Determination of the Vibro-Rotational Constants, the Dipole Moment’s Function and the Intensities of the HTO’s ν1 (ν3 by Usual Convention) Band

doi:10.4236/jmp.2012.312243

Journal of Modern Physics, 2012, 3, 1945-1957

http://dx.doi.org/10.4236/jmp.2012.312243 Published Online December 2012 (http://www.SciRP.org/journal/jmp)

Determination of the Vibro-Rotational Constants, the

Dipole Moment’s Function and the Intensities of the HTO’s

ν1 (ν3 by Usual Convention) Band

Modou Tine1, Diouma Kobor1, Ibrahima Sakho1, Laurent Coudert2

1Laboratoire de Chimie et de Physique des Matériaux (LCPM), Université de Ziguinchor, Ziguinchor, Sénégal

2Laboratoire Inter-Universitaire des Systèmes Atmosphériques, Créteil, France

Email: [email protected]

Received June 29, 2012; revised October 16, 2012; accepted October 30, 2012

ABSTRACT

In the first part of this paper, an analysis of the high-resolution spectrum of the HTO molecule ν1(ν3) band, from 3630 to

3950 cm–1, was undertaken. The rotational transition of this band was assigned using combination differences. Their

wavenumbers were analyzed with a least squares fit program in order to obtain spectroscopic constants. A perturbed

state has been evidenced. In the second part, with a view towards building a spectroscopic data base, a calculation of the

dipolar momentum function was undertaken.

Keywords: Infrared Spectrum; Energy Level; Rays Intensities; Perturbation

1. Introduction

HTO molecule is important for the environment. It is

found in discharges of nuclear power plants with 12.3

years for half-life. It is essential to know its infrared

spectrum in order to detect it remotely. It is also interest-

ing to have available database to use this molecule spec-

troscopy. These were the motivation of our work focused

on the fundamental v1(v3) looks near 3700 cm–1.The aim

of this paper is twofold, firstly we perform a comprehen-

sive analysis of high-resolution spectrum of the v1(ν3)

band, and secondly we determine the dipole moment’s

function which will permit us to calculate the rays inten-

sities in a wide range of quantum numbers.

We continued with an analysis of the transitions wave

numbers to determine the spectroscopic constants that are

essential to calculate the positions of such transitions.

2. Theoretical Calculations

2.1. The Vibration-Rotation Hamiltonian

Leaving from the approximations of Born Oppenheimer

[1,2] and of the harmonic that consists in separating the

movement of the electrons to that of the cores and in de-

scribing the molecule while developing only the potential

energy to the second order, with the help of the 3N-6

correctly chosen coordinates. The vibration-rotation Ham-

iltonian, under compact shape, is written, according to

Darling and Dennison [3], simplified by Watson [4,5],

by:



 

2

1

2

1

2k

k

H

JP JP

PVQUQ



 







 





(1)

where the indices α and β correspond to the xyz axes of

the mobile reference mark. Jα and pα quantities are re-

spectively the components of the total angular moment

and the internal angular moment of the vibration; μαβ

represents the component of the efficient inertia tensor’s

inverse; pk with 136KN



 the conjugated momen-

tum of the normal coordinate of Qk vibration; V(Q) the

potential energy (quadratic) and U(Q) a term of a poten-

tial energy type. For the zero-order this Equation (1) be-

comes:

0vr

H

HH





With



322

1

2

vkk

k

H

PQ







 (2)

By introducing the dimensionless normal coordinates,

we obtain:

14

12

k

kk

qQ



 and k

k

i

pq





The Equation (2) gets under the shape:



322

12

k

vkk

k

H

hcp q







 (3)

M. TINE ET AL.

1946

where

12

2π

k

kc





 is the k node energy in cm–1 and the c

constant, the light speed. The Equation (3) is the sum of

three harmonic oscillators. We deduce easily the wave’s

function as well as the vibrational energy:

 

3

123 1

,,

vvkk

k

vvvq







and

3

1

2

vk

k

Ehc v













 (4)

where Φv(q) is the harmonic oscillator wave’s function

for the vibrational quantum number v. From this equation,

the values of the H2O, HDO and HTO three normal

nodes energies are given in the Table 1. The calculation

has been done taking into account the molecule geometry

at the equilibrium state and the surface potential energy

of Patridge and Schwenke [6]. To compare our results,

some energy levels, coming from the results of the ref-

erences [7-12], were presented in the same table. We

represented the small displacements to the H2O isitopic

in Figure 1. The normal modes are labeled using the

IUPAC convention [13], followed in brackets by the usual

convention. Hr can be written by:

2

r

HBJ



 (5)

where α corresponds to the x, y, z axes and



1

2

1

cm 8πe

h

BcI





r

H

has been calculated choosing the reference related

to the O’xyz molecule while the tensor of inertia would

be diagonal (Figure 2).

For an asymmetric molecule type, as HTO, the use is

to designate by A, B and C the constants of the rotational

Bα, with A > B > C. The corresponding axes have been

noted a, b, c.

Table 1. Calculated and observed energies of the three

normal modes for the isotopic varieties of water.

Variety Mode Calculated Observed

ν1 3831 3655

ν2 1648 1594

H2O

ν3 3942 3755

ν1 3889 3707

ν2 2824 2726

HDO

ν3 1445 1403

ν1 3888 3716

ν2 2369 1648

HTO

ν3 1370 1332

Figure 1. The small displacements of the three normal

nodes of vibration [13] of H2O, HDO and HTO.

B

z

H

A

O

Oʹ

T

x

Figure 2. The axes systems of HTO molecule.

Following the Ir representation





,,

x

by cza

the Equation (5) becomes :

 



222

22

11

22

1

4

rzxy

z

HAJBJCJ

A

BC JBCJ

BC JJ







  





 

(6)

where 2222

x

yz

J

JJJ



 and .

x

y

J

JiJ 

To estimate the molecule’s asymetry, we introduce the

Ray parameter [14] κ, equal to –0.8. This means that the

HTO molecule is enough close to a stretching asymmet-

ric rotator (1





).

There is no analytical forms for the eigenvalues of the

Equation (5). To obtain the rotation energy, it is neces-

sary to diagonalize the matrix of this hamiltonian in the

basis ,,

J

KM of the simultaneous eigenstates of the

total angular momentum

J

2 and of its projections on

the laboratory Z axis and the variable z of the mobile

reference, respectively, JZ and Jz.

M. TINE ET AL.

1947



2,,1 ,,

,, ,,

Z

z

J

JKMJ JJKM

J JKMMJKM

J JKMKJKM





(7)

where J, K, M is the quantum numbers corresponding to

these three operators. The expression of the wave func-

tion ,,

J

KM depends on the chosen conventions. In

the present case we take the same convention of Bunker

[15] and we write it as the following way:



2

21

,, ,,

8π

J

KM

J

JKM D







 (8)

where



ii

,, ee

JJ

K

KM KM

Dd





 



The function



J

K

M

d is defined in the reference [16].

When the molecule is isolated, the rotational energy

doesn’t depend on M and, to simplify, we use the linear

combinations of the following ,

J

K functions:

,,

for0

,, 2

,0for 0

JKJ KK

JK

JK K



 









(9)

where 1



 . When 0K, only the linear combina-

tion with γ = +1 exists. The non zero matrix elements of

the rotational Hamiltonian (Equation (5)), could exist

only when the linear combinations and the Equation (9)

are in the same type. That means 0, 2K

and

0



. We can write the rotational Hamiltonian eigen-

functions as followed:

,,,

nn

vK

KaJK





 (10)

where n is a quantum number varying from 0 to 2J and

where ,

n

K

a



the related coefficients of the developping

wave’s functions. Rather than to use the quantum number

n to identify the rotation levels, we prefer to use the

pseudo quantum numbers a

K

and c

K

of the rotation

[17]. Although these latters are not real quantum num-

bers, all of the three quantum numbers

J

, a

K

and c

K

with 0,

a

K

J0c

K

J and

or 1

ac

KKJ J  (11)

permit to, unequivocally, identify the 21

J

 rotation

levels corresponding to a value of

J

. Indeed, we have

ac

nJKK .

r

E the rotation energy of the

J

, a

K

, c

K

level, we

can write:

,, ,,

ac ac

J

KK JKK

rr rr

HE

(12)

The Figure 3 shows the arrangement of first levels of

rotation of HTO. We will notice that the energy espe-

cially depends on

J

and a

K

. Two levels, character-

120

100

80

60

40

20

0

2

02

3

03

4

04

1

01

0

00

1

10

1

11

2

12

2

11

3

13

3

12

2

20

2

21

Figure 3. Diagram of rotations energies levels of HTO.

ized by the same values of these two quantum numbers

with different c

K

values, have slightly different ener-

gies because of the asymmetry duplication. This latter

increases with

J

but decreases with a

K

and it is re-

sulting from the fact that BC. So the Hamiltonian of

zero-order becomes:

123

,,,,

0123

,,,, ,ac

vv vJKK

ac rr

vv v JKK  (13)

2.1.1. Watson’s Hamiltonian

The above given results are only valid in the setting of

the zero order approximation. In order to consider the

centrifugal distortion effects, we will use the Watson’s

Hamiltonian [18-20]. However in this equation the rota-

tional energy is not the eigenvalue of the rotational Ham-

iltonian of the Equation (6) but the eigenvalue of the ef-

ficient Hamiltonian depending only on the angular mo-

mentum of the rotation.



222 4224

2226 24 42

642242 8

264462 8

624426

,

vvvv vv

rzxyKzKJzJ

vvv vv

Kz JxyKzKJzJKz

vvv vv

JKzKJzJxyKz

vvvv

KKJz KJz KJJz J

vv v v

Kz KJz JKz J

HAJBJCJJJJ J

JJJHJHJJHJJ

HJhJ hJJ hJJLJ

LJJLJJ LJJLJ

lJ lJJ lJJ lJ



 

 

 



  



2

102 84 664

82 10

826446282

,

xy

vvv v

KzKKKKJ zKKJ zKJJ z

vv

KJJJz J

vvv vv

K

z KKJz KJ zKJJz Jxy

J

PJPJ JPJJPJ J

PJJ PJ

pJpJJpJJ pJJ pJJ





 



(14)

M. TINE ET AL.

1948

where 222

x

yxy

J

JJ

and {,} is the ant-commutator as



,

A

BABBA.

In this equation, all parameters depend on the consid-

ered vibrationnal state as indicated by the exhibitor v.

The adopted resolution is the A-type which is more ade-

quate for an asymmetric molecule as HTO.

390 transitions have been assigned for J values below

15 (Table 2).

2.1.2. The Wave Numbers

For every transition, characterized by the rotationnal

quantum numbers ,,

ac

J

KK





and ,,

ac

J

KK

 for the

lower and higher levels, respectively, the wave number

cal



has been calculated by:



 

10

,,: ,,:

cal

racrac

vv

E

JK KPEJ KKP

EE



 



 (15)

Table 2. The transitions assigned in the infrared spectrum of the ν1(ν3) band of HTO.

J' a

K

 c

K

 J" a

K



 c

K



 Obs Cal Diff

2

1

6

3

2

5

0

4

5

1

8

1

4

1

3

2

11

2

6

10

9

8

1

8

6

5

4

9

8

7

6

5

2

0

1

0

2

0

1

2

1

2

0

2

1

0

1

5

0

2

5

6

5

6

0

5

6

5

1

3

4

1

6

3

2

5

4

0

4

5

3

0

7

1

3

1

3

2

7

6

2

5

6

5

4

3

5

3

2

1

4

3

0

1

2

0

1

4

7

5

4

3

2

3

1

2

3

2

6

4

3

5

4

1

4

5

4

2

8

2

5

2

3

2

3

11

2

5

10

9

8

1

8

6

5

3

9

8

7

6

5

2

1

0

1

3

1

3

1

2

0

1

2

0

5

1

3

5

6

5

6

1

5

6

5

2

3

4

2

5

4

3

4

1

3

4

2

1

6

2

4

2

0

3

6

7

1

2

5

6

4

5

3

4

5

2

3

0

3

4

1

0

2

1

0

1

6

4

3

4

3

2

1

3678.521(10)

3678.849(10)

3680.963(10)

3682.342(10)

3683.105(10)

3683.710(10)

3686.775(10)

3688.956(10)

3689.717(10)

3690.777(10)

3690.777(50)

3690.937(10)

3691.777(10)

3693.354(10)

3693.994(10)

3694.065(50)

3694.102(10)

3694.970(10)

3695.623(10)

3696.543(10)

3696.643(10)

3697.093(10)

3697.172(10)

3697.518(10)

3697.708(10)

3698.242(10)

3698.360(10)

3698.658(10)

3698.941(10)

3699.149(10)

3699.524(10)

3700.508(10)

3701.061(10)

3702.410(50)

3702.935(10)

3703.183(10)

3703.817(10)

3703.879(10)

3704.358(50)

3704.817(10)

3721.521

3658.849

3692.963

3704.342

3696.105

3669.710

3549.775

3660.956

3687.717

3698.777

3714.777

3693.937

4057.777

3679.354

3603.994

3688.065

3686.102

3694.970

3695.623

3678.543

3680.643

3680.093

3392.172

3665.518

3703.518

3714.708

3715.708

3655.242

3693.360

3703.360

3719.658

3665.941

3699.149

3702.149

3702.524

3697.508

3694.061

3694.410

3718.935

3664.935

3699.183

3709.817

3711.879

3738.358

3720.358

3731.817

3727.817

−43

20

−12

−22

−13

14

137*

28

2

−8

−24

−3

−362*

14

90*

6

8

0

18

16

17

305*

32

−6

−17

−18

43

5

−5

−21

33

0

−3

3

7

8

−16

38

4

−6

−8

−34

−16

−27

−23

6 2 4 5 3 3 3704.817(50) 3733.817 −29

M. TINE ET AL.

1949

Continued

3

0

4

6

8

3

7

5

6

1

5

1

0

4

2

3

1

3

1

3

1

2

0

1

5

6

3

5

4

1

4

2

1

4

6

8

3

7

4

6

2

5

2

0

4

2

3

1

3

2

3

0

2

1

0

4

5

2

4

2

3

2

3

3704.857(50)

3704.939(50)

3705.197(10)

3705.219(10)

3705.518(10)

3705.744(50)

3707.199(10)

3708.150(50)

3708.254(10)

3708.554(10)

3708.930(50)

3722.857

3684.939

3676.197

3734.197

3417.219

3714.518

3724.744

3713.199

3705.150

3723.254

3724.554

3584.930

−18

20

29

−29

288*

−9

−19

−6

3

−15

−16

124*

5

6

4

3

8

2

4

6

3

2

3

1

4

5

1

4

6

2

7

3

1

8

4

1

5

2

4

3

2

3

5

2

4

1

5

6

3

4

6

3

2

6

7

3

1

2

1

2

1

2

1

2

1

0

2

1

0

2

1

3

1

3

0

1

0

1

2

0

1

0

1

2

3

2

1

0

5

2

3

6

2

1

0

1

2

3

0

3

2

4

1

2

1

5

2

1

6

3

2

0

4

1

3

2

4

2

1

3

1

4

5

3

2

1

4

5

3

2

5

4

5

6

4

3

8

2

4

5

3

2

3

1

4

5

1

3

5

6

2

1

3

7

3

0

8

4

5

1

4

2

5

2

1

3

5

1

4

0

5

6

2

3

5

2

1

6

7

3

1

2

1

2

1

2

1

2

1

2

1

0

2

1

2

0

2

0

2

1

0

1

0

1

0

1

2

1

2

0

1

0

1

2

4

1

2

0

1

6

1

2

3

1

0

1

2

0

3

4

1

2

5

2

1

2

6

3

0

7

4

3

1

3

2

4

2

1

3

5

1

0

4

0

5

6

2

1

0

3

4

2

1

6

5

6

3708.930(50)

3709.119(10)

3709.373(10)

3709.430(50)

3709.685(50)

3709.695(50)

3710.114(10)

3710.775(10)

3711.338(10)

3711.866(50)

3712.651(10)

3713.257(50)

3713.496(10)

3713.832(50)

3714.137(10)

3714.756(10)

3716.657(10)

3717.367(10)

3719.221(10)

3719.327(10)

3719.865(10)

3720.452(50)

3723.518(10)

3724.079(10)

3724.568(10)

3725.042(10)

3728.149(10)

3730.842(10)

3731.131(10)

3721.577(10)

3733.357(10)

3734.056(10)

3734.942(50)

3735.299(50)

3736.028(10)

3737.204(10)

3737.549(10)

3738.615(10)

3739.500(10)

3740.513(10)

3741.318(50)

3743.338(10)

3746.505(10)

3747.373(10)

3747.668(10)

3748.046(10)

3748.544(10)

3748.608(10)

3749.199(10)

3750.541(10)

3752.710(10)

3753.185(10)

3753.185(50)

3754.397(10)

3754.468(10)

3720.930

3744.119

3716.373

3734.430

3700.685

3701.695

3737.114

3722.775

3638.338

3672.866

3699.257

3742.257

3739.496

3694.832

3714.137

3685.756

3692.657

3709.367

3728.221

3692.327

3700.865

3713.452

3698.518

3725.079

3716.568

3735.042

3712.149

3718.842

3740.131

3693.577

3725.357

3735.056

3748.942

3723.299

3719.028

3737.204

3744.549

3753.615

3723.500

3750.513

3770.318

3747.338

3762.505

3765.373

3759.668

3725.046

3738.544

3739.608

3755.199

3732.541

3764.710

3763.185

3753.185

3742.397

3754.468

−12

−35

−7

−25

9

8

−27

−12

73*

39

13

−29

−26

19

0

29

24

8

−9

27

19

7

25

−1

8

−10

36

12

−9

28

8

−1

−14

12

17

0

−7

−15

16

−10

−29

−4

−16

−18

−12

23

10

9

−6

18

−12

−10

0

12

0

5 2 3 5 1 4 3755.434(10) 3730.434 25

M. TINE ET AL.

1950

Continued

8

4

9

3

2

3

2

1

2

6

2

1

2

4

9

1

8

3

4

3

7

3

1

3

1

7

1

0

3

8

2

3755.919(10)

3755.994(50)

3756.024(50)

3757.218(10)

3757.948(10)

3758.901(10)

3759.322(50)

3746.919

3740.994

3746.024

3735.218

3773.948

3850.901

3764.322

9

15

8

22

−16

8

−5

4

5

4

7

2

3

5

10

7

4

2

5

3

6

11

4

5

8

5

4

10

5

6

5

9

5

12

6

5

6

8

6

7

6

7

6

2

6

7

2

6

5

7

4

3

5

7

6

8

7

2

0

1

2

1

4

2

1

2

3

1

2

5

3

1

2

1

0

2

3

2

0

4

2

3

1

2

1

3

2

3

0

5

2

3

0

2

1

2

3

1

2

1

3

0

3

1

3

0

3

2

5

4

6

0

3

1

2

8

6

3

1

3

2

5

2

1

8

4

7

5

3

7

3

6

2

3

4

6

4

10

6

5

4

3

5

6

3

2

5

4

7

4

6

1

3

7

0

5

2

7

4

1

2

1

3

7

4

8

5

3

4

3

7

2

4

10

6

3

2

4

3

5

11

3

4

8

4

10

4

5

9

4

12

5

4

5

6

8

5

6

5

7

6

5

1

6

1

5

6

4

3

5

6

7

2

1

0

1

0

4

1

2

1

3

1

5

2

0

2

0

1

2

1

4

1

2

1

0

3

1

2

0

5

2

1

2

0

1

2

1

2

0

3

2

0

2

1

2

1

4

3

7

1

2

0

1

9

5

2

1

4

3

1

0

9

3

8

4

8

2

5

1

2

5

7

3

11

5

4

3

2

6

5

2

1

4

5

6

3

5

0

4

6

1

4

3

6

3

2

3

2

4

6

5

7

6

3759.322(50)

3760.535(10)

3760.953(10)

3761.107(10)

3761.276(10)

3761.348(10)

3762.176(10)

3762.870(10)

3763.546(10)

3764.473(10)

3764.730(10)

3765.979(10)

3767.325(10)

3765.453(10)

3768.021(10)

3768.228(10)

3769.881(10)

3770.189(10)

3770.439(50)

3770.593(10)

3771.163(10)

3771.245(10)

3772.669(10)

3772.777(10)

3772.819(10)

3774.061(10)

3775.010(10)

3776.061(10)

3776.516(10)

3777.718(10)

3777.858(10)

3778.226(10)

3778.610(10)

3778.950(10)

3779.601(10)

3779.904(10)

3780.716(10)

3780.874(10)

3781.421(10)

3783.693(10)

3784.088(10)

3785.251(10)

3785.755(10)

3786.069(10)

3787.024(10)

3787.478(10)

3787.775(10)

3787.963(10)

3789.632(10)

3789.883(10)

3790.601(10)

3791.438(10)

3793.025(10)

3793.088(10)

3793.165(10)

3793.274(10)

3793.617(10)

3794.388(10)

3794.499(10)

3676.322

3743.535

3748.953

3757.107

3770.276

3783.348

3778.176

3780.870

3776.870

3749.546

3763.473

3779.730

3783.979

3773.325

3774.453

3777.021

3740.228

3805.881

3762.189

3799.439

3601.439

3768.593

3756.163

3675.245

3781.669

3779.777

3734.819

3800.061

3791.061

3608.010

3793.061

3786.516

3765.718

3781.858

3789.226

3789.610

3801.950

3450.601

3794.904

3735.716

3778.874

3446.421

3802.693

3677.088

3770.251

3784.755

3793.069

3800.024

3767.478

3809.775

3805.963

3789.632

3680.883

3802.601

3777.601

3781.025

3763.088

3791.165

3770.274

3796.617

3653.388

3793.499

83*

17

12

4

− 9

−22

−16

−18

−16

14

1

−15

−18

−6

−9

28

−36

8

−29

169*

2

15

96*

−9

7

38

−26

−17

167*

−17

−10

12

−4

−11

−23

329*

−15

45

2

335*

−19

107*

15

1

−7

−13

20

−22

−18

0

109*

−12

14

12

30

2

23

−3

141*

1

M. TINE ET AL.

1951

where i

v

E is the vibrationnal energy of the i state and



,, :

i

rac

EJKK P represents the rotational energy of

the J, Ka, Kc level, calculated with the efficient Hamilto-

nian given by the Equation (14) for the i

P spectro-

scopic parameter of the i vibrationnal state.

In this work, the 0

P spectroscopic parameters of the

fundamental state have been considered equal to the

Helminger’s values published in [21] and 0

v

E arbitrarily

fixed at zero. The 1

P spectroscopic parameters of the

vibrationnal state v1 = 1 has been determined by adjust-

ment using the least squares program. The results of the

analysis are in the Tables 2 and 3.

2.2. The ν1(ν3) Band Intensity

To calculate the transitions intensity, it is necessary for

us to determine, at first, the dipolar momentum function.

2.2.1. Determination of the Dipolar Momentum

Function

From the S1 and S2 coordinates corresponding respec-

tively to the following small variations ∆r31 and ∆r32 of

the inter atomic distances and S3 corresponding to small

variations of the valence angle ∆Φ, the Schrödinger

equation for the vibration [22] becomes:

33

,1 ,1

11

22

vijijijij

ij ij

H

GPPFSS







(16)

where i

i

PS









is the conjugated momentum of the co-

ordinate i

S, G and F correspond respectively to the

kinetic and potential energies and can be calculated nu-

merically from the Wilson Equation [22] once the mole-

cule’s equilibrium configuration is known. The F tensor

components can be calculated numerically using Patridge

and Schwenke [6] potential energy surface. The v

H

eigenenergies were obtained by solving the following

equation:

1

F

XGX





 (17)

where



and X are respectively eigenvalue and ei-

genvector.

Thus we can deduce the equation permitting the pas-

sage from normal dimensionless coordinates k

q to the

internal coordinates.

3

14

1

k

i

ik

k

X

Sq











 (18)

Then we try to obtain the dipolar momentum function

according to the internal coordinates:

3

0

1

ssi

x

xxi

iS

 







and

3

0

1

ssi

z

zzi

iS

 





 (19)

Table 3. The spectroscopic constants of the fundamental and the v1(v3) band of HTO.

Constant value to the ν1(ν3) band Constant value to the fundamental

E1 3716.5475(48) E0 0

A

B

C

21.8482(37)

6.58908(75)

4.97656(72)

A

B

C

22.61061

6.61116

5.01889

∆K

∆KJ

∆J

δK

δJ

13.796(810) × 10–3

2.306(130) × 10–3

0.1628(71) × 10–3

–6.710(320) × 10–3

16.670(390) × 10–6

∆K

∆KJ

∆J

δK

δJ

9.04872 × 10–3

1.61856 × 10–3

0.17385 × 10–3

1.71208 × 10–3

47.1697 × 10–6

HK

HKJ

HJK

HJ

hK

hKJ

hJ

888.900(680) × 10–6

127.500(160) × 10–6

–24.000(18) × 10–6

–0.224(64) × 10–6

–894.900(360 × 10–6

–35.500(270) × 10–6

–0.140(16) × 10–6

HK

HKJ

HJK

HJ

hK

hKJ

hJ

29.880 × 10–6

–2.110 × 10–6

1.316 × 10–6

17.337 × 10–9

17.947 × 10–6

0.5713 × 10–6

5.946 × 10–9

LK

LKKJ

LKJ

–0.277 × 10–6

0.0659 × 10–6

–0.018 × 10–6

lK

lKJ

–0.2713 × 10–6

–3.0395 × 10–9

PK 3.673 × 10–9

LK

LKKJ

LKJ

LKJJ

LJ

IK

IKJ

IJK

PK

PKKKJ

pK

–20.000(24) × 10–6

–11.130(50) × 10–6

1.906(120) × 10–6

0.0831(90) × 10–6

0.404(180) × 10–9

29.781(1600) × 10–6

0.924(80) × 10–6

89.840(760) × 10–9

238.400(300) × 10–9

115.200(100) × 10–9

–196.900(150) × 10–9

pK 0.969 × 10–9

M. TINE ET AL.

1952

where 0s





with or

x

z



 and

s

i





with 13i



are constants.

To determine the HTO dipolar momentum function,

we use the unvariability of this function with an isotopic

substitution when it is expressed by the internal coordi-

nates. However, it will be necessary to take to account

the way from which the molecule depending reference

mark O’xyz is going to change.

From the Equation (18) and by replacing the Si by the

qi in the Equation (19), we find the HTO dipolar mo-

mentum function expressed in the normal dimensionless

coordinates. Considering the Equation (18), the terms

k

i

X

and k



will have the values corresponding to the

HTO normal modes and the results are shown in the Ta-

ble 4.

2.2.2. The Absorption Band Intensity

The intensity of a molecular absorption band, in a gas at

the thermodynamic equilibrium [23], is given as fol-

lowed:

3

36

2

8π10 exp

3

1exp

BAAB A

A

AB

Z

g

hcE

ShcQ KT

hc AB

KT



















 









(20)

where

B

A

S is the intensity in cm–1/(molécule·cm–2) for a

transition from the lower level A to the up level B at the

T temperature; A

g

is the degenerancy due to the nu-

clear spin of the lower level A; AB



is the transition

wave number in cm–1;

A

E is the lower energy level in 1

cm; Q the distribu-

tion’s function; K the Boltzmann’s constant and

Z





is

the modified transition momentum in Debye. Its expres-

sion [23] is:

Z

xx zz





 (21)

where

x



and

z



are the dipolar momentum compo-

nents, in the reference mark related to the molecule, ex-

pressed according to the normal dimensionless coordi-

nates.

x

 and

z

 are operators [23] depending to the

vibrational coordinates which matrix elements are given

in the Tables 5 and 6.

From the Tables 3 and 5, the wave number and the

Table 4. Thevalues of the coefficients used in the develop-

ment of the dipolar momentum function.

Coefficients Valeur Coefficients Valeur

0

x



–1.74117 0

z



–0.84923

1

x



× 21/2 –0.01060 1

z



× 21/2 –0.03502

2

x



× 21/2 0.10583 2

z



× 21/2 –0.05269

3

x



× 21/2 –0.03039 3

z



× 21/2 0.04091

intensity of the authorized transitions have been calcu-

lated using Equations (15) and (20), for 11J



, for a

temperature of 296 K.

The Table 7 gives the portion of list covering the re-

gion 3630 to 3760 cm–1 for the transitions which inten-

sity is over





23 12

10cmmolecule cm

 

.

3. Results and Discussion

3.1. Analysis of Waves Numbers

In this analysis, the efficient hamiltonian spectroscopic

parameters of the Equation (14) and the vibrational en-

ergy have been considered.

The mean quadratic gap q = 0.018 cm–1 is enough

close to the experimental one for the observed wave

number: 0.010 cm–1. It is also possible to evaluate the

quality of the analysis with the standard deviation σ = 1.7.

Ideally, this value should be very close to 1. This excess

can be the consequence of the underestimation of the

experimental uncertainties and the use of an unadequate

efficient Hamiltonian. In this latter, it can come from a

perturbation of the superior vibrationnal state. This hy-

pothesis is confirmed by the results in the Table 2 for the

transitions presenting a big gap (Obs.-Cal.) or even those

that have been excluded in the analysis (in asterisk in the

Table 2).

One realizes that these transitions often share the same

superior rotationnal level and that the gap (Obs.-Cal.) is

practically independent of the low level.

The Table 8 illustrates well this fact for transitions ex-

cluded in the analysis for which the superior rotationnal

levels are 808 and 625.

Presumably, the vibrationnal state v1 = 1 is not isolated.

It exists another closed vibrationnal state which is cou-

pled whith it. To make a correct calculation of the vibra-

tionnal energy, it would be necessary to treat simultane-

ously these two states. A calculation based on Table 1

Table 5. The non zero elements matrix [23] operator Φz.

∆J ∆K Γ , , , ,

z

JKJ K







0 0









12

21 1KJ JJ





±1 0









12

22

–Km Km





Table 6. The non zero elements matrix [23] operator Φx.

∆J∆KΓ , , , ,

z

JKJ K







0±1











12

121 11

2JJKKJKKJJ 



±1 ±1









12

11

2JKm KKmKKm



  





M. TINE ET AL.

1953

Table 7. Waves numbers and lines intensities of v1(v3) band of HTO.

J' a

K

 c

K

 J" a

K



 c

K



 Cal S

2

1

7

2

9

3

5

11

7

10

11

8

5

8

0

10

9

4

5

1

11

8

11

4

1

9

11

4

1

3

2

8

11

6

2

9

11

10

9

8

1

2

0

1

2

0

4

3

4

3

2

3

0

4

3

0

1

2

1

0

2

1

0

4

3

2

1

0

1

3

8

5

2

0

3

6

5

6

8

5

9

6

1

6

2

8

3

5

8

5

7

6

4

5

0

6

4

5

3

0

11

7

11

4

1

6

9

3

1

3

2

5

1

0

7

6

5

2

7

6

5

6

5

4

3

2

1

5

4

0

1

3

2

3

2

6

3

9

2

5

10

8

11

10

7

4

7

1

11

10

4

5

4

2

10

8

7

10

3

2

10

11

5

2

3

2

3

9

11

5

2

8

11

10

9

10

9

10

8

1

2

1

3

1

2

1

5

2

5

4

3

4

1

3

2

1

3

1

2

3

2

0

3

1

2

0

2

8

5

3

1

4

6

5

6

8

5

10

6

2

3

7

0

4

5

6

10

6

3

1

4

1

9

3

4

2

1

8

6

4

8

1

2

7

8

4

2

0

3

5

2

1

6

7

2

1

4

5

6

5

6

4

5

3

4

3

2

4

5

1

0

2

3

3678.528

3678.564

3678.828

3679.121

3683.118

3683.043

3683.299

3683.695

3684.293

3684.665

3685.350

3685.575

3685.976

3686.796

3688.519

3688.927

3689.439

3689.711

3689.714

3690.785

3690.802

3690.941

3692.047

3691.788

3692.310

3692.615

3693.035

3693.340

3693.679

3693.883

3693.985

3694.058

3694.093

3694.968

3695.622

3696.172

3696.388

3696.552

3696.653

3697.186

3697.075

3697.116

3697.356

3697.359

3697.498

3697.537

3697.724

3697.725

3698.201

3698.205

3698.358

3698.371

3698.568

3698.679

6.47e−20

3.24e−20

2.22e−19

1.00e−22

6.79e−20

1.34e−21

7.21e−23

2.59e−19

1.31e−21

1.67e−20

2.63e−21

1.31e−21

6.99e−21

1.60e−20

7.03e−21

1.65e−19

2.45e−21

5.18e−21

3.32e−19

5.34e−21

1.65e−20

4.38e−20

3.26e−21

2.66e−21

1.28e−22

9.26e−22

2.21e−22

6.41e−20

4.01e−21

5.50e−22

5.86e−20

4.49e−20

3.70e−19

1.73e−20

1.67e−19

8.99e−21

3.63e−23

5.29e−22

1.96e−20

3.45e−19

5.87e−21

3.24e−22

1.08e−21

6.62e−22

1.29e−21

3.99e−23

2.11e−21

2.66e−23

2.39e−21

1

4

7

0

1

6

1

4

2

1

4

7

1

6

0

3

1

2

3698.907

3699.079

3699.123

2.46e−19

8.72e−21

4.26e−21

M. TINE ET AL.

1954

Continued

8

7

11

6

10

5

9

6

10

9

4

9

11

10

9

8

7

9

7

6

5

6

3

0

6

4

8

9

3

7

9

6

10

7

9

10

5

6

1

5

8

6

5

3

7

4

8

7

3

5

4

5

0

5

0

4

3

1

4

3

1

4

3

4

1

3

4

2

1

0

2

4

3

1

2

0

8

3

1

3

1

3

1

2

3

4

3

2

3

0

3

4

3

2

8

2

1

10

1

0

5

4

10

7

6

4

6

7

6

5

7

5

4

3

8

4

3

2

1

4

2

0

5

1

0

6

3

6

9

6

3

2

5

9

8

5

4

1

4

3

5

3

2

4

2

1

8

7

1

0

8

7

11

6

9

5

4

10

7

9

10

8

3

9

11

10

9

8

7

8

6

5

2

1

6

4

8

10

3

6

8

5

11

7

8

9

4

6

2

5

9

6

5

4

7

4

7

6

3

5

4

5

2

5

2

3

2

4

2

4

3

4

3

2

4

3

2

0

2

4

3

1

3

2

8

3

2

4

2

3

0

2

3

1

3

2

3

2

3

7

1

2

7

0

1

2

8

5

7

6

5

1

5

8

7

6

4

5

3

4

5

7

2

3

1

2

3

1

4

0

1

5

9

2

3

6

3

4

3

4

6

5

2

3

2

3

2

6

5

3

5

1

2

5

4

0

1

3699.147

3699.863

3699.864

3700.237

3700.500

3701.048

3701.056

3701.063

3701.368

3701.418

3701.470

3701.986

3702.401

3702.414

3702.529

3702.573

3702.899

3702.953

3703.180

3703.370

3703.825

3703.889

3703.941

3704.346

3704.376

3704.393

3704.842

3704.845

3704.847

3704.876

3704.918

3705.251

3705.227

3705.528

3705.534

3705.763

3706.308

3706.420

3706.744

3706.895

3707.206

3707.885

3707.903

3708.146

3708.269

3708.570

3708.964

3708.943

3709.016

3709.155

3709.242

3709.316

3709.370

3709.380

3709.456

3709.349

3709.746

3709.676

3709.687

3.94e−21

7.03e−21

6.63e−22

1.21e−21

1.21e−20

7.10e−21

2.00e−20

3.72e−22

3.95e−21

2.00e−20

2.13e−21

1.38e−21

5.92e−21

2.87e−20

2.72e−21

6.65e−22

1.38e−21

2.72e−21

2.47e−21

5.10e−21

9.14e−21

1.34e−22

1.36e−20

1.57e−20

2.62e−20

2.09e−20

2.33e−20

3.60e−20

9.30e−21

4.28e−20

4.79e−21

1.63e−23

1.43e−20

1.90e−20

1.41e−22

4.28e−22

3.13e−23

8.77e−21

4.60e−21

4.37e−21

3.01e−20

1.53e−20

9.17e−20

1.62e−20

2.57e−20

5.76e−21

1.53e−20

2.57e−20

4.58e−20

8.78e−21

4.21e−20

2.48e−22

3.66e−22

6.87e−20

M. TINE ET AL.

1955

Continued

8

2

4

9

8

6

3

7

8

2

7

10

2

3

6

1

10

8

4

5

8

7

10

8

5

6

5

8

1

11

9

11

4

6

7

11

9

2

5

9

10

7

2

5

7

3

7

3

10

9

4

10

3

1

2

3

1

2

1

4

2

7

1

5

2

3

1

5

3

2

7

2

3

2

3

2

1

3

7

3

1

2

1

2

1

2

7

4

0

2

3

2

1

4

2

5

2

5

2

3

6

8

6

4

2

5

7

4

1

0

9

6

0

1

3

1

5

7

2

3

2

1

6

5

7

5

3

4

3

6

0

9

3

2

8

3

2

4

5

10

7

1

3

8

4

3

4

2

1

2

5

2

3

1

9

5

2

8

2

4

9

7

5

7

3

8

6

9

2

7

9

11

2

3

7

1

11

10

7

4

6

8

9

8

9

5

7

6

9

1

10

9

11

3

5

6

10

2

6

8

10

8

1

3

6

7

3

8

4

9

10

5

11

3

1

2

3

2

1

2

1

2

3

2

7

3

4

2

1

4

3

2

1

7

1

0

4

1

2

0

2

0

1

4

7

3

2

1

2

3

1

0

3

7

3

1

2

1

3

1

3

4

0

1

6

1

2

7

5

3

7

1

8

4

7

0

1

6

7

1

2

6

0

8

4

3

6

1

2

9

8

6

8

4

7

4

9

1

6

2

3

9

2

5

4

7

10

2

6

5

3

4

5

1

2

5

6

2

6

4

6

5

11

3710.140

3710.787

3711.346

3711.777

3711.699

3711.826

3712.596

3712.637

3712.803

3713.116

3713.161

3713.286

3713.305

3713.361

3713.442

3713.523

3713.812

3713.825

3714.137

3714.491

3714.599

3714.328

3714.727

3714.819

3715.312

3715.443

3715.547

3716.434

3716.471

3716.633

3716.660

3717.007

3717.241

3717.358

3718.141

3718.468

3718.865

3719.231

3719.300

3719.846

3719.889

3719.979

3720.385

3720.445

3720.612

3721.537

3723.159

3723.377

3723.493

3724.080

3724.242

3724.560

3725.053

3725.511

3725.722

3726.797

3727.181

3727.190

3727.394

4.81e−21

2.45e−20

2.74e−20

2.50e−21

9.11e−21

2.42e−20

1.30e−20

4.53e−20

7.01e−21

1.60e−20

5.72e−21

7.74e−20

2.10e−21

1.15e−22

1.14e−21

7.74e−20

4.54e−20

1.79e−20

4.97e−20

1.14e−21

1.23e−21

1.55e−20

2.75e−20

2.10e−20

1.19e−21

3.39e−21

4.75e−23

4.50e−21

2.30e−23

1.64e−20

8.90e−23

2.10e−20

2.12e−23

5.01e−20

2.93e−21

6.41e−22

5.71e−22

4.78e−20

2.80e−20

9.53e−21

2.17e−20

8.08e−23

1.52e−21

2.51e−20

1.32e−22

1.10e−20

3.31e−22

3.31e−21

7.36e−21

9.64e−20

2.42e−20

1.99e−20

5.33e−21

1.51e−20

7.34e−21

2.86e−20

6.77e−21

1.71e−21

1.44e−22

6.52e−22

M. TINE ET AL.

1956

Continued

9

1

7

11

8

4

11

1

2

5

2

4

8

3

6

2

3

6

10

9

5

2

8

2

8

4

1

6

3

2

5

3

10

6

3

5

0

3

2

7

2

1

3

1

2

1

3

2

0

4

1

4

9

2

1

0

5

1

5

1

3

2

1

3

6

1

4

1

4

10

5

4

6

3

2

8

0

4

1

6

1

3

2

1

2

7

4

2

4

1

3

1

3

1

0

4

0

5

4

5

3

10

0

8

10

11

8

4

5

10

1

3

4

2

5

7

4

2

7

1

3

7

11

9

5

1

9

1

9

4

0

7

4

3

5

4

11

6

2

4

0

1

3

7

2

1

2

4

0

1

2

0

2

3

0

1

3

1

0

3

10

2

1

0

4

1

4

0

1

2

0

2

5

1

7

0

7

4

5

7

4

3

7

1

3

2

4

5

4

2

4

1

3

5

2

1

8

5

1

5

0

6

4

0

6

2

3

5

3

6

4

6

2

3727.680

3728.112

3729.425

3729.787

3729.968

3730.828

3731.141

3731.549

3731.999

3731.349

3733.700

3734.058

3734.957

3735.287

3735.852

3735.929

3736.011

3736.954

3737.204

3737.557

3737.943

3738.286

3738.589

3738.631

3739.483

3740.514

3740.524

3740.731

3741.348

3743.343

3743.732

3745.083

3746.205

3746.522

3746.720

3746.854

3746.897

3747.392

3747.680

1.71e−21

3.83e−20

2.69e−23

3.67e−21

1.63e−22

2.98e−21

9.50e−21

1.80e−20

3.12e−21

2.71e−19

1.85e−20

6.69e−20

3.84e−19

1.75e−20

1.98e−20

1.07e−22

1.89e−16

8.09e−21

8.04e−20

4.21e−19

8.08e−21

1.49e−23

1.51e−21

6.07e−21

7.26e−20

2.27e−21

5.01e−20

2.27e−21

3.91e−19

1.92e−19

2.61e−23

1.01e−20

4.39e−23

3.18e−19

9.99e−21

3.67e−22

3.86e−21

8.08e−20

Table 8. Putting in evidence of the perturbations in the v1(v3)

band of HTO spectrum.

J' a

K

 c

K

 J" a

K

 c

K Obs Cal Diff

8

6

0

2

8

5

9

8

7

6

7

5

6

1

0

1

0

3

2

1

3

2

1

9

7

4

6

5

4

6

3614.632

3616.432

3655.075

3794.388

3598.449

3627.371

3632.055

3661.947

3697.172

3705.219

3779.601

3759.632

3759.432

3821.075

3934.388

3741.449

3303.371

3295.055

3339.947

3390.172

3415.219

3448.601

–145

–143

–166

–140

–143

324

337

322

307

290

331

shows that the closest vibrational state is the state v2 = v3

= 1 (v1= v2 = 1), which energy is estimated to 3739 cm–1.

3.2. Discussion

The determination of the the dipolar momentum compo-

nents (Table 5) of HTO by the so-called isotopic substi-

tution method allowed us to calculate rays intensities in

the ν1(ν3) band.

The experimental rays intensities have not been deter-

mined simultaneously with the spectrum record.

For this reason only theoretical values are shown in

Table 7. In contrast, it should be noted that in the case

high excited levels transitions, the calculated energy

level has been replaced by its observed value.

In summary, to estimate the quality of the database

generated for HTO, there is shown, in Figure 4, some of

M. TINE ET AL.

1957

3710 3712 3714 3716 3718 3720

Figure 4. Spectrum observed (dash) and calculated (point)

of the v1(v3) band of HTO.

the observed spectra (dash) and calculated (point). It can

be noticed that the results are satisfactory.

4. Conclusions

The satisfactory analysis in terms waves rotational num-

bers of the ν1(ν3) band permitted us to make in evidence a

perturbation of the high vibrationnal state. Also, the

theoretical calculation of the dipole momentum function

allowed us to calculate the non measured intensities of

this band’s transitions.

Finally, as announced in the introduction, these results

permit us to create a HTO spectroscopy database.

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16O,” Molecular Physics, Vol.

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