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Determination of vibrational quantum numbers of diatomic molecules using a variational method
Volume 6.5, number 1
DETERMINATION
CHEhlICAL
OF VIBRATIONAL
USING A VARIATIONAL
PHYSICS LETTERS
1 August 1979
QUANTUM NUMBERS
OF DIATOMIC MOLECULES
hfEI-HOD
Received 2 April 1979
The method is based on a variational method xxhich gives vibrationat quantum nun~brrsfor 1, hich the measured term v&es are described by the quantum mechanical energy eisenvatues of a rotating vibrator. The method is applied to the AtE* states of the 4oCaz .md of the 6LiH molecuk.
I_ Introduction The assignment of rotational and vibrational qunntunl numbers in the andysis of eIectrouic transitions of diatomic molecules has been extensively discussed by Herzberg [I] _However. there are situations where the methods described do not work. This may occur. for example, if the minima of the potential curves involved are considerably shifted with respect to each other. In this case the Fran&-Condon fxtors for the first members of individual progressions are frequently SO small that the fxrst member \\ith u = 0 is diffcuh to identify_ In some cases the largest Fran&--Condon overlap of the vibrational Ievel with u = 0 may occur with the dissociation continuum of the other molecular state as in the case of the hfgl A-X system [2,3] _ (For these situations a way out is generally provided by the isotope shift.) Hoxtever, there are spectra hke the 4oCa, A-X system [4] where data of only one isotope combination are available and where the u’ numbering is uncertain_ This spectrum will be discussed below. There are also situations where the observed isotope shifts do not agree with the simple theory because of the breakdown of the Born-Oppenheimer approsimation. This was found in the spectra of some hydrides and deuterides and there is some question whether the existin g u assignments are correct_ The A I C+ state of the 6tiH molecule wviiibe discussed below [5,6] _
2. Method The procedure for determining the vibrationaf quantum numbers makes use of a variational method described by Vidal aud Scheingraber 171 \xhich is based on an inverted perturbation approach proposed by Kosman and Hinze [Sl_ Within the Born-Oppenheimer approximation a diatomic molecule can be described as a roraring vibrator which obeys the radial Schrddinger equation (H, + H,,)
tid (r) =
&J
&J
Ir)
(!)
7
where EuJ is the energy eigenvalue specified b) the vibrational and rotational quantum numbers u snd J, Ho
is the hamiltonian
Ho = (fi[-k~c)
oi the tl0tlrOtatirig
d2/dr2 + U. (t-) ,
molecule
(2)
r is the mternuclear
distance. ir is the reduced m.w of the nuclei and Llo(r) is the potential energy of the roniorat:ot;iess molecule_ ifrot describes rhf rotational titin w-. ihe nuclei. For 3 f S state. it is x iven bv2
H mf = (fi/4ixK)J(J
+ I )/, 7-_
(3)
By nle.xns of a variational method [7] UO nosy be deso that the eisenvalues EvJ agree in J least squares sense with the measured term values ruJ- For this purpose a correction function 5.V, is provideti which, most conveniently, is given by a linear combination OF orthonormal functions such as Legendre polynomials. Since typicaliy about 20 orthonormal termined
Sl
Volume
65. number
CHEMICAL PHYSICS LETI-ERS
i
functions are needed for describing several hundred me3sure~! term v3lues T _ “J, the system of linear equations for solving the coefficients of the orrhonormal functions is highly overdetermined and CZR be solved in a least squsres sense such that
approaches I minimum, where k stands for the different possible combinations of rorationai and vibntion31quantum nurllbers chancterizing the inithl and final states of t!ie measured line positions. The method for determining the vibrational qu3nturn numbers proceeds by llrst assigning the rotatianaI and vibmtiomd qutntum numbe-r, the vibr3tionat quantum nunlbcrs being given by ur + u, where u, is 3 running index and IJ is 3n rtrbitrxy addition31 integer which has to be determined in order to obtain the carrect vibr3tion31 numbering. In the nest step 3 Itxst wtuartr fit is performed of the measured line positions ~&ich are described
F= T’(u’.J3
by
- T”(u”,J.“),
lbhere for 3 tE stttc the term values T(u.s) 3re given by
T(u_J) =
2
--&(u) (u,
-I- u + i/y-
[J(J
•E
I)]”
_
(6)
in this nnnner one obtains for 3 reawn3ble range of u the corresponding sets of dix- (v) \\hich define the
term v&es T(u,J)_ TIE% term v&a art then used in the wri3tionaI method for crt!cuhting Ut,. Depending cm the 3ccurxy of the measurements dift-erent srandxd deviations s(c) are obtained fur diflisrent u xcwding IO eq_ (4). l’le correct vibrations1 numbering is giwn by ur + umptwh+re uupt is the v3Iue for which =!9) gwu throug!I 3 minimum_ Hence. the correct vibration31 quantum numbers 3re given for a wIur of u for which the mexared term v3lucs are most accuratcfy drucribcal by the quantum mccl~nicrti enerzy eigenvziues of 3 rot3ting vibrator 3.. defined by the hamittonisn in eq_ (I)_ The method m3y e3siIy be estended to more compIil;lted situlltions by 3dding the 3pproprhte terms in the h3miltonian, such 3s C1rsin 3 csse where the fine structure of multipIer spectra is significant _ In the following the method is applied to t\to different ases_
3-The A ‘Zz
1 Augst
1979
- X ‘22: system of the 4oCaz molecule
The 3bsorption spectrum of the A *2$-X t Ez system of the 3oC32 molecute w3s first analyzed by Balfour and Whitlock [4] who assigned 47 bands_ They were able to assign the u” quantum numbers from the corresponding u” progressions in the absorption b3nds but they encountered problems in assigning the u’ quantum numbers- Since they used only one isotopic species 3n evvluation of the vibntional isotope shift WJSnot possibIe. Hence the A r Zz state is 3ti interesting case for testing the new method_ Before applying the method of the previous section a problem h3s to be solved_ Balfour and Whitlock could not uniquely assign the J quantum numbers_ Their J numbering CJI~e3sily be verified by performing 3 least squares fit according to eqs. (5) and (6) For the correct numbering the standard deviation of the least squares tit goes through a minimum. This is the numerin1 3naIogue of the gnphical methods described by Heaberg [ 1 ] _ The results are shown in table ! for which J’ was changed in steps of 3 since for a L SC st3te of 3 molecule with zero nucIe3r spin J’ h3s to be odd_ It is importzmt to realize that 3 similar method is not c;lpable of finding the correct vibration31 numbering_ In 3 least squares fit of the measured line positions the tot31 standard deviation is ti~dcpemimt of the vibration numbering contrary to the rot3tion31 numbering_ Knowing the J quantum numbers a least squares fit ~3s performed for u = 0,l.Z where the ninnbering in tsbfe 3 of BaIfour 3nd Whitlock [4) ~3s used denoted by uB,,, and which cont3ins a!so the laser inTable 1 Stzuulxd deviation of rhe Isquares fit of thr”°Ca2 A-S sstsm for different J numberi~IB\y is the rotatiotul numbcrins by Balfour and Whitlock [4] ~--_ f ---
-
--
AF(cm-‘)
JB\V - -I
0.1174
JB\V - 2
0.0696
JBW
O.M6?
JB\V * 2
0.0649
JBW + 4
0.09s~
Volume 65. number 1
CHEMICAL
PHYSICS
Tabfe 2 Stsndxd deviation from the variational method for different vibrational numberings of the 4oCa2 A * X: and X ’ Z: states. UBQ is the vibrational numbering of Bnlfour and Whit&k [4]
LmERS
1 Au?rust 1979
Table 3 Standard deviations from the variational method for different vibrational numberings of the 6LiH A ’ \‘+ stdtc. 0~ is the vibrations1 numbering of Li and St~~alIe~ [Sl --___
v’or u”
AF(cm-‘)
”
-I)T; (cm-‘)
+W
O_O?-l2
“LS - *
0.635 0 Of6
f 1
0.0320
VLS
U’B&#-I-2 ,.
0.0557
“LS+
“BW
0.0353
U&V + 1 - ------
0.0776
u&r
uLS+?
--
duced fIuorescence lines measured by Vidal [9] _Wtth the Aj~ (u) the variational method w&s carried out and the results of rlF(u) are given in table 2 lchich shows that with u = 0 one obtains the correct vibrational numbering_ Table 2 contains also the results for the X * Zz state and verifies the assignment of Balfour and Whirl&k_ As an sdditionnl proof laser induced lluorescence of the %a, A-X system was performed where the measuredFran&-Condon factors clearIy confirm the assignment of table ?-The latter results will be described in a forthcoming paper (91 which also contains XII improved set of molecular constants AiX-_
5. Discussion It is interesting to compare the results ior the A states
of the 4oCa1 to tables
tnunl in AU(u)
of the 6LiH molecule
As n further test the A lZ+ state of the 6LiH molecule [5,6] WZISinvesrigdted. Already Crawford and Jorgensen [IO] noticed discrepancies of the isotope shifts for the lithium hydrides_ For rhe A tS+ state of the 6LiH_ 7LiH, 6LiD and 7LiD moIecules one obtains, for exunple. for utk+ 327_S, 2202.215.S and 236.3 cm-t_ respectively. whereas the X 1 Co+ state behaves quite normally and one obtains fo; .t&.+ 1319-4. lZGl9.4, 1319.5 and 1319.6 cm-t [6]_ Since again the minima of the A and X states rare strongly shifted with respect to each other and the u’ progressions camtot easily be folIowed to u’ = 0, some doubts may arise concerning the correct sssignment of the U’ quantmii numbers. Similar to the Cal molecule a least squares fit of the 6LiH A--X system WE performed using the vibrational numbering of Li and StwaIIey [5] and the u’
6.615
-
numbering WIS changed by u = --I ,O, 1.2. With these four different setsAik(u) the varirttional method [7] was carried out and the resulting AZ?(U) xe given in table 3. One cIearly recognizes that the u’ numbering of Li and StwalIey [?I] is correct_ Similar resuhs ha\e 31~0 been obtained for the other isotope combinations_ Details of the molecular constants and potential energy curves of the lithium hydrides will be presented in rl forthcoming paper [6] _
cording
4_ The A lZ*state
1.073
1
-
and the 6L~H -
2 and 3one
defining
notices
the cwrect
molecule_ that
Ac-
the mini-
u numbering
is
for the 6J_iH molecuIe titan for the 4oCa2 molecule. This is due IO the different sets of A, constants delkmg the measured term considerably
values
tnore
TUJ as expiained
In general, uniquely ues define
in the following.
An unperturbed
defined
nlolecule G, =E”
pronounced
the energy accrxding
eigenvdlues
(u + 1/2)i
The B, values are related
is
of :Iie roirtknless
_
(7)
to the expectation
values of
to
B, = (T/~;?~~)<~,,=~Iltr~1
= CAf, i
potential
to
J=O = ?_A,0
1 /r’? according
n~olecular
by the G, and B, values_ The G, val-
(U-I- 1/y
The lrttrer two relations
2~“,,= $
_ expressing
6) G, and B, in terms 53
Vohme 65, number 1
CHEhIICAL PHYSICS LE-I-I-ERS
of the A i. and -4iI coefficients, are not exact since the -_Ljkrareordy defined within the standard deviation of the measurements_ They rdso depend on the size of the dam Geld and the number of A, constmtts required for describing the measured line positions [7] _ In performing the Ieast sqturres fit according to eqs_ (5) and (6) one genemliy needs centrifugal distortion terms 0,. &_ L,, ___which are defined by rehttions similar to eqs_ (7) and (S) with k > 2 (see also Aibritton et rd. [ 1 I ])_ These centrifugal distortion terms, however, are redundant qurm:ities and do not add new physicai information on the electronic tmnsition of interestThe reason for this is that the line positions for J’ > 0 and J” > 0 can in principie also be calculated xcording to eq_ (5) where the term vslues are caicul~ted according to eqs_ (I)_ (2) and (3) and where U. is defined by the (;, and B, v&es_ -4 method of this kind was aIso proposed by Kirschner and Watson ] I?] _ It is now importrutt to reaiize that the method for determimng tfe vibrational quantum numbers rts proposed in this paper depends on the redundance of the centrifugal distortion terms_ In order to estmct the correct u numbering tkm the =lix (u) constxtts it tttrns out that the vMa!iot~l method [7] is the most powerful technique for esp!oiting this redundance_ It is, for esample_ by br superior to a technique which wou!d make use of the Dunham relations for De_ He7 ___and which_ in prxtice. is not unique since the measured term v&es cm frequently not be described by it perturbation approach which is the basis for the Dunham mIstions_ In compxing the A states of the 4OCa, - rutd of the 6LiH - molecule one notkxs that for-the mnge of measurements given in refs_ ]4.9] the 4oCa2 A state shows only ;f small degree of anharmonicity_ As a result, the m~~urements can be described by G:, B: and 0: terms only and the only redundant Aik consumts which are statistically signikmt, are AbZ,Ait and 11;~ [9] _ In the L;1seof “J._iH, however. the A state SOWS a very pronounced anharmonicity and for de-
1 August 1979
scribing the measurements a Iarge number of redundant centrifugal distortion terms is required, nameIy Ai (i< 71, Ai (i < 6) and Ai (i < 3) [6] _ It is this very strong redundance for the 6LiH A state which is the origin for the very pronounced minimum of U(u) in table 3_ In summary one should note that the method proposed in this paper is most powerful for situations which require a large number of redundant centrifugal distortion terms for describing the measured line positions and where one cannot make use of the isotope shift. The technique
presented
is based on a variation-
aI method which finds the vibrational which the measured by the quantum
numbering
for
term values TvJ are best described
mechanical
energy eigenvalues
EuJ of
a rotating vibrator_
References [ I] G. Hen&x%, Spectra of dhtontic molecules (Vxt Nostrand. Princeton. 1950). [2] WJ_ Bdfourand A.E. Dou$as, Can_J_ Ph)s_ 48 (1970) 901. 131 Ii_ Scheinzraber and CR. Vidal, J. Chem. Phys. 66
(1977) 3694. 141W.J_BaIfour and R.F_ WhitIock. Can. J. Phys. 53 (1975) 472_ 151 KC_ Li and W.C. Stwvalley.J. Yol. Spectry. 69 (197s) 294_ [ 6 1 C-R. Vi&I. KC_ Li. F-B_ Orth and W.C. Stwalley, J. Chem. Phyr, to be submitted for publication_ [?I CR. Vidal and Ii_ Scheingmber. J_ hIoL Spectry. 65 (1977) 46_ [St W.M. Kosman and J. Hinze, J_ Mol. Spectry. 56 (1975) 93. [QI CR_ Vidal. Chem Phys.. to be submitted for publication_ [ lOI FM. Crawford and T_ Jorgensen. Phys_ Re+_ 49 (1936) x5. [ 111 D-L. Albritton. WJ. Harrop. A-L_ Schmeltekopf and R-h’. Zare, J_ 1\!oLSpectry. 46 (1973) 25. [ 121 S-M. Kirschner and KG. Watson. J_ BloL Spectry. -li’ (1973) 334-5L(I974) 321. ,
DETERMINATION
CHEhlICAL
OF VIBRATIONAL
USING A VARIATIONAL
PHYSICS LETTERS
1 August 1979
QUANTUM NUMBERS
OF DIATOMIC MOLECULES
hfEI-HOD
Received 2 April 1979
The method is based on a variational method xxhich gives vibrationat quantum nun~brrsfor 1, hich the measured term v&es are described by the quantum mechanical energy eisenvatues of a rotating vibrator. The method is applied to the AtE* states of the 4oCaz .md of the 6LiH molecuk.
I_ Introduction The assignment of rotational and vibrational qunntunl numbers in the andysis of eIectrouic transitions of diatomic molecules has been extensively discussed by Herzberg [I] _However. there are situations where the methods described do not work. This may occur. for example, if the minima of the potential curves involved are considerably shifted with respect to each other. In this case the Fran&-Condon fxtors for the first members of individual progressions are frequently SO small that the fxrst member \\ith u = 0 is diffcuh to identify_ In some cases the largest Fran&--Condon overlap of the vibrational Ievel with u = 0 may occur with the dissociation continuum of the other molecular state as in the case of the hfgl A-X system [2,3] _ (For these situations a way out is generally provided by the isotope shift.) Hoxtever, there are spectra hke the 4oCa, A-X system [4] where data of only one isotope combination are available and where the u’ numbering is uncertain_ This spectrum will be discussed below. There are also situations where the observed isotope shifts do not agree with the simple theory because of the breakdown of the Born-Oppenheimer approsimation. This was found in the spectra of some hydrides and deuterides and there is some question whether the existin g u assignments are correct_ The A I C+ state of the 6tiH molecule wviiibe discussed below [5,6] _
2. Method The procedure for determining the vibrationaf quantum numbers makes use of a variational method described by Vidal aud Scheingraber 171 \xhich is based on an inverted perturbation approach proposed by Kosman and Hinze [Sl_ Within the Born-Oppenheimer approximation a diatomic molecule can be described as a roraring vibrator which obeys the radial Schrddinger equation (H, + H,,)
tid (r) =
&J
&J
Ir)
(!)
7
where EuJ is the energy eigenvalue specified b) the vibrational and rotational quantum numbers u snd J, Ho
is the hamiltonian
Ho = (fi[-k~c)
oi the tl0tlrOtatirig
d2/dr2 + U. (t-) ,
molecule
(2)
r is the mternuclear
distance. ir is the reduced m.w of the nuclei and Llo(r) is the potential energy of the roniorat:ot;iess molecule_ ifrot describes rhf rotational titin w-. ihe nuclei. For 3 f S state. it is x iven bv2
H mf = (fi/4ixK)J(J
+ I )/, 7-_
(3)
By nle.xns of a variational method [7] UO nosy be deso that the eisenvalues EvJ agree in J least squares sense with the measured term values ruJ- For this purpose a correction function 5.V, is provideti which, most conveniently, is given by a linear combination OF orthonormal functions such as Legendre polynomials. Since typicaliy about 20 orthonormal termined
Sl
Volume
65. number
CHEMICAL PHYSICS LETI-ERS
i
functions are needed for describing several hundred me3sure~! term v3lues T _ “J, the system of linear equations for solving the coefficients of the orrhonormal functions is highly overdetermined and CZR be solved in a least squsres sense such that
approaches I minimum, where k stands for the different possible combinations of rorationai and vibntion31quantum nurllbers chancterizing the inithl and final states of t!ie measured line positions. The method for determining the vibrational qu3nturn numbers proceeds by llrst assigning the rotatianaI and vibmtiomd qutntum numbe-r, the vibr3tionat quantum nunlbcrs being given by ur + u, where u, is 3 running index and IJ is 3n rtrbitrxy addition31 integer which has to be determined in order to obtain the carrect vibr3tion31 numbering. In the nest step 3 Itxst wtuartr fit is performed of the measured line positions ~&ich are described
F= T’(u’.J3
by
- T”(u”,J.“),
lbhere for 3 tE stttc the term values T(u.s) 3re given by
T(u_J) =
2
--&(u) (u,
-I- u + i/y-
[J(J
•E
I)]”
_
(6)
in this nnnner one obtains for 3 reawn3ble range of u the corresponding sets of dix- (v) \\hich define the
term v&es T(u,J)_ TIE% term v&a art then used in the wri3tionaI method for crt!cuhting Ut,. Depending cm the 3ccurxy of the measurements dift-erent srandxd deviations s(c) are obtained fur diflisrent u xcwding IO eq_ (4). l’le correct vibrations1 numbering is giwn by ur + umptwh+re uupt is the v3Iue for which =!9) gwu throug!I 3 minimum_ Hence. the correct vibration31 quantum numbers 3re given for a wIur of u for which the mexared term v3lucs are most accuratcfy drucribcal by the quantum mccl~nicrti enerzy eigenvziues of 3 rot3ting vibrator 3.. defined by the hamittonisn in eq_ (I)_ The method m3y e3siIy be estended to more compIil;lted situlltions by 3dding the 3pproprhte terms in the h3miltonian, such 3s C1rsin 3 csse where the fine structure of multipIer spectra is significant _ In the following the method is applied to t\to different ases_
3-The A ‘Zz
1 Augst
1979
- X ‘22: system of the 4oCaz molecule
The 3bsorption spectrum of the A *2$-X t Ez system of the 3oC32 molecute w3s first analyzed by Balfour and Whitlock [4] who assigned 47 bands_ They were able to assign the u” quantum numbers from the corresponding u” progressions in the absorption b3nds but they encountered problems in assigning the u’ quantum numbers- Since they used only one isotopic species 3n evvluation of the vibntional isotope shift WJSnot possibIe. Hence the A r Zz state is 3ti interesting case for testing the new method_ Before applying the method of the previous section a problem h3s to be solved_ Balfour and Whitlock could not uniquely assign the J quantum numbers_ Their J numbering CJI~e3sily be verified by performing 3 least squares fit according to eqs. (5) and (6) For the correct numbering the standard deviation of the least squares tit goes through a minimum. This is the numerin1 3naIogue of the gnphical methods described by Heaberg [ 1 ] _ The results are shown in table ! for which J’ was changed in steps of 3 since for a L SC st3te of 3 molecule with zero nucIe3r spin J’ h3s to be odd_ It is importzmt to realize that 3 similar method is not c;lpable of finding the correct vibration31 numbering_ In 3 least squares fit of the measured line positions the tot31 standard deviation is ti~dcpemimt of the vibration numbering contrary to the rot3tion31 numbering_ Knowing the J quantum numbers a least squares fit ~3s performed for u = 0,l.Z where the ninnbering in tsbfe 3 of BaIfour 3nd Whitlock [4) ~3s used denoted by uB,,, and which cont3ins a!so the laser inTable 1 Stzuulxd deviation of rhe Isquares fit of thr”°Ca2 A-S sstsm for different J numberi~IB\y is the rotatiotul numbcrins by Balfour and Whitlock [4] ~--_ f ---
-
--
AF(cm-‘)
JB\V - -I
0.1174
JB\V - 2
0.0696
JBW
O.M6?
JB\V * 2
0.0649
JBW + 4
0.09s~
Volume 65. number 1
CHEMICAL
PHYSICS
Tabfe 2 Stsndxd deviation from the variational method for different vibrational numberings of the 4oCa2 A * X: and X ’ Z: states. UBQ is the vibrational numbering of Bnlfour and Whit&k [4]
LmERS
1 Au?rust 1979
Table 3 Standard deviations from the variational method for different vibrational numberings of the 6LiH A ’ \‘+ stdtc. 0~ is the vibrations1 numbering of Li and St~~alIe~ [Sl --___
v’or u”
AF(cm-‘)
”
-I)T; (cm-‘)
+W
O_O?-l2
“LS - *
0.635 0 Of6
f 1
0.0320
VLS
U’B&#-I-2 ,.
0.0557
“LS+
“BW
0.0353
U&V + 1 - ------
0.0776
u&r
uLS+?
--
duced fIuorescence lines measured by Vidal [9] _Wtth the Aj~ (u) the variational method w&s carried out and the results of rlF(u) are given in table 2 lchich shows that with u = 0 one obtains the correct vibrational numbering_ Table 2 contains also the results for the X * Zz state and verifies the assignment of Balfour and Whirl&k_ As an sdditionnl proof laser induced lluorescence of the %a, A-X system was performed where the measuredFran&-Condon factors clearIy confirm the assignment of table ?-The latter results will be described in a forthcoming paper (91 which also contains XII improved set of molecular constants AiX-_
5. Discussion It is interesting to compare the results ior the A states
of the 4oCa1 to tables
tnunl in AU(u)
of the 6LiH molecule
As n further test the A lZ+ state of the 6LiH molecule [5,6] WZISinvesrigdted. Already Crawford and Jorgensen [IO] noticed discrepancies of the isotope shifts for the lithium hydrides_ For rhe A tS+ state of the 6LiH_ 7LiH, 6LiD and 7LiD moIecules one obtains, for exunple. for utk+ 327_S, 2202.215.S and 236.3 cm-t_ respectively. whereas the X 1 Co+ state behaves quite normally and one obtains fo; .t&.+ 1319-4. lZGl9.4, 1319.5 and 1319.6 cm-t [6]_ Since again the minima of the A and X states rare strongly shifted with respect to each other and the u’ progressions camtot easily be folIowed to u’ = 0, some doubts may arise concerning the correct sssignment of the U’ quantmii numbers. Similar to the Cal molecule a least squares fit of the 6LiH A--X system WE performed using the vibrational numbering of Li and StwaIIey [5] and the u’
6.615
-
numbering WIS changed by u = --I ,O, 1.2. With these four different setsAik(u) the varirttional method [7] was carried out and the resulting AZ?(U) xe given in table 3. One cIearly recognizes that the u’ numbering of Li and StwalIey [?I] is correct_ Similar resuhs ha\e 31~0 been obtained for the other isotope combinations_ Details of the molecular constants and potential energy curves of the lithium hydrides will be presented in rl forthcoming paper [6] _
cording
4_ The A lZ*state
1.073
1
-
and the 6L~H -
2 and 3one
defining
notices
the cwrect
molecule_ that
Ac-
the mini-
u numbering
is
for the 6J_iH molecuIe titan for the 4oCa2 molecule. This is due IO the different sets of A, constants delkmg the measured term considerably
values
tnore
TUJ as expiained
In general, uniquely ues define
in the following.
An unperturbed
defined
nlolecule G, =E”
pronounced
the energy accrxding
eigenvdlues
(u + 1/2)i
The B, values are related
is
of :Iie roirtknless
_
(7)
to the expectation
values of
to
B, = (T/~;?~~)<~,,=~Iltr~1
= CAf, i
potential
to
J=O = ?_A,0
1 /r’? according
n~olecular
by the G, and B, values_ The G, val-
(U-I- 1/y
The lrttrer two relations
2~“,,= $
_ expressing
6) G, and B, in terms 53
Vohme 65, number 1
CHEhIICAL PHYSICS LE-I-I-ERS
of the A i. and -4iI coefficients, are not exact since the -_Ljkrareordy defined within the standard deviation of the measurements_ They rdso depend on the size of the dam Geld and the number of A, constmtts required for describing the measured line positions [7] _ In performing the Ieast sqturres fit according to eqs_ (5) and (6) one genemliy needs centrifugal distortion terms 0,. &_ L,, ___which are defined by rehttions similar to eqs_ (7) and (S) with k > 2 (see also Aibritton et rd. [ 1 I ])_ These centrifugal distortion terms, however, are redundant qurm:ities and do not add new physicai information on the electronic tmnsition of interestThe reason for this is that the line positions for J’ > 0 and J” > 0 can in principie also be calculated xcording to eq_ (5) where the term vslues are caicul~ted according to eqs_ (I)_ (2) and (3) and where U. is defined by the (;, and B, v&es_ -4 method of this kind was aIso proposed by Kirschner and Watson ] I?] _ It is now importrutt to reaiize that the method for determimng tfe vibrational quantum numbers rts proposed in this paper depends on the redundance of the centrifugal distortion terms_ In order to estmct the correct u numbering tkm the =lix (u) constxtts it tttrns out that the vMa!iot~l method [7] is the most powerful technique for esp!oiting this redundance_ It is, for esample_ by br superior to a technique which wou!d make use of the Dunham relations for De_ He7 ___and which_ in prxtice. is not unique since the measured term v&es cm frequently not be described by it perturbation approach which is the basis for the Dunham mIstions_ In compxing the A states of the 4OCa, - rutd of the 6LiH - molecule one notkxs that for-the mnge of measurements given in refs_ ]4.9] the 4oCa2 A state shows only ;f small degree of anharmonicity_ As a result, the m~~urements can be described by G:, B: and 0: terms only and the only redundant Aik consumts which are statistically signikmt, are AbZ,Ait and 11;~ [9] _ In the L;1seof “J._iH, however. the A state SOWS a very pronounced anharmonicity and for de-
1 August 1979
scribing the measurements a Iarge number of redundant centrifugal distortion terms is required, nameIy Ai (i< 71, Ai (i < 6) and Ai (i < 3) [6] _ It is this very strong redundance for the 6LiH A state which is the origin for the very pronounced minimum of U(u) in table 3_ In summary one should note that the method proposed in this paper is most powerful for situations which require a large number of redundant centrifugal distortion terms for describing the measured line positions and where one cannot make use of the isotope shift. The technique
presented
is based on a variation-
aI method which finds the vibrational which the measured by the quantum
numbering
for
term values TvJ are best described
mechanical
energy eigenvalues
EuJ of
a rotating vibrator_
References [ I] G. Hen&x%, Spectra of dhtontic molecules (Vxt Nostrand. Princeton. 1950). [2] WJ_ Bdfourand A.E. Dou$as, Can_J_ Ph)s_ 48 (1970) 901. 131 Ii_ Scheinzraber and CR. Vidal, J. Chem. Phys. 66
(1977) 3694. 141W.J_BaIfour and R.F_ WhitIock. Can. J. Phys. 53 (1975) 472_ 151 KC_ Li and W.C. Stwvalley.J. Yol. Spectry. 69 (197s) 294_ [ 6 1 C-R. Vi&I. KC_ Li. F-B_ Orth and W.C. Stwalley, J. Chem. Phyr, to be submitted for publication_ [?I CR. Vidal and Ii_ Scheingmber. J_ hIoL Spectry. 65 (1977) 46_ [St W.M. Kosman and J. Hinze, J_ Mol. Spectry. 56 (1975) 93. [QI CR_ Vidal. Chem Phys.. to be submitted for publication_ [ lOI FM. Crawford and T_ Jorgensen. Phys_ Re+_ 49 (1936) x5. [ 111 D-L. Albritton. WJ. Harrop. A-L_ Schmeltekopf and R-h’. Zare, J_ 1\!oLSpectry. 46 (1973) 25. [ 121 S-M. Kirschner and KG. Watson. J_ BloL Spectry. -li’ (1973) 334-5L(I974) 321. ,