A s x s x s x

B s x s x s x

M q k k q k k q

M q k k k q k k q

.

Denote the combinations of elastic and geometric constants in the following way

A22

1 kcosksink B22

12 kcosksinkk2 .

As is customary, we seek the solution of this system of linear homogeneous differential equations with constant coefficients in the form

ti11xs eAq , ti

22xs eAq ,

where A1, A2 are some constants. Substitution of these functions in the equation system and cancellation by the term tie gives the system of linear homogeneous algebraic equations for the constants

21 2

21 2

( ) 0

0

A A A

A B B

M k A k A

k A M k A

.

In order to have solutions not equal to zero, the system determinant should be zero

2

2

0

A A

A A

A B

B B

k k

M M

k k

M M

.

Therefore

0MM

kk

M

k

M

k

BA

BA

B

B

A

A24

.

The equation has two roots

BA

BA

2

B

B

A

A

B

B

A

A2

MM

kk

M

k

M

k

M

k

M

k

2

1 ,


which define the frequencies of longitudinal vibrations. Since at that 0q 1xa , i.e. 14 uu ,

and

0)uu(M)uu(M 32B41A ,

we have 23 uu .

These conditions give the vibrations of two types ( ) ( ). 8. Conclusion We have developed the theory of molecule vibrations in the frame work of classical mechanical approach for tetra-atomic symmetric bent molecules. Compared to the previous calculations, where some approximations have been made [8], now we gave a revised and more complete version. As a result, we have found the frequencies of all five vibrations. In particular, the new symmetrical vibration along the direction normal to the longitudinal axis of the molecule is found. Besides the frequencies of normal and nearly normal vibrations differ from the values earlier obtained. At the same time, the frequencies of longitudinal vibrations did not change. This means that the vertical displacement of internal, usually more heavy atoms, do not influence on the longitudinal vibrations. References [1] A.I. Melker // Proc. SPIE 6597 (2007) 6597-01. [2] V.N. Kondratiev, Structure of Atoms and Molecules (Fizmatgiz, Moscow, 1959),

in Russian. [3] A.I. Melker, M.A. Vorobyeva // Proc. SPIE 6253 (2006) 625305. [4] A.I. Melker, M.A. Vorobyeva // Reviews on Advanced Materials Science 20 (2009) 14. [5] L.D. Landau, E.M. Lifshitz, Mechanis (Nauka, Moscow, 1988), in Russian. [6] A.I. Melker, M.A. Krupina // Materials Physics and Mechanics 9 (2010) 20. [7] M.A. Vorobyeva, A.I. Melker // Proc. SPIE 6253 (2006) 625304. [8] A.I. Melker, M.A. Vorobyeva // Proc. SPIE 6597 (2007) 659708.


theory of vibrations of tetra-atomic symmetric bent … · of this tetra-atomic molecule in the...

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