theory of vibrations of tetra-atomic symmetric bent … · of this tetra-atomic molecule in the...
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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 .
12 Alexander I. Melker, Maria Krupina, Vitaly Kotov
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
13Theory of vibrations of tetra-atomic symmetric bent molecules
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(
14 Alexander I. Melker, Maria Krupina, Vitaly Kotov
cossin)vv()uu(2cos)vv(sin)uu()l( 2121
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
15Theory of vibrations of tetra-atomic symmetric bent molecules
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
,
16 Alexander I. Melker, Maria Krupina, Vitaly Kotov
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
cossin)vv()uu(2cos)vv(sin)uu()l( 212122
2122
212
1 ,
cossin)vv()uu(2cos)vv(sin)uu()l( 343422
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
17Theory of vibrations of tetra-atomic symmetric bent molecules
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
18 Alexander I. Melker, Maria Krupina, Vitaly Kotov
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
,
19Theory of vibrations of tetra-atomic symmetric bent molecules
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 ,
20 Alexander I. Melker, Maria Krupina, Vitaly Kotov
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.
21Theory of vibrations of tetra-atomic symmetric bent molecules