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Study of the vibration of CO2 using effective normal coordinates : solution of the inverse problem
G. Amat
To cite this version:
G. Amat. Study of the vibration of CO2 using effective normal coordinates : solution of the inverse problem. Journal de Physique II, EDP Sciences, 1992, 2 (2), pp.147-161. �10.1051/jp2:1992120�.
�jpa-00247620�
J. Phys. ii France 2 (1992) 147-161 FEBRUARY 1992, PAGE 147
Classification Physics Abstracts
35.20P
Study of the vibration of CO~ using effective normal coordinates : solution of the inverse problem
G. Amat
Laboratoire de Physique Moldculaire et Applications (*), CNRS, Universitd Pierre et Marie Curie, Tour13, B-P. 76, 4 Place Jussieu, 75252 Paris Cedex05, France
(Received 3 September 199J, accepted 31 October 1991)
Rksumk. Nous £tablissons [es £quations qui permettent de calculer (au second orate
d'approximation) le potentiel vibrationnel de C02 h pant des niveaux d'dnergie expdrimentaux,
en utilisant le formalisme des coordonndes normales effectives.
Abstract. We establish the equations which make it possible to compute (up to second order of approximation) the vibrational potential of C02 from the experimental energy levels, in the
framework of the « effective normal coordinates » method.
1. Introduction.
The «effective normal coordinates» ~q~~ are linear functions of the standard normal coordinates q~~ of CO~, defined by
~l ~ a + bl *~l
, ~21 ~ ~2*~21, ~22 " ~2 *~22, ~3 ~ ~3 .~3
The coefficients a, bi, b~, b~ depend upon the value of the quantum number v~; they are
chosen in such a manner that the rotation-vibration Hamiltonian (I) of CO~, written in terms of effective normal coordinates and expanded in a power series with respect to them, has the
two following properties [Ii :
the vibrational part of Ho (zeroth order term in the power series expansion) is the sum of Harniltonians of harmonic oscillators, just as it is in the standard method,
(*) Laboratoire associ£ aux Universit£s P, et M. Curie et Paris-Sud.
(I) In the ENC method, the same potential V is used throughout the computation. It is defined by
molecular constants w~, k~~~ k~~~~, where c, d, e, f can assume the values 1, 2 or 3, with some restrictions due to symmetry. However the power series expansion V
= ,Vo + ~V, +,V~ depends upon the value of quantum number v~ : it involves « effective constants » ~w~~k~~~fi~~~~ whose values depend upon the value of v~.
the first order term Hi contains an operator proportional to ~qi whose matrix elements cancel out the matrix elements (v, v~[ ~ki~~~qi~q)[v~ ± I, v~) originating from the cubic
anharmonic potential, which are responsible for the bad convergence of the perturbation
calculation performed on the rotation-vibration Harniltonian [2].
In developing the « effective normal coordinates » (ENC) method, the first step has been to establish the algebraic equations required for the calculation of rotation-vibration energy
levels in terms of molecular parameters («direct problem»). The formulae giving the
expressions of the relevant matrix elements (2) of H' (transformed Hamiltonian) in terms of quantum numbers and molecular parameters (coefficients w~k~~~k~~~~ appearing in the potential and rotational constant B~) are given in reference [3] up to the second order of
approximation. Using then a reasonable potential and the formulae of reference [3], we calculated several vibrational levels v~ v~iv~ of ~~C ~iJ~, up to second order, with both the
standard and the ENC methods [4]. The calculation performed on levels 00°v~ with
v~ = 1, 5, lo showed that the difference between the results obtained by the two methods is large (as expected, it increases when v~ increases and reaches 632 cnl~ for v~ = lo). Since the
convergence of the perturbation calculation is better in the ENC method than in the standard method, a potential determined from the experimental vibrational energy levels using the
ENC method should be closer to the true potential than a potential obtained using the
standard method.
The purpose of this paper is to present an approach which would make it possible to
calculate the potential of CO~ from the experimental vibrational energy levels (« inverse
problem »), using the ENC method up to the second order of approximation.
2. Vibrational spectral term.
Referred to the minimum of the potential surface, the vibrational term in the diagonal matrix element of the transformed Hamiltonian H' can be written (3) as follows [5] :
*~i(Vi~2~~3) ~ WI (~l + + ~°2(~2 + ~) + ~°3 (~3 + +
~ t
(g~ + )~ ~~~
l ~
~ ~g~2(~2 ~ ~ *~~~
~ i
+*~~~~~~~~
t (g ~~ (~3~2~
(g~ + 1) + *X13 ' 2
~*~~~~~~~~
i f2 ~~C.
~ j~~~~ ~ i) (v3 + j ~ *~~~
In this formula ~g~~i~ includes a contribution from the rotational spectral term
(~g~~ = ~g[~ B~) the constants w~, w~, w~ are the same as in the standard calculation
~g~~ and the six parameters ~x depend upon the value of the quantum number v~ when
v~ - their limits are identical to the spectroscopic constants g~~ and x~~ used in the
2
(2) Diagonal, Fermi coupling, I-type coupling matrix elements.
(3) There is an obvious misprint in reference [5] : one should read + (and not x) at the end of line I and at the beginning of line 2 in equation (7a). Furthermore, in equations (13) of the same paper, one
should read ~K(~ instead of ~x~~.
N° 2 VIBRATIONAL POTENTIAL OF C02 149
standard calculation ; ~e is a function of v~ (symbol ~e designates the same quantity as symbol ,C" in references [5, 4]) :
~C = ~Ci + ~C2 + ~C3 (2)
3 ~kii~~ 7 ~k),1
*~~ 8 16~wj ~~~~
~ ~2
~~~ ~ ~~~ 8(~W
~~
~W2)
~~~~
3 ~k~~~~ 3 ~k)~~ ~w1
~~~ ~ ~ 16(4 ~W *~°)) ~~~~
Here again the parameters ~w and ~k are functions of v~ whose limits when v~ - are
2 identical to the corresponding standard parameters (defined by the same symbols without a star).
When referred to the lowest vibrational level, the vibrational spectral term
~Go(vi v~ ~v~)
= ~G(vi v~ ~v~) oG(00° 0) is given by the formula
*~i0(Vl~2~~3) ~ *~°)Vi + *£°IV2 + *£°I~3 + *X11~( +
+ *X22~( + *X~3~~ + *X12~l ~2 + *K~3Vi ~3
+ ~r(~ v~v~ + ~g~~i~ + ~G(00° 0) oG(00°0) (4) where
~t
o 13
*°'1 WI + *X11 + *X12 + f (5a)
~w( = w~ + 2 ~r~~ +
*~~~
+
*~~~ (5b)
2 2
*W~ # W~ + ~x)~ +
~~~
+ ~X)~ (5c)
2 and
oG(00° 0) is obtained by replacing all subscripts
~
by subscripts zero in the right hand side of
equation (6) : it designates the value of ~G(00°0) for v~ =
0.
3. Linear analysis.
3,I In CO~, the vibrational levels are distributed among Fermi polyads. Let M be the
multiplicity of a polyad containing the levels Xi X~... XM (the X's are the roots of the secular equation associated with the Fermi matrix). Due to the invariance of the trace of the Fermi matrix, the sum of the energies X~ of the levels belonging to the same polyad can be expressed
as a linear combination of the diagonal spectroscopic parameters appearing in the right hand side of equation (4). It is therefore possible to obtain from the experimental levels, through a
linear analysis, the values of several spectroscopic parameters [6]. If we compare it with the
standard method, there are two peculiarities in the ENC method: the spectroscopic
parameters are functions of v~ and a new term ~e appears in ~G(vi v~~v~). In the ENC method, we shall then perform a specific linear analysis for each value of v~. We shall
furthermore use
~Z(Mi)
=
I z ~G(ViV~~V~) ~G(0~t V~) (7)
'~
poiyad
From the general formula given in [6], we easily obtain (4) :
M-i (M-i)(M-2) of cM-ii+J~12
~z(M )=~v4 (8)
~ +?
3 +**°'~ ~* i
where
**ii ~ **£° + ~**£°~ + w~ll + 4 ~X22 (9a)
*~
" *X12 + ~*X22 (9b)
J~ ~ *X22 + *~22 (9C)
*~
~ *~ll + 4*X22 + *X12 (9d)
and
**~° ~
*~°)+ *X~3~3 (10a)
**~°~ ~ *W~ + *X(3V3 (10b)
The parameters (9) and ( lob) can be obtained (5), for each value of v~, from the experimental
energy levels :
~~i
=
2 ~Z~(20) (lla)
~C = 21~Z~(21) ~Z~(11) ~Z~(20)] (lib)
J~
=
~Z~(12) ~Z~(11) (llc)
~F = LZ~(30) 2 ~Z~(20)] (lid)
~~w(= [4~Z~(11) ~Z~(12)] (lie)
superscript e means that we use the experimental values of ~Z(Ml ). Parameters (9d), (9a) and (lob) can be conveniently replaced by
~Kii = / ~C (12a)
~°1+*X'3(~3+() "*M4~~**~°~~~*~ll~*~ (~~~)
w~ +
xj~(v~
+ = ~~w( ~C (12c)
2 2
respectively (we made use of relations (5a) (5b)).
Using the experimental values X of the vibrational spectral terms [7], it is possible to obtain the values of the molecular parameters (9b) (9c) (12a) (12b) (12c) for v~ = 0, 1, 2, 3. An
(4) The symbols used in the right hand side of equation (8) are similar to those used in reference [6].
(5) We have for each polyad £ ~Go = £X~ (conservation of the trace of Fermi matrices).
oj ~2 r
N° 2 VIBRATIONAL POTENTIAL OF CO~ lsl
extrapolation for v~- leads then to the values (6) of the standard parameters 2
wi, w~, xi~, C
= x~~ + 4 x~~, D = x~~ + g~~. The same extrapolation performed on the slopes of the tangents to the graphs of the functions of v~ associated with parameters (12b) and (12c)
leads to the values (6) of the standard parameters x~~ and x~~. The extrapolations for v~ - can be carried out graphically ; it is however less arbitrary to fit the data obtained
2
for v~= 0,1, 2, 3 (for the various spectroscopic parameters u) to polynomial functions
u = a + px + yx~ + 3x~ (where x
= v~ + ). wi w~ xii CD are given by coefficients
2
a while xi~ and x~~ are given by coefficients p. These 7 standard spectroscopic parameters are
the only ones which can be determined from the experimental energy levels, through a linear vibrational analysis involving an extrapolation for v~ -
j. The
next step in the inversion procedure will be to determine w~ and x~~ from the experimental energy levels.
3.2 In the extrapolation performed on the spectroscopic parameters, we have used
polynomials of degree 3. Another option would be to use polynomials of degree 2 and to
disregard the experimental values associated with v~ = 3.
3.3 It can actually be seen that the empirical graphs associated with the variation, with respect to v~, of the spectroscopic parameters considered above may present points of
inflection the variation of the quantities ~Z~(Mi) is more regular and one can find it more convenient to proceed as follows : the polynomial fit is made on the quantities ~Z~(Mi), with Mi
= II,12, 20, 21, 30, which exhibit a regular variation with respect to v~. Relations derived from relations (11) and (12) by suppressing all star subscripts make it possible to obtain wj w~x~i CDxi~ and x~~ from the new polynomial coefficients a and p.
If polynomials are used in order to carry out the extrapolations, the methods described in
(3,I) and (3.3) are equivalent and lead to the same results : the linear analysis and the calculation of up y3 are two linear operations which do commute.
3.4 A similar linear analysis can be performed on the rotational constants B. Their
experimental values will be designated by B~(vi v~ ~v~). In the ENC method [5], their diagonal
theoretical values are given by the formula
B~ is the standard value of the equilibrium rotational constant ; ~aj ~a~ ~a) are functions of
v~ whose limits when v~ - are equal to the standard parameters at a~ a~. Due to the
invariance of the trace of Fermi matrices, we can write :
~a~ = B~(01~ v~) + B~(00°v~) (14a)
~ai =
B~(I0°v~) B~(02°v~) + 2 B~(01~v~) (14b)
B~ ~a)
(v~ + =
~~~~~~~~ ~~~~~~~~
+ 2 B~(00°v~) (14c)
(6) In all numerical calculations, considering the accuracy on the experimental data, we found it reasonable to use 4 decimal digits in the vibrational parameters w~ x~d g22 CD and k~~~ k~~~~. This means that a rounding off to 4 decimal digits is performed when k~dek~def w~ x~~ are obtained in the course of the calculations. For the rotational parameters B~ a~, we use 7 decimal digits.
where ~ai has been obtained as (~ai + 2 ~a~) 2(~a~) and where B~(01~v~) designates the half sum of the two values of B~ associated with the two components of the I-type doublet. A
polynomial extrapolation for v~
- (similar to the ones performed in paragraphs 3.I or 2
3.3) makes it possible to determine, from the experimental B values given in [7], the standard
spectroscopic constants B~ ai a~ a~ which will be used in the next step of the vibrational
analysis.
4. Determination of w~ and x~~.
4,I We shall consider the set of levels 00°v~
; from equation (I), we can write :
G~(oo°v~)
= w~ (v~ + + ~x)~(v~ +
~
+ e + ~lC + J (v~ + + 5~ (15)
* 2 2 * 2
with
~K = ~x~~ + ~x~~ + ~x~~ (16)
4 2
j = ~x)~ + ~x(~ (17)
2
5~=-oG(00°0)+~wi+w~
=- [oe+oK+joJ+jw~+~ox)~j. (18)
5~ is a constant J can be obtained from the linear analysis, while ~e (given by Eqs. (2), (3)) and ~Jl cannot. Our purpose is to calculate w~ and x~~ (limit of ~x)~ for v~ - ) from the
2
experimental energy levels 00°v~ (these energy levels, being unaffected by the Fermi resonance, are equal to ~Go(00° v~)).
4.2 TYPE- I INVERSiON. In this first approach we shall make as much use as possible of the results of the linear analysis in order to eliminate terms from the right hand side of
equation (15) and make the inversion procedure as simple as we can. The quantity
~=j~v4+(~c-J~-~~F (19)
can be calculated from the experimental vibrational levels for each value of v~, using equations (I1) :
~ = ~Z~( ii ) ~Z~(12) + ( ~ze(20)
+ ~ze(21) ~ze(30) (20)
From equations (9), (12), (16), (17), we can see that
~y=~K+J(v3+-) -~g(2+fl+w~+B~w (21)
where we have used the relation ~g~~ = ~g]~-B~. Using the expressions of the effective
spectroscopic constants (?) given in reference [3], we can write :
*~~~ 2 * ~~~~ 4(2 ~w
~ + ~w1) ~~~~
(?) Our symbol ~g]~ is identical to symbol ~xj~j~ used in reference [3].
N° 2 VIBRATIONAL POTENTIAL OF C02 153
We see that ~g]~ is identical to ~e~ given by formula (3b). The quantity
~u = ~Go(00°V3) ~Y (23)
can be obtained from the experimental energy levels. From equations (23), (15), (21), (2), (3b), (22), we can write :
~U= w~(v~+ +~x)3(v~+ )~+~e'+5~' (24)
2 2
with
*C' = ~Cl + *C3 (~~)
and
5~'=5~-fl-w~-B~=-oG(00°0)-B~. (26)
Equation (24) is much simpler than equation (15). The standard constants w~ and x~~ can be obtained from the experimental energy levels, using an iterative method.
Run 0. Using formulae (23) and (20), we calculate the experimental values ~U~ of
~U for v~ = 0, 1, 2, 3. They are then fitted to a polynomial :
~U~ = a + px + yx~ + 3x~ (x = v~ + (27)
2
Comparison with formula (24) shows that we can write
~°3 W ~°)~~ " /~ X33 W X)i~ " Y (~8)
(we recall that ~x)~ - x~~ when v~ - ). In making the approximation
2
w~ = w)°~ x~~ = x))~ (29)
we neglect the linear and the quadratic coefficients in the Mac Laurin power series expansion
of ~e' with respect to x; these coefficients are indeed small compared to w~ and x~~ respectively.
Run J. Knowing the standard constants determined in section 3 using the linear analysis
and taking for w~ and x~~ the values (29) obtained in run 0, we can calculate the mising
standard spectroscopic constants (xj~x~~g]~) and the standard cubic and quartic coefficients
appearing in the potential (see Refs. [6, 8] and Fig, I). From at a~a~ cot w~w~ and
B~ we obtain the cubic coefficients k~~~ of the potential; from ki~~wjw~ CD and
B~ we obtain z =xj~-4x~~ which brings the last piece of information needed for the
calculation of all the standard spectroscopic constants x~~ and g]~; finally these spectroscopic
constants together with cot w~ w~ ki~~ki~~ki~~ and B~ give us the values of the coefficients
~cdef.
The next step is to compute, for each value of v~(0, 1, 2 or 3), the ENC coefficients
a bj~ b/~ [4] and then the effective parameters ~wi ~w~ ~ki~i~ki~~ ~kj
ii i~k~~~~ from which (8)
we can obtain ~x)~ and ~e'. We can now calculate (for v~
= 0, 1, 2, 3)
~Uo = ~U ~e' (30)
(8) At this point, we can also compute ~jj which will be considered in paragraph 4.3c.
JOURNAL DE PHYSIQUE II T 2, N' 2, FEBRUARY 1992
From equation (24) we obtain
i 2
*~° ~'~~~~~2~ ~*~~~~~~~2~ ~~' ~~~~
Instead of equations (30) and (31), we can use :
i 2
*~00 ~ *f~0 + (X33 *X33) V3
+ § (3~)
which can also be calculated for v~ =
0, 1, 2, 3 and
We shall now fit the four values obtained for ~Uo or ~Uoo with a polynomial similar to the one written in equation (27) and take as new values for w~ and x~~
w)~~ = p x)j~
= y (34)
Runs 2.3. The process is iterated : we perform run n (similar to run I described above) taking for w~ and x~~ the values w)~~~~ and x)(~~~ obtained at the end of run n I. The
solution is stabilized when two runs in succession give the same values for w~ and
x~~. The standard potential (9) obtained in the last run will be considered as the solution of the inverse problem defined in the introduction.
4.3 DlscussloN OF TYPE-I INVERSION.- In doing the calculation described in para-
graph 4.2, different choices can be made :
a) the polynomial fit can be made either on the quantities ~Uo or on the quantities
~Uoo. Our choice is to use ~Uoo
b) the formulae given in reference [3] show that
We can use this formula to simplify the expression (25) of ~e'. The equations (30), (23), (19), (20), (21) and (25) can be replaced respectively by the following ones :
~l' ~l' ~" (~fi)
~ o ~
~U' = ~Go(00° v~) ~y' (37)
~Y' " ~w4 J~ ~F (38a)
~y' = ~Z~(I1) ~Z~(12) + ~ ~Z~(20) ~ ~Z~(30) (38b)
~'
= ~ + ~Kii (39)
and
*C~ ~ *Cl + *C3 j *X11 (~0a)
(9) The values of kill and kjjii stay unchanged during the iteration because they depend only upon wj xii B~ ai which are known from the linear analysis.
