Cristina
Puzzarini
Dipartimento di Chimica “G. Ciamician”, Università di Bologna, Via Selmi 2, I-40126, Bologna, Italy. E-mail: criss@ciam.unibo.it
First published on 27th November 2002
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.
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.
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 compounds14,13 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.
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.
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 |
CCSD(T)/MT
(all electrons) |
CCSD(T)/CVQZ
(frozen core) |
CCSD(T)/CVQZ
(all electrons) |
CCSD(T)/aVQZb
(frozen core) |
B3LYP/VTZc | 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![]() ![]() ![]() ![]() ![]() ![]() |
||||||
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 |
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.
CCSD(T)/cc-pV∞Za | CCSD(T)+core/cc-pV∞Zb | Extrapolated geometry 1c | Extrapolated geometry 2d | Extrapolated geometry 3e | Recommended geometryf | Exp. r0g | |
---|---|---|---|---|---|---|---|
a Extrapolated equilibrium geometry to the valence correlation limit by applying the geometric extrapolation A![]() ![]() |
|||||||
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![]() |
43![]() |
43![]() |
43![]() |
43![]() |
43![]() |
43![]() |
B | 5![]() |
5![]() |
5![]() |
5![]() |
5![]() |
5![]() |
5![]() |
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![]() |
43![]() |
43![]() |
43![]() |
43![]() |
43![]() |
– |
B | 5![]() |
5![]() |
5![]() |
5![]() |
5![]() |
5![]() |
– |
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![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | (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.
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) |
CCSD(T)/ecc-pV∞Z | CCSD(T)+core/ecc-pV∞Z | Extrapol.e geometry 1 | Extrapol.e geometry 2 | Extrapol.e geometry 3 | Infinitef basis limit | Recommendedg 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![]() ![]() ![]() ![]() |
||||||||
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)) |
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:
Etot(X)![]() ![]() ![]() ![]() ![]() ![]() | (2) |
ESCF(X)![]() ![]() ![]() ![]() | (3) |
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)![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | (4) |
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 |
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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