chemical workbench user guide - kintech lab

FLUID WORKBENCH

Version 1.0

Model's

description

Kintech Lab

2018

2

Copyright:

Copyright© 2018 Kintech Lab. All rights reserved. No part of this text may be reproduced in any form or by any

means without express written permission from Kintech Lab.

Information in this document is subject to change without notice.

Kintech Lab Ltd.

12, 3rd Khoroshevskaya str.,

Moscow, 123298, Russia.

tel: +7 (499) 704-2581

url: http://www.kintechlab.com

e-mail: [email protected]

http://www.kintechlab.com/

3

CONTENT

INTRODUCTION ............................................................................................................. 4

About this program ....................................................................................................... 4

1. Calculation of chemical composition ................................................................... 5

Assumption of local thermodynamic equilibrium ............................................................................... 5

Frozen composition approach ............................................................................................................. 11

2. Transport properties ............................................................................................ 12

1.FWB can calculate the transport coefficients: ................................................................................ 12

2.How we calculate it ............................................................................................................................ 12

3. Collision integrals ............................................................................................................................. 16

4. Interaction potentials ........................................................................................................................ 17

5. Calculation of collision integrals for the interaction of atom with parent ion ............................ 20

References ............................................................................................................................................. 21

3. Absorption coefficient ......................................................................................... 23

Atoms. .................................................................................................................................................... 23

Spectral lines and Broadening mechanisms ...................................................................................... 23

Continuum ........................................................................................................................................... 26

Molecules ............................................................................................................................................... 27

Evaluation of absorption coefficients of diatomic molecules. ............................................................. 27

Absorption coefficient due to photodissociation ................................................................................. 38

Absorption coefficient due to photoionization ..................................................................................... 39

Quick Start FWB 1.0

4

INTRODUCTION

About this program

Fluid Workbench (FWB) is designed to calculate radiation and thermophysical properties of gases.

Following properties can be calculated in frame work of the code:

-Equilibrium composition, ni

-Constant pressure heat capacity Cp(T,P)

-Enthalpy H(T,P)

-Entropy, S(T,P)

-Viscosity coefficient μ(T,P)

-Binary diffusion coefficients Dij(T,P) for all pairs of particles (i,j)

- Thermal conductivity coefficient λ(T,P)

-Electrical conductivity σ(T,P)

-Absorption coefficient kabs(T,P)

-Net Emission Coefficient εnet(T,P)

-Radiative heat conductivity λrad

-Refraction index

-Radiation spectrum

Calculation can be done in suggestion of Local Thermodynamic Equilibrium (LTE) and for non-

equilibrium composition provided by user.

This text contains description of the models used for calculation of enumerated properties.

Model's descrition FWB 1.0

5

1. Calculation of chemical composition

Calculation of thermophysical and optical properties of gases demands definition of chemical

composition before calculation of these properties. There are two approaches used in FWB code:

- assumption of Local Thermodynamic Equilibrium (LTE)

- using of fixed (predetermined by user) chemical composition

Below, detailed description of both approaches is presented.

Assumption of local thermodynamic equilibrium

Local Thermodynamic Equilibrium (LTE) approach assumes that in every point of the space

chemical composition is determined by total chemical equilibrium between gaseous components.

It means that chemical composition is function of only :

- composition of chemical elements

- pair of thermodynamic parameters (PT),(VT),(HP),(UV)

where P,T,V,H,U – pressure, temperature, volume, enthalpy and internal energy

Thermodynamic equilibrium reactor model is used for calculation of chemical composition in

assumption of LTE.

Thermodynamically Equilibrium Reactor Model (TER )

The Thermodynamically Equilibrium Reactor Code (TER) is designed for the calculation of the

chemical equilibrium of multicomponent heterogeneous system. This thermodynamic system is

considered as self-contained and closed. In this system a state of thermodynamic equilibrium is

achieved by internal chemical and phase transformations. It means that the system is under

mechanical and energetic equilibrium. It is proposed also that the system under investigation is a

heterogeneous one and consists of several uniform phases. That is why, the gas components form

a separate gas phase, but condensed substances can form as separate phases as condensed

solutions. The condition of existence of the gas phase is regarded as an obligatory condition in this

model, but the condensed phase can be absent in this system.

The Thermodynamically Equilibrium Reactor Code uses the common principle of entropy

(S) maximum for the calculation of chemical and phase composition. In accordance with this

principle the equilibrium state is characterised by the uniform distribution of thermodynamic

parameters in system volume and chemical composition corresponds to maximum of probability

of energetic levels distribution for macro particles:

S S= max ; for Mj= const, U= const, v= const. ( 1)

where Mj is chemical elements mass, U - internal energy function, v- specific volume. Condition

( 1) does not put bound to possibilities of definition of equilibrium states in the code because any

Quick Start FWB 1.0

6

system can be regarded under equilibrium state only in the case where any external action can be

neglected.

The principle of entropy maximum permits us to calculate practically any thermodynamic

equilibrium state.

Equations of state for parameters calculation under thermodynamic equilibrium.

The entropy of multicomponent system consists of entropy of different separated

components and phases: gas neutral components and ions which have properties of ideal gas;

components formed pure condensed phases ( solid or liquid states) and condensed solutions.

The entropy of the gas phase can written as follows:

S S RRT

vM Mg i i i

i

k

= −

=

0

1

ln ( 2)

where: Si0 - conventional standard absolute entropy, v- specific volume of system, p = R T Mi/v -

pressure of I component, Mi - number of moles of specie i per one kg. of system.

The entropy of components which form pure separated condensed phase in accordance to

the additive principle is equal to:

S S Mc r r

r

R

==

0

1

( 3)

where: Si0 - entropy of condensed phase per one mole of substance, Mr - number of moles in one

kg of condensed phase, R- total number of separated condensed phases.

The third system consists as it was mentioned above, of condensed solutions. The presented

code can calculate the chemical composition of two solutions S1 and S2 at once and it is proposed

that types of substances in theses solution are known. For definition of entropy of solutions the

model of ideal solution is used. In the framework of this model it is proposed that enthalpy of

dissolution of one component in other and volume change are equal to zero. In this case the entropy

of dissolution is calculated as entropy of noninteracting particles. Thus, the entropy for the case of

two solutions has the following form:

S S M RMM

MS M RM

M

MS S n n n

n

Sn

N

n

N

n n nn

Sn

N

n

N

1 2 10

1 11

11 1

1

1 1

1

20

2 22

22 1

2

2 1

2

, ln ln= − + −== ==

( 4)

where: N1,N2 - the number of components in solution S1 and S2 respectively, Mn1 and Mn2 -

number of moles in solution S1 and S2.

Thereby the total entropy of system can be presented as:

S=Sg+Ss+SS1,S2 ( 5)


7

Lagrange function method.

The definition of equilibrium state parameters is a determination of values of all variables

including the number of components moles for the condition of entropy maximum. In the process

of calculation the following additional restrictions are used. These restrictions reflect the condition

of system existence.

• The value of total internal energy must be constant in all chemical and phase

transformations:

− + + + + == = = =

U M U M U M U M Ui i

i

k

r r

r

R

n n

n

N

n n

n

N

1 1

1 1

1 1

1

2 2

2 1

2

0 ( 6)

• In accordance with the conservation law of mass the number of moles of each

elements must be constant:

M n M n M n M n nr rj

r

R

i ij

i

k

n jn

n

N

n jn

n

N

j

= = = =

+ + + =1 1

1 1

1 1

1

2 2

2 1

2

, j=1,2…,m

( 7)

where: ni,j are stoichiometric coefficients in substances.

• For all the transformations in the system the law of charge conservation must be valid:

M ni ei

i

k

==

01

( 8)

where nei - is a sing and value of charge.

In this code it is proposed that condensed phases consist of electrically neutral components

only.

• The state of gas phase is determined by the equation of ideal gas mixture state:

pv RT M i

i

k

− ==

01

( 9)

This equation is the equivalent of proposing a small value of condensed phases volume fraction.

• Condition of moles conservation for the components in the solution S1 and S2:

M MS n

n

N

1 1

1 1

1

0− ==

; M MS n

n

N

2 2

2 1

2

0− ==

( 10)

Quick Start FWB 1.0

8

Thus, the problem of determination of state parameters of investigated system is reduced

to searching of entropy maximum ( 5) with additional restrictions on system parameters ( 6) - (

10).

For this purpose, the Lagrange method is used. In accordance with the Lagrange procedure

the subsidiary function L (Lagrangian function) is composed:

( ) ( )L f x x x x x xn s s n

s

= +1 2 1 2, , ... , , , ... , ,

where: f(x1,x2,..,xn) - function under extremum searching; s - restriction for variables x1,x2,..,xn,

s - Lagrange multipliers, s - the number of restricting equations.

Consequently, after substitutions the following expression for Lagrange function can be

written:

L S RRT

vM Mi i i

i

k

= −

=

0

1

ln + S Mr r

r

R0

1=

+

S M RMM

MS M RM

M

Mn n n

n

Sn

N

n

N

n n nn

Sn

N

n

N

10

1 11

11 1

1

1 1

1

20

2 22

22 1

2

2 1

2

− + −== ==

ln ln -

− + + + +

= = = =

U M U M U M U M Ui i

i

k

r r

r

R

n n

n

N

n n

n

N

u

1 1

1 1

1 1

1

2 2

2 1

2

+

pv RT M i

i

k

pv−

+

=

1

M ni ei

i

k

e

=

1

+

j

m

=

1


r

R

i ij

i

k

n jn

n

N

n jn

n

N

j j

= = = =

+ + + −

1 1

1 1

1 1

1

2 2

2 1

2

+

M MS n

n

N

S1 1

1 1

1

1−

+

=

M MS n

n

N

S2 2

2 1

2

2−

=

( 11)

where: j(j=1,2,..,m), e, u, pv, S1,S2 - Lagrange multipliers.

To found an extremum of enthalpy S, it is necessary to take the derivatives of L for the all

independent variables. These derivatives must be equal to zero in point of function extremum. This

set of equations permits us to connect all unknown variables by means of a simple algebraic

relations. In our case the Lagrange function has the following variables: Mi (I=1,2,…k), Mr

(r=1,2,…R), MS1, Mn1 (n1=1,2,…N1), MS2, Mn2 (n2=1,2,…n2), j(j=1,2,..,m), e, u, pv, S1,S2,

P, T, v, U. The total set of algebraic equation after procedure of derivatives taking can be written

in the following form:

SI

TR M R

RT

vn ni

ii ij j ej e

j

m0

1

0− − − + + ==

ln ln ; i=1,2…,k ;

SU

Tnr

rjr j

j

m0

1

0− + ==

; r=1,2,…,R ;


9

SU

TR

M

Mnn

n n

S

jn j

j

m

10 1 1

1

1

1

0− − + ==

ln ; n1=1,2,..,N1 ;

SU

TR

M

Mnn

n n

S

jn j

j

m

20 2 2

2

2

1

0− − + ==

ln ; n2=1,2,…,N2 ;


r

R

i ij

i

k

n jn

n

N

n jn

n

N

j

= = = =

+ + + =1 1

1 1

1 1

1

2 2

2 1

2

; j=1,2…,m ;

pv RT M i

i

k

− ==

01

;

M ni ei

i

k

==

01

;

M MS n

n

N

1 1

1 1

1

0− ==

;

M MS n

n

N

2 2

2 1

2

0− ==

;

S=Sg+Ss+SS1,S2 ;

− + + + + == = = =

I I M I M I M I Mi i

i

k

r r

r

R

n n

n

N

n n

n

N

1 1

1 1

1 1

1

2

2 1

2

0 ;

− + + + + == = = =

U U M U M U M U Mi i

i

k

r r

r

R

n n

n

N

n n

n

N

1 1

1 1

1 1

1

2

2 1

2

0 ;

( 12)

In total, this set consists of k+R+N1+N2+m+7 equations and k+R+N1+N2+m+9 variables:

k- the number of moles of gas phase components (Mi); R- the number of moles of condensed

components in pure phase (Mr); N1 - the number of moles in the first solution (Mn1) and N2 - in

the second solution (Mn2); m - the number of unknown Lagrange multipliers j and e, P, T, MS1,

MS2, I, U, S too. To define the parameters of equilibrium state it is necessary to introduce the two

characteristics of the system and the initial chemical elements composition. The calculation can

be provided using a set of equations ( 12) for the given thermodynamic functions of enthalpy Ii0

and entropy Si0 of substances. To obtain the closed set of equations for any type of thermodynamic

problem the two equations for the two additional thermodynamic parameters must be introduced:

T1=V1 ; T2=V2 ( 13)

where: T1 and T2 the first and second thermodynamic parameters, V1 and V2 initial approximation

of these parameters.

The set of equations ( 12),( 13) is a transcendental set. That is why this set is resolved by

means of numerical method. Thermodynamic Equilibrium Code uses the Newton-Raphson

Method as is one of the oldest and most widely used numerical techniques for solving a set of non-

linear equations. In this method to obtain the linear set of equations the linear approximation in

the neighbourhood of initial estimation of problem decision is used:

Quick Start FWB 1.0

10

( ) ( ) ( ) ( )f x x x f x x x x xx

f x x xn n i i

i

n x1 20

10

20 0

1 2 0, , .. , , , .. , , , .. , + −

As a result, the linear set of equations can be resolved by this well-known method. But

obtained solutions can not be regarded as final because of arbitrary initial approximations x1,

x2,..,xn. That is why the calculation is reiterated with new approximation up to the moment when

the difference between two consequent approximations is less than the error of calculation. One

can see that the problem of convergence depends strongly on the choice of initial approximation.

That is why the problem of convergence is resolved in this code by means of introducing new

variables and damping restrictions. The following new variables are used:

x Mi i= ln ; i=1,2,..,k; ( 14)

x Mr r= ln ; r=1,2,..R;

w M Mn n S1 1 1= ln ; n1=1,2,.., N1;

w M Mn n S2 2 2= ln ; n2=1,2,.., N2;

y RT v= ln ;

Z1= ln MS1;

Z2= ln MS2;

These variables allow us to eliminate the possibility of operating with negative values of

system parameters and straighten out the area of the parameters numerical values. After the

transformation of the non-linear set of equations ( 12) to the linear set with new variables ( 14),

the thermodynamic parameters can be found by means of a standard matrix method. The iteration

procedure is repeated for each step several times and a relative deviation from the initial state is

determined. The calculation is terminated when maximal relative errors becomes less than 10-6.

The choice of initial approximation of system state is based on calculating the results of

the two preceding iterations:

fitern = f0 = fn-1 + f sign(fn - fn-1) for fn - fn-1>f; ( 15)

fitern = f0 = fn when fn - fn-1 f,

where: fitern = f0 - initial approximation of f ; fn , fn-1 the unknown values for the n and n-1 step of

iteration procedure respectively; f - maximum value of function deviation.

The calculation procedure described above proved to be good for the simulation of a wide

spectrum of thermodynamic systems for different phase and substance compositions. This

procedure provides a reliable means of solving problems.

The polynomial interpolating from seven terms is used in TER for convenience of

introducing on the thermodynamic properties of substances. In the range from 298.15 K up to

20000 K this interpolating provides a high accuracy of calculation. In a wide temperature range


11

several interpolating polynomial are used. Due to dependence between thermodynamic functions

common interpolating coefficients are applied:

G T x x x x x x( ) ln= + + + + + +− − 1 2 32

41

5 62

73

; (15)

S T x x x x x( ) (ln )= + + − + + +− 1 2 32

5 62

731 2 3 4 ;

C T x x x xp ( ) = + + + +− 2 32

5 62

732 2 6 12 ;

where: x=10-4 T, G(T) is Gibbs function of free energy, Cp(T) - heat capacity at constant pressure.

However, the equations for the G(T), S(T), Cp(T) in neighbourhood of phase

transformation points have break of derivations. That is why the algorithm described above, must

be modified in cases when temperature T is a problem variable. In this case the approximation of

thermodynamic function break by linear function of temperature is used:

( )S T S T TS T T

TT T T( ) ( * )

( * )*= − +

−− +

2; ( 17)

( )( )C T C T T C T T Tp p( ) ( * ) * /= + − − 2 ;

T T T T T* *+ − ,

where 2T is approximation range.

Generally, the value of approximation range is small and has an order of error of phase

transition temperature definition.

Frozen composition approach

This approach is rather simple in comparison of LTE model. In frame work of these approach it is

supposed that chemical composition is predetermined by user and does not change with variation

of thermodynamic parameters P,T,V,H,U. However, thermodynamic functions and

thermodynamic properties are functions of thermodynamic parameters.

Quick Start FWB 1.0

12

2. Transport properties

1.FWB can calculate the transport coefficients:

• viscosity μ() is calculated by the accurate formulas of Chapman-Enskog theory;

• electrical conductivity (ξ) is calculated by the accurate formulas of Chapman-Enskog theory;

• thermal conductivity ( ) ( ) inttr += ;

here tr is the translational thermal conductivity, calculated by the accurate formulas of

Chapman-Enskog theory, int is the input to thermal conductivity due to the transfer of internal energy modes (rotational,

vibrational), calculated by the semi-empirical Eucken-Hirschfelder formula or by Mason-

Monchick theory.

• for the LTE case, the effective (or total) thermal conductivity is calculated also:

eff (ξ) = (ξ) + r , here r is the reactive thermal conductivity.

2.How we calculate it

The transport coefficients are calculated under the assumptions of the rarefied gas or plasma, only

binary collisions of particles are considered. Transport coefficients μ, ( )tr , are calculated on

the basis of the Chapman-Enskog (CE) method [1,2,3]. The gas flow is assumed continuous media

in the thermal (non-LTE) or both in the thermal and chemical equilibrium (LTE). External

magnetic field is assumed small enough. Also, we do not account for direct influence of chemical

reactions and ionization on transport properties. The gas and plasma are considered as ideal ones,

the ideal gas equation of state is used.

The exact formulas of the CE method are used to calculated μ(), ( )tr , (ξ) with account for

higher approximations ξ.

2.1. The viscosity μ() is calculated in the second approximation ξ=2 by the formulas [1,2]:

00mx

0

mx

det

1

2

5)(

ss

1,1rs

0,1rs

rr1,0

rs0,0

rs

−=

−−−

−

qq

qq

q


13

1,1rs

0,1rs

1,0rs

0,0rs

det

−−−

−

=

qq

qq

q

, r,s=1,...,N,

here, for each pair of indexes m,p (m,p=0,...,-1), mp

rsq

is square matrix of the order NN with the

elements mp

ijq

(i,j=1,...,N):

( )

( )( )

.bmm

mm

xm

mx

kT

mq

tlpm

s

)s,l(ik

p,mlts

m

t/pm

ki

/])([tkj

N

k

m

l

jkl

ijk

p

j

ii

jmpij

l

+

−+

=

−++

==++

−−+

=

+

=

+

2

1023

2112

1

1

1

1

1

182

Numerical values of the coefficients p,m

ltsb can be found in [2a].

The main notations used here and further are as follows:

xi, mi - molar fraction and molar mass of i-th substance; N - the total number of substances; - the

number of approximations in the Chapman-Enskog theory, i.e. the number of retained terms in

Sonine polynomial expansions of the distribution functions; )s,l(

ij - collision integral (CI) for the

pair of particles with the indexes (i,j), (l,s) are the indexes for the order of the CI;

2.2. The electrical conductivity (ξ) is calculated in the second or third approximation (ξ=3) by

the formula [2]:

=

=N

)ei(i

eiiii )(DZmnmkT

e)(

1

2

here Zi - charge number, ni - number density of i-th ion.

Dei(ξ) are the multicomponent diffusion coefficients for the pairs (electron, i-th ion).

Dij(ξ) are calculated by the general formula [2]:

Quick Start FWB 1.0

14

000

0

0

0

2

23

111101

111101

101000

−

=

−−−−

−

−

is

,rs

,rs

,rs

,rs

,rs

,rs

rirj,

rs,

rs,

rs

ij

iij

~~~

~~~

~~~

~detmm

kTmn)(D

qqq

qqq

qqq

q

(i, j) = 1,...,N; (r,s) =1,...,N.

here mprs

~q (m,p=0,...,-1) is the square matrix of the order NN with the elements mpijq~ (i,j=1,...,N)

that are linear combinations of the collision integrals, the formula for mpijq~ could be found in [2].

2.3. The translational thermal conductivity ( )tr is calculated in the second or third non-zero

approximation (ξ=4) by the formulas [2-5]:

1111

1111

0

0

000

8

75

−−−

−

−=

,rs

,rs

,rs

,rsr

s

tr

x

x

det

k)(

qq

qq

q

here mprsq (m,p=1,...,-1) is the square matrix of the order NN with the elements mp

ijq (i,j=1,...,N)

that are linear combinations of the collision integrals, the formula for mpijq could be found in [5].

Note that here ( )tr is the so-called "true" thermal conductivity, not the "instantaneous" one - see

[5] for the details.

2.4. Calculation of int for gas mixture consists of two steps: first, values of int

i - the input to int

for the i-th molecule, are calculated for each substance; second, int is calculated by the summation

of inti with Eucken-Hirschfelder formula. Values of int

i are calculated either by the formulas of

Mason-Monchick theory [6-9] or by the simple Eucken formula.

2.4.1. The inti is calculated by the Mason-Monchick theory [6-9] provided the necessary input

parameters (rotZ , T0) are available in Kintech Database for the i-th molecule of the mixture and

the option "Use Mason-Monchick theory" is True. The formulas to calculate inti are as follows:


15

DcZ

c/c rotrot

rot

vibrotrotinti

−

−+=

2

1111

2

512

+=

T

T..

)Z(

Z

rot

rotrot 0

43

55241221 when *crossT

T

T

0

,

−−+=

3211

904402701

rotrotrot

rot

Z

.

Z

.

Z

. when *

crossTT

T

0

1

23

2322

12

311

24

1

21

−

+

+

+

+=

TTTZZ rotrot , 0T/TT =

),(

),(*A

11

22

115

6

5

6

== ,

+

+=

tr

rotrot

rot c

c

Z 2

521

In these set of formulas the low index "i" is omitted everywhere except inti . The notations are as

follows: D=Dii (1) - binary diffusion coefficient; crot = rotic , cvib = vib

ic - the inputs of rotational and

vibrational modes to the specific heat capacity at constant volume; T0 - the characteristic

temperature; rotrot Z,Z - rotational collision number at temperature T, and high-temperature

asymptotic value for rotational collision number for i-th molecule.

i,rotZ and T0i are the necessary

input data, they are available from literature for some well-investigated molecules.

2.4.2. In case when the data

i,rotZ , T0i are not presented in the Kintech Database, inti for this

molecule is calculated by the modified Eucken formula [15]:

(1)int

,

int

iiiiVi nDmc=

here int

i,Vc - specific heat capacity at constant volume for i-th molecule.

The binary diffusion coefficients Dij(1) are calculated by the formula:

),(ijij

ijm

kT

nD

11

12

16

3)1(

= ,

ji

jiij

mm

mmm

+= , i,j =1,...N

2.4.3. The total input to thermal conductivity due to the internal energy modes for all the molecules

in the mixture is calculated by Eucken-Hirschfelder formula: 1

1 1 (1)

(1)1

−

==

+=mN

i

N

ik,k iki

iikinti

int

Dx

Dx

here Nm is the number of molecules in the mixture.

Quick Start FWB 1.0

16

2.5. For the LTE case, if the option "Calculate the reactive thermal conductivity" is True, r is

calculated by the formula [4,5] :

0

1)(

2 T

T

qm

qmb

b kk

iiik

ik

rdetTk

−=

It is the advanced version of Butler-Brokaw formula [10] with account for the influence of thermal

diffusion and higher approximations of Chapman-Enskog method.

Then the effective thermal conductivity is calculated as the sum: ( ) ( ) r

inttr

eff ++=

The square matrix bik is of the order R*R, its elements bik are expressed through the diffusion

resistance coefficients ik(ξ) and stoichiometric coefficients ij; here i,k = L+1,...,N; L and R are

the number of chemical elements forming the mixture and the number of independent reactions in

the mixture.

( ) == ==

++−

+−

=L

j

jikjjkij

L

j

jl

L

l

klij

L

j

jj

kjijiki

i

ikik

xxb

11 11

; i,k = L+1,...,N;

=

=N

k

ikki x1

Value Tiiqm is the molar heat of i-th reaction, modified in order to account for the influence of

thermodiffusion:

−−=

=

Tj

L

j

ijTiiiTii kTqmqm

1

, ( ) ( ) iTiTiTi xk == ; i = L+1,...,N;

here kTi is the thermodiffusion ratio for i-th substance.

The diffusion resistance coefficients ik(ξ) and the thermodiffusion ratios kTi(ξ) are calculated by

the formulas [3-5] in the 2nd or 3rd approximation.

2.6. For the non-LTE case, the option "Calculate the reactive thermal conductivity" should be

False, and the total thermal conductivity is calculated as the sum: ( ) ( ) inttr +=

2.7. In future, on demand, the additional gas/plasma properties calculated with FWB can be

presented, such as binary diffusion coefficients Dij(1); thermodiffusion coefficients DTi (and/or

thermodiffusion ratios kTi); multicomponent diffusion coefficients Dij.

3. Collision integrals

To apply the exact formulas of the CE theory, the set of collision integrals )s,l(

ij is necessary for


17

all pairs of particles. We use the Hirschfelder form [1] of the collision integrals (CI):

−−++

+=

+

−

d)g(Qe])(l[)!s(

)l( )l(ij

s

l)s,l(

ij32

0

2

1121

14;

kT

mg

ij

2= ;

here i,j - the indexes of colliding particles; upper indexes denotes the order of CI; g is the initial

relative velocity; Q)l(

ij - scattering cross sections that are calculated by the interaction potential

Vij(r):

dbbcosgQ ijl)l(

ij

−=

0

)1(2)( ;

−−

−=

mr

ij

ij

ij

r

b

gm

)r(V

r/drb)g,b(

2

2

2

2

21

2

For calculation of the transport coefficients in the 4-th approximation (ξ=4), it is necessary to use

16 collision integrals (for each pair of substances i,j) with the orders l,s: 1 l 4; s = l, l+1,...,8-

l. This full set of CI is calculated numerically or determined by the approximation formulas.

In case of simple model potentials (e.g. Lennard-Jones (12-6) and Born-Mayer) the CI are

calculated by the well-known rather accurate approximation formulas.

For complicated potentials such as HFD-B, there are no approximation formulas, and the CI are

calculated numerically based on O'Hara and Smith method [11,12]. This numerical calculation is

built in the FWB code.

In some cases (e.g. for the collision of electrons with atoms or molecules) the CI tables for the

pairs of substances are provided in the Kintech Database and are used as the input data instead of

the parameters of model interaction potentials.

4. Interaction potentials

Interaction potentials for all pairs of mixture species are used for calculating the transport

coefficients (except the cases when the CI tables are included in the Database). For each model

interaction potential the appropriate parameters are downloaded as the input data from the Kintech

Database.

4.1. The following model potentials can be used in calculations for elastic interaction of neutral

particles:

Lennard–Jones 12–6 potential

−

=

612

4rr

)r(V

Quick Start FWB 1.0

18

σ – the character length, the effective collision diameter, V(σ)=0;

ε – the depth of potential well, V(r = rmin) = - ε.

Lennard–Jones m-6 potential

−−

−=

6

66

6

r

d

m

m

r

d

m)r(V

m

d - the character length, equilibrium distance; the distance to V(r) minimum, i.e. dV/dr=0 at r=d.

Buckingham–Corner exp–6–f potential

( )

m

mm

mm

rr,)r(f

rr,r

rexp)r(f

,)r(fr

rr/rexp

/)r(V

=

−−=

−−

−

=

1

14

16

61

3

6

rm - the distance to V(r) minimum;

- dimensionless parameter.

Born–Mayer (exponential repulsive) potential

( )rexpA)r(V −=

= 1/ is the character length.

Stockmayer potential for polar gases

36121

4

−

−

=

r)(F

rr)r(V ijij

ijijijij

3ijij

jiij

4

=

)cos(sinsincoscos2)(F jijijiij −−=

here i, j - dipole moments of the molecules (in Debye).

HFD-B (high accuracy potential for noble gases [14])

( )

Dx,)x(F

Dx,x

Dexp)x(F

r/rx,x

C)x(FxxexpA)r(V m

nnn

=

−−=

=

−−−=

=

1

1

2

5

3222


19

rm is the character length.

Pirani (Lennard-Jones like phenomenological potential [13])

e

m)x(n

r/rx,xm)x(n

)x(n

xm)x(n

m)r(V =

−−

−=

11, n(x) = + 4x2 ;

( ) 0950

3131

7671.

ji

/j

/i

e .r

+= ,

jiij

ss ++==

56 , k=i,j;

31 /kkk Ms = - for atoms only; 31 /

kks = - for molecules;

6720

e

d

r

C.= ;

ejjeii

jid

N/N/.C

+

= 715

here k, Nek - polarizability and effective number of electrons which contribute to polarization of

the k-th neutral particle (atom or molecule), Mk - ground state spin multiplicity of the k-th atom, re

- equilibrium distance, m=6 for the interaction of neutral particles.

4.2. For the elastic interaction of neutral particles with ions, the following potentials can be used:

Pirani potential [13]

e

m)x(n

r/rx,xm)x(n

)x(n

xm)x(n

m)r(V =

−−

−=

11, n(x) = + 4x2 ;

( ) 0950

3131

117671

.ni

/n

/i

e/

.r+

+= ,

niin

ss ++==

56 ;

31 /kkk Ms = - for atoms and atomic ions; 31 /

kks = - for molecules and molecular ions; k=i,

n;

+

= 1254

2

e

n

r

z. ;

( ) n/

ni

i

/z +

=

322 21; i - ion, n - neutral particle.

m=4 for the ion-neutral type of interaction.

Polarization potential [15] (inverse power attractive potential) is used for the interaction of the ion

(charge Z) and the neutral particle with polarizability :

Quick Start FWB 1.0

20

4

2

2r

Ze)r(V

−= (SGS system)

4.3. For the interaction of charged particles, the screened Coulomb potential is used [15]

)(

2

Dji

r/rexpr

ZZe)r(V −=

This potential depends on the Debye length rD that can be written in the two different ways -

a) with, or b) with no account for ions in screening effect:

( )224 ii ie

DZnen

kTr

+= case a), FWB option "Use Debye length correction=False"

24 en

kTr

e

D

= case b), FWB option "Use Debye length correction=True"

Here ne, ni are the number densities of electrons and i-th ions; Zi is the charge number of i-th ion.

The choice of a) or b) has non-negligible effect at very high temperature, when multi-charged ions

become the predominant species. Values of plasma electrical conductivity and thermal

conductivity are higher for the case b).

For example, the choice (a) is recommended in [2b,15,16], the choice (b) - in [2c,17,18].

Which case is more accurate? No answer at present, due to the reasons:

there are no accurate experimental data for plasma parameters for very high temperatures;

the screened Coulomb potential is semi-empirical one; it could not provide exact results.

5. Calculation of collision integrals for the interaction of atom with parent ion

In calculation of CI of the odd order l for the interaction of atom with parent ion, it is necessary to

account for the resonant charge-exchange process (non-elastic effect); it is impossible to use any

model potential in such calculation. In this case FWB and Kintech Database provide the two

possibilities to calculate )s,l(exch− of the odd order l:

5.1. The use of Devoto formula for charge-exchange transport cross-section [2b] with the two input

parameters A, B: ( )( ) ( ) ( ) 221 glnBA/gQ l −=

here ( )( )gQ l - scattering cross section (or total transport cross section) with the odd l; g is the

initial relative velocity. The use of this formula makes possible to calculate )s,l(exch− for odd values

of l and arbitrary s by the analytic formulas [15].

Calculation of )s,l( for even values of l can be made with use of ordinary interaction potentials

(e.g. Pirani or polarization potential) with no account for charge-exchange effect.

5.2. The use of the ready table of CI as functions of T for all (l,s), including odd l, solves the


21

problem, if this table is included in the Kintech Database for the necessary pair of particles. For

some pairs of this type (atom, parent ion) the tables of CI are found in literature and included in

the Database. Then the CI for necessary values of T are determined by interpolation across the

table.

References

1. Hirschfelder J.O., Curtiss C.F., Bird R.B. Molecular Theory of Gases and Liquids. Wiley,

New York, 1964

2. Devoto R.S. a) Transport properties of ionized monatomic gases. Phys. Fluids. 1966, vol. 9,

N.6, pp.1230-1240. Also: b) Transport coefficients of partially ionized argon, ibid. 1967,

vol.10, No.2, pp. 354-364. c) Transport coefficients of ionized argon, ibid. 1973, Vol.16,

No.5, pp. 616-623.

3. Kolesnikov A.F., Tirskii G.A. Equations of hydrodynamics for partially ionized multi-

component mixtures of gases, employing higher approximations of transport coefficients.

Fluid Mechanics - Soviet Research, vol. 13, No. 4, 1984, pp. 70-97. Scripta Technica Publ.

4. Vasil'evskii S.A., Sokolova I.A., Tirskii G.A. Exact equations and transport coefficients for

a multicomponent gas mixture with a partially ionized plasma. Journal of Applied

Mechanics and Technical Physics. 1984, Vol.25, No.4, pp.510-519. Also: Definition and

computation of effective transport coefficients for chemical-equilibrium flows of partially

dissociated and ionized gas mixtures. Ibid., 1986, Vol.27, No.1, pp.61-71.

5. Tirskii G.A. The hydrodynamic equations for chemically equilibrium flows of a

multielement plasma with exact transport coefficients. J. Appl. Maths. Mechs. 1999, vol.63,

issue 6, pp.841-861.

6. Mason E.A., Monchick L. Heat conductivity of polyatomic and polar gases. Chem. Phys.

1962, V.36, No.6, pp. 1622-1639.

7. Monchick L., Pereira A.N.G., Mason E.A. Heat conductivity of polyatomic and polar gases

and gas mixtures. J. Chem. Phys. 1965. V.42. No.9. 3241-3256

8. Brau C.A., Jonkman R.H. Classical theory of rotational relaxation in diatomic gases. J.

Chem. Phys. 1970, V. 52, Issue 2, pp. 477-484.

9. Uribe F.J., Mason E.A., Kestin J. A correlation scheme for the thermal conductivity of

polyatomic gases at low density. Physica A. 1989, Vol. 156, Issue 1, pp. 467-491.

10. Butler J. N., Brokaw R. S. Thermal conductivity of gas mixtures in chemical equilibrium.

Journal of Chemical Physics, vol. 26, No. 6, 1957, pp. 1636-1643. See also: ibid., vol. 32,

No. 4, 1960, pp. 1005-1006

11. O’Hara H., Smith F.J. The efficient calculation of the transport properties of a dilute gas to

a prescribe accuracy. J. Comput. Phys. 1970, 5, 328-344.

12. O’Hara H., Smith F.J. Transport collision integrals for a dilute gas. Comput. Phys.

Commun., 1971, 2, 47-54.

13. Laricchiuta A., Colonna G., Bruno D., Celiberto R., Gorse C., Pirani F., Capitelli M.

Classical transport collision integrals for a Lennard-Jones like phenomenological model

potential. Chemical Physics Letters. 2007, Vol.445, pp.133–139.

14. Aziz R.A. Accurate thermal conductivity coefficients for Argon based on a State-of-the-Art

interatomic potential. International Journal of Thermophysics. 1987, Vol.8, No.2, pp.193-

203

15. Capitelli M., Bruno D., Laricchiuta A. Fundamental Aspects of Plasma Chemical Physics.

Transport. Springer, N.Y. 2013.

16. Andre P, Bussiere W, Rochette D. Transport coefficients of Ag-SiO2 plasmas. Plasma

Chem. Plasma Process. 2007. Vol. 27, No.4, pp. 381–403.

Quick Start FWB 1.0

22

17. Murphy A.B. Transport coefficients of hydrogen and argon–hydrogen plasmas. Plasma

Chem. Plasma Process. 2000. Vol.20, No.3, pp.279–297.

18. Write M.J. Recommended Collision Integrals for Transport Property Computations, Part 1:

Air Species. AIAA J. 2005. Vol. 43, No. 12, pp. 2558-2564.


23

3. Absorption coefficient

The program allows calculating the absorption coefficients (AC) of radiation in case of a local

thermodynamic equilibrium. Program requires the following data:

1. The composition of thermodynamically equilibrium mixture of substances depending on

the temperature.

2. The database of spectral characteristics of atomic, molecular levels and lines.

3. The additional information on atoms and molecules.

Atoms.

Spectral lines and Broadening mechanisms

Probability of a bound-bound radiative transition nl → n’l’’ is given by

( )2'''0

3

2

0

2

)'',()12(3

2)''',(

ln

nlRllClRymc

elnnlW +=

, (1)

where '''0 lnnl EE −= and ''' ln

nlR is dimensionless. Taking into account the spectral line

broadening with a normalized profile ( − 0) ( 1= d ) we can present corresponding

contribution into the emission coefficient

)()''',()( 00

'''

−= lnnlWNnl

ln

nl . (2)

Natural broadening is due to the finite lifetime of atom (ion) on the radiating level. The

line profile is dispersive

22

0

0)2/()(

2/)(

w

w

+−=−

(3)

with the width

='''

)''',(

ln

nl lnnlWw , (4)

where the sum over all levels for which dipole transition is possible.

Doppler broadening is due to the motion of particles. If the Maxwellian distribution

Quick Start FWB 1.0

24

function is valid, the Doppler line profile is

−−

=−

2

00 exp

1)(

DD

,

M

T

сD

20 = , (5)

where M is the atom (ion) mass.

Resonance broadening occurs in collisions where radiator and perturber species are

identical and if a dipole transition exists between upper or lower level of the considered line and

the state of the perturber. It leads to a dispersive profile (3). The resonance broadening width is

determined approximately by the following type of expression

2

( ) ( )

2

jires resij ij j

ij

e fw C N

m= , (6)

where fji is the oscillator strength, )(res

ijС is a constant, depending on the angular moments of the

ground and excited states (as well as on multiplicity) and should include contributions as due to

potential scattering along with dipole-dipole interaction potential as due to the transfer of

excitation from the target to the projectile. If to assume statistical independence of collisions of

the radiator in the particular excited states i with atoms of the same species in the different states

the expression for the resonance broadening widths for the transition → , in the case of

Boltzmann distribution of populations could be represented in the more general form

2

//( ) ( )( )

2

E TE Tres resresffe

w N C e C em

−−

= +

, (7)

where N is total density of neutral atoms. So, in this situation of lack of complete accurate data it

is obvious that to perform large scale calculations one needs to select some one reasonable

expression for the constant )(res

ijС for all transitions. Traving recommends for )(res

ijС the following

formula

ij

res

ij ggС /48.5)( = , (8)

where 12 += ii Jg and 12 += jj Jg are the statistical weights of corresponding levels.

The contribution to the line width due to Van der Waals part of interaction potential could

be evaluated from the adiabatic theory as the velocities of heavy neutral particles usually are small.

In the impact limit the result for the width is [4]

( ) ( ) ( )2/52/5 3/10( , )

6, 6,2 3 / 8 16 /w impij g i jw N T M C C = − , (9)


25

where Ng is the density of atoms in the ground state. Here it is taken into account that the interaction

is considered between atoms of the same species. For the coefficients C6 the approximation

2

2 2 2 26 0 , 0

1

1, / /

2

Z

p e e s es

C r r a i r i a=

=

(10)

can be used. Here Z designates the number of atomic electrons for the neutral with the nuclear

charge Z, p is the polarizability of the atom in ground state. Matrix element <r2> could be

approximately estimated from the hydrogen-like approximation

2

2 2 20/ 5 1 3 ( 1)

2

effeff

nnl r nl a n l l = + − +

, (11)

where )/( nlioneff EERyn −= is the effective principal quantum number.

On the other hand the Van der Waals quasistatic width conventionally determined as the

frequency shift, corresponding to the mean distance between particles, is proportional to the square

of atomic density 2

( , ) 26, 6,

4

3

w quasij i j gw C C N

= −

. (12)

Electron impact Stark width of non-hydrogenic neutrals could be represented by the

following expression

( , ) 3 2 20

2

2 2 1.33

2 5/30 0

2

8/ ( ) ( ) ( ) ,

27

1 2.27 0.487( ) ( ) ( ) , ( ) ln 1 ,

0.153

/ 3 , (

ii

ii

impact e nij a e w ii ii jj w jj jj

i j

xi i w

ii i i

V

Ryw N a R f R R f R

T

xR r dr R r r R r f x e

xa x

E E T r d

−

−

= +

= = + +

+

= −

2) ( ) 1ir R r =

(13)

where /2Rya = , Ri(r) is the radial function of the discrete state i, normalized to unity.

The electron impact Stark width in this case of non-hydrogenic singly charged ion could

be represented in the form

( , 3 2 20

3 34/ ( ) ,

2 23 3 ii

impact e i e eij a e eff jj eff

iii je jj

T TRyw N a R g R g

T

−

= +

(14)

where effg is the effective Gaunt factor that represents itself the averaged over Maxwell

distribution.

Quick Start FWB 1.0

26

The electron impact width for doubly and triply charged ions could be approximated along

with by the following formula

,,

33

( , ) 3 2 20 2 2

, 0 , 0

2 2

, , 0

1

334/ ( )

3 3 4 4

3 3

2 2

j ri r

ii

ii ii

i i

eeimpact e iij a e eff jj eff

i n j ne r r

e eeff eff

i i nii ii

l l

T nT nRyw N a R g R g

T Ry z Ry z

T TR g R g

−

=

= −

= + +

+ +

( ),

2

0 , 011

1/2

2

, 0 ( ),1

3

2

3 1.1 2, ( ) 0.7 ( ),

2

j ji i

i j r

j j

ejj eff

n j n jjl ll l

ejj eff eff eff

j n jj r r i j rl l

TR g

T RyR g g x g x n

z I E

= == + = +

== +

+

+ − + = −

(15)

The total Stark profile could be approximated by the Lorentz profile with the width, taking

into account ion broadening

( , )/ 1 1.75 (1 )impact e iij a ijw w A R −= + − . (16)

For neutral radiator it was prescribed to substitute take = 0.75 and for charged ones =1.75.

Parameter A is determined via polarizabilities i,j of the upper and lower levels

3/4

20

( , )

i j

impact e iij

FA

w

−

− =

, (17)

where 2/3

2/30 ,

4

2 15s i s

s

eF z N

=

(18)

is the normal Holtsmark field and

2 1/322 4

, , 1/ ,3 3

ii

ii

i ii i i D ei

ReE E E R r N

E

= = −

(19)

where rD is the Debye length.

Continuum

Calculating the absorption coefficient (5), we sum the photoionization contribution of

levels nl for which the ionization energy less than . However, for highly excited frequently

spaced levels the sum can be changed by an integral. Let us denote by E0 the boundary value of


27

the energy above which the sum is changed by the integral. The integral part of kbf() can be

combined with kff():

)(),()()()( int

ffbf

0

kpNkkkEE

nl

nl

+=+= −

, (20)

)()()( ffbf

intint kkk += . (21)

Here and below, energy is reckoned from the ideal continuum threshold (all Enl < 0). The total

integral term is traditionally expressed in terms of the corresponding absorption coefficient of

hydrogen (or hydrogen-like ions) in the approximation of the Kramers-Unsold )(H

int k and

Biberman function int() which takes into account nonhydrogenic spectrum of many-electron

atoms [13]:

)()()( int

H

intint kk = , (22)

=

kTNGN

kTm

Zek ie

0

33

245H

int exp33

216)(

. (23)

Here 0 = min{, |E0|/} and G is the Gaunt factor averaged over the electron energy spectrum

corresponding to the integral absorption coefficient –0 < E < .

Molecules

Evaluation of absorption coefficients of diatomic molecules.

The general expression for Einstein coefficients of the quantum transition between

states → in the diatomic molecular spectra [1-7] does not differ from the similar one for the

atomic transitions (compare with [4, 8])

3

22

34

3

eA D

c

=

. (1)

Each of the letters , designate in fact the sets of the electron, vibrational and rotational

energy levels of diatomic molecules. Using the conventional designations [1-7] in the case of

electron-vibrational-rotational transitions besides the electronic upper “i” and lower “j’ energy

terms the initial (upper) {v, J} and final (lower) {v’, J’} vibrational and rotational sublevels should

be additionally specified, so that , v, , v ,i J j J → → . Therefore the above expression

Quick Start FWB 1.0

28

could be rewritten in the Born-Oppenheimer approximation [1-7], assuming separation of different

freedom of motion, in the form [1-7]

32

3 2

,v, ; ,v', ' ,v, ; ,v ', ' '

4| v(i) ( ) v'(j) | ,

3i J j J i J i J ij JJ

i

eA D R S

g c

=

(2)

2

'

, '

| | | ' ' ' |JJ

M M

S J n J= , (3)

where R is the internuclear distance between the atoms of diatomic molecule, ( )ijD R is

the reduced matrix element of the electronic dipole operator due to the transition between the

electron states labeled by the set of quantum numbers of the electron potential curves of the upper

i and lower j terms; the vibrational initial and final wave functions v(i) , v (j) , according

to vibrational quantum numbers{v(i), v’(j)} related to the initial and final electronic potential terms

( ), ( )i jU R U R , the total angular momenta ,J J and their projections on the OZ axis of the lab

reference frame ,M M , the total spin S,S and its projection on the internuclear axis {, ’},

the electronic angular momentum L,L and its projection of on the internuclear axis {, ’}, the

sum of spin and electronic angular momentum projections on the intenuclear axis , { =+,

’ =’+’} ( to be exact the case “a” of Hund’s rules is assumed); n

is the unit vector along the

direction of the total electronic dipole moment in the lab reference frame; (2S +1) is the spin

multiplicity of the electronic terms involved into the transition (the spin number is conserved);

<SJJ’> is the Hönl-London factor [1-7]; gi= (2-0,) (2S+1)(2J+1), gj= (2-0, ’)(2S’+1)(2J’+1),

the factor (2-0,) being due to -doubling [1-7]; is the Kronecker delta. One should keep in

mind that the factor (2-0,) (2S+1) refers to the statistical weight of electronic state as a whole,

and the factor of spin multiplicity enters the definition of the statistical sum over electronic states,

whilst the factor (2J+1) refers to the statistical weight of the rotational state. This separation was

firstly clearly formulated by Tatum [9]. The -doubling factor is the same for all sublevels

belonging to the same electron state (potential term). The summation in the formular for <SJJ’> is

extended over all possible combinations of M and M’. All quantities entering these equation are

expressed in atomic units. It should be reminded that in (3) we could designate summation only

over one magnetic quantum number due to selection rules of dipole one-photon transitions, only

which are considered here. Remind, that the procedure, descibed here, is in fact conventional

operations, escorting evaluations of the process pobability, of summing over all final states and

averaging over all the initial states, that is expressed via division on the statistial weight of the

initial state [8].


29

The functions ( )ijD R could be obtained either by rather complicated ab initio quantum-

chemical calculations or from the experimental measurements [10-11]. The database of ( )ijD R for

different diatomic molecules and corresponding electron transitions is rather spare up to now [7,

10]. There are several commercial packages that in principal allow to perform ab initio calculations

of ( )ijD R in various approximations [11], and to our knowledge only one free access database

RADEN [10] with some collection of data from published literature. This database also has

information about various electronic terms ( ), ( )i jU R U R , which are necessary for calculations of

vaious vibrational wave functions ( )

v ( )j R , related to the corresponding electronic term ( )jU R

and vibrational number v , that are involved in calculations of matrix elements with account of

vibrational transitions.

As information about ( )ijD R as was already metioned is quite spare, it is conventional

to use the Frank-Condon approximation when the square of matix element between vibrational

states v(i) and v'(j) is approximated by factorizing the square of the effective value of

electronic dipole moment ( )ij effD R [1-7]

22 2| v(i) ( ) v'(j) | ( ) | v(i) v'(j) | ,ij ij effD R D R (4)

22 (i) (j)

v vv(i) v'(j) ( ) ( )dR R R , (5)

where the lower expresssion is called Frank-Condon factor. The tables of Frank-Condon

factors for the particular molecules and their electronic and vibrational transitions are available in

literature [5].

Derivation of vibrational wave functions in uniform quasiclassical approximation

It is known from literature [1-7] that the accuracy of quasiclassical approximation is

quite sufficient for description of the vibrational motion in diatomic molecules. On the other hand

the vibrational-vibrational transitions according to the Frank –Condon principle occur mostly

nearby to turning points, where rigourously speaking the apllicability of quasiclassical

approximation is violated [12-14]. This difficulty was overcome by the elaboration of uniform

quasiclassical approximation which provides the corrected approximate wave functions that are

applicable in the turning points as well [12-14]. Here we would not touch sutleties and derivation

of this method [13] but just describe instructions of its implementation. The main issue here is that

the approximate solution of quasiclassical equations is constructed in terms of the Airy [12-14]

functions applicable in front of turning point, in the turning point and behind it in the classically

forbidden region of motion. This solution provides appropiate accuracy of various calculations,

that would be achievable as well with the use of exact solutions. The uniformity of asymptotic

Quick Start FWB 1.0

30

solution means its universality, invariability and applicability in any small region around the

turning point and in the limit of 0 → .

The Shreodinger equation for the vibrational motion for the diatomic molecules in the

electronic potential ( )iU R could be reduced to the form (in the Born-Oppenheimer appoximation

[12])

2 ( )

( )vv v2 2

( ) 2( ) ( ) 0,

/ ( ).

ii

i

a b a b

d RE U R R

d R

M M M M

+ − =

= +

(6)

For construction solutions in the uniform quasiclassical approximation first of all we

introduce the impulse function

( ) ( )

v v2

2( ) ( )i i

ik R E U R = − . (6)

Remind that in turning points the function ( )

v ( )ik R =0.

Firstly let us consider the case of finite motion between two turning points with respect

to the location of the minimum or point of equilibrium of the potential curve ( )iU R , designated as

( )i

eR . Now let us designate the left turning point as ( ) ( )

v

i i

eR R , and the right turning point as

( ) ( )

v

i i

eR R . It is known also that the value of vibrational energy could be approximately represented

with the accuracy sufficient for the calculation of transition probability in the form

( ) ( )

v (v 1 / 2)i i

eE + . (7)

Traditionally even in the case of just quasiclassical approximation the I-IV ranges of

deviation of R values are introduced. First the region I of classically forbidden motion is behind

the left turning point ( )

v

iI R R . Then the region II of motion is between the left turning point

and the point of equlibrium ( ) ( )

v

i i

eII R R R . The region III of motion is between the point of

equlibrium and the right turning point ( ) ( )

v

i i

eIII R R R . And the region IV of classically

forbidden motion is behind the right (larger) turning point ( )

v

iIV R R . The next step is the

introduction of 4 functions ( )

v,I IV ( )iS R− , which represents themselves the integrals over R of the

impulse function ( )

v ( )ik R defined for each of the 4 introduced regions of deviation of R, and

derived from them functions ( )

v, ( )i

I IVZ R− :


31

( )v

( )v

( )

3/2( ) ( ) ( ) ( ) ( )

v,I v v v,I v,I

3/2( ) ( ) ( ) ( ) ( ) ( )

v,II v v v,II v,II

( ) ( ) ( )

v,III v

( ) (s), , ( ) (3 / 2) ( ) ,

( ) ( ), , ( ) (3 / 2) ( ) ,

( ) ( ),

i

i

ie

R

i i i i i

R

R

i i i i i i

e

R

R

i i i

e

R

S R ds k R R Z R S R

S R ds k s R R R Z R S R

S R ds k s R R

= = −

= =

=

( )v

3/2( ) ( ) ( )

v v,III v,III

3/2( ) ( ) ( ) ( ) ( )

v,IV v v v,IV v,IV

, ( ) (3 / 2) ( ) ,

( ) ( ), , ( ) (3 / 2) ( ) .i

i i i

R

i i i i i

R

R Z R S R

S R ds k s R R Z R S R

=

= = −

(8)

The functions ( )

v,I IV ( )iS R− and ( )

v,I IV ( )iZ R− are dimensionless.

Introducing at last the functions ( )

( ) v

v

2( 1)

ii e

a

Cm

= − ( where

( )i

e is the zero order

vibrational frequency of the bound electronic term “i”) now it is possible to write down the

expressions for the vibrational wave functions in the uniform quasiclassical approximation via

Airy functions [13]. By the definition they have the same expression in the merging regions with

respect to particular turning point, ie no additional phase factors will appear during crossing the

tuning point value

1/4( ) ( ) ( ) ( ) ( ) ( )

v v v v v,I v,I

1/4( ) ( ) ( ) ( ) ( ) ( ) ( )

v v v v v,II v,II

( ) ( ) ( ) ( ) ( ) ( )

v v v v v,III

, ( ) / ( ) ( ) Airy ( ) ,

, ( ) / ( ) ( ) Airy ( ) ,

, ( ) / ( )

i i i i i i

i i i i i i i

e

i i i i i i

e

R R R C k R Z R Z R

R R R R C k R Z R Z R

R R R R C k R Z

= −

= −

= 1/4

( )

v,III

1/4( ) ( ) ( ) ( ) ( ) ( )

v v v v v,IV v,IV

( ) Airy ( ) ,

, ( ) / ( ) ( ) Airy ( ) .

i

i i i i i i

R Z R

R R R C k R Z R Z R

−

= −

(9)

To obtain the corresponding vibrational-vibrational factor or the electron-vibrational

probability of the transition the above procedure should be performed for upper and lower

vibrational levels that could belong to different electronic terms. The above functions are

normalized obviously on the unit length.

The case of photoionization, when the final state is rotationally and vibrationally bound

but in the molecular ion could be considered in the similar way. Just the upper electronic state in

fact belongs to the different diatomic molecule – its molecular ion.

In the case of the description of molecular dissociation the final vibrational state is

related to infinite motion along with the electronic term ( )jU R . In this case the wave function

( )

( )( )R

j

ER could be constructed in the uniform quasiclassical approximation as well, but there

are no the equilibrium and right turning points. Then the “vibrational” energy ( )RE is not quantized

and takes continuous values, decribing the relative translational motion of atoms. The resulting

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32

functions of the upper term could be represented in the form

( )v

( )v

3/2( ) ( ) ( ) ( ) ( )

v,I v v v,I v,I

3/2( ) ( ) ( ) ( ) ( ) ( )

v,II v v v,II v,II

( ) (s), , ( ) (3 / 2) ( ) ,

( ) (s), , ( ) (3 / 2) ( ) .

i

i

R

i i i i i

R

R

i i i i i i

e

R

S R ds k R R Z R S R

S R ds k R R R Z R S R

= = −

= =

(10)

For the case of photodissociation the vibrational wave functions of finite motion on the

lower term are defined as above

1/4( ) ( ) ( ) ( ) ( ) ( )

v v v v v,I v,I

1/4( ) ( ) ( ) ( ) ( ) ( ) ( )

v v v v v,II v,II

( ) ( ) ( ) ( ) ( ) ( )

v v v v v,III

, ( ) / ( ) ( ) Airy ( ) ,

, ( ) / ( ) ( ) Airy ( ) ,

, ( ) / ( )

i i i i i i

i i i i i i i

e

i i i i i i

e

R R R C k R Z R Z R

R R R R C k R Z R Z R

R R R R C k R Z

= −

= −

= 1/4

( )

v,III

1/4( ) ( ) ( ) ( ) ( ) ( )

v v v v v,IV v,IV

( ) Airy ( ) ,

, ( ) / ( ) ( ) Airy ( ) .

i

i i i i i i

R Z R

R R R C k R Z R Z R

−

= −

(11)

while the wave function of the infinite motion on the upper term coventionally is the

expanding radial one dimensional wave, normalized here on delta-function versus impulse of

translational motion (wave vectors k related to energy value ( )RE

( ) ( ) ( )

( )

( ) ( ) ( )

( )

1/4(j) ( ) ( ) ( )

( ) ,I ,I

1/4( ) ( ) ( ) ( )

( ) ,II ,II

2 1, ( ) ( ) Airy ( ) ,

m ( )

2 1, ( ) ( ) Airy ( ) .

m ( )

R R R

R

R R R

R

i j j

jE E E

E

j j j j

jE E E

E

R R R Z R Z Rk R

R R R Z R Z Rk R

= −

= −

(12)

Dipole matrix elements between molecular rotational states and electron-

vibrational-rotational ones

The complexity of calculation of those matrix elements are stipulted firstly by necessity

to calculate components of dipole moment defined in the lab reference frame, while the rotational

wave functions are defined in the molecular frame of coordinates. So, one has to introduce rotation

of coordinates axes to take this into account [1-7]. The other complexity is due to that one has to

follow total angular electron momentum, which represents itself the vector sum of three angular

vector moments – electron orbital L , rotatoinal R and spin S . The result depends on the sequence

of their summation, which is defined by the type of Hund’s coupling a, b, c, d, e what is more

appropriate for the given molecule and the particular electron transition. So, because of rotation of

the coordinate axes one could at once deduce that we would have 3-j symbols in the expression of

this matrix element, while summation of three angular moments will lead at once to appearance of

6-j symbols in its expression too [8]. The detailed formulas for calculation of Heonl-London factor


33

are presented in [15], while corrections to them are given in [16]. The calculations of Heonl-

London factors in some simple cases are considered in Landau and Lifshits monograph [12]. The

free access programs for their calculations could be found in [17, 18]. On the other hand the

rotational spectra complexity is caused by the often met difference of “Hund’s coupling” of the

iniftial and final levels involed into transitions, which could be not known beforehand for a given

transition, and by their multiplicity which in its turn increase the total number and types of the

rotational bands [1, 9, 19]. The necessity of verification of the type of Hund’s coupling for the

rotational trasition transforms the task of the calculation of Heonl-London factors into a very

complex separate study, with increasing difficulties due to levels multiplicity. Here we present two

formulars, related to rotational dipole transitions within the Hund’s coupling case “a” and “b”

(compare[20-22]). In the case“a” the total moment J is equal to the vector sum J R= + , where

R is the moment of rotation of nuclei

2( , ) (2 1) 1, ( , ) ( , )a aS J J J J J S S

= + − . (13)

In the case “b” the vector orbital momentum of electrons firstly is summed with the vector

of the moment of rotation of nuclei N L R= + , J N S= +

2

2( , ) (2 1)(2 1)(2 1) 1, ( , )b b

S N JS J J J N J N N S S

1 J N

= + + + −

.

(14)

In (13)-(14) 1 1 2 1,j m j m m jm− are the Clebsh-Gordan coefficients and 1 2 3

4 5 6

j j j

j j j

is 6-j

symbol [8, 12]. In the case when the transition takes place between states with different Hund

coupling b a the corresponding Heonl-London factor could be calculated from

2

2

2

2 2

2

1( , ) ( , , )(2 1)(2 1)(2 1)

,

2 0, 1 0, 0( , , ) ,

1,

b a

J JS J J H J J N

J N S

orH

in all other cases

= + + + + − − −

− −

= = = + = =

(14a)

where 1 2 3

1 2 3

j j j

m m m

is 3j-symbol.

The complexity of ( , )S J J data generation is illustrated by the presented equations (13)-

(16) – they are presented in each of the papers [20-22], but there are some mistakes (or typos) in

any of them, either in the equations appearance either in their explanations. Up to date the

Quick Start FWB 1.0

34

calculations of Heonle-London factors for variety of fixed conditions were performed and

described many times [23-27]. Moreover, recently several program packages were created that

allow to calculate synthetic spectra of various molecules and patrticulaly to generate ( , )S J J for

many types of rotational bands related to all Hund’s coupling cases and different multiplicities of

transitions [19, 28, 21, 29, 30, 31].

However, taking in mind that if we are mostly interested by UV and optical region of

spectra and rather high density and high tempearature plasmas we can approximate the real spectra

by summation over rotational trasitions using for analysis only electron-vibrational spectra

characteristics. This is because the differencies between the rotational levels are very small in

comparison with the carrier frequency of the electronic transitions. If one would look at the

rotational bands in these conditions they would have symmetric appearance, with considerably

smoothed and rather uniform population of different rotational levels due to smallness of

parameter rot T [1]. In this case the percularity of rotational bands spectra of homonuclear

molecules --intensity alternation in the rotational bands [1, page 209], [2] due to the non-zero value

of nuclear spin I would not be noticeable like at low temperatures [1]. BTW remind, that pure

dipole vibrational-rotational transitions of homoniclear molecules are forbidden due to zero dipole

moment [1-7].

Calculation of Einstein coefficients of electron-vibrational transition

It is important to establish the hierarchic realations between the probabilities of the

electron-vibrational-rotational Einstein coefficients and the electron–vibrational ones. The last one

coefficient (for emission) has the form (compare [1-7])

32

0, ' 3 2

,v; ,v ' ,v; ,v '

0,

(2 )4| i v ( ) jv ' |

3 (2 )i j i j ij

eA D R

c

+− =

−

. (15)

Remind that here i v , jv ' means the vibrational wave functions with vibrational wave

numbers v , v ' in the field of electron potentials ( ), ( )i jU R U R correspondingly. The equation (15)

is obtained by the direct summation over the all possible J’ values whilst J is being fixed and

using the normalization of the Hönl-London factors [23-27]

, ' , ' '

(2 1)(2 1)JJ 0, + '

p p J

S (2 ) S J = − + +

, (16)

and neglecting the difference in 3

,v, ; ,v ', 'i J i J due to the rotational structure. The set of {p,

p’} designate counting over -and ’-doublets. It is important that the value of total probability


35

,v; ,v 'i jA , of electron-vibrational-rotational transitions from the (i, v, J), level (i,v) and (j,v’) being

fixed, does not depend on J ! For zero spin (see [1, 9]) this summation takes into account

transitions belonging to the all different branches of the rotational transitions : P, Q, R. Equation

(16) is symmetrical with respect to changing summation over J’ on one over J

0, '

, ' , '

(2 - )(2 1)(2 ' 1)JJ'

p p J

S S J +

= + + , (17)

that is necessary in the case of calculation of the absorption coefficient [1-7]. Then in the

denominator in (15) the factor (2-0,’) should appear [1-7, 9, 23-27]. It is worthy to mention that

the term “Hönl-London factor” is not widely used in the literature because many researchers at

the same time contributed into the study of its properties.

Now one can remember that the v→v‘ band is formed by at least 3 branches of electron-

vibrational-rotational transitions with fixed v,v’ (P, Q, R), depending on the value of J = J-J’

during the transition. So, all realizable J values belonging to the lower electron-vibrational state

may contribute to the total probability of the electron-vibrational emission band. And now the

notion of the electron-vibrational band probability would depend on whether the J -sublevels of

the electron-vibrational state could be considered degenerate or not? If the first statement would

be true, then the total probability of the electron vibrational transition would be formed by the

value from equation (15) multiplied by the number of realizable J states of the upper (lower - in

the case of absorption ) electron-vibrational level with the accuracy due to the weak dependence

of (15) on the difference between the energies of various rotational states with respect to the

difference of the vibrational energy and even less with respect to the difference of the electron

energy. In this case it would be supposed that all those states are equally populated. But although

the rotational “quantum” “Bv’’” or “Be” [1-7] is much less than aforementioned values, usually the

rotational sublevels could not be considered as degenerate over J even at the room translational

temperatures of heavy particles in the discharge. So, each J sublevel is occupied with some

probability that is much less than unity, while the summation over all sublevels would give

obviously unity. That is why in this last case the value from (15) could be considered as integral

probability of the electron-vibrational band, if to substitute in (15) some mean value of 3

;i,v,J j,v ,J

. On the other hand the (2J+1) subsublevels of the sublevel with fixed J are usually degenerate

and that is why the statistical weight neglecting -doubling is equal to (2J+1). This consideration

shows that the expression (15) is the key quantity in the description of the intensity distribution of

the molecular spectra [1-7]. So, in the case of evaluating the upper bound for the absorption

coefficient value k0 in the band one has to find the maximum in the rotational levels population

distribution and evaluate k0 for this case. In the Fig. 1 the probability distribution of emission in

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36

the electron-vibrational bands under the electron transition 3 1

0 0A X+ +

+ → for InI molecule

is presented. These data are obtained using the electronic energy terms and dipole moments ( )ijD R

, calculated within ab initio quantum chemistry approach and the uniform quasiclassical

approximation for evaluating the vibrational wave functions. As in the literature the information

on values of the Einstein electron-vibrational coefficients for InI was absent before present

calculations the accuracy of the method could be approved by comparison of results obtained for

AlCl molecule, for which such data is available in [32]. In the Table I below one can verify by

comparison with similar data from [32] that the accuracy of presented procedure that incorporates

ab initio quantum chemical calculations of dipole moment is high and quite good.

Fig.1. The emission probability distribution in the electron-vibrational bands of

3 1

0 0A X+ +

+ → electron transition of InI molecule.

Table I. Comparison of electron-vibrational Einstein coefficients for AlCl transition A1Π-

X1Σ+.

Upper vibrational

number

Lower vibrational

number

A from [6],

1/s

A calculated,

1/s

Discrepancy

0 0 1.93E+08 1.99E+08 3.2%

1 1 1.89E+08 1.95E+08 3.2%


37

2 1 7.93E+05 8.57E+05 7.8%

2 2 1.83E+08 1.89E+08 3.2%

Absorption cofficients of electron-vibrational bands in molecular spectra

Suppose we know the broadening mechanisms of the electron-vibrational rotational

transitions under consideration in the emission and absorption. Also suppose that the profiles

( , )

,v, ; ,v ', ' ,v, ; ,v ', '( )e a

i J i J i J i J − in the emission and absorption are equal to each other [8, 33]

and normalized to unity

( ) ( )

,v, ; ,v ', ' ,v, ; ,v ', ' ,v, ; ,v ', ' ,v, ; ,v ', '( ) ( )e a

i J i J i J i J i J i J i J i J − = − (18)

( , )

,v, ; ,v ', ' ,v, ; ,v ', ' ,v, ; ,v ', '( ) ( ) 1e a

i J i J i J i J i J i Jd +

−

− − = (19)

where 0 ,v, ; ,v ', 'i J i J= is the unperturbed circular frequency of the radition or absorption

and is the current circular frequency of the radiation or absorption, k0 is the absorbtion

coefficient in the line center. In fact the integration in (18) should be performed over the spectral

range of the isolated spectral line. However, taking into account that as a rule the line intensity

decreases fast in the line wings, the integration could be extended up to the infinity. In order to

eliminate constants the reduced dimensionless profile ( )

0( )a − in absorption is used

( ) ( ) ( )

0 0( ) ( ) / (0)a a a− = − . The absorption coefficient ,v, ; ,v ', '(0)i J j Jk in the center of the

particular electron-vibrational-rotational line with account of induced radiation transitions could

be expressed as follows (compare [8])

2 2,v, ,v ', ' ,v, ( )

,v, ; ,v ', ' ,v, ; ,v ', ' ,v', ' ,v, ; ,v ', '2

,v, ; ,v ', ' ,v ', ' ,v, ,v ', '

(0) 1 (0)i J j J i J a

i J j J i J j J j J i J i J

i J i J j J i J j J

g g Nck A N

g g N

= −

,

(20)

where Ai…j… is the Einstein coefficient of transition i…-j…, ,v,i JN , ,v', 'j JN are the

densities of molecules occupying the upper or the lower levels accordingly with statistical factors

gi… , gj…. The dimension of the absorption coefficient is obviously the inverse length. Introducing

charateristics of electron-vibrational transitions (20) could be represented in the form as

2 2,v ',v ,v ( )

,v; ,v ' ,v; ,v ' ,v' ,v; ,v2

,v, ,v ' ,v ' ,v ,v '

(0) 1 (0)ji i a

i j i j j i i

i j j i j

gg Nck A N

g g N

= −

. (21)

Confronting expressions (2), (15), (20)-(21) it is seen that the ,v; ,v '(0)i jk could not be simply

Quick Start FWB 1.0

38

related to ,v, ; ,v ', '(0)i J j Jk .

Absorption coefficient due to photodissociation

According to chosen strategy we shall consider photodissociation due to absorprion in

electron-vibrational transitions [34], which includes the vibrational-vibrational trabsitions as well.

The cross section of photodissociation due to transition from the particular electron-vibrational

state , vi of the lower electronic term i might be expressed as follows

( )

2 ( ) 2

, ; ,v,v

4( ) | , | ( ) | , v |

3R

Ri

i E jj

gi E D R j

g c

= , (22)

where is the incident frequency of the photon, ( ), |Ri E is the one-dimensional continuous state

of relative motion of the upper level, /k c= is the photon wave vector value. Here the

normalization of the continuous states is chosen on the -function of energy (or velocity as was

chosen above), while the normalization factor of the bound vibrational state is the reciprocal of the

square root from unit length. The degeneracy factor is defined by

0, '

,v 0, '

(2 )

(2 )

i

j

g

g

+

−=

−, (23)

, being the electron orbital momentum projections on the internuclear axis for the upper and

lower electronic states. This result is obtained by summation over all possible transitions between

the rotational states of the upper electron-vibrational level and the fixed rotational state of the

lower electron –vibrational level, that after using the summation rule for Hönl-London SJJ’ factors

does not depend on the rotational quantum numbers JJ’. During this procedure the summations

over magnetic numbers and over two independent directions of the photon polarization are also

performed that leads to the appearance of the factor 1/3.

The photoabsorption might occur in three ways: 1. the upper electronic i state is unbound; 2.

the upper electronic state is bound, but transferred energy is large enough and above the

dissociation limit of the upper state iD ; 3. the upper bound electronic state interact with another

unbound one – and the predissociation takes place. Here for simplicity only the first channel is

addressed.

The rate constant for the photodissociation in the photon flux with intensity ( )I in cm-2 per

unit frequency interval and per unit time taken as second too for the transition ( )v , Rj i E→ is

( ) ( )v ', ; ,v , ; ,v( ) ( ) ( )R Rji E j i E j

k T N d I

= (24)

where v 'jN ’ is the density of the molecules in the state vj ’ and is the dissociation efficiency


39

for the frequency (some kind of branching ratio).

At last the absorption coefficient due to photo-dissociation photodissociationk is.

( )

2 ( ) 2

v ' v ', ; ,v,v

4( ) ( ) | , | ( ) | , v |

3R

photodissociation Rij ji E j

j

gk N N i E D R j

g c

= = (25)

Absorption coefficient due to photoionization

The absorption coefficient due to photoionization could be represented in a similar way to (25).

This reaction could be realized in several ways as well: 1. the upper electronic i state of molecular

ion is unbound, that means additional dissociation of products; 2. the upper electronic state of

being generated molecular ion is bound; 3. the upper bound electronic state of molecular ion

interact with another unbound one – and the predissociation takes place. The simplest second

choice is considered.

An example of consideration of various processs in molecular plasma of OH is presented in [35].

References

1. Herzberg G., Molecular Spectra and Molecular Structure I. Spectra of Diatomic Molecules, 2nd

edition Van Nostrand Company Inc 1950.

2. Townes C.H., Schawlow A.L., Microwave Spectroscopy, McGraw-Hill Pub;ishing Company,

1955.

3. Brown J.W., Carrington A., Rotational Spectroscopy of Diatomic Molecules, Cambridge

University Press, 2003.

4. Bernath P.F., Spectra of Atoms and Molecules, Oxford University Press, 2005.

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6. Kuznetsova L.A., Kuz’menko N.E., Kuzyakov Yu.Ya., Plastinin Yu.A., Probabilities of Optical

Transitions of Diatomic Molecules, Nauka, Moscow, 1980 (in Russian), 319 p. (in Russian)

7. Kuz’menko N.E., Kuznetsova L.A., Kuzyakov Yu.Ya., Frank-Condon factors of diatomic

molecules, MGU 1984 (in Russian).

8. I. Sobelman, Introduction to theory of atomic spectra, Pergamon Press, 1970.

9. J.B. Tatum, Astrophysical Journal Supplement 14, 21 (1967).

10. L.A. Kuznetsova, E.A. Pazjuk, A.V. Stoljarov, Radiative and Energetic Characteristics of

Diatomic Molecules (Data Bank –RADEN), Russian Journal of Physical Chemistry 67, 2271

(1993), http://www.elch.chem.msu.ru:8082/raden/index.htm

11. MOLPRO Quantum Chemistry Package: http://www.molpro.net/

12. Landau L.D., Lifshits E.M., Quantum Mechanics. Non-relativistic Theory, Pergamon Press,

1991.

Quick Start FWB 1.0

40

13. A.H. Nayfeh, Perturbation methods, John Wiley & Sons, Inc., N.Y., 1973.

14. Nikitin E.E, Umanskii S.Ya., Theory of Slow Atomic Collisions, Springer-Verlag, Berlin,

1984.

15. Kovacs I., Rotational Structure in the Spectra of Diatomic Molecules, Akademiai Kiado,

Budapest, 1969, 320 p.

16. Whiting E.E., Paterson J.A., Kovacs I., R.W. Nicholls, J. of Molecular Spectroscopy 47, 84

(1973).

17. Whiting E.E, NASA TN D-7268, Washington, 1973.

19. Lino da Silva M.,The line-by-line radiative code SPARTAN

v.2.6:http://esther.ist.utl.pt/spartan/ ;Lino da Silva M., Lopez B., Espinho S., SPARTAN 2,6

User’s Manual (2016).

20. Hornkohl J.O., Parigger C.G., American Journal of Physics 64, 623(1996).

21. Hornkohl J.O. , Parigger C.G., Nemes L., Applied Optics 44, 3686 (2005).

22. Parigger C.G., Hornkohl J.O., International Review on Atomic and Molecular Physics 1, 25

(2010).

23. Hougen, J.T., NBS Monograph 115, 1970.

24. Whiting E.E, Nicholls R.W., Astrophysical Journal Supplement Series 235, 1 (1974)

25. E. Whiting, A. Shadee, J. Tatum, J. Hougen, R.W. Nicholls, J. of Molecular Spectroscopy 80,

249 (1980).

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27. Hansson A., Watson J.K.G., Journal of Molecular Spectroscopy 233, 169 (2005)

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