Theoretical calculations of the vibrational transition probabilities in hydrogen selenide

Chemical Physics 122 (1988) 375-386 North-Holland. Amsterdam

THEORETICAL CALCULATIONS IN HYDROGEN SELENIDE

OF THE VIBRATIONAL TRANSITION PROBABILITIES

J. SENEKOWITSCH, A. ZILCH, S. CARTER, H.-J. WERNER ‘, P. ROSMUS Fachbereich Chemie der Universitiit. D-6000 Frankfurt, FRG

and P. BOTSCHWINA Fachbereich Chemie der Universitiit, D-6350 Kaiserslautern. FRG

Received 4 December 1987

Three-dimensional ab initio dipole and potential energy functions for H$e have been calculated from highly correlated SCEP CEPA wavefunctions. First-order relativistic corrections according to the Cowan-Griffin approach, which retains only the mass velocity and one-electron Darwin terms, have been applied. These data have been used in perturbation and variational calculations of anharmonic vibration-rotation term values and wavefunctions. Radiative transition probabilities between vibrational levels up to about 10000 cm-’ have been calculated from electric dipole transition matrix elements. It is found that the radiative lifetimes vary in a mode-specific way.

1. Introduction

The vibration-rotation transitions of hydrogen selenide and its isotopic species have been investigated in several studies. For the vibrational ground state and the v,, v2, 2v2 and v3 states, high-resolution data of HIsOSeare known [ l-4 1. A set of equilibrium spectroscopic constants [ 5,6 ] and an anharmonic quartic force field [ 6 ] have been evaluated from experimental vibrational band origins and rotational term values. Recently, Halonen and Carrington [ 71 reported semi-empirical variational calculations of vibrational term values for H2Se and D,Se. To our knowledge, there have been neither experimental nor theoretical studies of radiative transition probabilities for H2Se published so far (cf. also a recent review on absolute infrared intensities; ref. [ 8 ] ). Only the dipole moments in the vibrational ground state of D,Se and HDSe have been determined from Stark effect measurements [ 9- 111. ’ Present address: Fakultlt fIir Chemie, UniversitPt Bielefeld, D-4800 Bielefeld, FRG.

It is one of the objectives of our current research to develop a technique for the prediction of accurate transition probabilities for vibrational and rovibrational states of triatomic molecules. The value of such predictions to the spectroscopist is obvious, and it is to be anticipated that the assignments of many hitherto unrecorded transitions would follow. In this paper, we investigate the use of highly correlated electronic wavefunctions including relativistic corrections in the construction of triatomic potential and dipole moment surfaces which are subsequently analysed by a variational procedure to produce converged vibrational frequencies and transition probabilities. We also investigate the accuracy of results obtained by perturbation theory by comparison with the accurate variational results. This is of considerable relevance, since in larger molecules perturbation methods appear the only means of predicting. spectroscopic properties from potential energy surfaces. In the present work the vibration-rotation spectrum of HzSe has been calculated from ab initio threedimensional potential energy and electric dipole mo-

0301-0104/88/$03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

J. Senekowitsch et al. / Vibrational transition probabilities in HzSe

376

ment surfaces using highly correlated CEPA (coupled electron pair approximation) [ 121 electronic wavefunctions. To date, only for Hz0 [ 13,141 and O3 [ 15 ] a similar full three-dimensional treatment of the vibrational transition probabilities has been investigated by ab initio techniques which employ highly correlated electronic wavefunctions. The experimental determination of absolute transition intensities is a long and rather difficult task; moreover, the available data are usually restricted to a limited number of transitions. On the other hand, theoretically calculated potential energy and dipole moment functions yield information which can be used in a compact form for the evaluation of all radiative transition probabilities, and their variations within the vibrational levels can thus be determined in a straightforward way. In the present study, we have evaluated all transition probabilities in vibrational states of H,Se up to about 10000 cm- ‘. We have found that the radiative relaxations of vibrational levels vary in a mode-specific way. The computational methods are discussed in the next section, and in sections 3 and 4 the numerical results of this study are summarized.

functions, we have made use of the coupled electron pair approximation (CEPA) [ 121 for the calculation of highly correlated electronic wavefunctions. In this approach, all single and double substitutions with respect to the closed-shell Hartree-Fock determinant are included, and the most important higher substitutions are taken into account in an approximate way. The CEPA equations were solved by means of the selfconsistent electron pair (SCEP) technique [ 16,171, and version 1 of CEPA was employed. All valence electrons have been correlated. Furthermore, according to Cowan and Griffin [ 18 1, the energy was corrected for relativistic effects. They pointed out, that the most important relativistic contributions to the energy stem from the mass velocity and the one-electron Darwin terms of the Breit-Pauli Hamiltonian. In their approach, the expectation values over the relativistic operators are evaluated from nonrelativistic wavefunctions. As has been demonstrated earlier [ 19,201, such a simple perturbation treatment can be applied not only to the SCF approximation, but also to correlated wavefunctions. For instance, the CEPA energy at rseH=2.7 au and ar,seH= 90” was - 240 1.1432 au and the relativistic correction yielded - 27.4783 au.

2. Electronic structure calculations 3. Anharmonic vibration-rotation term values The Gaussian basis set used in the calculations of the potential energy and the electric dipole moment functions comprised 82 contractions and is explicitly given in table 1. Starting with Hartree-Fock wave-

The calculated relativistic CEPA energies for 53 geometries in the vicinity of the equilibrium geometry were fitted to various polynomial expansions in bond stretching and angle bending coordinates:

Table 1 GTO basis set a) used in the CEPA calculations Se s: 560600.0(0.217), 84010.0( 1.696), 19030.0(8.83), 5419.0(35.435), 1792.0,659.5,262.0, 108.6,33.66, 14.45, 4.378, 1.876,0.45,0.18,0.072 p: 4114.0(0.1414), 980.4( 1.1924), 310.7(6.2668), 114.2, 46.26, 19.87,8.309, 3.598, 1.522,0.45,0.18,0.072 d: 94.03(0.23848), 26.79( 1.4131), 9.336(3.69572), 1,145, 0.35 H s: 68,16(0.00255), 10.2465(0.01938), 0.67332,0.22466,0.08222 p: 1.2,0.3 a) Contraction

coeffkients

2.34648(0.0928),

given in parentheses.

For the stretches ( Ql and Q,), three different types of coordinates have been investigated: (i) displacement coordinates Q=R = r- rref; (ii) Simons-Parr-Finlan [ 2 1] coordinates Q=R,,,= 1-rref/r, (iii) Morse coordinates [ 221 Q=RM= { 1-exp [ -j?(r/rref- 1)]}//3, withp= cf;,/2D,)‘/*. The bending coordinate Q3 was expressed either as an angle displacement 8= (Y- aJef according to (i) or in terms of a cubic expansion in 8 as @CH=AO@[email protected] latter coordinate satisfies the boundary conditions Q,( 180” ) = 1 and dQ, ( 180 o )/da = 0. The A,, parameter was roughly

J. Senekowitsch

et al. / Vibrational

optimized with respect to the total rms (root mean square) error; Al and A2 then follow from the restrictions above. This type of coordinate considerably improves the asymptotic behavior of the bending potential in the vicinity of linear structures as well as for small angles. A similar coordinate has recently been introduced by Carter and Handy [ 23 ] in their calculations of the rovibronic energy levels of water. Expectedly, due to their correct asymptotic behavior, SPF or Morse coordinates yield faster and better fits than the simple displacement coordinate. In the final calculations we have used a sextic expansion referred to the equilibrium geometry in RM and Sc, coordinates, which is given explicitly in table 2. This expansion reproduced the calculated energies to within a few wavenumbers (rms error 1.O cm- I ). The energy range covered by these points reaches up to about 20000 cm-’ and the function is valid in the geometry range Rs,..=2.1 to 3.7 au and crHSeH=600 to 150”. For H**‘Se there are fourteen band origins known from experiments. This set of data has been used by Hill and Edwards [ 51 to derive harmonic frequencies o, and the anharmonic constants Xijby treating some of the vibrational states as Darling-Dennison resonances. Also, the equilibrium rotational constants (Y,have been evaluated. An empirical anharmanic quartic field has been derived by Mills [ 61, Table 2 Expansion coefftcients a) of the three-dimensional rium potential energy surface of H,Se (in au) C,,,, -2428.622401

in H$e

371

who used the nine (Y,values to determine the set of cubic force constants. The most important higher constants were adapted such as to match the xij constants of Hill and Edwards. From the expansion coefficients in table 2 we have calculated an internal force field which is compared with the empirical force field in table 3. The overall agreement for large constants is satisfactory. Also the quartic constants neglected by Mills are found to be small. The largest deviations are found in constants involving the bending coordinate, which has been traced as a residual deficiency of the ab intio potential (cf. below). The internal CEPA force field has been transformed by L-tensor algebra [ 241 to the force fields of HZsOSeand its deuterated species in dimensionless normal coordinates, from which spectroscopic constants have been calculated by second-order perturbation theory [ 25 ] (cf. table 4 ) . The calculated equilibrium distance is larger than the experimental value by 0.001 8, and the equilibrium angle agrees to within 0.1 O.The relativistic corrections reduced the equilibrium distance by 0.003 8, and had virtually no effect on the equilibrium angle. The theoretical (Y,and TVconstants are in fair agreement with the values of Hill and Edwards [ 5 1. The CEPA harmonic frequencies for the symmetric and Table 3 Comparison of the quartic force fields (values are given in al/A”)

near equilib-

C, ,o -0.006488

Cm C ,I, C 310 C 202 COOS

0.096564 -0.113604 -0.031192 -0.116905 0.006161

C,,, -0.009823 C,, -0.017202 C,,, -0.028066 C ,,_,3 0.018384 C w6 0.014882

C2CQ C 300 C 102

0.853560 - 1.309781

C lo, 0.026565 C,,, -0.016565

-0.066386 0.085494 0.089876 -0.902238

GO0 1.142040 C,, , 0.004785 C,, 0.018935 c,, 0.355314

C220 C 112 C500

transition probabilities

a) The calculated total energies for fifty-three geometries are available on request. The internal coordinates are Q, =RM and Qz=RM with f,,=3.48 aJ/A2 and D,=3.?9 eV and Qs=8cn withA~=0.95,Al=0.01473rad-‘andA2=-0.1364rad-2;the surface has been expanded at the calculated equilibrium geometry (cf. table 4); Cj,k=C,a.

CEPA

Empirical ‘)

3.485 0.760 -0.013 0.075 - 17.390 -0.315 -0.019 -0.110 - 0.220 -0.355 -75.761 - 1.074 -0.133 0.356 -0.077 0.148 - 0.646 0.324 0.184

3.507 0.710 - 0.024 0.130 -16.7 -0.7 0.0 -0.3 0.1 -0.2 63.0 -0.09 0.3 0.3 0.0 0.0 -1.8 0.0 0.0

J. Senekowitsch et al. / Vibrational transition probabilities in HrSe

378 Table 4 Spectroscopic

constants

for H,?Se,

HD*“Se and D2*‘Se H,Se

RF” (A) CX~~“’(deg)

1.461 90.62 8.120 7.749 3.965 0.822 2433.5 1099.2 2442.9

A, (cm-‘) B, (cm-‘) C, (cm-‘) Ka’

(cm-‘) wz (cm-‘) wj (cm-‘) d) :I:, d, w,

62, d, rARA,rA (cm-‘) rBBBB(cm-’ ) T~~.(.~.(cm-’ ) 7AABB (cm-’ ~BSC.~. (cm hAA

(cm-‘)

bsAB

(cm-‘)

) - ’1

(cm-‘) a< (cm-‘) a: (cm-‘) cr;R (cm-‘) of (cm-‘) (YT (cm-‘) cu:‘ (cm-‘) a? (cm-‘) cx:‘ (cm-‘) cxf

xl1 (cm-‘) k2 (cm-‘) + (cm-’ ) xl2 (cm-‘) xl3 (cm-‘) ~23 (cm-’ ) ye) (cm-‘) G(OO0) (cm-‘) a) b’ c’ d’ ‘)

Asymmetry parameter. Values refer to the vibrational 02+2x22. Coriolis constant. Darling-Dennison resonance

-0.0216 -0.9998 -0.00445 -0.00355 -0.00017 0.00262 -0.00030 - 0.00038 - 0.00067 0.116 - 0.244 0.154 0.109 -0.141 0.073 0.059 0.048 0.040 -21.3 -5.8 -21.0 -17.2 -84.1 -20.5 -41.2 2945.4

Ref. [6] 1.46 90.53

2439.0 1057.6 2454.2

Ref. [5]

HDSe

1.46 90.57 8.139 7.782 3.972 0.79241 2438.66 1053.16” 2453.77

1.465 90.80 7.941 4.014 2.666 -0.489 1735.3 954.1 2438.1 0.5751 -0.0188 -0.8179 -0.00139 -0.00034 - 0.00008 0.0000 -0.00015 -0.00016 -0.00148 0.003 -0.185 0.224 0.079 - 0.064 0.001 0.03 1 0.026 0.023 -21.2 -4.4 -41.9 -10.1 -0.9 -18.6

-0.00473 -0.00419 0.00320

0.123 - 0.227 0.148 0.091 -0.195 0.065 0.042 0.047 0.047 -21.4 -2.4 -21.7 - 17.7 -84.9 -20.2 -41.6

-0.00088 0.082 -0.271 0.127 0.142 -0.142 0.110 0.054 0.043 0.047 -21.43 -21.71 - 17.69 - 84.90 - 20.20

b’

b, b, b,

b1

2539.5

DzSe

4.163 3.878 2.008 0.735 1731.3 782.6 1738.9

I

- 0.0064 -0.9999 -0.00117 -0.00089 - 0.00004 0.00067 -0.00008 - 0.00009 -0.00017 0.041 -0.088 0.057 0.040 -0.051 0.025 0.020 0.017 0.015 - 10.8 -2.9 - 10.7 -8.8 -42.6 - 10.3 -20.9 2104.9

ground state.

parameter.

asymmetric stretching modes deviate from the empirical values by 5 and 11 cm- ‘, respectively. The calculated nonrelativistic frequencies w, and w3 are about 10 cm-’ larger. A larger deviation of 46 cm-’ is found for the bending harmonic frequency, a deticiency found also in similar ab initio potentials for Hz0 [ 13,261 and H2S [ 271. The reason is probably the absence of higher angular momentum basis func-

tions in the ab initio calculations [ 281. The relativistic effect on the bending frequency is negligible. The only previous SCF effective core calculation by Miiller et al. [ 291 yielded harmonic frequencies which were more than 100 cm-’ higher than the experimental values. The CEPA x, constants are also in reasonable agreement with the empirical values. The hitherto unknown constants for the deuterated species are

J. Senekowitsch et al. / Vibrational transition probabilities in HzSe

given in table 4. We note that the v3 frequency for HTSe (2345.3 cm-‘) is about 0.8 cm-’ larger than the corresponding frequency for HDSe (2344.5 cm-’ ), while the harmonic o3 value for HTSe is smaller than for HDSe (2438.01, 2438.07 cm-‘, respectively). This unusual isotope effect has been found previously for H20, H2S, H2Cl+, and H2Br+ and has been explained in refs. [ 29,301. The vibration-rotation term values have been evaluated in two different ways: (i) from the spectroscopic constants in table 4, and (ii) from independent variational calculations. We have used the variational approach of Carter and Handy [ 3 l-34 1, in which the Hamiltonian of the nuclear motion in the coordinates r,, r, and (Yis represented by vibration-rotation basis functions consisting of products of harmonic-oscillator eigenfunctions for the asymmetric stretching modes, Morse-oscillator eigenfunctions for the symmetric stretching modes, linear combinations of associated Legendre functions for the bending modes and rotational functions (cf. refs. [ 3 l-34 ] for a detailed description ) . In large basis sets the integration over the vibronic parts of the matrix elements was done numerically and over the rotational part analytically. The variational term values for high vibrational states were converged with respect to the basis set size to within 2 cm-‘; for low lying states the accuracy was better than 0.1 cm- ‘. In table 5 the vibrational band origins calculated variationally are compared with existing experimental values for HZgOSe.The overall agreement between the variational values and the experimental data is good. The stretching frequencies agree to within about 10 cm- ’ , the term values involving the bending modes to within 35 cm- ‘. The highest observed band origin in HZgOSelies at 8894.6 cm- ’ and has been assigned as a 301 mode. For this band origin the CEPA potential yields 8874.5 cm-‘, being in excellent agreement with the experiment. For comparison, band origins calculated from the spectroscopic constants of table 4 (i.e. from the CEPA quartic force field obtained from the expansion in table 2 and standard nondegenerate second-order perturbation theory) are given together with the differences between the corresponding variational values obtained with the complete expansion of the potential in table 2. For the pure stretching modes, a remarkable agreement of the perturbation theory and the variational

319

calculations has been reached. The modes involving bending vibrations show a systematic deviation. With increasing bending quantum numbers the variational band origins become increasingly lower in energy than the perturbational values. In the variational calculations we have also obtained the vibration-rotation term values up to J= 9. Since the Hamiltonian includes the full vibration-rotation coupling, this procedure allows us, for instance, to calculate microwave transitions in a straightforward way. In table 6 we have compared some purely rotational transitions in the vibrational ground state with the experimental data of Helminger and DeLucia [ 11. For these transitions, the agreement can also be regarded as very satisfactory.

4. Radiative transition probabilities in the electronic ground state of H2Se and its deuterated species The principal axes of inertia a, b, c have been identified with the molecular axis system y, x, z according to a near symmetric rotor (case III’ ) [ 35 1. The corresponding three-dimensional CEPA dipole moment functions were expanded in a similar way as the energy, but R and 8 displacement coordinates have been used. These expansion functions are given in table 7 and are valid in the same geometry range as the potential. The dipole moment functions have been used to calculate numerically the dipole moment matrix elements (J=O) from the anharmonic three-dimensional vibrational wavefunctions. In the analysis of radiative transition probabilities by the contact transformation approach to the dipole moment operator [ 36 1, it is convenient to expand the dipole moment functions in terms of dimensionless normal coordinates. Such an expansion up to quadratic terms is given in table 8. In this case the Eckart frame has been used as a coordinate system and the derivatives were calculated along the x and y axes. Similar expansions have been given by Camy-Peyret and Flaud [ 361 for the electronic ground states of HZ0 and H2S. An inspection of the data in table 8 shows that the first derivatives of the dipole moment functions are an order of magnitude larger than the second derivatives, suggesting an approximate electric harmonicity in the transition matrix elements. For the symmetric stretching mode the first derivatives

J. Senekowitsch et al. / Vibrational transition probabilities in Hse

380 Table 5 Comparison

of calculated

and observed

vibrational

CEPA/V 010 020 100 001 030 110 01 I 040 120 021 200 101 002 050 031 130 111 210 012 060 041 140 121 220 022 201 102 300 003 070 051 150 131 230 032 211 112 310 013 080 061 160 141 240 042 221 122 301 202 320 023 400 103

1068.8 2126.1 2340.2 2348.7 3171.9 3391.8 3397.1 4206.1 4431.8 4433.9 4604.4 4604.8 4688.8 5228.7 5459.2 5460.3 5635.9 5636.1 5719.3 6239.8 6472.9 6477.2 6655.4 6655.9 6738.6 6781.5 6781.5 6938.6 6957.0 7239.4 747s. 1 7482.6 7663.5 7664.0 7746.5 7793.9 7794.0 7953.8 7965.9 8227.4 8465.8 8476.4 8659.9 8660.3 8743.2 8794.8 8794.8 8874.5 8874.5 8957.5 8963.3 9121.2 9122.7

e’

band origins for H,%e CEPA/P 1069.1 2127.6 2340.2 2348.7 3176.1 3391.4 3396.9 4214.8 4432.3 4434.6 4604.8 4605.3 4688.9 5343.8 5462.4 5463.4 5634.6 5634.9 5718.8 6263.2 6480.6 6485.0 6653.8 6654.6 6738.6 6783.1 6783.2 6939.3 6957.9 7273.1 7489.2 7497.1 7663.4 7664.5 7749.0 7791.6 7791.8 7952.1 7965.1 8273.1 8488.3 8499.7 8663.5 8664.6 8750.3 8790.1 8790.3 8878.2 8878.7 8955.1 8962.2 9123.8 9125.0

=)

CEPA/V-CEPAfP

Exp. b,

-0.3 - 1.5 0.0 0.0 -4.2 0.4 0.2 -8.7 -0.5 -0.7 -0.4 -0.5 -0.1 -15.1 -3.2 -3.1 1.3 1.2 0.5 -23.4 -7.7 -7.8 1.6 1.3 0.0 -1.6 -1.7 -0.7 -0.9 -33.7 - 14.1 - 14.5 0.1 -0.5 -2.5 2.3 2.2 1.7 0.8 -45.7 -22.5 -23.3 -3.6 -4.3 -7.1 4.7 4.5 -3.7 -4.2 2.4 1.1 -2.6 -2.3

1034.21 2059.96 2344.36 2357.66 3361.72 3371.81

4615.33 46 17.40

5613.72 5612.73

6798.23 6798.15 6953.6

8894.6

J. Senekowitsch

et al. / Vibrational

transition probabilities

CEPA/V ‘)

CEPA/P ‘)

CEPA/V-CEPA/P

9203.9 9209.2 9444.9 9458.7 9644.8 9644.9 9728.5 9784.0 9784.0 9868.1 9949.2 9949.9

9266.3 9211.2 9478.1 9493.3 9655.3 9654.3 9742.5 9779.0 9779.2 9864.7 9949.8 9948.9

-62.4 -2.0 -33.2 -34.6 - 10.5 -9.4 - 14.0 5.0 4.8 3.4 -0.6 1.0

in H$e

381

Table 5 (continued)

090 004 071 170 250 151 052 231 132 311 033 330

Exp. b’

a) CEPA/V refers to variational calculations, CEPA/P to calculations by standard second-order perturbation theory from the spectroscopic constants of table 4. b, Refs. [5,6]. Table 6 Observed ‘) and variationally calculated rotational transitions in the vibrational ground state of H,‘%e (in cm-‘) Transition

CEPA

Observed

Transition

CEPA

Observed

101-l10 %0-l,, 211-220 212-221 202-ZI I 111-202 101-212

4.22 13.05 4.61 12.64 11.55 19.37 19.78 5.25 13.17 11.10 6.16 13.86 10.66

4.2688 12.0710 4.7423 12.7910 11.5064 19.3834 19.8738 5.5119 13.4278 11.0073 6.6304 14.2530 10.5461

541-550 542-551 532-54,

7.39 14.68 10.35 8.91 15.59 10.27 10.69 10.54 12.58 11.24 14.44 12.43

8.1308 15.238 1 10.2699 9.9849 16.3401 10.3204 12.1084 10.8160 14.3177 11.8444 16.4165 13.4513

321-330 322-33, 312-32, 431-440 432-441 422-43,

65,460 652461 642-651

761-770 752-761 871-880 8,,-8, I 981-990 972-981

a) Ref. [l]. Table 7 Expansion coefftcients a) of the x and y components b, of the dipole moment function (in au) x component ( C,,, = C,, ) Coo0 0.279029 C ,,. -0.031277 C2,0 0.004482 CM),-0.135745 C,,, 0.008759 C ,03 0.087924 y component ( C,,k= -C,,k) Cloo - 0.086904 c,,, -0.0114743 C3,0 0.002686 c,os -0.016810

C,,, -0.080121 CIO, -0.095594 C,,, -0.034859 C4oo 0.005585 C,, , 0.032479 Coo4 0.026623

Coo, 0.171458 Coo2 0.111632 C,,, -0.028498 C,,, 0.018486 c,,, -0.033439

C,,, c,,, c,o2 C,,, C,,2

C2M1-0.004889 C,,, -0.058022 C,,, -0.019729

C lo, 0.116471 C ,02 0.037829 C,,, 0.023131

C,,, 0.003908 C4oo 0.005476 C,,, -0.035226

0.005396 -0.000177 -0.021159 -0.023695 -0.057263

a) The internal coordinates are Q, CR,, Q2=R2, and Q3= 8; both functions have been expanded at the calculated equilibrium geometry. b1 See text for the definition of the axes system.

J. Senekowitsch et al. / Vibrational transition probabilities in HSe

382

of the puycomponent have different sign for Hz0 and H2S. In HzSe this quantity has the same sign as in H2S but is an order of magnitude larger [ 371, even larger than in H20. For the bending and asymmetric stretching mode the values for H,Se are also larger than in H2S; for the asymmetric stretching mode the first derivative of the dipole moment changes its sign relative to Hz0 and H2S. Of all these functions the one for H2S is the flattest for all displacements. In table 9 the dipole matrix elements and the rates of spontaneous emission for the ten lowest vibrational states of H,Se are given. Two very different values of the dipole moment in the vibrational ground state are found in the literature. From Stark effect measurements of D$e, Jache et al. [ lo] obtained ~~~0.24 D, Mirri et al. [ 91 0.627 + 0.002 D, and Veselago [ 111 0.62 D for HDSe. Rauk and Collins [ 38 ] calculated a dipole moment of 0.913 from Hartree-Fock wavefunctions. Our calculated CEPA values in the vibrational ground states are 0.701 D (H,Se), 0.702 D (HDSe) and 0.704 D (D,Se). Obviously, the value of 0.24 D can be ruled out. The remaining deviation of about 0.07 D could be caused by relativistic effects. At the equilibrium geometry the dipole moment calculated as derivative from the relativistic energies using finite electric fields is 0.66 D. This can be compared to the dipole moment of 0.72 D obtained from the nonrelativistic energies, and the expectation value of 0.71 D. Thus, within this approximation, the relativistic contribution shifts the dipole moment by about 0.05 D towards the experimental value. The relativistic correction has not been made for the whole dipole moment surfaces, since this Table 8 First and second derivatives of the Eckart frame a) dipole moment with respect to dimensionless normal coordinates (in au) x component @, -0.0247

cc;, -0.0012

/L’i -0.0066

PC; 0.0305

fig

0.0040

/l;s

0.0

/lc;

j&

0.0021

j&

0.0

y component jIu; 0.0

/l(;,

0.0

p;z

0.0

/I;

0.0

fly*

0.0

jly,

0.0004

/I;

0.0275

&

0.0

,& - 0.0049

0.0

ai) The Eckart system is defined with respect to the molecular fixed axes x, y and z; ,u’=O.2796 au; p, corresponds to the ith mode.

G (cm-‘)

1069.1 2127.6 2340.2 2348.7 3176.1 3391.4 3396.9 4214.8 4432.3 4434.6 4604.8 4605.3 4688.9 5243.8 5462.4 5463.4 5634.6 5634.9 5718.8 6263.2 6480.6 6485.0 6653.8 6654.6 6738.6 6783.1 6783.2 6939.3 6957.9 7273.1 7489.2 7497.1 7663.4 7664.5 7749.0 7791.6

010 020 100 001 030 110 011 040 120 02 I 200 101 002 050 03 1 130 111 210 012 060 041 140 121 200 022 201 102 300 003 070 051 150 131 230 032 211

absorption

VlW3

HzSe

Table 10 Integrated a) (atm-

0.279+2 0.136+1 0.484 + 2 0.590+2 0.279- 1 0.227+ 1 0.144+ 1 0.632- 3 0.257- 1 0.124- 1 0.109+ I 0.989+0 0.287+0 0.109-3 0.319-3 0.708-2 o.l84+0 0.272- 1 0.162-3 0.149-7 0.144-4 0.472 - 5 0.251-2 0.937-4 0.246 - 3 0.231- 1 0.450 - 1 0.132- 1 0.536-2 0.108-7 0.194-S 0.708- 5 0.216-4 0.362- 3 0.121-3 0.977-Z

S

intensities

1023.7 509.4 123.2 99.7 335.6 103.4 91.6 247.0 90.1 84.6 60.0 58.5 52.5 192.8 78.2 80.3 53.6 54.9 49.2 156.0 72.4 72.8 49.8 51.2 46.2 41.9 41.9 39.8 34.3 129.2 67.1 66.5 46.6 48.3 43.6 39.1

T

’ cm-‘)

010 020 100 001 030 110 011 040 120 021 200 101 002 050 130 031 210 I 11 012 060 140 041 220 121 022 102 201 300 003 070 051 150 131 230 032 211

vlu2v3

DaSe

and radiative

767.3 1529.1 1684.1 1691.1 2285.7 2442.4 2447.9 3037.3 3195.3 3199.3 3331.8 3332.9 3376.0 3785.5 3943.1 3945.5 4079.8 4080.3 4123.3 4530.0 4686.0 4686.6 4822.4 4822.5 4865.5 4935.7 4935.7 5012.8 5027.4 5288.3 5424.5 5426.2 5559.6 5559.8 5602.6 5673.0

G (cm-‘) O.l49+2 0.5os+o 0.252+2 0.302+2 0.733-2 0.827+0 0.553+0 0.113-3 0.579-2 0.325-2 $388fO 0.341 +o 0.127+0 0.152-4 0.132-2 0.799-4 0.667-2 0.488 - 1 0.192-3 0.192-9 0.652-6 0.205 - 5 0.816-5 0.436 - 3 0.717-4 0.126- 1 0.568-2 0.361-2 0.121-2 0.695 - 7 0.287 - 8 0.668-6 0.461-5 0.511-4 0.136-4 0.187-2

S 3734.4 1861.7 457.5 376.4 1233.3 391.2 343.6 915.9 343.7 315.5 222.1 215.5 196.5 718.3 307.7 291.1 204.2 199.7 185.3 584.0 279.3 269.5 190.1 186.4 175.2 151.2 151.1 147.1 128.1 449.8 249.8 258.3 175.2 178.7 166.1 142.2

T 010 100 020 001 110 030 01 l 200 120 040 101 021 210 130 050 002 111 300 031 220 140 060 012 201 121 310 041 230 150 102 070 022 400 211 131 320

VIv2v3

HDSe

lifetimes b, (ms) for HssoSe, DZ8’Se and HD*‘Se

C) 931.1 1687.3 1854.1 2344.6 2608.1 2769.3 3256.7 3332.4 3520.8 3676.9 403 1.o 4160.8 4242.7 4425.8 4577.1 4606.0 4932.2 4935.9 5057.2 5 144.9 5323.2 5469.9 5498.6 5676.4 5826.2 5835.4 5946.1 6039.5 6213.3 6294.9 6357.6 6383.3 6500.3 6573.6 6712.7 6727.4

G (cm-‘)

(continued

7 1

3

1

3

1

1746.0 417.9 885.9 108.9 336.5 595.2 99.6 218.1 282.9 446.9 84.2 92.4 192.3 244.1 355.5 57.9 81.0 141.8 84.5 177.0 214.3 292.9 54.5 69.3 73.1 142.6 81.5 161.8 190.3 49.1 245.3 51.8 116.6 64.3 68.7 135.7

t

on next page)

0.216+2 0.274+2 0.723+0 0.543+2 0.552+0 0.256o.l7l+l 0.373+0 0.886-2 0.2670.103+0 0.169-l 0.3400.195-2 0.113-4 0.111+1 0.130-2 0.868-2 0.221-2 0.4320.925-5 0.265 0.937 0.315-2 0.558-3 0.110-2 0.105-4 0.351-4 0.142-b 0.577-2 0.301-9 0.112-2 0.710-3 0.118-3 0.517-4 0.168-4

S

7791.8 7952.1 7965.1 8273.0 8488.3 8499.7 8663.5 8664.6 8750.3 8790.1 8790.3 8878.2 8878.7 8955.1 8962.2 9123.8 9125.0 9211.2 9266.3 9478.1 9493.3 9654.3 9655.3 9742.5 9779.0 9779.2 9864.7 9865.3 9948.9 9949.8

112 310 0.412-3 0.626-3 0.270-3 o.ooo+o 0.422-9 0.280-9 0.164-5 0.131-7 0.208-7 0.243-3 0.128-5 0.196-2 0.519-2 0.316-4 0.139-5 0.701-6 0.652-3 0.118-2 o.ooo+o o.ooo+o 0.123-g 0.707-9 0.364-6 0.206-6 0.110-5 0.266-4 0.623-3 0.512-5 0.102-4 0.181-7

S

T 39.1 36.2 33.2 108.9 62.1 61.2 43.9 45.9 41.2 36.9 37.0 33.5 33.5 33.5 32.1 30.3 29.1 27.1 92.2 57.5 56.6 41.5 43.8 39.1 35.2 35.3 31.5 31.5 31.4 31.1 5673.0 5751.9 5763.6 6045.6 6158.0 6163.3 6291.7 6292.2 6335.0 6405.1 6405.1 6485.7 6494.6 6496.9 6497.1 6619.6 6622.7 6667.9 6863.8 6906.7 6927.1 7021.2 7021.8 7064.7 7132.1 7132.2 7214.7 7220.4 7223.6 7223.8

G (cm-‘) 0.588-4 0.136-3 0.391-4 0.248-7 o.ooo+o 0.752-9 0.188-6 0.516-g 0.231-g 0.275-4 0.289-6 0.363-5 0.627-6 0.363-3 0.109-2 0.400 - 5 0.106-3 0.210-3 0.426-7 0.544-9 0.787-7 0.889-9 0.245-7 0.533-g 0.300-6 0.259-5 0.914-6 0.355-6 0.853-4 0.337-6

S

mode, and the HSe stretch, respectively.

112 310 013 080 061 160 141 240 042 221 122 320 023 301 202 400 301 004 090 071 170 I51 250 052 231 132 330 033 311 212

VIvi%

D2Se

a) S(at 297Kincm-*atm-‘)=10.245xAE(cm-‘)xR*(D’) b, r,=1/,?IAAi,;Ao(s-‘)=3.l368xlO-7xAEJ(cm-’)xR2(DZ). c’ v,, vz and V~correspond to the DSe stretch, the bending

013 080 061 160 141 240 042 221 122 301 202 320 023 400 301 004 090 071 170 151 250 052 231 132 311 212 330 033

G (cm-‘)

01VA

H2Se

Table 10 (continued)

f 142.4 136.6 124.0 365.8 230.8 242.2 165.4 169.1 157.7 134.8 135:o 128.0 120.0 118.2 118.2 111.9 106.6 99.4 201.9 209.0 260.6 158.1 162.1 150.9 128.6 128.9 121.0 116.2 112.3 112.3 003 051 240 160 112 080 032 301 410 221 141 330 013

V,V~V)c,

HDSe

7404.5 7467.7 7592.4 7612.5 7676.7

6788.3 6827.9 6926.6 7096.1 7190.2 7238.5 7260.7

G (cm-‘) 0.359- 1 0.224 - 6 0.470-6 0.395-9 0.526-3 o.ooo+o 0.119-3 0.956-4 0.762-4 0.426- 5 0.278-6 0.205-S 0.503-2

S

r

f ZF z F

110.3 60.2

; 6 a P g k P % g: 2 ‘D a g s z?. D E’

P I% 6

209.3 49.5 59.6

65.0 129.3 37.3

Ir

40.5 77.3 148.7 170.3 45.3

J. Senekowitsch et al. / Vibrational transition probabilities in H$e

would have made the ab initio calculations much more expensive. To date, there are neither experimental nor theoretical transition probabilities known for H,Se. For the absorption processes from the vibrational ground state we have calculated integrated absorption intensities which are given in table 10. As expected from the shape of the dipole moment function, the fundamental transitions are an order of magnitude stronger than the overtones. The largest integrated absorption intensity of the fundamental transitions is calculated for the asymmetric stretch (59.0 atm- ’ cme2), followed by the symmetric stretching mode (48.4 atm- ’ cm-*) and the bending mode (27.9 atm-’ cm-*). In H20, the bending transition is the most intense followed by asymmetric and symmetric stretch transitions. It is also interesting to note the comparison of the calculated absorption intensities and the experimentally detected vibrational band origins (cf. table 5 ). It can be seen, that for those overtone transitions, which have been observed, we also calculate larger intensities than for those which are so far unobserved. This finding strongly suggests that the shape of our dipole moment function is correct. In table 10 the radiative lifetimes for H,Se, D,Se and HDSe are listed. For these species there are large differences in lifetimes of the particular modes. For instance, in the levels with one quantum of pure ul, v2 or u3, the lifetimes in H,Se are 123, 1024 and 100 ms, respectively. These differences, of course, parallel the absorption intensities. It is well known that the transition intensities are sensitive to the mode which is excited in the particular transition. However, as follows from the data in table 10, the lifetimes also vary rather mode-specifically in higher levels. For instance, they decrease among the 100 and 300 levels from 123 to 40 ms, among the 010 and 070 levels from 1024 to 129 ms and among the 00 1 to 003 levels from 100 to 34 ms. Similarly, smooth variations are also found in the combination levels. The reason for this behavior is the approximate harmonic character of the vibrational wavefunctions and the electric harmonicity. In such cases the sums of the Einstein coefficients of spontaneous emission are dominated by the emission rates for Au= 1 transitions, which means that for combination levels there can be two or three dominant terms. Other transitions in H,Se are found to have almost negligible influence. Mainly because

385

the energy separations of the vibrational levels in HDSe and D,Se are smaller than in H,Se the radiative lifetimes increase dramatically relative to H,Se. The low lying bending levels, for instance, are calculated to have lifetimes on the seconds scale. This result emphasises that the mode-characters of the vibrational states, rather than their energies, should be used to rationalize the variations of the vibrational radiative decay.

Acknowledgement This work was supported by the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie. JS thanks Land Hessen for a Fellowship.

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J. Senekowitsch et al. / Vibrational transition probabilities in H$e

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