- Email: [email protected]
Improvement of the RKR procedure obtained by semi-classical calculations
Volume 96, number 3
IhlPROVEMENT
8 April 1983
CHEMICAL PHYSICS LEXTERS
OF THE RKR PROCEDURE
OBTAINED
BY SEMI-CLASSICAL CALCULATIONS
G. GOUJZDARD and J. VIGUE Loboratoire de Spectroscopic Hertzienne de 1‘ENS *. 24 Rue Lhomond. 75231 Park Cedex 05. France
Received 31 January 1983
In this letter we present a new technique to improve RKR molecular potential curve calculations. The potential curve obrained gives calculated G(v) and B(v) values extremely close to the measured ones. The aim of these calculations is the same as that of the inverted perturbation approach of Vidaland by using Watson’s semi-classical inversion procedure.
1. Introduction Experimentally the G(u) and B(u) values of a given molecular state may be obtained for a large number of vibrational states. The molecular potential curve V(r) is then approximated by the RKR technique, which gives VRKR(r) over a possibly large range of r values. The VRKR(r) curve may then be used to calculate quantum-mechanical G,-,(u) and Be(u) values [l] _ It appears that the deviations AC,(u) = G(u) - GO(u) and A&(u) = B(u) -Be(u) are small, showing that VRKR is quite a good approximation potential_
are calcukted
As a second approach one may quote the inverted perturbation approach of Vidal and Scheingraber [4] _ There the deviation ~Iv(r) = V(r) - VRKR(~) is espanded on a basis of known functions&(r):
By first-order perturbation E(v, J) -ERKR(u,J)
theory:
= AE(v,J)
of the true v(r)
However great precision in the experimental determination of G(u) and B(u) leads to a need for a better approximation to the potential V(r). A number of techniques have been designed to improve over the plain RKR potential. A first set of efforts has been devoted to the calculations of the higher orders in the semi-classical appro_ximation (the first order giving the RKR potential) [2] ; while these techniques are quite interesting in themselves, their practical use is limited by the amount of work to be done, except for the wellknown and very simple Kaiser [3] correction. However, a trial [2] to converge to the true quantummechanical potential seems not to have been conclusive. * Laboratoire
Sheingraber, but the potential-well corrections
A least-squares
fit method is then used to fit the cj values; the resulting potential i giving improved
calculated
fully quantum-mechanical very efficiently.
molecular constants.
method
This
indeed converges
Our proposal is to apply Watson’s semi-classical inversion procedure [S] to this particular problem. Let us
recall Watson’s paper: given a known molecular property X, whose vibrational and rotational variations are known, K(u, J) = Xv +XD,J(J+ 1) + ___,it is possible to go semi-classically to the X(r) variation by essentially inregrations over the RKR potential:
Associt au CNRS. LA No. 18.
0 009.2614/83/0000-0000/S
03.00 0 1983 North-Holland
293
\‘01umc 96. numbsr
.\‘,r: _ - -- .YIfi =
3
CHEMICAL
a
2kz
’ /
_S(u’)(G,-Cu.)-“’
PHYSICS
LETTERS
1983
8 April
associated with interpolations_ extrapolations, and integrations have all to be conducted very carefully; certainly improvements in the results may come from better methods_
du’.
Umin
3. Results
-
l’nun
v,1111 k = ir/Sn~pc*. hcrc rl JIKI r, are the turning points &Iirh- ~Ihr.I~ionJl levrl u.-r-; the dsrivatives i3ri/au. and _\; = V(r
= f,).
The .U(r)
11~ ~nte~polJrIon v.Jnt
here
to
vdriJtioI1
brtwcen
tilt
is then
RKK
turning
reconstructed points.
We
the general S(r) quantit> with Lt~,,(‘). tl~z eqw ~ienie #ving Ihen ?u, = AuG(u) This IS the semi-classical counter.lIlLl _\-,>,. = -l,fl(~1. p.Irt 01’ rhe Il’A tcshnique: the IPA uses firts-order pcrturiwtml thw~ IO find AI/(r). wbiie our proposal I\ 10 lid .I aili-clJssicJ1 Jppro~iInJtioIl of AL’(r).
1’1Isr. y!rn tjhscrvcd G(u) and U(u) values. we calC~IIJIC~hc RKR pc~~t~~ltlal curse I’,,,(r). Then. by a
technique.
\UI~CIL~~
01‘tliis
the “eigenvalues”
poIcnriJi
Jrc‘
Go(u) qI.IJIltUIll-
nI~~lI.~r~~call~
caiculated . Ths deviations JGu(u)
Jnd -1Ut,(u)
= G(u)
inverted
- U,(v)
FI\~ the
we
and
= G(u) - G,(u)
then
semi-
AVt(f) to the initi4 IeKhK(r): l’,(r)= FRKR(r) + Ak’,(r). Then Cl(u) .III~ B,(u) .rre cJla~l.wd. compared IO G(v) and B(u) .md 51)on to obtain the desired approximation_ A probk1I1 ,rrrses because AG,(u) and As,(u) 11lust be interpoi.ned b\ smooth iunctions of u (polynomials for c~mple). Our experience is that AGo and A&, are
\rl.~ss~~,~ll~ tar
rndeed
s~iwotiIl~
AC;, .~nd ABi
are:
rderitii)
2. Scheme of the rxlculations
ii,,(U)
This technique has been tried for a number of wellknown molecular potential curves: the CO X1 Z+ state whose constants are very well established [l] _ the X state of SOSe,. t30Te2, NaZ, I,, . .. . Using very recent and precise constants [6]. very good results have been obtained for Se? (0 < u d 35). The rms deviations between observed and calculated (RKR) vaiues
vJryiI1,. 0 but after
sllow
1s theret*ore more
corrections
3 and
clear oscillating
more difficult. to ghe a non-zero.
several
iteratiorls
variation.
the fit
Consequently
11x nIc11wd tends but weak. quad~.tttc dskttion between real and calculated constants_ .Ilso. tire qu~~iIu~n-IlieclIanical calculations require ~‘\~IJpol~tioIls of tile potential outside the RKR r.rnge; these also are rather critical for good converonce 31 hgll u v.tlues. The numerical calculations >
= 0.77 x 1O-6
ZT0=0.09cni-*,
mtJ
After live corrections
one obtains
z,
= 0.16 X lO-3
c111-~
.
z?,
= 0.16 x 10-7
cm-1
.
The improvements
are therefore
On the other
the resulting
B&]E~
hand
10-G )
=JGmz,
Clll--’
.
respectively
precision
oecomes
= lo-‘_
On figs_ la and
and AC,,
1b are displayed respectively AGO and ABo and LB5 ; scale changes show by
themselves the improvement_ Resides AGs(u) and ABi(u) indeed show some oscillatory character; nevertheless, the obt-ained precision [AGs(u) < 10m3 c111-~] is well beyond spectroscopic precision. The calculated corrections API(r) show that the RKR potential is faulty mainly on the rapidly varying repulsive branch of the potential. The convergence may be tested by observing the successive A Vi(r), which are indeed but rather slowly decreasing; the maximum AV,(r) is = lo-? cm-l. One can therefore accept that the resulting potential is then defined to better than lo-? cm--l. For CO X 1 Z+ [l ] the improvements are respec- _ tively
Volume 96, number 3
s-
..
.
.
.
.
.
.
.
.
.
.
i%i?, = 2.8 X 1O-3 cm-l,
.-
XB, = 1.1 X 10”
+
a .
cm-
cm-l.
These results are somewhat less convincing and the reason is not yet known: perhaps for a light molecule, quantum effects are important_ This semiclassical improvement of the RKR potential curves may be quite useful as an alternative to the IPA approach. At the moment it is not possible to separate purely calculational, or genuine quantum, limitations of the method. Nevertheless, the first results obtained here indeed show the legitimacy of tlris technique_
I
AB(v)110’
S April 1983
CHEhllCAL PHYSICS LETTERS
References 111 A-W. hlantz,J_I(.G_\\‘atson,
b
i
.
.
i-5
b
--__:-
.
fig. I _ (a) Deviations between observed and calculated G(u) values: (0) M&(u), (+) 100 XAG,(u). (b) Deviations between observed and calculated B(u) values: (0) A&(v): (+) 10 X 4lVj(tJ)-
K. Nanhari Rao, D-l_. Albritton, A-L_ Schmeltekopf and R.N. Zare, J- Mol. Spectp- 39 (1971) 1so; 1. Tellingkuisen. J. hlol. Spectry. 44 (1971) 194; AS. Dickinson, J. Mol. Spectry. 4-Z (1971) 163; SM. Kirschner and J.K.G. Watson, J. hlol. Spectry. 47 (1973) 234. S-hi. Kirscbner and J.E.G. Watson. J. Mol. Spectry. 51 (1974) 321. L31 E.W. Kaiser, J. Chem. Phys. 53 (1970) 1686. VI C-R_ Vidal and H. Scbeingraber, J. Mol. Spectry- 65 (1977) 46_ 151 J-KG. Watson, J. hlol. Spectry. 74 (1979) 319. 161 S-l. Presser, RF_ Barrow, C. Effantin, J. D’lncan and J. Verges, J. Phys. B15 (1982) 41SI.
295
IhlPROVEMENT
8 April 1983
CHEMICAL PHYSICS LEXTERS
OF THE RKR PROCEDURE
OBTAINED
BY SEMI-CLASSICAL CALCULATIONS
G. GOUJZDARD and J. VIGUE Loboratoire de Spectroscopic Hertzienne de 1‘ENS *. 24 Rue Lhomond. 75231 Park Cedex 05. France
Received 31 January 1983
In this letter we present a new technique to improve RKR molecular potential curve calculations. The potential curve obrained gives calculated G(v) and B(v) values extremely close to the measured ones. The aim of these calculations is the same as that of the inverted perturbation approach of Vidaland by using Watson’s semi-classical inversion procedure.
1. Introduction Experimentally the G(u) and B(u) values of a given molecular state may be obtained for a large number of vibrational states. The molecular potential curve V(r) is then approximated by the RKR technique, which gives VRKR(r) over a possibly large range of r values. The VRKR(r) curve may then be used to calculate quantum-mechanical G,-,(u) and Be(u) values [l] _ It appears that the deviations AC,(u) = G(u) - GO(u) and A&(u) = B(u) -Be(u) are small, showing that VRKR is quite a good approximation potential_
are calcukted
As a second approach one may quote the inverted perturbation approach of Vidal and Scheingraber [4] _ There the deviation ~Iv(r) = V(r) - VRKR(~) is espanded on a basis of known functions&(r):
By first-order perturbation E(v, J) -ERKR(u,J)
theory:
= AE(v,J)
of the true v(r)
However great precision in the experimental determination of G(u) and B(u) leads to a need for a better approximation to the potential V(r). A number of techniques have been designed to improve over the plain RKR potential. A first set of efforts has been devoted to the calculations of the higher orders in the semi-classical appro_ximation (the first order giving the RKR potential) [2] ; while these techniques are quite interesting in themselves, their practical use is limited by the amount of work to be done, except for the wellknown and very simple Kaiser [3] correction. However, a trial [2] to converge to the true quantummechanical potential seems not to have been conclusive. * Laboratoire
Sheingraber, but the potential-well corrections
A least-squares
fit method is then used to fit the cj values; the resulting potential i giving improved
calculated
fully quantum-mechanical very efficiently.
molecular constants.
method
This
indeed converges
Our proposal is to apply Watson’s semi-classical inversion procedure [S] to this particular problem. Let us
recall Watson’s paper: given a known molecular property X, whose vibrational and rotational variations are known, K(u, J) = Xv +XD,J(J+ 1) + ___,it is possible to go semi-classically to the X(r) variation by essentially inregrations over the RKR potential:
Associt au CNRS. LA No. 18.
0 009.2614/83/0000-0000/S
03.00 0 1983 North-Holland
293
\‘01umc 96. numbsr
.\‘,r: _ - -- .YIfi =
3
CHEMICAL
a
2kz
’ /
_S(u’)(G,-Cu.)-“’
PHYSICS
LETTERS
1983
8 April
associated with interpolations_ extrapolations, and integrations have all to be conducted very carefully; certainly improvements in the results may come from better methods_
du’.
Umin
3. Results
-
l’nun
v,1111 k = ir/Sn~pc*. hcrc rl JIKI r, are the turning points &Iirh- ~Ihr.I~ionJl levrl u.-r-; the dsrivatives i3ri/au. and _\; = V(r
= f,).
The .U(r)
11~ ~nte~polJrIon v.Jnt
here
to
vdriJtioI1
brtwcen
tilt
is then
RKK
turning
reconstructed points.
We
the general S(r) quantit> with Lt~,,(‘). tl~z eqw ~ienie #ving Ihen ?u, = AuG(u) This IS the semi-classical counter.lIlLl _\-,>,. = -l,fl(~1. p.Irt 01’ rhe Il’A tcshnique: the IPA uses firts-order pcrturiwtml thw~ IO find AI/(r). wbiie our proposal I\ 10 lid .I aili-clJssicJ1 Jppro~iInJtioIl of AL’(r).
1’1Isr. y!rn tjhscrvcd G(u) and U(u) values. we calC~IIJIC~hc RKR pc~~t~~ltlal curse I’,,,(r). Then. by a
technique.
\UI~CIL~~
01‘tliis
the “eigenvalues”
poIcnriJi
Jrc‘
Go(u) qI.IJIltUIll-
nI~~lI.~r~~call~
caiculated . Ths deviations JGu(u)
Jnd -1Ut,(u)
= G(u)
inverted
- U,(v)
FI\~ the
we
and
= G(u) - G,(u)
then
semi-
AVt(f) to the initi4 IeKhK(r): l’,(r)= FRKR(r) + Ak’,(r). Then Cl(u) .III~ B,(u) .rre cJla~l.wd. compared IO G(v) and B(u) .md 51)on to obtain the desired approximation_ A probk1I1 ,rrrses because AG,(u) and As,(u) 11lust be interpoi.ned b\ smooth iunctions of u (polynomials for c~mple). Our experience is that AGo and A&, are
\rl.~ss~~,~ll~ tar
rndeed
s~iwotiIl~
AC;, .~nd ABi
are:
rderitii)
2. Scheme of the rxlculations
ii,,(U)
This technique has been tried for a number of wellknown molecular potential curves: the CO X1 Z+ state whose constants are very well established [l] _ the X state of SOSe,. t30Te2, NaZ, I,, . .. . Using very recent and precise constants [6]. very good results have been obtained for Se? (0 < u d 35). The rms deviations between observed and calculated (RKR) vaiues
vJryiI1,. 0 but after
sllow
1s theret*ore more
corrections
3 and
clear oscillating
more difficult. to ghe a non-zero.
several
iteratiorls
variation.
the fit
Consequently
11x nIc11wd tends but weak. quad~.tttc dskttion between real and calculated constants_ .Ilso. tire qu~~iIu~n-IlieclIanical calculations require ~‘\~IJpol~tioIls of tile potential outside the RKR r.rnge; these also are rather critical for good converonce 31 hgll u v.tlues. The numerical calculations >
= 0.77 x 1O-6
ZT0=0.09cni-*,
mtJ
After live corrections
one obtains
z,
= 0.16 X lO-3
c111-~
.
z?,
= 0.16 x 10-7
cm-1
.
The improvements
are therefore
On the other
the resulting
B&]E~
hand
10-G )
=JGmz,
Clll--’
.
respectively
precision
oecomes
= lo-‘_
On figs_ la and
and AC,,
1b are displayed respectively AGO and ABo and LB5 ; scale changes show by
themselves the improvement_ Resides AGs(u) and ABi(u) indeed show some oscillatory character; nevertheless, the obt-ained precision [AGs(u) < 10m3 c111-~] is well beyond spectroscopic precision. The calculated corrections API(r) show that the RKR potential is faulty mainly on the rapidly varying repulsive branch of the potential. The convergence may be tested by observing the successive A Vi(r), which are indeed but rather slowly decreasing; the maximum AV,(r) is = lo-? cm-l. One can therefore accept that the resulting potential is then defined to better than lo-? cm--l. For CO X 1 Z+ [l ] the improvements are respec- _ tively
Volume 96, number 3
s-
..
.
.
.
.
.
.
.
.
.
.
i%i?, = 2.8 X 1O-3 cm-l,
.-
XB, = 1.1 X 10”
+
a .
cm-
cm-l.
These results are somewhat less convincing and the reason is not yet known: perhaps for a light molecule, quantum effects are important_ This semiclassical improvement of the RKR potential curves may be quite useful as an alternative to the IPA approach. At the moment it is not possible to separate purely calculational, or genuine quantum, limitations of the method. Nevertheless, the first results obtained here indeed show the legitimacy of tlris technique_
I
AB(v)110’
S April 1983
CHEhllCAL PHYSICS LETTERS
References 111 A-W. hlantz,J_I(.G_\\‘atson,
b
i
.
.
i-5
b
--__:-
.
fig. I _ (a) Deviations between observed and calculated G(u) values: (0) M&(u), (+) 100 XAG,(u). (b) Deviations between observed and calculated B(u) values: (0) A&(v): (+) 10 X 4lVj(tJ)-
K. Nanhari Rao, D-l_. Albritton, A-L_ Schmeltekopf and R.N. Zare, J- Mol. Spectp- 39 (1971) 1so; 1. Tellingkuisen. J. hlol. Spectry. 44 (1971) 194; AS. Dickinson, J. Mol. Spectry. 4-Z (1971) 163; SM. Kirschner and J.K.G. Watson, J. hlol. Spectry. 47 (1973) 234. S-hi. Kirscbner and J.E.G. Watson. J. Mol. Spectry. 51 (1974) 321. L31 E.W. Kaiser, J. Chem. Phys. 53 (1970) 1686. VI C-R_ Vidal and H. Scbeingraber, J. Mol. Spectry- 65 (1977) 46_ 151 J-KG. Watson, J. hlol. Spectry. 74 (1979) 319. 161 S-l. Presser, RF_ Barrow, C. Effantin, J. D’lncan and J. Verges, J. Phys. B15 (1982) 41SI.
295