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Observed adiabatic corrections to the born-oppenheimer approximation for diatomic molecules with ten valence electrons
Chcmiczl Physics 67 (1982) 133-138 North-Hollmd Publishing Company
OBSERVED ADEABATXC CORRECTIONS FOR CUTOMC
TO THE BORN-OPPENHEIXER
APPROXMATION
MOLECULES WiTH TEN VALENCE ELECTRONS
E. TIEMANN
H. MST, Inditut
W.U. STIEDA, T. TOMUNG and J. HOEFT
~Tl?lr Moie.%illpIrysik, Freie UniversitPr Berlin, D.1000 Berlin 33, FRG
Received 18 December 1981
Us& millimeter-wave (mw) spectroscopq’ pure rotationnl transitions were measured with very high precision in scvcrzl vibration4 srates for many compounds of the goup III/VII and IV/VI dktomic molecules. The spectra were fitted to the usual Dunham espznsion adoptin? the normal mass relations for the Y/‘!x_ except for Yol in order to combine alI dnro of different isotopes for the sxne compound. For Yo1 the atomic mzss relation given by Watson is used which introduces phcnom. enological pxvneters 4el, Ab for molecule .4B bking the od;sbaric and nonndhbatic corrections to the Born-Oppenheimeropprosimstion into ayu;t. All observed spectra are well described by such 2 procedure. From these calculations the correctIon parameters Aol ~ s.O1 were ohtaked with an accuracy of = 10% or better. Using known values of the rotational,q fxtor and of the el?ctric dipole moment tl:e nonadhbatic part ws calculated and with this result the 3diaSatic put wzs evaluated from Ani for each atom. The ndiabatic correction does not change very much for one specific ztom bY V&Ting the chemical counterpart, and in geneml it is less :han 30% of the total correciion for this clan of molecules. The only exceptions are InI and the Tl md pb compounds for which the zdkbbntic corrections me obtained ten to hundred times Luger than those of the other compounds. No esplznntion is known for rhis behavior in the published literature.
1. Introductioil In recent years several papers [ 1,2] on the breakdown of the Born-Opper&eimer approximation appeared in the literature describing in much detail the correction terms to the Born-Oppenheimer approximation for the vibrational and rotational ener,gy of diatomic moiecules in the case of ‘11 electronic states. This theoretical analysis demonstrates that the ener_q eigenvalues of the vibrational-rotational motion of a diatonic molecule can be written as a power series in the vibrational quantum number u and the rotational quantum numberJ with Dunham.coefficients Ylk as parameters not only within the BomOppenheimer approximation but also outside this approximation. The difference of both representations appears in different mass relations of the Yi, for chulging the isotope combination of the same molec0301.0104/82/GOOO-0000/S
02.75 0 1982 North-Holland
ular compound. Within the Born-Oppenheimer approximation the Ylk are only functions of the reduced mass .U of the molecule M but in the general case the Yrk are functions of the atomic masses of atom A and atom 9, separately, including the adiabatic and nonadiabatic contribution from the breakdown of this approsimation. The new mars formulas were applied to the interpretation of observed rotational spectra only in few cases because of the lack of data which are precise enough to discriminate between the normal mass fu~lction within the Born-Oppenheimer approximation and the other function. First examples are given in ref. [la] using the measurements on CO, HCI and IX1 from ether laboratories. Combining infrared measurements and mw measurements very detailed resu!ts were obtained for the rotational constant as well as for vibrational constants in the case of CO
is
E. Tknann et al./il.diabatic corrections to the 80 approximation
molecule [3]. The analysis is extended in ref. [lb]. Heavier molecules GeS, GeSe, and PbS were studied in refs. [4,5] with pure mw spectroscopy. In these two publications unespecred large differences of the adiabatic corrections to the Bo_n-Opperzheimer approximation were found. No systematic stud>J concerning the variation of the correction terms within a group of similar molecules exists in the literature. Therefore the interpretation of the ma_nitude of the adiabatic correction is impossible or would be very speculative This paper describes miliimeter-wave spectra of several diatomic compounds of the group IV/VI continuing the work of refs. [LF,5j and it extends the systematic study to other diatomics with ten valence electrons like the thallium halides which belong to the group 111/W. These resuits will allow to compare the !arge adiabatic correction, observed for PbS [4], with similar molecules.
quency. The sample in the heated absorption celi was carefully outgassed to reduce perturbing gases and the vaporization temperature was chosen as low as possible for a detectable signzl. In this mamler the pressure broadening was minimized and a typical line width of ZOO-300 kHz was observed which $ves an increase in resolution up to a factor of 10 compared to earlier measurements on the same molecules from this laboratory (se2 references in ref. [7]).
3. Summary of the applied theory The theory for describing the breakdown of the Born-Oppenheimer approrcimaticn in Cbrarional and rotationa! enerm of a l X electronic state is carefully reviewed in ref. [la] and extended in ref. [lb]. The energy formulas used for fitting the obsen;ed spectra (selection rules AJ = 1~ 4u = 0) are: E(u, J) = h z
Y&J
-i ;)‘[J(J
+ I)]”
,
(1)
2. Experiment and To study the mass function of the rotarional parameters the rotarional transirion frequencies must be measured with a relative precision of the order of lo-G-IO-:, and to separate the simple mass variation due to the rotation--vibration interaction the measurements must be extended to several vibrational states and different rotational transitions. Because th2 espected additional ~nass shifts are small, it is necessary to observe as many isotopes as possible for a
good check of the consistency of the data. Consequently in order to invest&at2 highly excited vibratic& states and isotopes of low natural abundance, a spectrometer in the 100 GHz region with h&h sensitivity was desgined using the saturation modulation developed by Tarring [6] which makes the modulation effect nearly indzpendent of the rotational quantum numberJ. For this purpose th2 mw source is frequency modulated producing sidebands which ar2 far enough from the carrier to allow phase synchronisation of the mu’ source for precise frequency measurements. The heated absorption cell was constructed as a cylindrical wave guide for high transmission rate in the 100 GHz region. The cross ssction is 0.75 cmz and therefore large enou&h to avoid saturation broadening by typical input powers of ZiO mW for the carrier fre-
x [ 1 f (??Z,/N~)4;
+ (rn,,&j
j 4;9
.
(2)
Eq. (1) is the usual Dunham expansion of the energy in the vibrational state u and rotational stat2 J. Eq. (7) @ves the mass function of the Dunham coefficients Ylk, where p is the reduced mass of molecule _AB?Uik are the isotope independent mo!ecular parameters which are directly related to the Dunham potential coefficients oi (compare ref. [lb]), m,,MA andMB the masses of the electron, atom A and atom B, respectively, and 4; and 4Fk the correction terms for the specitic atom A or B. These A’s include the Dunham correcticn, which is normally very small, and the adiabatic and nonadiabatic contributions to the Born-Oppenheimer approximation. For pure rotational spectra only the parameter Yol is determinable with a precision which is high enough to derive the 4 paramerers. The higher expansion coefficients Ylk are normally obtained with a relative accuracy of less than 10-O which excludes the study of the A’s for these parameters. The 4,, can explicitly be separated into the adiabatic and nonadiabatic part [l] when the lowest Durham potential-coefficients, the rotationalg, factor, and the
135 electric dipole moment of the molecule is known (in previous papers the parameters d, , d, were used instead of A&, Ai ; they are connected by the relation AtiB = -3_d,*,): ~1A $”
$1
(wJ)~
= @&)“d f -,neBe’ + - ?tl
>
(3)
P
where (AtiB)ad is the pure adiabatic part to the Born-Oppenheimer approximation; AGT) the Dunham correction to the rotational constant B, (compare eq. (7) in ref. [la]); ?nP the mass of the proton; and (r.rgJ)x or tj.rgJ)B the isotopically independent values of pgJ referred to the nucleus -4 or B as origin, respectively (compare eq. (22) of ref. [la]). To calculate (K~)~ or (ggJ), fromgJ, one needs the electric dipole moment of the molecule. Because at present we are not interested in the inner structure of the correction terms we tise the abbreviations: Dunhx Au1 = A;; f Ao1
+ A;yld
,
for eqs. (3) and (4) with different parameters for the different atoms throuhout this paper. The isotopically invariant constant Uol is related to the rotational constant BE0 and can be used directly to determine the internuclear distance e” as model parameter within rhe Born-Oppenheimer approsimarion:
(5) and (rBo)2 e
!I
z-z-
ii
8r2~B,Bo
(6)
8n’ Uol .
From the measurements which will be discussed in the next section, we will derive the adiabatic contribution to the Born-Oppenheimer approximarion for the rotational constant Yal and the correction free equilibrium internucIear distance under the assumption that theAlk for the hi_&er expansion coefficients Ylk can be neglecred, therefore these Y,, are functions of the reduced mass p.
4. Measurements
and analysis
Fcr the diatomic compounds of the group IvjvI, new measurements were performed on SnS, SnSe, SnTe, PbSe and PbTe, and for the group HI/VII on GaF, InI, TIF, TIC!, TIBr, and T!I. Table 1 shows for each molecule the structure of the data set which is important for the correlation of the derived molecular parameters. The frequency lists are contained in refs. [8,9] and will not be reproduced here to save space. Most of the transition frequencies were recorded severaI times to get an estimate of‘ tfre statisticaI error for these measurements. The strong transitions (S/?J better than IO) were obtained with an absolute accuracy of better than 20 kHz, the
Table 1 Oveniew of measured transitions hiolccule
Number of isotopes
Mxx. vibr. state
Inten~al in rot. quantum nunrbor.l
;iumber of transitions
Typical XCUIJCY
s
6
snse SIlTe PbSc PbTe
8 13 11 10
8 6 11 12
7-13 19-29 30-44 23-36 37-58
97 58 114 75 178
2 x 10-y 2.5 x 10-7 2.4 x 10-1 25 x 10-T 2.5 x 10-T
GaF In1 TIF TIC1 TlBr TlI
2 2 2 3 4 2
3 16 4 7 8 13
O-5 30-51 2-8 18-21 25-44 42-69
19 88 4E 24 178 214
4 4 3 2.8 2.5 3
sns
x x x x x x
10-7 10-T 10-7 10-7 10-y 10-T
136
E. Tienmn er cl/Adicbdc
coirecrions to rhe 60 appro.xinwlion
ing eq. (1) nt different powers of the vibrational and rotational quantum numbers. The selected molecular parameter set describes the cbservations well within the experimental error, the standard deviation of the fit is about one half the primary error and the magni-
relative accuracy given in table 1 is ;u1 average over the whole data set. The measurements for ea-h molecule were fitted to eq. (1) in a combined least-squares procedtire for all isotopes, using the mass relations of eq. (2). Only for the parameter Y,, , the coefficients AA and AB
tude
were allowed
generated
to vary.
Table 2 Derived paramctcrs
pxxwrers
Various
fits were tried,
for the breakdown
truncnt-
of the Born-Oppenheimer
of each parameters standard
of tie
parameter.
spprosimaiion
belonging to atoms of the group III and VI -1.8454(36)“)
co/c CSIC SiS/Si GcSjGe G&e/Cc SnS/Sn SnSc/Sn SnTeJSn PbSjPb PhSc/Pb PbTe/Pb
-2.06 l(34) 2) -2.596(49)C) -1.392(593c) -1.463(7035) -1.612(46)6) -1.76(19) -1.555 (84) -1.749 (97) -12.94(1Sl>h) -i1.86(92) -11.98(51)
-o.o1526(6)b) -0.0148 -0.0106 + 0.0008 -0.0086 f 0.0058 -0.0076 -0.0114 +0.0090 -0.0236 -0.0366
-2.4758 -1.1760f) -1.2244 -1.2812 -1.0608 -1.1290 -1.1962 -1.2056 -1.1726 -1.1596
Gal=/Ga GZI/G3. InI/In TlF/Tl TlClpI TlBrjTl TU.Rl
-0.60 (30) -0.706(96)i) -2.68{27) -18.76(110) -18.96(200) -15.61(46) -14.68(47)
-0.093 -0.0182 -0.0066 +0.0100 -0.0040 -0.0216 -0.017’
-0.4384 d) -0.50(3O)W -0.30 (30) k) -0.1516 d) -0.lS60 d) -0.05 (30) k) -0.05(30) k,
pxamc:ers
is at least two times the
deviation
bclanginp
d) fI f) f) f) f) f) f) f)
-0.197(35) -0.10.5(55) -0.2G5(69) -0.239(70)
-0.3X(46) -0.70(19) -0.419(84) -0.541(97) -11.74(141) -10.66(92) -10.78(81)
1.1282291(14) 13348224(23) 1.9292639(19) 2.Oi20431(10) 2.1346018(10) x2089829(22) 3.3255738(11) 25227979(13) 2.2867844(40) 2.402!776(24) 25949253(27)
-0.07(30) -0.19(30) -X37(45) -18.61(110) -18.76(200) -15.55 (75) -14.61(75)
1.7743351(25) 2.5746263141)) 2.753618(!0) 2.084386(10) 2.484739(10) 2.6181114(15) 2.813614(10)
to atoms of rhe proup VI and VII
co/o CSIS SiS/S GeSiS SnS/S PbS:S G&/Se SnSc/Se PbSc/Se SnTz/Tc PbTe/Tc
aiis(47)a) -2.223 (98) C) -1.870(65)‘) -1.871(45) %) -1.821(65) -l.997(71)h) -2.014 (69)X) -2.134(50) -2X0(76) -1.653(97) -1.79-I(110)
-O.O1526(6)b) -0.0148 -0.0106 + 0.0008 + 0.0058 + 0.0090 -0.0086 -0.0076 -0.0236 -0.0114 -0.0366
-1.8072(36)b) -1.9446 d) -1.5494 f) -1.6%4 0 -1.6602 0 -1.8594 0 -1.6028 f) -1.63400 -1.7430 0 -1.55764 -1.5978 fj
-0.296(50) -0.264 (110) -0.310 (75) -0.233(45) -0.167 (65) -0.147 (71) -0.403(69) -0.482(50) -0.353(76) -0.034(97) -0.160(110)
TICI/CI TlBr/Br
-1.243(49) -1.138(64)
-0.0040 -0.0126
-0.9426 d) -o.i64(300)k)
-0.296 (50) -0.361(300)
3) Ref. [3]. b) Ref. [la). c, Rotational frequencies from ref. [lj]. d) g factor and dipole moment from ref. [ 121. e) Rotntional frequencies from ref. j16]. ‘)g factors from ref. [ll] and elccrricdipolc moments from ref. [12]. g) Rotational frequencies from ref. is]. h) Rotational frequencies from mf. [4]. I) Rotational frcqucncics from ref. [?7]. k) Es:ima?ed values using estrapoked go factors or/and e!ectric dipolc moments; for further detaiis see text.
E. Tiemanx et aI.f;ldiabatic
corrections
Because this report concentrates on the systematic study of the adiabatic corrections, the final results for the parameters Ylk will be given in a forthcoming paper, in order to discuss more extensively the potential coefficients Gi, which are derivable from the rotational constants Y,. The Aol parameters are obtained with an accuracy of 19% or better except for GaF. But one should note, that such detailed analysis relies completely on the precision of the atomic masses. The calculations were carried out with atomic masses from the tables of Wapstra and Bos [IO], the relative precision of which is at least &n/m = 1O-7, and therefore should be sufficient for the analysis of our data with an WZTage precision of 6 v/v= 3 X 1W7, according to table 1. For all molecules observed in thk report, the rotationalgJ factors [ 11,121and the electric dipole moments [I?] are known or can be estimated from systematic trends within groups of similar molecules. Therefore, the nonadiabatic part of eqs. (3) and (4) can be calculated, and the small Dunham correction is derivable from the parameters Yik which were obtained by the least-squares fit of the rotational frequencies. .4s the final result, we get the adiabatic contributions to the Born-Oppenheimer approximation. .&ll these evaluations are combined in table 2, which also shows the model parameter $” as explained in the theoretical section; the conversion factor 505379.045 &!Hz A2 amu is used. Table 2 (upper part) reports the derived values for the metalic atoms, group IV and III, and (lower part) for the nonmetalic atoms, group VI apd VII, which in general is the negatively charged side of the molecule. Table 2 includes results from calculations, using measurements from other authors in order to give a complete overview of the diatomics with ten valence electrons.
5. Discussion The group N/VI compounds are most extensively studied and will give the best possibility for systematic comparison. In all cases, the Dunhan corrections are small compared to the observed isotope shifts Aol, and the uncertainty of this correction do& not contribute significantly to the accuracy of the evaluation of the adiabatic correction term.
137
to the Bi3 approxirrztion
The nonadiabatic part can be calculated with the known values of the go factor and the electric dipole moment with an accuracy better than I%, which is higher than that ofthe total isoto c shift Afil. There8. fore the adiabatic correction Ai IS determinable with nearly the same absolute error as bl in the case of group IV/VI compounds. Looking for systematic trends, one immediately realizes that A$ is not much dependent on the chemical binding partner in the molecule, i.e., A;: is more like an atomic parameter than a molecular one within the accuracy of our experiments. The average values are:
A;;
C
Si
Ce
Sn
Pb
-0.15
-0.21
-0.28
-0.55
-11.0
and
Agf :
0
s
-0.30
-0.24
Se
.___Te -0.41 -0.12
Neglecting the very hi& value of pb, there is no much variation of these parameters between the atoms at all. This behavior was predicted by Watson [la] I assuming that the matrix elements of the total electronic momentum and the nuclear momentum for the electronic wavefunction have a similar dependence on the internuclear distance. Taking the wobble-stretch theory [ 131 for an approximation of the adiabatic correction terms and ihe estimation [la] of Zhe matrix element of the operator (Lz +L;) of the electronic angu!ar momentum+ one fmds:
x (z&W:
iZB&
- g#np)
,
(7)
where izAv is the ener,7 difference of the electronic grour,d state to first excited statzs R = 1 with the same parity as the ground state and Z,, ZB are the nuclear charges of atoms i+,and B, respectively. Eq. (7) gives the right order of magnitude as it is observed except for the Pb-atom. That value is about 20 times larger than the expected and the other observed ones. There is no obvious reason, why the estimation of the adiabatic correction through the wobble-strerch model should breakdown in the case of Pb. On the other hand, Watson [ 1b] indicited that the A parameters can become unusually large, if the
related parameter U is accidentally brnall due to the combination of the potential coefficients ai. But this cannot happen for the rotational parameter U,, , bccauseit is determined by the moment of inertia of the molecule and not by the potential coefficients ui_ Before comparing the systematic trends just described with those in the group III/VII compounds, some remarks on the evduatj.on of the nonadiabatic correction terms are necessaX~. Only for GaF, TIF and Tic1 theg, factors and the electric dipole moments are known f:om the literature. The missing parameters for the other compounds were estimated using model descriptions like tlrose given by Hoeft et al. [7] for the electric dipole moment of the group IV/VI compounds or comparing thegJ factors with those of the ionic model. The assumed values are GaI:
pcl = 7_.9(5) D;
gJ = -C.15(15)
Inl:
pel = 3.8(5) D; gJ =-0.01
gJ= -0.01(l)
TII:
gJ = -0.006(6)
Acknowledgement This work has been performed within the research program of the Sonderforschungsbereich 16 1 “Hyperfeinwechselwirkungen” which is supported by Deutsche Forschungsgemeinschaft.
References
[ 1] (a) J.K.G. Watson, J. ?&I. Spectry. 45 (1973) 99;
(1))
TlSr: y,! from ref. [l?]; pcl from ref. [ I?];
,
the adiabatic correction in those cases. In a paper by Schlembach and Tiemann [ 141 the influence of the finite nuclear size on the Coulomb potential for the electrons will be discussed as an explanation of the unusually large values of At: _
i .
The error limiz are chosen large enough to account for systematic model errors. .ils far as one can speak from any systemaiics within the few examples of the group HI/VII, the similarity between the results of this group to those of the group IV/VI is striking. For Ga, Cl and Br we fmd the same magnitude as it is explainable with the simple re!ation (7). tndium shows a slight increase of 4;:) but for thallium the value of 4;: is more than a factor of 10 larger than rhe expected one. Therefore such an increase of 4;;’ seems to be a more general behavior for the heavy atoms. In conclusion, the adiabatic correction to the Born-Oppenheimer approsimacion is smaller than the nonadiabatic one in most cases and shows little influence of the chemical binding partner. The values i’o: diffwxt atoms do not vary much. But for the atoms Pb and Tl a great increase of 4;; is observed and this behavior is unexplained by the existing theory. It suggests that an assumption in developing the effective hamiltonian cr in writing up the ab initio hamilronian [l ,z] is no longer valid and the isotope shift encountered in 4:: has no connection with
(b) J&G. Watson, J. Mol. Spectry. 80 (19SO) 411. [Z] P.R. Bunker, J. Mol. Spectra. 68 (19i7) 367, and references therein. [3] RM. Dole; If. Hermq J.W.C. Johns, A.R.W. >$,cKeUar, S.Nsgler and I.K_M.Srmthy, Can. J. Phys. 57 (1979) 677. [4] E. Tiemann, W.U. Stieda, T. Tarring and J. Hoeft, Z. Naturforsch. 3Oa (1975) 1606. [j] W.U. Stieda, E. Tiemann, T. Tiirriq, J. Hoeft, Z. Naturforsch. 313 (1976) 374. [6] T. T&ring, J. Xlol. Spectry. 48 (19731 148. [7] J.Hocft,FJ.Lovas,E.TiemannandT.TGrring,J.Chem. Phys. 53 (1970) 2736. [S] W.U. Sticdq Tl~esis, Freie Universit~t Berlin, Germmy (1976). [9] H. Amst,Thesis, Freie UniversitSt Berlin, Germany (1977). [IO] AH. Wnpstra and K. Bos, At. Dntn Nucl. Data Tables 13 (1957) 177. [ 1l] R. Honerjger and R. Tischer, Z. Naturforsch. 32a (1977) i. [ 121 F-T. Lovx and E. Tiemann, J. Phys. Chem. Ref. Data 3 (1974j 609. [ 131 B. Rosenblum, AH. Nethercot and C.H. Townes, Phys. Reu. 109 (1958) 400. [ 141 J. Schirmbnch, E. Tiemann, Chcm. Phys., to be published. [ 151 R. Bustreel, C. Demnynck-Marl&e, J.L. Destombes and G. Joumel, Chem. Phys. Letters 67 (1973) 178; M. Bogey, C. Demnynck and J.L. Destombes, Chem. Phys.Letters 81 (1981) 256. [16] E. Tiemann, E. Rewanz, J. Hoeft and T. TBrring, Z. Naturforsch. 2?a (1972) 1566. [ 171 K9.R. Nair, H.-U. SchUtzePrthLmmarm end J. Hozft, Chem. Phys. Letters 80 (1981) 149.
OBSERVED ADEABATXC CORRECTIONS FOR CUTOMC
TO THE BORN-OPPENHEIXER
APPROXMATION
MOLECULES WiTH TEN VALENCE ELECTRONS
E. TIEMANN
H. MST, Inditut
W.U. STIEDA, T. TOMUNG and J. HOEFT
~Tl?lr Moie.%illpIrysik, Freie UniversitPr Berlin, D.1000 Berlin 33, FRG
Received 18 December 1981
Us& millimeter-wave (mw) spectroscopq’ pure rotationnl transitions were measured with very high precision in scvcrzl vibration4 srates for many compounds of the goup III/VII and IV/VI dktomic molecules. The spectra were fitted to the usual Dunham espznsion adoptin? the normal mass relations for the Y/‘!x_ except for Yol in order to combine alI dnro of different isotopes for the sxne compound. For Yo1 the atomic mzss relation given by Watson is used which introduces phcnom. enological pxvneters 4el, Ab for molecule .4B bking the od;sbaric and nonndhbatic corrections to the Born-Oppenheimeropprosimstion into ayu;t. All observed spectra are well described by such 2 procedure. From these calculations the correctIon parameters Aol ~ s.O1 were ohtaked with an accuracy of = 10% or better. Using known values of the rotational,q fxtor and of the el?ctric dipole moment tl:e nonadhbatic part ws calculated and with this result the 3diaSatic put wzs evaluated from Ani for each atom. The ndiabatic correction does not change very much for one specific ztom bY V&Ting the chemical counterpart, and in geneml it is less :han 30% of the total correciion for this clan of molecules. The only exceptions are InI and the Tl md pb compounds for which the zdkbbntic corrections me obtained ten to hundred times Luger than those of the other compounds. No esplznntion is known for rhis behavior in the published literature.
1. Introductioil In recent years several papers [ 1,2] on the breakdown of the Born-Opper&eimer approximation appeared in the literature describing in much detail the correction terms to the Born-Oppenheimer approximation for the vibrational and rotational ener,gy of diatomic moiecules in the case of ‘11 electronic states. This theoretical analysis demonstrates that the ener_q eigenvalues of the vibrational-rotational motion of a diatonic molecule can be written as a power series in the vibrational quantum number u and the rotational quantum numberJ with Dunham.coefficients Ylk as parameters not only within the BomOppenheimer approximation but also outside this approximation. The difference of both representations appears in different mass relations of the Yi, for chulging the isotope combination of the same molec0301.0104/82/GOOO-0000/S
02.75 0 1982 North-Holland
ular compound. Within the Born-Oppenheimer approximation the Ylk are only functions of the reduced mass .U of the molecule M but in the general case the Yrk are functions of the atomic masses of atom A and atom 9, separately, including the adiabatic and nonadiabatic contribution from the breakdown of this approsimation. The new mars formulas were applied to the interpretation of observed rotational spectra only in few cases because of the lack of data which are precise enough to discriminate between the normal mass fu~lction within the Born-Oppenheimer approximation and the other function. First examples are given in ref. [la] using the measurements on CO, HCI and IX1 from ether laboratories. Combining infrared measurements and mw measurements very detailed resu!ts were obtained for the rotational constant as well as for vibrational constants in the case of CO
is
E. Tknann et al./il.diabatic corrections to the 80 approximation
molecule [3]. The analysis is extended in ref. [lb]. Heavier molecules GeS, GeSe, and PbS were studied in refs. [4,5] with pure mw spectroscopy. In these two publications unespecred large differences of the adiabatic corrections to the Bo_n-Opperzheimer approximation were found. No systematic stud>J concerning the variation of the correction terms within a group of similar molecules exists in the literature. Therefore the interpretation of the ma_nitude of the adiabatic correction is impossible or would be very speculative This paper describes miliimeter-wave spectra of several diatomic compounds of the group IV/VI continuing the work of refs. [LF,5j and it extends the systematic study to other diatomics with ten valence electrons like the thallium halides which belong to the group 111/W. These resuits will allow to compare the !arge adiabatic correction, observed for PbS [4], with similar molecules.
quency. The sample in the heated absorption celi was carefully outgassed to reduce perturbing gases and the vaporization temperature was chosen as low as possible for a detectable signzl. In this mamler the pressure broadening was minimized and a typical line width of ZOO-300 kHz was observed which $ves an increase in resolution up to a factor of 10 compared to earlier measurements on the same molecules from this laboratory (se2 references in ref. [7]).
3. Summary of the applied theory The theory for describing the breakdown of the Born-Oppenheimer approrcimaticn in Cbrarional and rotationa! enerm of a l X electronic state is carefully reviewed in ref. [la] and extended in ref. [lb]. The energy formulas used for fitting the obsen;ed spectra (selection rules AJ = 1~ 4u = 0) are: E(u, J) = h z
Y&J
-i ;)‘[J(J
+ I)]”
,
(1)
2. Experiment and To study the mass function of the rotarional parameters the rotarional transirion frequencies must be measured with a relative precision of the order of lo-G-IO-:, and to separate the simple mass variation due to the rotation--vibration interaction the measurements must be extended to several vibrational states and different rotational transitions. Because th2 espected additional ~nass shifts are small, it is necessary to observe as many isotopes as possible for a
good check of the consistency of the data. Consequently in order to invest&at2 highly excited vibratic& states and isotopes of low natural abundance, a spectrometer in the 100 GHz region with h&h sensitivity was desgined using the saturation modulation developed by Tarring [6] which makes the modulation effect nearly indzpendent of the rotational quantum numberJ. For this purpose th2 mw source is frequency modulated producing sidebands which ar2 far enough from the carrier to allow phase synchronisation of the mu’ source for precise frequency measurements. The heated absorption cell was constructed as a cylindrical wave guide for high transmission rate in the 100 GHz region. The cross ssction is 0.75 cmz and therefore large enou&h to avoid saturation broadening by typical input powers of ZiO mW for the carrier fre-
x [ 1 f (??Z,/N~)4;
+ (rn,,&j
j 4;9
.
(2)
Eq. (1) is the usual Dunham expansion of the energy in the vibrational state u and rotational stat2 J. Eq. (7) @ves the mass function of the Dunham coefficients Ylk, where p is the reduced mass of molecule _AB?Uik are the isotope independent mo!ecular parameters which are directly related to the Dunham potential coefficients oi (compare ref. [lb]), m,,MA andMB the masses of the electron, atom A and atom B, respectively, and 4; and 4Fk the correction terms for the specitic atom A or B. These A’s include the Dunham correcticn, which is normally very small, and the adiabatic and nonadiabatic contributions to the Born-Oppenheimer approximation. For pure rotational spectra only the parameter Yol is determinable with a precision which is high enough to derive the 4 paramerers. The higher expansion coefficients Ylk are normally obtained with a relative accuracy of less than 10-O which excludes the study of the A’s for these parameters. The 4,, can explicitly be separated into the adiabatic and nonadiabatic part [l] when the lowest Durham potential-coefficients, the rotationalg, factor, and the
135 electric dipole moment of the molecule is known (in previous papers the parameters d, , d, were used instead of A&, Ai ; they are connected by the relation AtiB = -3_d,*,): ~1A $”
$1
(wJ)~
= @&)“d f -,neBe’ + - ?tl
>
(3)
P
where (AtiB)ad is the pure adiabatic part to the Born-Oppenheimer approximation; AGT) the Dunham correction to the rotational constant B, (compare eq. (7) in ref. [la]); ?nP the mass of the proton; and (r.rgJ)x or tj.rgJ)B the isotopically independent values of pgJ referred to the nucleus -4 or B as origin, respectively (compare eq. (22) of ref. [la]). To calculate (K~)~ or (ggJ), fromgJ, one needs the electric dipole moment of the molecule. Because at present we are not interested in the inner structure of the correction terms we tise the abbreviations: Dunhx Au1 = A;; f Ao1
+ A;yld
,
for eqs. (3) and (4) with different parameters for the different atoms throuhout this paper. The isotopically invariant constant Uol is related to the rotational constant BE0 and can be used directly to determine the internuclear distance e” as model parameter within rhe Born-Oppenheimer approsimarion:
(5) and (rBo)2 e
!I
z-z-
ii
8r2~B,Bo
(6)
8n’ Uol .
From the measurements which will be discussed in the next section, we will derive the adiabatic contribution to the Born-Oppenheimer approximarion for the rotational constant Yal and the correction free equilibrium internucIear distance under the assumption that theAlk for the hi_&er expansion coefficients Ylk can be neglecred, therefore these Y,, are functions of the reduced mass p.
4. Measurements
and analysis
Fcr the diatomic compounds of the group IvjvI, new measurements were performed on SnS, SnSe, SnTe, PbSe and PbTe, and for the group HI/VII on GaF, InI, TIF, TIC!, TIBr, and T!I. Table 1 shows for each molecule the structure of the data set which is important for the correlation of the derived molecular parameters. The frequency lists are contained in refs. [8,9] and will not be reproduced here to save space. Most of the transition frequencies were recorded severaI times to get an estimate of‘ tfre statisticaI error for these measurements. The strong transitions (S/?J better than IO) were obtained with an absolute accuracy of better than 20 kHz, the
Table 1 Oveniew of measured transitions hiolccule
Number of isotopes
Mxx. vibr. state
Inten~al in rot. quantum nunrbor.l
;iumber of transitions
Typical XCUIJCY
s
6
snse SIlTe PbSc PbTe
8 13 11 10
8 6 11 12
7-13 19-29 30-44 23-36 37-58
97 58 114 75 178
2 x 10-y 2.5 x 10-7 2.4 x 10-1 25 x 10-T 2.5 x 10-T
GaF In1 TIF TIC1 TlBr TlI
2 2 2 3 4 2
3 16 4 7 8 13
O-5 30-51 2-8 18-21 25-44 42-69
19 88 4E 24 178 214
4 4 3 2.8 2.5 3
sns
x x x x x x
10-7 10-T 10-7 10-7 10-y 10-T
136
E. Tienmn er cl/Adicbdc
coirecrions to rhe 60 appro.xinwlion
ing eq. (1) nt different powers of the vibrational and rotational quantum numbers. The selected molecular parameter set describes the cbservations well within the experimental error, the standard deviation of the fit is about one half the primary error and the magni-
relative accuracy given in table 1 is ;u1 average over the whole data set. The measurements for ea-h molecule were fitted to eq. (1) in a combined least-squares procedtire for all isotopes, using the mass relations of eq. (2). Only for the parameter Y,, , the coefficients AA and AB
tude
were allowed
generated
to vary.
Table 2 Derived paramctcrs
pxxwrers
Various
fits were tried,
for the breakdown
truncnt-
of the Born-Oppenheimer
of each parameters standard
of tie
parameter.
spprosimaiion
belonging to atoms of the group III and VI -1.8454(36)“)
co/c CSIC SiS/Si GcSjGe G&e/Cc SnS/Sn SnSc/Sn SnTeJSn PbSjPb PhSc/Pb PbTe/Pb
-2.06 l(34) 2) -2.596(49)C) -1.392(593c) -1.463(7035) -1.612(46)6) -1.76(19) -1.555 (84) -1.749 (97) -12.94(1Sl>h) -i1.86(92) -11.98(51)
-o.o1526(6)b) -0.0148 -0.0106 + 0.0008 -0.0086 f 0.0058 -0.0076 -0.0114 +0.0090 -0.0236 -0.0366
-2.4758 -1.1760f) -1.2244 -1.2812 -1.0608 -1.1290 -1.1962 -1.2056 -1.1726 -1.1596
Gal=/Ga GZI/G3. InI/In TlF/Tl TlClpI TlBrjTl TU.Rl
-0.60 (30) -0.706(96)i) -2.68{27) -18.76(110) -18.96(200) -15.61(46) -14.68(47)
-0.093 -0.0182 -0.0066 +0.0100 -0.0040 -0.0216 -0.017’
-0.4384 d) -0.50(3O)W -0.30 (30) k) -0.1516 d) -0.lS60 d) -0.05 (30) k) -0.05(30) k,
pxamc:ers
is at least two times the
deviation
bclanginp
d) fI f) f) f) f) f) f) f)
-0.197(35) -0.10.5(55) -0.2G5(69) -0.239(70)
-0.3X(46) -0.70(19) -0.419(84) -0.541(97) -11.74(141) -10.66(92) -10.78(81)
1.1282291(14) 13348224(23) 1.9292639(19) 2.Oi20431(10) 2.1346018(10) x2089829(22) 3.3255738(11) 25227979(13) 2.2867844(40) 2.402!776(24) 25949253(27)
-0.07(30) -0.19(30) -X37(45) -18.61(110) -18.76(200) -15.55 (75) -14.61(75)
1.7743351(25) 2.5746263141)) 2.753618(!0) 2.084386(10) 2.484739(10) 2.6181114(15) 2.813614(10)
to atoms of rhe proup VI and VII
co/o CSIS SiS/S GeSiS SnS/S PbS:S G&/Se SnSc/Se PbSc/Se SnTz/Tc PbTe/Tc
aiis(47)a) -2.223 (98) C) -1.870(65)‘) -1.871(45) %) -1.821(65) -l.997(71)h) -2.014 (69)X) -2.134(50) -2X0(76) -1.653(97) -1.79-I(110)
-O.O1526(6)b) -0.0148 -0.0106 + 0.0008 + 0.0058 + 0.0090 -0.0086 -0.0076 -0.0236 -0.0114 -0.0366
-1.8072(36)b) -1.9446 d) -1.5494 f) -1.6%4 0 -1.6602 0 -1.8594 0 -1.6028 f) -1.63400 -1.7430 0 -1.55764 -1.5978 fj
-0.296(50) -0.264 (110) -0.310 (75) -0.233(45) -0.167 (65) -0.147 (71) -0.403(69) -0.482(50) -0.353(76) -0.034(97) -0.160(110)
TICI/CI TlBr/Br
-1.243(49) -1.138(64)
-0.0040 -0.0126
-0.9426 d) -o.i64(300)k)
-0.296 (50) -0.361(300)
3) Ref. [3]. b) Ref. [la). c, Rotational frequencies from ref. [lj]. d) g factor and dipole moment from ref. [ 121. e) Rotntional frequencies from ref. j16]. ‘)g factors from ref. [ll] and elccrricdipolc moments from ref. [12]. g) Rotational frequencies from ref. is]. h) Rotational frequencies from mf. [4]. I) Rotational frcqucncics from ref. [?7]. k) Es:ima?ed values using estrapoked go factors or/and e!ectric dipolc moments; for further detaiis see text.
E. Tiemanx et aI.f;ldiabatic
corrections
Because this report concentrates on the systematic study of the adiabatic corrections, the final results for the parameters Ylk will be given in a forthcoming paper, in order to discuss more extensively the potential coefficients Gi, which are derivable from the rotational constants Y,. The Aol parameters are obtained with an accuracy of 19% or better except for GaF. But one should note, that such detailed analysis relies completely on the precision of the atomic masses. The calculations were carried out with atomic masses from the tables of Wapstra and Bos [IO], the relative precision of which is at least &n/m = 1O-7, and therefore should be sufficient for the analysis of our data with an WZTage precision of 6 v/v= 3 X 1W7, according to table 1. For all molecules observed in thk report, the rotationalgJ factors [ 11,121and the electric dipole moments [I?] are known or can be estimated from systematic trends within groups of similar molecules. Therefore, the nonadiabatic part of eqs. (3) and (4) can be calculated, and the small Dunham correction is derivable from the parameters Yik which were obtained by the least-squares fit of the rotational frequencies. .4s the final result, we get the adiabatic contributions to the Born-Oppenheimer approximation. .&ll these evaluations are combined in table 2, which also shows the model parameter $” as explained in the theoretical section; the conversion factor 505379.045 &!Hz A2 amu is used. Table 2 (upper part) reports the derived values for the metalic atoms, group IV and III, and (lower part) for the nonmetalic atoms, group VI apd VII, which in general is the negatively charged side of the molecule. Table 2 includes results from calculations, using measurements from other authors in order to give a complete overview of the diatomics with ten valence electrons.
5. Discussion The group N/VI compounds are most extensively studied and will give the best possibility for systematic comparison. In all cases, the Dunhan corrections are small compared to the observed isotope shifts Aol, and the uncertainty of this correction do& not contribute significantly to the accuracy of the evaluation of the adiabatic correction term.
137
to the Bi3 approxirrztion
The nonadiabatic part can be calculated with the known values of the go factor and the electric dipole moment with an accuracy better than I%, which is higher than that ofthe total isoto c shift Afil. There8. fore the adiabatic correction Ai IS determinable with nearly the same absolute error as bl in the case of group IV/VI compounds. Looking for systematic trends, one immediately realizes that A$ is not much dependent on the chemical binding partner in the molecule, i.e., A;: is more like an atomic parameter than a molecular one within the accuracy of our experiments. The average values are:
A;;
C
Si
Ce
Sn
Pb
-0.15
-0.21
-0.28
-0.55
-11.0
and
Agf :
0
s
-0.30
-0.24
Se
.___Te -0.41 -0.12
Neglecting the very hi& value of pb, there is no much variation of these parameters between the atoms at all. This behavior was predicted by Watson [la] I assuming that the matrix elements of the total electronic momentum and the nuclear momentum for the electronic wavefunction have a similar dependence on the internuclear distance. Taking the wobble-stretch theory [ 131 for an approximation of the adiabatic correction terms and ihe estimation [la] of Zhe matrix element of the operator (Lz +L;) of the electronic angu!ar momentum+ one fmds:
x (z&W:
iZB&
- g#np)
,
(7)
where izAv is the ener,7 difference of the electronic grour,d state to first excited statzs R = 1 with the same parity as the ground state and Z,, ZB are the nuclear charges of atoms i+,and B, respectively. Eq. (7) gives the right order of magnitude as it is observed except for the Pb-atom. That value is about 20 times larger than the expected and the other observed ones. There is no obvious reason, why the estimation of the adiabatic correction through the wobble-strerch model should breakdown in the case of Pb. On the other hand, Watson [ 1b] indicited that the A parameters can become unusually large, if the
related parameter U is accidentally brnall due to the combination of the potential coefficients ai. But this cannot happen for the rotational parameter U,, , bccauseit is determined by the moment of inertia of the molecule and not by the potential coefficients ui_ Before comparing the systematic trends just described with those in the group III/VII compounds, some remarks on the evduatj.on of the nonadiabatic correction terms are necessaX~. Only for GaF, TIF and Tic1 theg, factors and the electric dipole moments are known f:om the literature. The missing parameters for the other compounds were estimated using model descriptions like tlrose given by Hoeft et al. [7] for the electric dipole moment of the group IV/VI compounds or comparing thegJ factors with those of the ionic model. The assumed values are GaI:
pcl = 7_.9(5) D;
gJ = -C.15(15)
Inl:
pel = 3.8(5) D; gJ =-0.01
gJ= -0.01(l)
TII:
gJ = -0.006(6)
Acknowledgement This work has been performed within the research program of the Sonderforschungsbereich 16 1 “Hyperfeinwechselwirkungen” which is supported by Deutsche Forschungsgemeinschaft.
References
[ 1] (a) J.K.G. Watson, J. ?&I. Spectry. 45 (1973) 99;
(1))
TlSr: y,! from ref. [l?]; pcl from ref. [ I?];
,
the adiabatic correction in those cases. In a paper by Schlembach and Tiemann [ 141 the influence of the finite nuclear size on the Coulomb potential for the electrons will be discussed as an explanation of the unusually large values of At: _
i .
The error limiz are chosen large enough to account for systematic model errors. .ils far as one can speak from any systemaiics within the few examples of the group HI/VII, the similarity between the results of this group to those of the group IV/VI is striking. For Ga, Cl and Br we fmd the same magnitude as it is explainable with the simple re!ation (7). tndium shows a slight increase of 4;:) but for thallium the value of 4;: is more than a factor of 10 larger than rhe expected one. Therefore such an increase of 4;;’ seems to be a more general behavior for the heavy atoms. In conclusion, the adiabatic correction to the Born-Oppenheimer approsimacion is smaller than the nonadiabatic one in most cases and shows little influence of the chemical binding partner. The values i’o: diffwxt atoms do not vary much. But for the atoms Pb and Tl a great increase of 4;; is observed and this behavior is unexplained by the existing theory. It suggests that an assumption in developing the effective hamiltonian cr in writing up the ab initio hamilronian [l ,z] is no longer valid and the isotope shift encountered in 4:: has no connection with
(b) J&G. Watson, J. Mol. Spectry. 80 (19SO) 411. [Z] P.R. Bunker, J. Mol. Spectra. 68 (19i7) 367, and references therein. [3] RM. Dole; If. Hermq J.W.C. Johns, A.R.W. >$,cKeUar, S.Nsgler and I.K_M.Srmthy, Can. J. Phys. 57 (1979) 677. [4] E. Tiemann, W.U. Stieda, T. Tarring and J. Hoeft, Z. Naturforsch. 3Oa (1975) 1606. [j] W.U. Stieda, E. Tiemann, T. Tiirriq, J. Hoeft, Z. Naturforsch. 313 (1976) 374. [6] T. T&ring, J. Xlol. Spectry. 48 (19731 148. [7] J.Hocft,FJ.Lovas,E.TiemannandT.TGrring,J.Chem. Phys. 53 (1970) 2736. [S] W.U. Sticdq Tl~esis, Freie Universit~t Berlin, Germmy (1976). [9] H. Amst,Thesis, Freie UniversitSt Berlin, Germany (1977). [IO] AH. Wnpstra and K. Bos, At. Dntn Nucl. Data Tables 13 (1957) 177. [ 1l] R. Honerjger and R. Tischer, Z. Naturforsch. 32a (1977) i. [ 121 F-T. Lovx and E. Tiemann, J. Phys. Chem. Ref. Data 3 (1974j 609. [ 131 B. Rosenblum, AH. Nethercot and C.H. Townes, Phys. Reu. 109 (1958) 400. [ 141 J. Schirmbnch, E. Tiemann, Chcm. Phys., to be published. [ 151 R. Bustreel, C. Demnynck-Marl&e, J.L. Destombes and G. Joumel, Chem. Phys. Letters 67 (1973) 178; M. Bogey, C. Demnynck and J.L. Destombes, Chem. Phys.Letters 81 (1981) 256. [16] E. Tiemann, E. Rewanz, J. Hoeft and T. TBrring, Z. Naturforsch. 2?a (1972) 1566. [ 171 K9.R. Nair, H.-U. SchUtzePrthLmmarm end J. Hozft, Chem. Phys. Letters 80 (1981) 149.