Accurate structure and torsional barrier height of disilane

Abstract

Very accurate structural parameters and torsional barrier heights of disilane have been determined using the CCSD(T) method with correlation consistent basis sets. Not only has the basis set incompleteness been taken into account using the appropriate extrapolation schemes, but also the core-correlation corrections have been considered.

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1 Introduction

Silicon-compounds play an important role in the semiconductor industry, thus there is great interest in their structural and energetic properties. Furthermore, disilicon compounds have interesting structural properties since they do not exhibit the typical properties of the analogous dicarbon compounds, for instance they do not form multiple bonds.

Few studies have been carried out in the recent past on the equilibrium structure and/or rotational barrier of disilane. In particular, two ab initio and density functional studies of the molecular structure and torsional barrier of disilane and other related compounds were published in 1997: Urban et al. performed a joint experimental–theoretical study on C₂H₆, Si₂H₆, SiGeH₆ and Ge₂H₆,¹ while Cho and coworkers carried out a theoretical investigation on Si₂H₆, Si₂F₆ and Si₂(CH₃)₆.² In both works, MP2 and B3LYP methods with basis sets of double- and triple-zeta quality were employed. An earlier study on the rotational barrier of ethane congeners, published in 1992 by Schleyer and coworkers,³ was carried out at a lower level of theory.

As far as molecular structure is concerned, the most accurate calculations were performed by Feller and Dixon in a theoretical investigation on heats of formation of small silicon containing compounds.⁴ Their geometry optimizations of disilane were made at CCSD(T) level employing the aug-cc-pVXZ (X = D,T,Q) bases of Dunning. In the meantime, B3LYP computations with bases of double- or triple-zeta quality of structural properties and heats of formation for silicon–hydrogen compounds were published by Jursic.⁵ The latter study is the only one in which the complete basis set limit for structural properties was evaluated.

Although the equilibrium structure and the torsional barrier of disilane have been theoretically investigated in the recent past,^1,2,4,5 a thorough analysis of the molecular structure and barrier height of this important molecule is still missing. In particular, the basis set incompleteness error and core-correlation effects were not previously taken into account. This lack stimulated the present work; thus, we have decided to perform an highly accurate study of both the equilibrium geometry and the torsional barrier.

2 Methods and computations

The equilibrium geometry and the torsional barrier have been evaluated at coupled cluster level; we have used the CCSD(T)^6–8 method, which is a good approximation to take into account triple excitations. Throughout the present work only valence electrons have been correlated in CCSD(T) calculations (frozen core approximation), except for a few geometry optimizations and single point calculations in which all electrons have been correlated; these computations have been performed in order to evaluate the extent of core–valence correlation effects. Nondynamical electron correlation does not seem to be significant, as judged from the coupled cluster T₁ diagnostic^9,10 which is always lower than 0.01 (it should be lower than 0.02) for all employed basis sets.

The computations have been performed using different bases. We have employed the correlation consistent double-zeta (cc-pVDZ), triple-zeta (cc-pVTZ) and quadruple-zeta (cc-pVQZ) basis sets of Dunning,¹¹ which comprise 66, 152 and 298 contracted Gaussian-type orbitals (cGTOs), respectively. In addition, since the cc-pVXZ basis sets are not suitable for calculations in which core electrons are correlated, the completely uncontracted cc-pVTZ basis, denoted as uVTZ, the Martin–Taylor set, denoted as MT, and the core–valence quadruple-zeta basis of Dunning (cc-pCVQZ)¹² have been used. In detail, the MT basis has been obtained by adding one large-exponent p function, two large-exponent d and f functions to the uVTZ. The exponents for silicon atoms have been evaluated as explained in ref. 13, while for H atoms the regular uVTZ set has been used. For the cc-pCVQZ basis (CVQZ), the weighted core–valence cc-pwCVQZ for silicon has been employed.¹² The uVTZ, MT and CVQZ sets comprise 214, 268 and 398 GTOs/cGTOs, respectively. It should be noted that the MT basis has been used since it was found to recover essentially the full effect of including core-correlation in geometry optimization of first-row compounds^14,13 and to perform very well for second row elements.¹⁵

All quantum-chemical calculations have been carried out using the MOLPRO suite of programs.^6,16–18 As far as basis sets are concerned, spherical harmonics have been used in all calculations.

Geometry optimizations have been performed using numerical gradients, as implemented in MOLPRO.^19,17 The step sizes used are the default ones: 0.005 Å for bond distances and 1° for angles. The largest internal gradient components at the stationary points are always lower than 2 × 10⁻⁵ a.u. The energy difference between the staggered and eclipsed conformations optimized at a certain level of theory gives the torsional barrier at that level.

3 Results and discussion

3.1 Equilibrium geometry

The structure of the molecule is determined by three parameters, namely, the Si–H and Si–Si bond lengths and the angle SiSiH. Since the two silyl groups are allowed to rotate around the Si–Si axis, the internal rotation angle τ has to be defined. It is the dihedral angle between the planes SiSiH of two different silyl groups. The staggered conformation is defined for τ = 0°, the eclipsed one for τ = 60°.

In general, CCSD(T) calculations give very reliable results for equilibrium geometries and very systematic errors. More precisely, CCSD(T) computations employing cc-pVXZ (X = D, T, Q, ...) bases correlating only valence electrons give overestimated bond lengths (see for example refs. 20–22). The optimized structures both for staggered and eclipsed conformations at the CCSD(T) level employing different basis sets are reported in Table 1. On the basis of the vast literature on these topics, we can assume an accuracy of 0.001–0.003 Å and 0.05–0.1° for CCSD(T)/cc-pVQZ structures. In general, the minimum value is referred to single bonds, while the maximum discrepancy is for multiple bond lengths, even if single bond lengths involving second row atoms are affected by larger uncertainty.

Table 1 Equilibrium geometry for staggered (τ = 0°) and eclipsed (τ = 60°) conformations of disilane computed at CCSD(T) level employing different basis sets.^a Comparison with previous ab initio calculations. Distances in Å, angles in degrees

	CCSD(T)/VDZ (frozen core)	CCSD(T)/VTZ (frozen core)	CCSD(T)/VQZ (frozen core)	CCSD(T)/uVTZ (frozen core)	CCSD(T)/uVTZ (all electrons)	CCSD(T)/MT (frozen core)
Staggered
Si–Si	2.3662	2.3519	2.3437	2.3510	2.3481	2.3463
Si–H	1.4941	1.4860	1.4832	1.4850	1.4831	1.4815
∠SiSiH	110.31	110.33	110.26	110.34	110.34	110.35
Eclipsed
Si–Si	2.3774	2.3632	2.3552	2.3623	2.3597	2.3579
Si–H	1.4939	1.4858	1.4829	1.4848	1.4829	1.4813
∠SiSiH	110.51	110.51	110.44	110.52	110.51	110.52

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	CCSD(T)/MT (all electrons)	CCSD(T)/CVQZ (frozen core)	CCSD(T)/CVQZ (all electrons)	CCSD(T)/aVQZ^b (frozen core)	B3LYP/VTZ^c	MP2/TZ(2df,2pd)^d
a VDZ, VTZ, VQZ and CVQZ mean cc-pVDZ, cc-pVTZ, cc-pVQZ and cc-pCVQZ, respectively. uVTZ and MT mean completely uncontracted cc-pVTZ and Matin-Taylor sets, respectively (see text). b Ref. 4. CCSD(T) optimization employing the aug-cc-pVQZ basis set. c Ref. 5. CBQS-MP2/6-31G(d′): Si–Si = 2.340 Å Si–H = 1.487 Å and ∠SiSiH = 110.4°. d Ref. 1.
Staggered
Si–Si	2.3367	2.3411	2.3294	2.346	2.354	2.349
Si–H	1.4777	1.4813	1.4765	1.484	1.486	1.478
∠SiSiH	110.35	110.26	110.26	110.2	110.4	110.1
Eclipsed
Si–Si	2.3483	2.3527	2.3410	—	—	2.363
Si–H	1.4775	1.4810	1.4762	—	—	1.477
∠SiSiH	110.53	110.44	110.45	—	—	110.3

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In Table 1 the CCSD(T) optimized geometries obtained with the uVTZ, MT and CVQZ basis sets correlating both valence and all electrons are also reported. First of all, as expected, the CCSD(T)/uVTZ and, particularly, CCSD(T)/MT frozen core geometries are a little bit shorter than the CCSD(T)/cc-pVTZ; this is due to the absence of contraction error in the valence shell. In general, as far as frozen core calculations are concerned, we notice that by enlarging the basis the distances shorten independently of the kind of set considered, minimal or not on core orbitals. Furthermore, as expected, including the core correlation there is a further shortening of the bonds; for example, the distances obtained at CCSD(T)/MT level with all electrons correlated are smaller than those at CCSD(T)/cc-pVQZ level. On the other hand, from Table 1 it comes out that on varying the basis set the bond angles change just a little: less than 0.1°.

By comparing valence and all electrons computations with uVTZ but especially with the MT and CVQZ bases, we can deduce the extent of core-correlation effects. First of all, it seems that there are no effects on bond angles, while there is a small correction for the Si–H bond (−0.005 Å with MT, −0.007 Å with CVQZ) and a slightly larger effect for the Si–Si bond length (−0.007 Å with MT, −0.014 Å with CVQZ).

In Table 1, all our evaluated structures are compared with previous ab initio data; from Table 1 it comes out that the CCSD(T)/aug-cc-pVQZ frozen core optimized structure by Feller and Dixon⁴ is just a little bit worse than the CCSD(T)/cc-pVQZ, i.e. the bond lengths are few thousandths of Å longer; the same happens for all the other CCSD(T)/aug-cc-pVXZ and CCSD(T)/cc-pVXZ (X = D,T) optimized geometries. Although discrepancies of the order of a few thousandths of Å are too small to draw conclusions, it seems that not only the additional computational effort due to inclusion of diffuse functions is not justified but also it is disadvantageous.

Using CCSD(T)/cc-pVXZ (X = D,T,Q) bond distances it is possible to extrapolate to the valence correlation limit by applying the geometric extrapolation A + B/Cⁿ, where n is 2 for X = D, 3 for X = T and so on. This extrapolation scheme was first proposed by Feller²³ and successfully employed by Martin and Taylor (see for example refs. 13 and 24). The extrapolated CCSD(T)/cc-pV∞Z geometries, both for staggered and eclipsed conformations, are reported in Table 2. By comparing the CCSD(T)/cc-pVQZ and CCSD(T)/cc-pV∞Z results we can remark that, while the Si–H bond distance is practically converged at CCSD(T)/cc-pVQZ level, the variation of Si–Si bond is relevant: ∼0.01 Å.

Table 2 Extrapolated equilibrium geometries of staggered (τ = 0°) and eclipsed (τ = 60°) conformations of disilane. Comparison with experiment. Distances in Å, angles in degrees and rotational constants in MHz

	CCSD(T)/cc-pV∞Z^a	CCSD(T)+core/cc-pV∞Z^b	Extrapolated geometry 1^c	Extrapolated geometry 2^d	Extrapolated geometry 3^e	Recommended geometry^f	Exp. r0^g
a Extrapolated equilibrium geometry to the valence correlation limit by applying the geometric extrapolation A + B/C^n; see text. b Extrapolated equilibrium geometry to the valence correlation limit plus core correlation correction: additivity assumption of eqn. (1); see text. c Extrapolated equilibrium geometry obtained from eqn. (1) employing the MT basis set; see text. d Extrapolated equilibrium geometry obtained from eqn. (1) employing the CVQZ basis set; see text. e Extrapolated equilibrium geometry obtained from eqn. (1) employing the uVTZ basis set; see text. f Recommended equilibrium geometry; see text. g Ref. [31].
Staggered
Si–Si	2.333	2.323	2.334	2.332	2.341	2.333(1)	2.3317(15)
Si–H	1.482	1.478	1.479	1.478	1.481	1.480(1)	1.4874(17)
∠SiSiH	110.3	110.3	110.3	110.3	110.3	110.3(1)	110.66(16)
A	43 259.5	43 494.0	43 435.2	43 494.0	43 318.0	43 377.(60.)	43 195
B	5 101.8	5 144.0	5 100.2	5 108.8	5 071.6	5 103.(4.)	5 093
Eclipsed
Si–Si	2.345	2.335	2.346	2.344	2.353	2.345(1)	–
Si–H	1.481	1.477	1.479	1.478	1.481	1.479(1)	–
∠SiSiH	110.4	110.4	110.4	110.4	110.4	110.4(1)	–
A	43 374.1	43 609.3	43 491.5	43 550.3	43 374.1	43 492.(60.)	–
B	5 054.1	5 095.7	5 051.8	5 060.2	5 023.6	5 056.(4.)	–

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Assuming additivity of the effects of basis set extension beyond cc-pVQZ and core correlation, Martin and Taylor suggested the following expression to obtain an extrapolated equilibrium geometry:¹³

r ≃ r(cc − pVQZ, valence) + r(MT, all) − r(MT, valence), (1)

where valence and all mean CCSD(T) computations correlating only valence and all electrons, respectively. This estimation has also been performed employing the optimized structures obtained with the CVQZ basis set. Since the MT set has been constructed starting from the uVTZ basis and, in the case of molecules without H atoms or containing only few hydrogens, the number of functions added is large, we decided to test the possibility of using the uVTZ instead of the MT set in the extrapolation described by eqn. (1).

The structures resulting from eqn. (1), evaluated both for staggered and eclipsed conformations, are denoted as “Extrapolated geometry 1”, “Extrapolated geometry 2” and “Extrapolated geometry 3” according to whether the MT, CVQZ or uVTZ sets have been employed, respectively; they are reported in Table 2. By comparing them we can deduce that they are very similar (differences of the order of a few thousandths of Å), except for the Si–Si bond of “Extrapolated geometry 3” which is a little bit longer.

Moreover, the additivity assumption expressed by eqn. (1) has also been applied to the CCSD(T)/cc-pV∞Z geometry in order to try to recover the core correlation effects affecting this structure. The resulting geometry, denoted as “CCSD(T) + core/cc-pV∞Z”, is compared with the other extrapolated structures in Table 2. First of all, we can point out that for the Si–H distance they agree perfectly with one another: the changes are of the order of 2–4 thousandths of Å. On the other hand, larger deviations have been found for the Si–Si bond; more precisely, the value of the “CCSD(T) + core/cc-pV∞Z” structure seems to be underestimated, while that of “Extrapolated geometry 3” is overestimated. According to the literature on this topic, we can assume an accuracy of at least 0.001 Å for bond lengths and 0.1° for bond angles for the CCSD(T)/cc-pV∞Z, “Extrapolated geometry 1” and “Extrapolated geometry 2” geometries, from which our best estimation of molecular structure has been deduced. Thus, we suggest as the recommended equilibrium geometry (also reported in Table 2): r(Si–Si) = 2.333(1) Å, r(Si–H) = 1.480(1) Å and ∠(SiSiH) = 110.3(1)° for the staggered conformation, and r(Si–Si) = 2.345(1) Å, r(Si–H) = 1.479(1) Å and ∠(SiSiH) = 110.4(1)° for the eclipsed conformation.

Finally, by comparing our extrapolated equilibrium geometries with the experimental r₀, we notice a very good agreement for the Si–Si bond length, while the experimental Si–H distance seems to be overestimated. This is expected since we are performing a comparison between the equilibrium geometry and the vibrational ground state structure, which are only approximately comparable. In particular, the bonds containing light atoms like hydrogen are those affected by the largest vibrational contribution.

3.2 Torsional barrier

As previously mentioned, the torsional barrier has been evaluated as the energy difference between the staggered and eclipsed conformations. The energies employed in this determination are the values corresponding at the optimized geometries; thus, as for the geometry optimization, the torsional barrier has been computed at CCSD(T) level using different basis sets. The values obtained are reported in Table 3, where they are compared with previous theoretical data. For the cc-pVXZ series of bases, we can notice that by enlarging the set the barrier height increases: ∼17 cm⁻¹ going from cc-pVDZ to cc-pVTZ basis and ∼5 cm⁻¹ from cc-pVTZ to cc-pVQZ. On the other hand, as far as frozen core computations are concerned, the MT set gives a value higher by about 5 cm⁻¹ than cc-pVQZ.

Table 3 Rotational barrier evaluated at CCSD(T)^a level employing different basis sets.^b Rotational barrier for the extrapolated equilibrium geometries reported Table 2. Infinite basis set limit and recommended value of the barrier height. Comparison with previous ab initio calculations and experiment. Values in cm⁻¹ and kcal mol⁻¹ (given in parenthesis)

	CCSD(T)/VDZ (f.c.)	CCSD(T)/VTZ (f.c.)	CCSD(T)/VQZ (f.c.)	CCSD(T)/uVTZ (f.c.)	CCSD(T)/uVTZ (a.e.)	CCSD(T)/MT (f.c.)	CCSD(T)/MT (a.e.)	CCSD(T)/CVQZ (f.c.)	CCSD(T)/CVQZ (a.e.)	MP2/^cTZ(2df,2pd)	B3LYP/^d6-31G*
Rotational barrier/cm^−1(kcal mol^−1)	337.10 (0.96)	353.92 (1.01)	358.76 (1.03)	355.43 (1.02)	356.16 (1.02)	363.50 (1.04)	368.29 (1.05)	364.85 (1.04)	370.53 (1.06)	– (1.12)	– (0.91)

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	CCSD(T)/^ecc-pV∞Z	CCSD(T)+core/^ecc-pV∞Z	Extrapol.^e geometry 1	Extrapol.^e geometry 2	Extrapol.^e geometry 3	Infinite^f basis limit	Recommended^g value	Exp.^h
a The acronyms f.c. and a.e. mean frozen core and all electrons calculations, respectively. b VDZ, VTZ, VQZ and CVQZ mean cc-pVDZ, cc-pVTZ, cc-pVQZ and cc-pCVQZ, respectively. uVTZ and MT mean completely uncontracted cc-pVTZ and Matin–Taylor bases, respectively (see text). c Ref. 1. DFT/TZ(2df,2pd): 0.99 kcal mol^−1. d Ref. 2. MP3/6-31G* and CISD/6-31G*: 1.04 and 1.01 kcal mol^−1, respectively; they both are single point calculations at MP2/6-31G* geometry. e Single-point calculations: computations performed at the various extrapolated geometries (reported in Table 2) employing the CCSD(T) method (correlating all electrons) and the MT basis set (see text). f Value obtained by 1/(X − 1)^3 extrapolation scheme. ΔE∞ = 359.91 cm^−1: value obtained by 1/X^3 extrapolation scheme. See text. g Recommended value: extrapolated value to the infinite basis set limit plus the core correlation correction; see text. h Ref. 32: from Raman spectrum of gaseous disilane. Ref. 33: 1.0 kcal mol^−1, value obtained indirectly from combination bands in the mid-infrared spectrum.
Rotational barrier/cm^−1 (kcal mol^−1)	369.9 (1.06)	369.0 (1.05)	369.3 (1.06)	369.1 (1.06)	369.9 (1.06)	360.79 (1.03)	366.5 (1.05)	439(10) (1.26(3))

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In order to take into account the basis set truncation error, the CCSD(T)/cc-pVXZ energies both for the staggered and eclipsed conformations have been extrapolated to the infinite basis limit following two extrapolation schemes. The first one is based on the extrapolation of the total energy by the formula:

E^tot(X) = E^tot_∞ + A/(X − 1)³, (2)

proposed and tested by Lee and Park.^25–27 The second estimate has been obtained in a more sound way, i.e., by extrapolating the CCSD(T)/cc-pVXZ correlation contributions to the barrier height by 1/X³ (ref. 28 and 29) and adding the Hartree–Fock infinite basis limit of the barrier, evaluated by the formula:^23,30

E^SCF(X) = E^SCF_∞ + A exp(−BX). (3)

The two extrapolation schemes give very similar results, which differ by less than 1 cm⁻¹. They are reported in Table 3 and compared with the other determinations, previous theoretical values and experiment. By this comparison, it comes out that the barrier height is practically converged at CCSD(T)/cc-pVQZ level. In fact, the value obtained at that level is only ∼1–2 cm⁻¹ lower than the extrapolated ones. On the other hand, we can see that the values determined employing MT and CVQZ sets are higher than the extrapolated limits.

The effects of core-correlation on the torsional barrier height have been investigated by performing CCSD(T) calculations with all electrons correlated, employing the uVTZ, MT and CVQZ bases. By comparing the barrier heights obtained at the same level (same method, same basis) correlating valence and all electrons, we notice that the core-correlation effects seem to be small: of the order of 1%. More precisely, including the core-correlation the torsional barrier increases by a few cm⁻¹. Thus, if we assume the additivity of these effects, our best estimate can be determined by adding this correction to the extrapolated value. In this way, a value of ∼366 cm⁻¹ (also reported in Table 3) is obtained; it is interesting to note that this value is intermediate between those determined correlating and not the core electrons with MT or CVQZ bases.

Moreover, in order to test how much small variations of the optimized geometries can affect the barrier height, we have performed single point calculations at CCSD(T)/MT (all electrons) level for the various extrapolated geometries. The values obtained are reported in Table 3; we can see that they differ by less than 1 cm⁻¹ (∼0.1–0.2%). We can conclude that small variations in the bond lengths (less than 0.02 Å for Si–Si, a few thousandths of Å for Si–H) have negligible effect on the torsional barrier, which, on the other hand, essentially depends on the method and basis set used.

Finally, the potential energy function for internal rotation, V(τ), expressed as the Fourier series:³⁴

V(τ) = V₃/2(1 − cos 3τ) + V₆/2(1 − cos6τ) + V₉/2(1 − cos9τ) + ...., (4)

has been evaluated at CCSD(T)/cc-pVQZ level. The list of energies for various values of the τ angle (the other structural parameters have been fixed at the CCSD(T)/cc-pVQZ optimized geometry values) and the expansion coefficients obtained fitting these energies with expression (4) are reported in Table 4. Since the expansion coefficient V₉ has been found to be very small (see Table 4), the higher order terms have been neglected. As the V₆ term in eqn. (4) vanishes for the eclipsed conformation, and the V₉ term is negligible, the expansion coefficient V₃ (367.1 cm⁻¹) becomes, to a good approximation, equal to the barrier height. By comparing this value with our other determinations, it should be noted that it is very close to our best estimate of 366 cm⁻¹, obtained by adding the core-correlation correction to the infinite basis set extrapolated value. Thus, it is just a little bit lower than the values evaluated at CCSD(T)/MT or CCSD(T)/CVQZ level with all electrons correlated.

Table 4 Torsional potential energy function evaluated at CCSD(T)/cc-pVQZ level: list of the energies for various values of τ^a and expansion coefficients (see text)

τ ^a/°	Energy/Eh
60.0	−581.718595782356
57.5	−581.718603068998
55.0	−581.718624795533
52.5	−581.718660570446
50.0	−581.718709746887
47.0	−581.718785185544
44.0	−581.718876685611
40.0	−581.719019579489
30.0	−581.719439431491
20.0	−581.719855628701
17.0	−581.719962703925
15.0	−581.720026999365
12.0	−581.720111111611
10.0	−581.720158100136
7.5	−581.720205743539
5.0	−581.720240337680
2.5	−581.720261323606
0.0	−581.720268358177

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Coefficients
a The other structural parameters have been fixed at the values of the staggered geometry optimized at CCSD(T)/cc-pVQZ level.
V 3/cm^−1	367.05
V 6/cm^−1	−1.62
V 9/cm^−1	0.03

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4 Conclusion

Although our best estimates for the equilibrium geometry and the rotational barrier, reported in Tables 2 and 4, respectively, still remain in error from the exact values due to incomplete treatment of electron correlation and intrinsic approximations of the extrapolation schemes employed, the present results can be considered as benchmarks for the molecular structure of disilane in view of the high quality of the computations performed and the completeness of the study.

Acknowledgements

This work has been supported by MURST (ex-40%), CNR and University of Bologna (funds for selected research topics and ex-60% funds). The author would like to thank Prof. K. A. Peterson for providing the core–valence basis sets for silicon prior to publication.

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