Four-dimensional dipole moment surfaces and local mode vibrational band intensities of GeH4

23 November 2001

Chemical Physics Letters 349 (2001) 131±136 www.elsevier.com/locate/cplett

Four-dimensional dipole moment surfaces and local mode vibrational band intensities of GeH4 Sheng-Gui He a

a,* ,

Hai Lin

b,1

, Walter Thiel b, Qing-Shi Zhu

a

Open Laboratory of Bond Selective Chemistry, University of Science and Technology of China, Hefei 230026, PR China b Max-Planck-Institut fur Kohlenforschung, Kaiser-Wilhelm-Platz 1, D-45470 Mulheim an der Ruhr, Germany Received 11 September 2001 Dedicated to Prof. Hans B urger on the occasion of his 65th birthday.

Abstract The local mode vibrational band intensities of GeH4 were calculated employing four-dimensional dipole moment surfaces (DMS) and local mode potential energy surfaces. Two kinds of DMS were tested: purely ab initio DMS from coupled cluster CCSD(T) calculations and e€ective DMS where selected higher-order terms in the former were optimized by ®tting experimental intensities. Both DMS reproduce the intensities for transitions below 6500 cm 1 well. The e€ective DMS leads to better agreement for higher excitations and reproduces the intensity anomaly in the fourth local mode manifold. Ó 2001 Elsevier Science B.V. All rights reserved.

1. Introduction The X±H stretching vibrational energy levels of tetrahedral XH4 (X ˆ Si, Ge, Sn) molecules are well described by local mode vibrational theory, for instance, the anharmonically coupled anharmonic oscillator (ACAO) models [1±3] with three adjustable parameters in the empirical potential energy surface (PES). There are two such ACAO models available which are de®ned by the following Hamiltonians:

*

Corresponding author. Fax: +86-551-360-29-69. E-mail address: [email protected] (S.-G. He). 1 On leave from Open Laboratory of Bond Selective Chemistry, University of Science and Technology of China.

Hˆ

4  X 1

2

iˆ1

‡

4 X

 Grr pi2 ‡ De yi2

…Grr0 pi pj ‡ Frr0 ri rj †;

i
Hˆ

4  X 1

 Grr pi2 ‡ De yi2

2  4  X 1 Grr0 pi pj ‡ 2 Frr0 yi yj : ‡ a i
…1†

…2†

Here yi ˆ 1 exp… ari †, ri is the X±H stretching displacement, Grr ˆ 1=mH ‡ 1=mX , mH (mX ) is the mass of the H (X) atom, Grr0 ˆ 1=3mX , pi is the momentum conjugate to ri , De and a are Morse parameters, and Frr0 is the inter-bond potential coupling coecient. The models in Eqs. (1) and (2)

0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 1 1 7 9 - 4

132

S.-G. He et al. / Chemical Physics Letters 349 (2001) 131±136

are denoted as ACAO I and ACAO II, respectively. Based on the ACAO models, the band intensities of local mode manifolds in SiH4 and GeH4 have been computed employing one-dimensional dipole moment surfaces (DMS), either derived empirically from ®tting experimental data [3,4] or calculated from density functional theory [5], 2

I ˆ 3K~ mjhN jMz j0ij :

…3†

Here, j0i is the vibrational ground state and jN i the excited state with F2z symmetry of the Td point group, m~ is the band center in cm 1 , K ˆ 4:1623755  10 19 cm2 D 2 , and Mz is the dipole moment function. The calculations succeeded for overtones but failed for combinations due to the neglect of inter-bond cross terms in the DMS [5]. More recently, we have studied band intensities of local mode manifolds in SiH4 employing a fourdimensional (4D) ab initio DMS [6]. We demonstrated that the local mode combination bands gain their intensities mainly from the cross terms in the DMS, and rationalized the intensity anomaly that the (3000) [7] overtone is weaker than the (4000) overtone and the (2100) combination [6]. A similar intensity anomaly is present in GeH4 where the (4000) overtone is weaker than the (3100) combination [4]. This e€ect is investigated in the present work using an ab initio 4D stretching DMS as well as e€ective DMS that are partly derived from the experimental intensities.

2. Intensities employing ab initio DMS The ab initio DMS calculations were based on the CCSD(T) [8±10] coupled cluster method and employed the correlation-consistent polarized valence quadruple zeta (cc-pVQZ) basis set [11,12]. Using MO L P R O 2000 [13,14] the computations for GeH4 were carried out in complete analogy to those for SiH4 [6], except for the substitution of Si by Ge, and the reader is therefore referred to [6] for technical details. The optimized equilibrium  The calcuGe±H bond length is Re ˆ 1:54086 A. lated DMS data points are available from the authors upon request. The Cartesian DMS model developed in [6] was applied in the least-squares ®tting of the ab initio data points. The determined DMS expansion coecients up to third-order are denoted as 4DDMS1 and are listed in Table 1. A good ®t is indicated by the small root-mean-square (RMS) of the ®tting residual, 3:22  10 4 D (1D ˆ 3:33564  10 30 C m). We adopted the PES parameters from an earlier  and study [15]: De ˆ 34965:9 cm 1 , a ˆ 1:41396 A, 2 1  0 for ACAO I model and Frr ˆ 617:1 cm A  and Frr0 ˆ De ˆ 34977:6 cm 1 , a ˆ 1:41417 A, 2 1  626:2 cm A for ACAO II model. The vibrational wavefunctions were calculated variationally. The maximum total vibrational quantum number [7] Vmax of the basis was 12. Further details can be found in literature [3,16] and will not repeated here.

Table 1 Four-dimensional stretching vibrational dipole moment surfaces of GeH4 nF2 Cabcd 2 CF1000 F2 C2000 2 CF1100 F2 C3000 2 C1F 2100 2 C2F 2100 2 CF0111

…D (D (D (D (D (D (D

 A  A  A  A  A  A  A

RMS (D) a

1

) ) 2 ) 3 ) 3 ) 3 ) 3 ) 2

4D-DMS1 a

4D-DMS2 b

4D-DMS3 c

)1.74899(25)

)1.74899

)1.74899

)0.35530(45) 0.20842(40) 0.5409(28) )0.2203(21) 0.1292(22) 0.0686(19)

)0.35530 0.20842 0.445(31) )0.440(79) 0.1292 0.0686

)0.35530 0.20842 0.447(31) )0.437(77) 0.1292 0.0686

±

±

0.000322

DMS obtained by ®tting to ab initio data points. The value given in parentheses is one standard error in the last signi®cant digit, this also applies to 4D-DMS2 and 4D-DMS3 . b Empirically adjusted DMS for ACAO I Hamiltonian model. See text for details. c Empirically adjusted DMS for ACAO II Hamiltonian model. See text for details.

S.-G. He et al. / Chemical Physics Letters 349 (2001) 131±136 Table 2 Local mode vibrational band centersa (cm 1 ) and intensitiesb (10 Band (1000) (2000) (1100) (3000) (2100, 1F2 ) (2100, 2F2 ) (4000) (3100, 1F2 ) (3100, 2F2 ) (5000) (4100, 1F2 ) (4100, 2F2 ) (6000) (5100, 1F2 ) (5100, 2F2 ) (7000) c

D

m~obs

m~cal1

m~cal2

22

cm) of

74

GeH4

4D-DMS1

Iobs

4D-DMS2

4D-DMS3

IACAOI

IACAOII

IACAOI

IACAOII

0.637E + 6 0.592E + 4 0.151E + 3 0.142E + 2 0.214E + 1 0.223E + 1 0.138E + 1 0.179E + 0 0.642E ) 1 0.567E + 0 0.110E ) 1 0.576E ) 2 0.110E + 0 0.861E ) 3 0.552E ) 3 0.192E ) 1

0.638E + 6 0.571E + 4 0.162E + 3 0.229E + 2 [0.156E + 2]

0.638E + 6 0.572E + 4 0.151E + 3 0.230E + 2 [0.151E + 2]

0.451E + 0 [0.936E + 0]

0.452E + 0 [0.932E + 0]

0.311E + 0 [0.627E ) 1]

0.312E + 0 [0.639E ) 1]

0.670E ) 1 [0.509E ) 2]

0.674E ) 1 [0.526E ) 2]

0.121E ) 1

0.122E ) 1

1.07

0.53

0.52

2111.14 4153.82 ± 6128.58 6263.70 ± 8035.84 ± 8241.62 9875.78 ± ± 11647.23 ± ± 13352.66

2110.42 4153.45 4221.83 6128.37 6264.45 6266.28 8035.62 8240.74 8241.61 9875.12 10149.04 10149.43 11646.85 11989.38 11989.74 13350.82

2110.70 4153.48 4221.59 6128.49 6264.60 6264.80 8035.69 8239.94 8240.09 9875.07 10147.44 10147.59 11646.63 11987.13 11987.27 13350.38

0.551E + 6 0.738E + 4 ± 0.196E + 2 [0.472E + 1]

± ± ± ±

0.637E + 6 0.591E + 4 0.162E + 3 0.141E + 2 0.315E + 1 0.126E + 1 0.139E + 1 0.484E ) 2 0.230E + 0 0.569E + 0 0.542E ) 2 0.101E ) 1 0.110E + 0 0.545E ) 3 0.728E ) 3 0.192E ) 1

±

±

±

±

1.10

0.461E + 0 [0.190E + 1] 0.265E + 0 [0.959E ) 1]

133

a

The two (n100) bands with lower and higher centers are labeled by (n100; 1F1 ) and (n100; 2F2 ), respectively. Experimental data are taken from [15,21,23±28]. m~cal1 and m~cal2 are calculated band centers from ACAO I and II models, respectively. b Experimental data are taken from [4,17,18]. The value in the square brackets is the summed intensity of (n100; 1F2 ) and (n100; 2F2 ) bands. See text for details. c Logarithmic deviation, see text.

The evaluated absolute intensities of 74 GeH4 are listed in the sixth and seventh columns of Table 2 for ACAO I and ACAO II models, respectively. It should be pointed out that there are two components within the same (n100) combinational manifold when n > 1. In this work, (n100; 1F2 ) and (n100; 2F2 ) are adopted to label the bands with lower and higher centers, respectively. The experimental absolute intensities were computed from the relative intensities [4,17] using the absolute intensity of the fundamental [18] as reference. They are also included in Table 2 for comparison. 3. Intensities employing e€ective DMS The optimization of the e€ective DMS was carried out in a least-squares procedure, in which 2 ndata  1 X Ii …cal† D2 ˆ ln …4† ndata iˆ1 Ii …obs† is minimized. Here D is the logarithmic deviation [19], ndata is the number of experimental data,

Ii …cal† and Ii …obs† are calculated and observed band intensities, respectively. As the reported experimental intensities are the sum of individual components for the (n100) manifolds [4,17], we ®tted the summed intensities of the whole manifold (given in square brackets) rather than intensities of the individual components. The theoretical intensities were evaluated using the same general approach as in Section 2 (PES parameters from [15], Cartesian DMS model). The ®rst-order and second-order coecients in the DMS were constrained to ab initio values. This was also done for 2 the CF0111 term [6] because of its small contribution 2 to the transition moment and for the C2F 2100 term 1F2 because of its strong correlation with C2100 . Only 2 2 CF3000 and C1F 2100 were optimized. Higher-order terms were constrained to 0. The ®tted e€ective DMS in the ACAO I and ACAO II models are denoted as 4D-DMS2 and 4D-DMS3 , respectively. They are listed in Table 1. Obviously 4D-DMS2 and 4D-DMS3 are similar 2 to each other. In both cases, the optimized CF3000 deviate by only about 18% from the ab initio value

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S.-G. He et al. / Chemical Physics Letters 349 (2001) 131±136

Table 3 Ge±H local mode band intensities (10 b

22

cm) of

74

GeHn D4

n

(n ˆ 1, 2, 3) calculated from the optimized dipole moment surfacea

Band

IGeHD3

IGeH2 D2

IGeH3 D

(100) (200) (110) (300) (210) (400) (310) (500) (410) (600) (510) (700)

0.160E + 6 0.144E + 4 ± 0.591E + 1 ± 0.109E + 0 ± 0.775E ) 1 ± 0.167E ) 1 ± 0.305E ) 2

0.320E + 6 0.288E + 4 0.254E + 2 0.117E + 2 0.250E + 1 0.220E + 0 0.154E + 0 0.155E + 0 0.106E ) 1 0.336E ) 1 0.875E ) 3 0.611E ) 2

0.479E + 6 0.430E + 4 0.760E + 2 0.174E + 2 0.754E + 1 0.335E + 0 0.465E + 0 0.233E + 0 0.318E ) 1 0.505E ) 1 0.263E ) 2 0.916E ) 2

a b

The 4D-DMS3 and ACAO II model are used. The 4D-DMS2 and ACAO I model give almost identical results. See text for details. (n1 n2 n3 ) for GeH3 D, (n1 n2 ) for GeH2 D2 , and (n1 ) for GeHD3 . The summed intensities are given for each manifold.

2 while C1F 2100 is about twice as large. The evaluated intensities are listed in the eighth and ninth columns of Table 2. The optimized DMS (4D-DMS2 and 4D-DMS3 ) were also used to predict the Ge±H local mode band intensities in other isotopomers. Both DMS give almost identical results and thus only the 4D-DMS3 values are shown in Table 3.

4. Discussion The ab initio DMS achieves reasonably good agreement between the calculated and observed intensities for V ˆ 1, 2, 3. For higher values of V, the agreement is less satisfactory. This is consistent with previous ®ndings [20]: as excitation increases (e.g., when V > 2), contributions from higher order such as the third-order DMS terms become more and more important, and while the CCSD(T) calculations give highly accurate low-order DMS terms, the high-order terms may still not be determined with sucient accuracy. This suggests the combination of ab initio data and some ®tting to experimental intensities in order to arrive at more realistic e€ective DMS. This strategy has been followed when optimizing 4D-DMS2 and 4DDMS3 in Section 3. It is interesting to ®nd that the predicted intensities of (n100; 1F2 ) from ACAO I are rather close to those of (n100; 2F2 ) from ACAO II for n ˆ 3, 4, 5 (and vice versa). This implies that the two models give two similar eigenfunctions for

the (n100) manifolds, but with reversed order of the almost degenerate eigenvalues. Such switching of order occurs also within each model if the center atom 74 Ge is substituted by other isotopes (kinetic e€ect) or if the potential parameters vary. The ACAO I model is very sensitive to such changes while the ACAO II model is more stable. These ®ndings underline the importance of eigenfunction-related spectroscopic information such as intensities in the assignment of close-lying transitions. The reversal of order in the ACAO models is also critical for the prediction of physical quantities based on the eigenfunctions of combination states. For instance, some of the Coriolis parameters of the (n100) manifolds given in Table 2 of [21] should be exchanged. Even after taking the switching problem into consideration, the ACAO I and II models predict rather di€erent intensities for the components of (2100) manifold. However, the summed intensities are almost the same. This is due to the fact that as the inter-manifold coupling is negligible compared with the intra-manifold coupling, the summed intensity must then remain invariant to variational wavefunction mixing within a given manifold. On the other hand, it is dicult to determine the respective intensities experimentally since the two (n100) components are almost degenerate. It may be more convenient in such a case to compare only the summed intensities with observation when deriving the e€ective DMS, as we have done in this work.

S.-G. He et al. / Chemical Physics Letters 349 (2001) 131±136

135

Table 4 Transition moments (D) from di€erent terms in the DMS expansion for the (4000), (3100, 1F2 ), and (3100, 2F2 ) bandsa 2 CF1000 2 CF2000 C21100 2 CF3000 1F2 C2100 2 C2F 2100 2 CF0111

Totalb a b

(4000)

(3100, 1F2 )

(3100, 2F2 )

)0.3979E ) 3 )0.1987E ) 3 0.1561E ) 5 0.2582E ) 3 0.7700E ) 5 )0.1110E ) 5 )0.7398E ) 8

)0.1902E ) 5 0.8216E ) 6 0.1839E ) 4 0.1591E ) 5 0.7596E ) 4 )0.1564E ) 4 )0.1826E ) 6

0.1303E ) 6 0.3897E ) 6 0.2467E ) 4 )0.8570E ) 7 )0.5672E ) 4 )0.2122E ) 4 )0.2455E ) 6

0.6712E ) 4

0.7903E ) 4

)0.5308E ) 4

The 4D-DMS3 and ACAO II models are used. See text for details. Calculated total transition moment.

Replacing the purely ab initio 4D-DMS1 by e€ective 4D-DMS2 and 4D-DMS3 reduces the logarithmic deviation (D) signi®cantly (from 1.10 and 1.07 to 0.53 and 0.52, respectively). Moreover, the intensity anomaly that the (4000) overtone is weaker than the (3100) combination band is now successfully reproduced. For illustration, the transition moments from di€erent terms in 4DDMS3 are listed in Table 4 for the (4000), (3100, 1F2 ), and (3100, 2F2 ) bands. It is evident that the transition moment of the (4000) overtone mainly 2 arises from the CFn000 (n ˆ 1, 2, 3) terms in the DMS. However, these contributions cancel each other signi®cantly and thus result in a weak (4000) band. A similar cancellation has been found for the Si±H stretching motion in SiHF3 [20]. On the other hand, the large transition moments of the (3100, 1F2 ) and (3100, 2F2 ) combinations mainly 2 come from the inter-bond coupling terms (CF1100 , 1F2 2F2 C2100 , and C2100 ). Neglecting such cross terms in multi-dimensional DMS will thus tend to underestimate the intensities of combination bands [5]. This point has also been demonstrated for P±H stretching in PH3 [22]. In conclusion, the intensity anomaly in GeH4 is caused by the weakening of the (4000) overtone through cancellation e€ects and the strengthening of the (3100) bands through large inter-bond cross terms in the DMS. Acknowledgements This work was supported by the National Project for the Development of Key Fundamental

Sciences in China, the National Natural Science Foundation of China, the Chinese Academy of Science, and EC (Research Training Network No. HPRN-CT-2000-00022). References [1] L. Halonen, M.S. Child, Mol. Phys. 46 (1982) 239. [2] L. Halonen, M.S. Child, J. Chem. Phys. 79 (1983) 4355. [3] L. Halonen, M.S. Child, Comput. Phys. Commun. 51 (1988) 173. [4] H. Lin, D. Wang, X.Y. Chen, X.G. Wang, Z.P. Zhou, Q.S. Zhu, J. Mol. Spectrosc. 192 (1998) 249. [5] H. Lin, L.F. Yuan, Q.S. Zhu, Chem. Phys. Lett. 308 (1999) 137. [6] H. Lin, S.G. He, X.G. Wang, L.F. Yuan, H. B urger, J.F. D'Eu, N. Reuter, W. Thiel, Phys. Chem. Chem. Phys. 3 (2001) 3506. [7] Local mode notation (n1 n2 n3 n4 ) where ni denotes the quantum number of the excitation of the ith X±H bond. V ˆ n1 ‡ n2 ‡ n3 ‡ n4 is the total vibrational quantum number. [8] G.D. Purvis, R.J. Bartlett, J. Chem. Phys. 76 (1982) 1910. [9] M. Urban, J. Noga, S.J. Cole, R.J. Bartlett, J. Chem. Phys. 83 (1985) 4041. [10] K. Raghavachari, G.W. Trucks, J.A. Pople, M. HeadGordon, Chem. Phys. Lett. 157 (1989) 479. [11] T.H. Dunning, J. Chem. Phys. 90 (1989) 1007. [12] D.E. Woon, T.H. Dunning, J. Chem. Phys. 98 (1993) 1358. [13] MO L P R O 2000 is a package of ab initio programs written by H.-J. Werner and P.J. Knowles, with contributions from R.D. Amos et al. [14] C. Hampel, K. Peterson, H.-J. Werner, Chem. Phys. Lett. 190 (1992) 1; The program to compute the perturbative triples corrections has been developed by M.J.O. Deegan, P.J. Knowles, Chem. Phys. Lett. 227 (1994) 321, and references therein. [15] F.G. Sun, X.G. Wang, J.L. Liao, Q.S. Zhu, J. Mol. Spectrosc. 184 (1997) 12.

136

S.-G. He et al. / Chemical Physics Letters 349 (2001) 131±136

[16] H. Lin, L.F. Yuan, S.G. He, X.G. Wang, Chem. Phys. Lett. 332 (2000) 569. [17] H. Lin, Ph.D. Thesis, University of Science and Technology of China, 1998. [18] A.M. Coats, D.C. McKean, D. Steele, J. Mol. Struct. 320 (1994) 269. [19] T.K. Ha, M. Lewerenz, R.R. Marquardt, M. Quack, J. Chem. Phys. 93 (1990) 7097. [20] H. Lin, H. B urger, E.B. McKadmi, S.G. He, L.F. Yuan, J. Breidung, W. Thiel, T.R. Huet, J. Demaison, J. Chem. Phys. 115 (2001) 1378. [21] X.H. Wang, H. Lin, X.Y. Chen, X.G. Wang, D. Wang, K. Deng, Q.S. Zhu, Mol. Phys. 98 (2000) 1409. [22] S.G. He, J.J. Zheng, S.M. Hu, H. Lin, Y. Ding, X.H. Wang, Q.S. Zhu, J. Chem. Phys. 114 (2001) 7018.

[23] F.G. Sun, X.G. Wang, Q.S. Zhu, C. Pierre, G. Pierre, Chem. Phys. Lett. 239 (1995) 373. [24] S.Q. Mao, R. Saint-Loup, A. Aboumajd, P. Lepage, H. Berger, A.G. Robiette, J. Raman Spectrosc. 13 (1982) 257. [25] Q.S. Zhu, H.B. Qian, B.A. Thrush, Chem. Phys. Lett. 186 (1991) 436. [26] Q.S. Zhu, B.A. Qian, B.A. Thrush, Chem. Phys. Lett. 150 (1989) 181. [27] X.Y. Chen, H. Lin, X.G. Wang, D. Wang, K. Deng, Q.S. Zhu, J. Mol. Struct. 517±518 (2000) 41. [28] A. Campargue, J. Vetterh o€er, M. Chenevier, Chem. Phys. Lett. 192 (1992) 353.