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Third-order RKR procedure
JournalofMoleculorStructure, 142(1986)319-322 Elsevier Science PublishersB.V., Amsterdam -F'rinPrintedinThe Netherlands
319
A. PARCO,J.J. CAMACHO,J.M.L. POYATO AND J.I. FERNANCEZ-ALONSO. Departamento de QuCmica-Ffsicay Qufmica-Cu&nt.ica, Facultad de Ciencias, UniversidadAut6normde Madrid.28349+ladrid(Spain).
Potential energycurvesfor the X'T+state of6LiH, TLH and 'LiD, ? LID molecules have been calculatedby the third-order RKR inversion procedure b including tk Kaiser correction. The resultsare in agreementwith previously obtainedcurvesby other autbrs using differentsmethods.As a check,the exact vibrational eigenfunctions,appropriateto these potentials, are obtainedby directnumericalsolutionsof the radialSchrtiingerequation. 1NIRoDuCr1oN Tk
Rydbergqleing-Reesmethod has been a very available means for
determiningenpiriml potentialcurvesfor diatomicmoleculesfrom experimental data for the vibrational term valuesC(v) and rotational ccnstants B(v). Hcwever, the RKRmethod is based upon tk first+rder WKB approximation,and while in mst cases the potentialCurvesit yields are adequate (electronic statesof CO, Lie, Na,, etc.),there are situationsin which their defficiencies are unacceptable (stateswith high Yoo values, i.e., A'x+ statesof alcali hydrides). Thz purposeof the presentstudyistoexamine a recenthigher+rder RKRtype inversion procedure (ref.11which is exact within the third+rder WKB approximtion.We have appliedthis method for the X'L+state of the6LiH,?LlH andCLiD, ?LiD
moleculessince it is neccesaryto use high qualityexperimental
data (ref.2) for the twoisotcpes.BecausethisprocedureassUnesthattk
two
isotopeshave exactlythe same potential(thepotentialenergy functiondepends only on the n&ions of the electronsand the Coulombrepulsionsof the nuclei), this does that whanitis appliedtotk
A’T+steke,
where Born-Oppenheimer
approximationbreakdown, thz selfccnsistence of ths obtainedresults is not energy of adequate. Thz mass differenceaffectsonly the vibrational-rotational 0022-2860/86/$03.50 01986 Elsevier 6ciencePublishenrB.V.
320
the molecule.If the methodis appliedto isotopicspecieswith a minor relation of atomic massesthan between6Li andTLi, it is expectedthat the obtained potentialsare more accurate. MIRD-ORDERRKRME!tHCO Third+rder RKR mathxlisa recentsemiclassical procedureto obtain the classical turningpoints, r+ and r_, of the vibrationalmotion. For a more complete formulationof the WKB approximation,with respectto ordinaryfirstorder RKR procedure,it is necessaryto have into accounta secondintegral
rt
l/2
dr [E -U(r)] r_
+ (p/%n)
-3/2
dr U"(r) [E - U(r)]
(I)
. where /3'=4'/2 is the reducedsass,W(r) is the secondderivativeof the PY effectivepotentialenergy U(r) and the contourof integraticnrencloses the portion of the real line for which U(r)< E. If there are data for two isotopic specieswith effectivemassesrlandp2, same potential, the
and ths two isotopeshave exactlythe
classical turningpointsare obtainedby means of the
followingequations:
v,(E)
I
r+ - r_ = [2/$/(1-/+/&)I
‘
l/2 dv,'/ [E(v$ - E(v$l
-
v0
l/2
va 08
[2/3Jo+-
111 v~
(2)
dv; ! [E(vZ)- E(vQl
/
and l/2 l/r_ - l/r+ = [2/li)i(l
dv,’ Bi(v() / [E(vi I - E(vi)l
d”h Be’“;’
/ (E(v2) - E(v;))
ID
wkrarethe integrations limitsare definedby EI[vI(E)I=Ee[va(E)I, Ei(C$4 CO% -l/2 - Yo$ /tit)
(3) (or
(ref.31and (v'~+l/2)=[(J1,,$"a .(~~+1/2).(i;,2+1/2)~~1~
(i=l,2).The upper limitsof integration vi(E)and Q(E) causesa singularityin quadrature tetiique was the evaluationof these integrals. A Gauss-Legendre employedto cvercomsthe difficultyat the upper limits.
321
TABLE 1 The third-order RKR potentialenergycurve and eigenvaluesof the6LiB and7LiB molecules,X'Z+state.re"=l.59527 A
G(v)+Ydaf’) By&y;
7
534.33(a) 524.74(b) 1581.82 1553.78 2062.93 2557.37 3593.06
4:10750 4.24679 4.09730 4.15279 4.00825 4.o6004
z;; 4489:72 5511.90
3.92037 3.96849 3.83362 3.87810
ZxG 6324h.I Ez
3.74794 3.78879 3.66327 3.70043
)
r
Ci)
1.46268 1.46378 1.37915 1.m84 1.32743 1.32943 1.28857
1.29078 1.25713 1.25948 I.23064 1.23309 1.20772 1.21025 1.18754 l.lgo14 1.16954 1.17218
r
(OA)
1.75262 1.75107
1.88635 I.8833 l.sB912 I.98480 2.07908 2.07437 2.16413 2.15746 2.24437 2.23653 2.32201 2.31D 2.39795 2.38774 2.47282 2.46139
Energyeigenvalues E(v) B(v) 4.34211 534.53 4.18758 525.01 4.24686 1582.43 4.09736 1554.37 4.15284 2604.07 2558.47 3599.68 z*;; 4491:89 5514.18 zzk; 6327:89 7330.04
3.69757
;*zz$ ;*z$? 8196186 8201.60 7209.94 3148864 3:49653 8068.99 8064.83 _______“““__‘__________“--__________~~~~~~~~~~~~~___~~~~~~_______-----~------Potentialfor minimumzone (~(4.5): U(r) = f(r-re)";f=26075.223 cni'Ae2 Potentialfor asymtoticzone (v> 8), with Ai in units of an-‘.A& U(r) = A0 + A6 / rc + A8 / r8 + ...... + A22 /ro2 8
AO=21499.813, A6=-2.354OEBE7, A8=2.899974438, AlO=-1.670396E9,Al2=5.@34316E9 Al4=-1.1886536E10, Al6=1.596664310, Al8=-1.327497310, A20=6.23%4539 A22=-1.26957739,rms error=1.8783-5
a)6LiB; b) fLiB TABLE 2 The third-order RKR potentialenergy urve and eigenvaluesof the of tk6LiH re"=1.59556 TLiH molecules,X',??state. 1
u 1 4 7
G(v)+Yoo(cm-') BP&y:' ) 7140666 707O.W;; 2078:56 2057.70 5927.66 58n.25 9387.21
7.34111
E%
$577
r (li)
r (I)
Energyeigenvalues B(v) G(v)
1.44561 I.44491
1.77793 I.77895
69.17 705.38
7.40699 7.56054
1.35207
1.93859 1.93660 2.27993 2.27516 2.57839
2079.16 2058.28 5930.35 5873.89 9393.78
7.34133
1.35313
I.22016 1.22157 1.14724
and
;*;;g $5gg
5&l 2.57073 9306.33 1.14875 5:98832 5.50129 2.87518 12475.54 5.50031 1.09749 5.40987 2.86445 12371.38 5.4wTl 1.09999 4.88591 3.19138 15181.09 4.88436 1.06104 13 15176:61 4.81487 3.17598 15064.34 4.81322 1.06251 15059.96 ____________________~~~~~_~~~~~~~~~~~~~~~~~~-~--~~~-~-------------------------IO
93ce.94 -w&g
Potentialfor minimumzone (v<-O.5): U(r) = f(r-refif=26268.73cm’1-a Potentialfor asymtoticzone (~713) with Ai in units of cm A U(r)=AO+A6/r6+A8/re+......+A22/r"'
~0~2212.945, A6=-2.5802689!~7, A8=3.O98Olol~8,AIO-1.6641666E9, Al23.0359839
Al4=-9.295430539, A161.07l0521E10, Al8=-7.540771E9, A20=2.974581E9 A22=3.047143E8,rms error=2.285E-5 a)6LiH; b)zLiH
322
RESULTSANDDISCUSSICN Tk
third-orderRKR inversion procedure was used to calculate the
potential energy curves for tk molecules. In tables 1 and 2
X'Z+state of 'LIH, ' LiH and 'LID, irLID the curves of tkse
X'L+states
are
summarized. Altl-oughthe constructions of the potentialmay be correct, one camot be sure a priori that this potentialrepresentsthe actualpotentials of ths molecular state under investigation.A good qualitative test on the accuracy of the third-order RKR potentialis to solve the radialwave equation using this potential, and then examinehow well the calculatedeigenvaluesE(v) and the expectationvaluesof ,
and hence B(v) values,are consistent
with the experimental C(v) and B(v) values. Tb
potentials functions employedin our calculationsare obtained by
connecting tk
turningpointswith a smoothcurve generatedby seventh-order
Lagrangian interpolation. For the tninimun zone (~(-0.5) we used a harmonic oscillator U(r) = f(r - re?
and for the asymptoticzone, inversepower was
considered U(r) = A0 + A6/r6 + AB/r* + AlO/&' + ............... where the Ai constantswere determinedfrom ths last third-orderRKR turning points. It can be seen from the tablesthat ths presentmethod, appliedto these real systems, not sin@ficantly yieldsbetterresultsthan either the simple first-orderRKR procedure. In othar paper (ref. 4), we describean accurate Huffaker-RKR-vander Waal potentialwhich gives very good results applied tc alcalihydrides.
This researchhas been supportedby the Ykomisicin Asesorade Investigaci6n Cienti'fica y T&cnicaw,projectNo. 1203. REFERENCES 1 2 3 4
C. Schwartzand R.J. L-eRoy, J. Chem. Phys.,81 (1984)3996-4001. C.R. Vidal and W.C. Stwalley,J. Ckm. Phys.,77 (1932) 883-89. E.W. Kaiser,J. Chum. Phys.,53 (197'0) 1686-1073. A. Pardo,XVII EuropeanCongresson MolecularSpectroscopy.
319
A. PARCO,J.J. CAMACHO,J.M.L. POYATO AND J.I. FERNANCEZ-ALONSO. Departamento de QuCmica-Ffsicay Qufmica-Cu&nt.ica, Facultad de Ciencias, UniversidadAut6normde Madrid.28349+ladrid(Spain).
Potential energycurvesfor the X'T+state of6LiH, TLH and 'LiD, ? LID molecules have been calculatedby the third-order RKR inversion procedure b including tk Kaiser correction. The resultsare in agreementwith previously obtainedcurvesby other autbrs using differentsmethods.As a check,the exact vibrational eigenfunctions,appropriateto these potentials, are obtainedby directnumericalsolutionsof the radialSchrtiingerequation. 1NIRoDuCr1oN Tk
Rydbergqleing-Reesmethod has been a very available means for
determiningenpiriml potentialcurvesfor diatomicmoleculesfrom experimental data for the vibrational term valuesC(v) and rotational ccnstants B(v). Hcwever, the RKRmethod is based upon tk first+rder WKB approximation,and while in mst cases the potentialCurvesit yields are adequate (electronic statesof CO, Lie, Na,, etc.),there are situationsin which their defficiencies are unacceptable (stateswith high Yoo values, i.e., A'x+ statesof alcali hydrides). Thz purposeof the presentstudyistoexamine a recenthigher+rder RKRtype inversion procedure (ref.11which is exact within the third+rder WKB approximtion.We have appliedthis method for the X'L+state of the6LiH,?LlH andCLiD, ?LiD
moleculessince it is neccesaryto use high qualityexperimental
data (ref.2) for the twoisotcpes.BecausethisprocedureassUnesthattk
two
isotopeshave exactlythe same potential(thepotentialenergy functiondepends only on the n&ions of the electronsand the Coulombrepulsionsof the nuclei), this does that whanitis appliedtotk
A’T+steke,
where Born-Oppenheimer
approximationbreakdown, thz selfccnsistence of ths obtainedresults is not energy of adequate. Thz mass differenceaffectsonly the vibrational-rotational 0022-2860/86/$03.50 01986 Elsevier 6ciencePublishenrB.V.
320
the molecule.If the methodis appliedto isotopicspecieswith a minor relation of atomic massesthan between6Li andTLi, it is expectedthat the obtained potentialsare more accurate. MIRD-ORDERRKRME!tHCO Third+rder RKR mathxlisa recentsemiclassical procedureto obtain the classical turningpoints, r+ and r_, of the vibrationalmotion. For a more complete formulationof the WKB approximation,with respectto ordinaryfirstorder RKR procedure,it is necessaryto have into accounta secondintegral
rt
l/2
dr [E -U(r)] r_
+ (p/%n)
-3/2
dr U"(r) [E - U(r)]
(I)
. where /3'=4'/2 is the reducedsass,W(r) is the secondderivativeof the PY effectivepotentialenergy U(r) and the contourof integraticnrencloses the portion of the real line for which U(r)< E. If there are data for two isotopic specieswith effectivemassesrlandp2, same potential, the
and ths two isotopeshave exactlythe
classical turningpointsare obtainedby means of the
followingequations:
v,(E)
I
r+ - r_ = [2/$/(1-/+/&)I
‘
l/2 dv,'/ [E(v$ - E(v$l
-
v0
l/2
va 08
[2/3Jo+-
111 v~
(2)
dv; ! [E(vZ)- E(vQl
/
and l/2 l/r_ - l/r+ = [2/li)i(l
dv,’ Bi(v() / [E(vi I - E(vi)l
d”h Be’“;’
/ (E(v2) - E(v;))
ID
wkrarethe integrations limitsare definedby EI[vI(E)I=Ee[va(E)I, Ei(C$4 CO% -l/2 - Yo$ /tit)
(3) (or
(ref.31and (v'~+l/2)=[(J1,,$"a .(~~+1/2).(i;,2+1/2)~~1~
(i=l,2).The upper limitsof integration vi(E)and Q(E) causesa singularityin quadrature tetiique was the evaluationof these integrals. A Gauss-Legendre employedto cvercomsthe difficultyat the upper limits.
321
TABLE 1 The third-order RKR potentialenergycurve and eigenvaluesof the6LiB and7LiB molecules,X'Z+state.re"=l.59527 A
G(v)+Ydaf’) By&y;
7
534.33(a) 524.74(b) 1581.82 1553.78 2062.93 2557.37 3593.06
4:10750 4.24679 4.09730 4.15279 4.00825 4.o6004
z;; 4489:72 5511.90
3.92037 3.96849 3.83362 3.87810
ZxG 6324h.I Ez
3.74794 3.78879 3.66327 3.70043
)
r
Ci)
1.46268 1.46378 1.37915 1.m84 1.32743 1.32943 1.28857
1.29078 1.25713 1.25948 I.23064 1.23309 1.20772 1.21025 1.18754 l.lgo14 1.16954 1.17218
r
(OA)
1.75262 1.75107
1.88635 I.8833 l.sB912 I.98480 2.07908 2.07437 2.16413 2.15746 2.24437 2.23653 2.32201 2.31D 2.39795 2.38774 2.47282 2.46139
Energyeigenvalues E(v) B(v) 4.34211 534.53 4.18758 525.01 4.24686 1582.43 4.09736 1554.37 4.15284 2604.07 2558.47 3599.68 z*;; 4491:89 5514.18 zzk; 6327:89 7330.04
3.69757
;*zz$ ;*z$? 8196186 8201.60 7209.94 3148864 3:49653 8068.99 8064.83 _______“““__‘__________“--__________~~~~~~~~~~~~~___~~~~~~_______-----~------Potentialfor minimumzone (~(4.5): U(r) = f(r-re)";f=26075.223 cni'Ae2 Potentialfor asymtoticzone (v> 8), with Ai in units of an-‘.A& U(r) = A0 + A6 / rc + A8 / r8 + ...... + A22 /ro2 8
AO=21499.813, A6=-2.354OEBE7, A8=2.899974438, AlO=-1.670396E9,Al2=5.@34316E9 Al4=-1.1886536E10, Al6=1.596664310, Al8=-1.327497310, A20=6.23%4539 A22=-1.26957739,rms error=1.8783-5
a)6LiB; b) fLiB TABLE 2 The third-order RKR potentialenergy urve and eigenvaluesof the of tk6LiH re"=1.59556 TLiH molecules,X',??state. 1
u 1 4 7
G(v)+Yoo(cm-') BP&y:' ) 7140666 707O.W;; 2078:56 2057.70 5927.66 58n.25 9387.21
7.34111
E%
$577
r (li)
r (I)
Energyeigenvalues B(v) G(v)
1.44561 I.44491
1.77793 I.77895
69.17 705.38
7.40699 7.56054
1.35207
1.93859 1.93660 2.27993 2.27516 2.57839
2079.16 2058.28 5930.35 5873.89 9393.78
7.34133
1.35313
I.22016 1.22157 1.14724
and
;*;;g $5gg
5&l 2.57073 9306.33 1.14875 5:98832 5.50129 2.87518 12475.54 5.50031 1.09749 5.40987 2.86445 12371.38 5.4wTl 1.09999 4.88591 3.19138 15181.09 4.88436 1.06104 13 15176:61 4.81487 3.17598 15064.34 4.81322 1.06251 15059.96 ____________________~~~~~_~~~~~~~~~~~~~~~~~~-~--~~~-~-------------------------IO
93ce.94 -w&g
Potentialfor minimumzone (v<-O.5): U(r) = f(r-refif=26268.73cm’1-a Potentialfor asymtoticzone (~713) with Ai in units of cm A U(r)=AO+A6/r6+A8/re+......+A22/r"'
~0~2212.945, A6=-2.5802689!~7, A8=3.O98Olol~8,AIO-1.6641666E9, Al23.0359839
Al4=-9.295430539, A161.07l0521E10, Al8=-7.540771E9, A20=2.974581E9 A22=3.047143E8,rms error=2.285E-5 a)6LiH; b)zLiH
322
RESULTSANDDISCUSSICN Tk
third-orderRKR inversion procedure was used to calculate the
potential energy curves for tk molecules. In tables 1 and 2
X'Z+state of 'LIH, ' LiH and 'LID, irLID the curves of tkse
X'L+states
are
summarized. Altl-oughthe constructions of the potentialmay be correct, one camot be sure a priori that this potentialrepresentsthe actualpotentials of ths molecular state under investigation.A good qualitative test on the accuracy of the third-order RKR potentialis to solve the radialwave equation using this potential, and then examinehow well the calculatedeigenvaluesE(v) and the expectationvaluesof ,
and hence B(v) values,are consistent
with the experimental C(v) and B(v) values. Tb
potentials functions employedin our calculationsare obtained by
connecting tk
turningpointswith a smoothcurve generatedby seventh-order
Lagrangian interpolation. For the tninimun zone (~(-0.5) we used a harmonic oscillator U(r) = f(r - re?
and for the asymptoticzone, inversepower was
considered U(r) = A0 + A6/r6 + AB/r* + AlO/&' + ............... where the Ai constantswere determinedfrom ths last third-orderRKR turning points. It can be seen from the tablesthat ths presentmethod, appliedto these real systems, not sin@ficantly yieldsbetterresultsthan either the simple first-orderRKR procedure. In othar paper (ref. 4), we describean accurate Huffaker-RKR-vander Waal potentialwhich gives very good results applied tc alcalihydrides.
This researchhas been supportedby the Ykomisicin Asesorade Investigaci6n Cienti'fica y T&cnicaw,projectNo. 1203. REFERENCES 1 2 3 4
C. Schwartzand R.J. L-eRoy, J. Chem. Phys.,81 (1984)3996-4001. C.R. Vidal and W.C. Stwalley,J. Ckm. Phys.,77 (1932) 883-89. E.W. Kaiser,J. Chum. Phys.,53 (197'0) 1686-1073. A. Pardo,XVII EuropeanCongresson MolecularSpectroscopy.