Hyperfine and isotopically invariant parameters of the GeS microwave spectrum

JOURNAL OF MOLECULAR SPECTROSCOPY 103,408-416

(1984)

Hyperfine and Isotopically Invariant Parameters of the GeS Microwave Spectrum A. C. LE FLOCH AND J. MASSON Dkpartement de Physique, Part de Grandmont, Universite’ de Tours, 37200, France

The complete set of microwave data for the nine isotopic species of the GeS molecule, available in literature, is reduced to molecular parameters by a nonlinear least-squares fit. All the hfs lines due to 73Ge(I = 9/2) and “S (Z = 3/2) are included in this fit. Watson’s expressions of Dunham coefficients, including corrections to the Born-Oppenheimer approximation, are used. Contributions from off-diagonal terms are taken into account in the hfs Hamiltonian which is numerically diagonal&d. A set of 12 hyperfine and isotopically invariant parameters is obtainedand compared with previousdeterminations.It is shown that the accuracy of the hfs parameters is improved by one order of magnitude by the global treatment of all the data. 1. INTRODUCTION

High-resolution microwave spectroscopy provides precise rotational molecular energy levels. From these data, one obtains an accurate and consistent set of molecular parameters, provided that appropriate energy level formulas are used. Recently, Watson (1) derived expressions of Dunham coefficients in terms of mass-independent parameters which include corrections to the Born-Oppenheimer approximation. Using these expressions, it is possible to reduce simultaneously the data involving all the isotopic species and to obtain isotopically invariant parameters U,, and the associated corrections from a direct fit of the data. Watson’s expressions have been successfully tested for several molecular species in various ranges of the spectrum: with infrared laser spectroscopy for CO (2, 3), by FTS measurements for HCI and CO (4), in optical spectra for LiH (5), and also with numerous microwave measurements (6-8). Most of the above studies involved ‘Z electronic states and ignored hypefine structure. Recently, however, Bogey et al. (9), combining infrared, millimeter, and submillimeter data for several isotopic species of the CS molecule, have been able to determine a highly reliable set of molecular parameters for the ground state of CS. Because of the small value of the hyperline structure parameter (eqQ N 13 MHz) of CS, off-diagonal elements of the hfs energy matrix can be neglected and Casimir’s formula with a diagonal spin-rotation contribution is an appropriate description of the hypetine interaction. In this note, we investigate the case of GeS, a molecule which has a large hfs structure. The GeS molecule is a good case since its hfs parameter (eqQ N 187 MHz) is more than 10 times larger than that for CS. Contributions from off-diagonal terms can be estimated from perturbation theory calculations: one finds, for example, that the AJ = +2 term gives a contribution of about 24 kHz to the J = 0 level. Such contributions are not negligible compared with the experimental uncertainties (the 0022-2852184 $3.00 Copyright0 1984 by Academic f’res, Inc. All rightsof reproductionin any form mrved.

408

hfs AND

DUNHAM

COEFFICIENTS

OF GeS

409

WRMS is ~10 kHz) and should therefore be taken into account. Instead of using perturbation theory calculations, we have numericahy diagonalized the complete hfs Hamiltonian as it has been done already by Nair and Hoeft ( 10) for the GaI molecule. The microwave spectrum of the GeS molecule was observed by Hoeft (II) and Hoeft et al. (12). Later, Hoeft et al. (13) resolved and analyzed the hyperlme structure due to 73Ge (I = 9/2) and 33S (I = 3/2). More recently, Stieda et al. (14) measured with high accuracy 57 new frequencies. On the whole 130 experimental frequencies involving nine isotopic species are presently available. They are collected in Table I. Up to now, there has been no treatment of the complete set of data. The most recent analysis was carried out by Stieda et al. (14) who used 34 lines from Hoeft et al. (12) in addition to their own 57 new measurements, but did not include the 24 hfs lines. Furthermore, 15 earlier measurements of Hoefi (II) were also discarded, although they include about half of the data dealing with rare (34S) isotopic species. Stieda et al. were able to extract the adiabatic part of the Born-Oppenheimer correction from the primary observation on the rotational constants Yr,,. A direct fit of the complete set of data using the expressions of Dunham’s coefficients given by Watson seems a more efficient and reliable procedure. The present note describes this approach and compares its results with the parameters obtained previously. 2. THE EFFECTIVE

HAMILTONIAN

AND

ITS ELEMENTS

For a ‘Z electronic state and for a single spinning nucleus, the Hamiltonian be written as (10, 15, 16) H = Z&&u, Z) + &V(Z) +

f&Ro(z)

+ *

l

‘9

may

(1)

where Z&t is the usual rotational term, H,,(Z) the nuclear quadrupole interaction, and H&Z) the nuclear spin-rotation interaction. I, J, F, respectively, designate the nuclear spin angular momentum, the total angular momentum without the nuclear spin, and the total angular momentum: F = I + J. Z, J, and Fare the corresponding quantum numbers and u is the vibrational quantum number. Matrix elements of this Hamiltonian, in case (b& coupling with IZVASJZF)basis functions (I 7), are listed by Cheung et al. (16) for Z states (of any multiplicity) and are also given by Nair and Hoeft (IO) for a ‘Z state. ZZ,, is diagonal in J and F. Its matrix elements, with the preceding basis functions (where A = 0, S = 0, and N = J may be suppressed), can be expressed as Dunham (IS) expansions

(JZFjH,tlJZF) = 2 Yk,

*[J(J+ l)]‘,

(2)

kl

where the Dunham mass-dependent expression as

coefficients Yk,are given (in MHz) by Watson’s

where ~1is the reduced mass of the isotopic molecule AB under consideration, calculated (in amu) with the aid of the Wapstra-Bos atomic data tables (I 9). I& is the fundamental isotopically invariant parameter, and A$ and A$ its corrections (of values near unity)

410

LE FLOCH AND MASSON TABLE I Frequencies of All the GeS Microwave Spectrum Lines J + 1 - J Which Have Been Observed PIOLECLILEB’

73

Go 32

V

S

:

:

x:: s/2 s/2 P/2 s/2 7/2 1 lZ2 7/2 11/s

: 1

: 1 1

: 1 1

z: 7~2 11/2 982 s/2 13~2 7/2 s/2 s/2 3/2

0 0

0 0

70

t4

S

0

0

:

:

11’2 7/2 s/2 s/2 s/2 7/2 11f2 7/2 7/2

. 4 . a 4 . 4 a a. . a a a 4 . a 4

3/2 3/2

10927.356 10929.097

1. 4

11356.200 11310.130 11264.828 22712.335 22620.166 22527.978 22343.525 34068.366 33930.125 33791 a770 68135.146 67859.596 67S82.012 79490.1 IS 79167.480 70844.792 76522.074 90844.651 90475.934 90107.163

b b b b b b b b b c d cl d d d d d d d d

e i 1

1

: 0 1

: 2 2

i

:

: 0 I

: 6 6

i

6

e3 7” 7 72

Cc 32

B

:

: c : d : Note : b

S

:

7

0 1

0

8” 1 :

z1

4 1

1 2

i 2 0 1 3 0 1 2 3 0 1 3

i 2 5 5 5 6

1 :

: 6 7 7 7

and

Hoeft.

Tischer,

and

end

Hooeft

Stied?..

Lovas.

a . a a

11192.616 11200.420 11234.855 11147.512 11155.272 11109.51e 22393.400 22410.155 22410.189 22412.101 22415.521 22435.647 22444.590 22444.624 22452.428 22303.153 22319.808 22319.841 22321.823 22325.144 22345.159 22361.846 40 60

Tiemann.

Tarring

(131,

10927.423 10929.134 113S6.199 11310.11e 11264.015 22712.340 22620.162 22S27.970 22343.535 34060.363 33930.096 33791.888 68135.130 67858.603 67S02.02S 79490.102 79167.477 78044.602 78522.069 90844.6SS 90475.940 90107.166

15 1s IS 1S 1S

11257.343 b 11211.853 b 11166.350 b 22514.563 b 22423.630 b 22332.619 b 22241.628 b 22150.600 b 3363s. 300 b 11120.663 b 33771.777 b 33496.740 c 67541.920 d 67268.991 d 66722.993 d 78798.030 d 78479.620 d 76161.152 d 77642.626 d 90053.747 d 89689.641 d 88961.858 d

Hoeft, Lows. Tismann, and Tiemann (271.

H0eft

FREPUENCIES w*t******** Error Calculated Dif.

11192.630 11200.410 11234.855 11147.535 11155.203. 111a9.530 22393.413 22410.164 22410.164 22412.169 22415.515 22435.668 22444.608 22444.609 22452.419 22303.165 22319.838 22319.830 22321.040 22325.136 22345.135 22361.04s

s/2 s/2

1

s

**********x* Mersur*d

9/2

: 1

G8 33

F”

7/2 11/2 P/2 7/2 11/e P/2 7/2 11/2 P/2 s/2 13/2 7/2 11/2

0

74

32

F’<___

J

:z 30 30 22: 100 10 1S 16 27 27 25 20 16 17 21

with

T-wring

11257.324 11211.837 11166.342 22514.590 22423.615 22332.626 22241.621 22150.596 33635.276 11120.639 33771.741 33490.794 67541.922 67268.995 66723.008 78790.033 78479.617 78161.152 77842.630 98053.740 89689.634 88961.046

recomnended

errors

14 -10 2: 11 12 z -2s -: 21 16 -16 -9 :2’ -11 !g7 -24 -3 -67 -37 2: !; 4 -1: 2: -38 -; -13 13 3 -10 5 1; -1

:6’ 6 -7 1s -7 7 2: :z -54 -2 -;: -3 3 0 -4 7 7 12

from

Lovas

(121.

[III. Tismann.

The calculated [see tert.1.

Tarring, frequenciee

have

been

(14). obtained

from

8 direct

and single

fit

411

hfs AND DUNHAM COEFFICIENTS OF GeS TABLE I-Continued. VOLECULES

‘V

74 GI

0 1 ;

32 8

70 G* 34 S

***********a Measured (MHZ,

0

11163.743 11118.815 ii073.890 11028.965 22327.412 22237.559 22147.717 22057.943 33490.982 33356.264 66980.413 66710.880 66441.322 6S982.046 65632.3% 78142.9515 77028.589 77514.014 77199.461 76884.842 99305.196 88945.721 80596.3W 88226.811 87967.232 109466.793 mfa62.498 99658.141 99253.71s 111179.754 118729.497 1102610.194 109939.621 109391.100 33221.510 33696.650

b b b b b b b b b b U d d d d d d d d d U (1 (1 u u U d d d (I d U d d e c

11075.030 11030.638 19996.259 22149.997 22961.210 21972.4% 33224.859 33991.6% 32958.540 66449.174 77522.933 77211.316 76900.577 88595.493 88249.407 99669.512

b b b b b b b c c d d d d d d d

6S390.055 65130.97S mese.698 32696.130

d d c c

1:; 100

65399.856 65139,877 19999.792 32696.159

64797.699 64541.1% 10799.900 32399.500

d d c c

9 14 100 100

64797.601 6454l.lSS 10799.910 32399.518

-1 3 -10 -19

64236.056 63982.947 19706.340 32119.650

d d c c

11 14 10e 100

64236.6% 63982.943 10796.314 32118.734

264 -84

0

: 0 1

:

:

0 1 2 3 4

76 Ge 32 S

J

5 6

2 6 6

s

: 3

2 2

0 1

0 0

i 1 i

0 1

: 0 0 1 B’

; s 6 6 6

kl

:e

0 i

5 5

2 :

0 72 Cr 34 S

74 cm 34 s

0

5

A 0

z 2

0

5

: 0

: 2

**I********* FREQUENCIES Error Calculatrd Qlf. (MHz> (KHZ, 15 15 1S 1S 15 15 :i IS 20 11 13 14 16 21 13 16 :5 20 12 12 10 12 33 13 13 13 17 27 35 40 59 3s :: 1s 2: IS 15 20 1: 190 :: g I7 14 15

11163.734 11118.814 11673.886 11029.950 22327.412 22237. S70 22147.715 22857.043 33490.976 33356.213 66989.418 66719.991 66441.322 65992.03e 65632.310 78142.959 77929.510 77514.812 77199.4ss QZXX %94s:731 z!!!E 1%4%:792 ‘~~:~ 992S3:712 111118.131 119729.463 11e2w.997

9 1 4 15 -1: ; 6 -9 -S -11 : -2 r: 2” -3 3 -16

1 -2 : 1

: 34 -2:

:zzx;: 33221:439 339%. 623 ll97S.924 11039.639 199%. 245 22149.992 22%1.2% 21972.434 33224.949 33991.6% 329S9.511 66449.1% 77S22.035 77211.3% 76999.579 %%5.493 99249.491 99669.504

et 27 6 1: -1: -4 11 -10 29 -12 -2 -14 -1 : e -1 -12 -29

-2

due to the breakdown of the Born-Oppenheimer approximation. A$ and A& are isotopically invariant; they are respectively associated with the atom A = Ge (B = S) of atomic mass MA (Ma). The electron rest mass is noted m,. Matrix elements of the nuclear quadrupole interaction Hamiltonian Ho,(Z) are diagonal in F but are no longer diagonal in J. Their values are (10, 16)

412

LE FLOCH AND

MASSON

(J’ZF(HQ,IJZF) = (- l)‘fF X a eqvQ X [(2J’ + 1)(2J + l)]‘/*

%Q

where (:

:

:);{:

is

the :

:I

nuclear

quadrupole

interaction

parameter

and

the 3j and 6j Wigner symbols, arising from the addition

of angular momenta (20, 21). The first of these 3j symbols will be nonvanishing if J + J’ is even. As the triangular inequality imposes (J - 21 G S < J + 2, we deduce that nonvanishing off-diagonal elements of Z&,(Z) are obtained for J’ = J and S = J f 2. Let us remark that for J’ = J, that is, for the diagonal term, Eq. (4) yields to the well-known Casimir function (IS, 22). Practically, the dimensions of the Hamiltonian matrix can be reduced, since only off-diagonal terms for J < J’MAX+ 2 give nonnegligible contributions to the energy eigenvalues. The structure of the actual matrix is shown in Table II. The nine offdiagonal terms included in calculations connect diagonal terms with AF = 0 and AJ = +2. Their numerical expressions are given in Table II. The overall matrix, of dimension 17, can obviously be factored into blocks of smaller dimensions. The experimental data set contains only two hfs lines arising from ‘5 (I = 3/2). A crude estimation of the off-diagonal elements shows that they are negligible in this case and therefore the usual diagonal hfs Hamiltonian has been used for these two lines. As we have only one spinning nucleus, Z&&Z) is diagonal for all quantum numbers (10). Its matrix elements are simple and given by (JZFI~~,,~IJZF) = k C,,[F(F + 1) - z(z + 1) - J(J + I)],

where C,, is the nuclear spin-rotation

(5)

interaction parameter.

3. FITTING

PROCEDURE

The parameters have been obtained by fitting the experimental data to the line frequencies calculated from the appropriate Hamiltonian described in the previous section. This requires a nonlinear least-squares fitting procedure. Several techniques have been used for similar cases. We have written a computing program following the method described by Kotlar et al. (23). Every line was included in the fit, with a statistical weight equal to the square of the reciprocal of the experimental uncertainty. Corrections to initial parameters were obtained by using Marquardt’s algorithm (24) and the procedure is repeated until convergence is achieved. The progress of the fit toward convergence is measured by the dimensionless weighted variance &* = (N - p)-’ ; (F, -f;)‘/af I=1

(6)

hfs AND DUNHAM

413

COEFFICIENTS OF GeS

TABLE II Hyperfine Structure Energy Matrix for I = 9/2

r-T---r . : .* . . .

I

.*

.

.

.

.

.

.

.

.

.

.

.

.

.*

.

.

*

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

-*

.

.

.

.

.

.

.

.

.

.

.

. .*

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

\

,'y;7p97

4

3

f

5552 1

2

f

0

J

Nore: All off-diagonal elements are zero, except the AF = 0, AJ = k2 elements which are the following nine: (2 9/2 [Hal 0 9/2) = eqQ X 0.15138251 (3

l/2 lHpl 1 712) = eqQ X 0.08291562

(3

912 [HaI 1 912) = eqQ X 0.12747548

(3 1l/2 lHal I 1l/2) = eqQ X 0.12747548 (4

5/2 IHal 2 5/2) = eqQ X 0.04414404

(4

l/2 IHal 2 712) = eqQ X 0.08312047

(4

912 IHQI2 912) = eqQ

X

0.11636866

(4 I l/2 lHpl 2 1 l/2) = eqQ X 0.13176156 (4 13/2 IH&2 13/2) = eqQ X 0.11293848 The matrix is symmetric; only the upper triangle has heen given. The observed lines involve levels up to J’=2andF’= 1312.

where N = 130 is the number of input data and p = 12 the appropriate total number of parameters. Fiis the measured transition frequency, ci its error, andfi’ the calculated value with the current set of parameters.

414

LE FLOCH AND MASSON 4. RESULTS AND DISCUSSION

The choice of the appropriate parameters is as usual a delicate matter. We carried out numerous trial calculations also investigating the effects of high-order Akl corrections. This led us to choose a total number of 12 parameters which are listed in Table III. Instead of B, the rotational energy involves six parameters: UO, (with two corrections $i= and A&), U,, , U2,, and US, ; while the distortion term includes two parameters U02 and Vi,. The hfs energy arising from 73Ge is expressed in terms of three parameters: eqoQ, eq,Q for the nuclear quadrupole interaction, and a single nuclear spin-rotation parameter C (the u-dependent term is unsignificant within the TABLE III Hyperfme and Isotopically Invariant Parameters for the GeS Molecule.

I .I

(01 UOl

124436.78

(15,

114416.575(51)

124436.91112)

A 01 (0.1 Aol I s I

_1.4411971

_1.471(36,

_1.465,70,

J.544156)

_1.994(1Ol

_1.471(451

Ull

_23&7.919(69J

UZI

_0.65,14,

US1

_O.l95l47l

_0.200,56,

Uoa

_1.1797118l

_1.1771(13,

Ull

-0.0060~39l

_o.o07l,as,

147.755

(351

146.749I54)

_0.000116)

33

5

b.B5(171

WoQ

a

:

Stisda

b

:

Hoeft

c

:

Tiemann

:

hfs

parameters

are

dimensionless.

Note

cant

et et

al.

[14).

al. et

figure

6.96130)

(131, al.

and Lovas

and Tiemann

(27).

[71. are

quoted.

in

MHz,

Llkl

Uncertainties

In

k/Z

+ 1

units

of

MHz x (amu) C(J) are

In

, the

,lGe 01 last

and

s dOl

slgnifi-

hfs AND DUNHAM

COEFFICIENTS

OF GeS

415

experimental uncertainty). Only one parameter (eqOQ)could be determined from the hfs due to 33S. The correlation coefficients between parameters have also been considered in the choice of the parameters to be fitted. Five couples of parameters have been found to be highly correlated’: (U,,,, A$,“;)= 0.63; (UO,, A&) = 0.83; (U1,, I&) = 0.94; (U,, , U3,) = 0.90; and ( UzL, U,,) = 0.98. The last coefficient indicates that the value obtained for UjLmay be dubious. Including U,, terms in Dunham’s expansion seems, however, to be necessary in order to get reliable values for the other parameters. Actually, the vibrational parameter U,,(weG) calculated from UOIand UOZ(2) is in much better agreement with the optical value w, (for 74Ge32S)(25, 26) if U,, terms are included in the rotational energy expressions. The best set of parameters obtained in this work are given in Table III. The Dunham parameters previously calculated by Stieda et al. (14) and the hfs parameters given by Hoeft et al. (13) and by Lovas and Tiemann’s table (27) are also included in Table III for comparison. The final dimensionless variance (Eq. 6) is I? = 0.388 with a WRMS = 0.010 MHz. Since the value of G is smaller than 1, we conclude that most transitions are fitted to within their experimental uncertainties. Table I shows that the maximum value for the ratio IFi - fi‘l/a; is 1.8. This justifies the inclusion of all the 130 lines in the fit and demonstrates that the global treatment is adequate. It should be noticed that the parameters obtained in the present treatment agree with previous results within their respective uncertainties. The uncertainties (a) given originally by Stieda et al. are systematically smaller than ours; this is, however, not conclusive since the reduction of the Stieda et al. data recently performed by Tiemann et al. (7) yields uncertainties similar to ours. Conversely, the present treatment improves the accuracy of the hfs parameters by one order of magnitude. In conclusion, the global and direct fit of the high-quality and medium-quality data leads to a self-consistent set of microwave frequencies of the GeS molecule, while the hfs parameters have greatly gained in accuracy from the high quality of other measurements. ACKNOWLEDGMENTS The authors thank Dr. J. Rostas for a critical reading of the manuscript and for helpful suggestions. RECEIVED:

July 25, 1983 REFERENCES

1. J. K. G. WATSON, J. Mol. Spectrosc.80, 411-421 (1980). 2. A. H. M. Ross, R. S. ENG, AND H. KILDAL, Opt. Commun. 12,433-438 (1974). 3. R. M. DALE, M. HERMAN,J. W. C. JOHNS,A. R. W. MCKELLAR,S. NAGLER,AND I. K. M. STRATHY, Canad. J. Phys. 57, 677-686 (1979). 4. G. GUELACHVILI,P. NIAY, AND P. BERNAGE,J. Mol. Spectrosc. 85,27 l-28 1 ( I98 1); G. GUELACHVILI, D. DE VILLENEUVE,R. FARRENQ,W. URBAN, AND J. VERGES,J. Mol. Spectrosc. W,64-79 (1983).

’ Correlations between the U,, and 4, parameters are not too high and allow a direct determination of these parameters. An attempt to linearize Eq. (3) by Tiemann’s expression (28) yields the same results.

416

LE FLOCH AND MASON

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