1 Introduction

The search for organic materials for non-linear optics (NLO) has revealed quite many compounds with high efficiency for frequency doubling of laser light, but they suffer from a lack of transparency. This is due to the generally accepted design of a NLO material which consists in connecting a donor and an acceptor group by a conjugated bridge which leads to highly coloured materials. To avoid the problems with this approach, we started to develop compounds whose hyperpolarizability relies on the existence of weak bonds. Here, the silatranes represent a well-studied class of stable molecules with a weak Si-N bond. Silatranes (2,8,9-trioxa-5-aza-1-silabicyclo[3.3.3]undecanes) are a special case of pentacoordinate silicon with unusal properties which have gained considerable interest over the last decades [1, 2, 3, 4]. Interestingly, the silicon-nitrogen bond is ignored in the name used by Chemical Abstracts (it should rather be a tricyclic system).

We introduced chirality as only compounds which crystallize in non-centrosymmetric space groups give nonvanishing hyperpolarizabilities as bulk materials, e.g. in crystalline form, which is important for applications in devices. Only very few silatranes with a chiral backbone are known up to now [5, 6]. Starting with (1S)-β-pinene as the source of chirality, we have prepared such a chiral silatrane and characterized it by X-ray structure analysis [7].

Figure 1: X-Ray structure of our chiral silatrane

2 Experimental

2.1 Synthesis of the chiral triethanolamine: (1R,2S)-2-[N,N-bis(2-hydroxyethyl)- aminomethyl]-6,6-dimethyl-2-hydroxybicyclo[3.1.1]heptane

A mixture of diethanolamine (1.05 g, 10 mmol) and (1R,2S)-β-pinene epoxid [8] (1.52 g, 10 mmol) was stirred at 120^{∘}C for 2 days. Volatile impurities were distilled off at 120^{∘}C (6mm). The residual yellow oil was used without further purification.

Yield 95 %; HPLC-ESI: m/e 257 (M^{+}); calculated for C_{14}H_{27}NO_{3}: 257.37.

Figure 2: Synthesis of the chiral silatrane

Table 1: ^{1}H NMR (CDCl_{3}) for 2( for numbering see figure 3)

δ (ppm) mult. J (Hz) Int.  assignment
0.95  s  3H 9 (anti to C-7)
1.25  s  3H  8 (syn to C-7)
1.51  d  9.15  1H 7 (anti to CMe_{2})
1.66  m  1H  6
1.78  m  1H  5
1.80  m  1H  6
1.89  m  1H  4
1.93  m  1H  5
2.16  m  1H  2
2.17  m  1H 7 (syn to CMe_{2})
2.54  d  14.0  1H  10
2.70  d  14.0  1H  10
2.64, 2.76 m, m  4H  12 and 14
3.58, 3.68 m, m  4H  11 and 13
3.60  s (br)  3H  OH

Table 2: ^{13}C NMR (CDCl_{3}) for2( for numbering see figure 3)

δ (ppm) mult.  assignment
23.83  CH_{3}  9
25.06  CH_{2}  5
27.25  CH_{2}  7
27.46  CH_{3}  8
29.48  CH_{2}  6
38.06  C_{q}  3
40.96  CH  4
50.30  CH  2
59.01  2 × CH_{2} 12 and 14
60.05  2 × CH_{2} 11 and 13
66.78  CH_{2}  10
76.21  C_{q}  1

2.2 Synthesis of the methylsilatrane: (1R,2S)-1’,6,6-trimethyl-spiro[bicyclo[3.1.1]heptane-2,3’-5-aza-2,8,9-trioxa-1-silabicyclo[3.3.3]undecane]

A mixture of trichloro-methylsilane (374 mg, 2.5 mmol), acetic anhydride (776 mg, 7.8 mmol) and NaOAc (50 mg, 0.6 mmol) was kept at 60^{∘}C for 4 h. Acetyl chloride and excess of acetic anhydride was distilled off at 6 mm. The residue was dissolved in CHCl_{3} (10 ml), cooled to 0^{∘}C, and then a solution of the chiral triethanolamine (643 mg, 2.5 mmol) in CHCl_{3} (20 ml) was added dropwise within 30 min. The reaction mixture was stirred at room temperature for 12 h, the solvent evaporated, and the residue chromatographed on silica gel/CH_{2}Cl_{2}:ethylacetate=10:1. The corresponding product was obtained as a white solid. X-ray quality crystals were grown by slow evaporation of a solution in CH_{2}Cl_{2}/Et_{2}O at room temperature.

Yield 270 mg (36 %); HPLC-ESI m/e 297 (M^{+}); calculated for C_{15}H_{27}NO_{3}Si: 297.47.

Table 3: ^{1}H NMR (CDCl_{3}) for1( for numbering see figure 3)

δ (ppm) mult. J (Hz) Int.  assignment
-0.14  s  3H  15
0.90  s  3H 9 (anti to C-7)
1.22  s  3H  8 (syn to C-7)
1.62  d  10.4  1H 7 (anti to CMe_{2})
1.76  m  1H  5
1.87  m  1H  6
1.88  m  1H  4
1.93  m  1H  2
1.95  m  1H  5
2.07  m  1H  6
2.15  m  1H 7 (syn to CMe_{2})
2.75  d  13.5  1H  10
2.78  m  3H  12 or 14
2.85  d  13.5  1H  10
2.91  m  1H  12 or 14
3.73  m  2H  11 or 13
3.74  m  2H  11 or 13

Table 4: ^{13}C NMR (CDCl_{3}) for 1( for numbering see figure 3)

δ (ppm) mult. assignment
-0.85  CH_{3}  15
23.64  CH_{3}  9
25.09  CH_{2}  5
27.30  CH_{2}  7
27.40  CH_{3}  8
34.30  CH_{2}  6
37.54  C_{q}  3
40.29  CH  4
54.11  CH_{2}  12 or 14
54.72  CH  2
54.72  CH_{2}  12 or 14
57.70  CH_{2}  11 or 13
57.90  CH_{2}  11 or 13
64.55  CH_{2}  10
76.58  C_{q}  1

Figure 3: Structure and numbering scheme for 1.

3 Topological analysis of the electron density

3.1 Introduction

In contrast to almost all other derivatives of silicic acid, silatranes are remarkably stable towards hydrolysis. This feature was traced back to the hypervalency of the silicon which has not only three bonds to oxygen and one to carbon, but shows, according to crystal structures, a quite low distance to nitrogen which might be interpreted as a fifth bond. The structure with a low Si-N distance (endo-structure, around 2.2Å) is expected to be more stable than the corresponding exo-species (Si-N distance around 2.6Å), where a direct interaction between nitrogen and silicon should be much weaker. The nature of this weak Si-N bond was the target of a great number of calculations [9]. Csonka and Hencsei have shown that at sufficient level of theory the pentacoordinate endo-species is the only energetic minimum on the potential surface, [10, 11], although the surface seems to be very flat in the direction of the Si-N vector. Gordon performed a Bader analysis for several silatranoles [12] finding an ionic interaction between silicon and nitrogen, as well as between silicon and oxygen. These calculations were performed on AM1-optimised structures, which give only a rough approximation of solid state properties. In particular, they give considerably longer Si-N distances than observed in crystal structures (but more close to the experimental gas phase values).

We performed a number of calculations on the X-ray geometry of our chiral silatrane derived from (S)-β-pinene using the AIMPAC^{1} package [13, 14] for topological analysis. To our knowledge, it is the first topological analysis of a chiral silatrane.

For a detailed explanation of the theory of topological analysis and the terminology used, see Bader’s website^{2}.

3.2 Gradient field analysis

The analysis of the electron density as introduced by Bader gives a mathematically stringent definition of a chemical bond. The existence of a bond is indicated by the presence of a so-called bond critical point (bcp). This is a saddle point (∇ρ(r) = 0) in the electron density between two nuclei, where the Hessian (the second derivative matrix) of the electron density has one positive eigenvalue in the bond axis and two negative eigenvalues perpendicular to the bond axis. From this follows that there exists a path of maximum electron density from one nucleus to another passing through the bond critical point [15]. This path is called “the bond”.

We determined some bond critical points and bond paths for our chiral silatrane. The bond paths are visualised in figure 4 and the properties of the critical points are summarised in table 5. Figure 4 shows clearly the bond path between silicon and nitrogen, as well as the path between silicon and oxygen. The bond critical points are the crossing points of the bond path (blue) and the interatomic surface (zero-flux surface, green) (for a definition, see below).

1http://www.chemistry.mcmaster.ca/faculty/bader.html

2http://www.chemistry.mcmaster.ca/faculty/bader.html

Figure 4: Gradient field of the core of our chiral silatrane in a plane defined by Si(1), O(2) and N(5) (red: gradient paths, blue: bond path, green: zero-flux path)

3.3 Critical points in the electron density

Further analysing the first and second derivatives at the critical points validate the assumption of ionic bonding in the silatrane.

Table 5: Properties of the critical points in the chiral silatrane

Si(1) O(2)  bcp (3,-1)
coordinates  X = 4.24113875 Å Y = -1.48450393 Å Z = -0.45686378 Å
distance from Si 1.27 Å O 1.87 Å
ρ(r)  0.1203 e/a_{0}^{3}
∇^{2}(ρ(r))  0.964 e/a_{0}^{5}
Si(1) N(5)  bcp (3,-1)
coordinates  X = 3.15303269 Å Y = -0.135561143 Å Z = 0.158368394 Å
distance from Si 1.56 N 2.54
ρ(r)  0.0497
∇^{2}(ρ(r))  0.148

The low electron density at the critical points and the positive Laplacian (∇^{2}ρ(r)) are clear evidence that the bonds are more ionic than covalent. As one compares these results to those obtained by Gordon [12] there is a good agreement with results obtained on 6-31G(d) optimised structures, showing that the X-ray structure is a valid geometry for doing ab-initio calculations.

To connect the analysis to the concept of the bond order (which is more familiar to the preparative chemist), Bader has derived an expression to calculate the bond order n from the electron density ρ at the bond critical point [16]:

n = exp 6.458(ρ − 0.2520)  (1)

Comparing the bond orders derived by Mulliken population analysis and calculated using equation (1) one finds that the silicon-oxygen bond is attributed a much more ionic character in the Bader analysis (table 6).

Table 6: Bond Order

bond  Mulliken derived Bader
Si(1)-O(2) 0.897  0.427
Si(1)-N(5) 0.237  0.270

3.4 Atomic charges

As Bader has shown in his “Theory of Atoms in Molecules”^{3} [13] one can define an atom via the zero-flux surface of the electron density: Looking at the gradient vector field, one finds that the all gradient vector trajectories terminate at only one nucleus each. This allows to define an atomic basin as the space which is crossed by the gradient vector trajectories starting at the corresponding nucleus (cf. 4). The border of such a basin is the zero-flux surface. This type of definition of an atom yields quantum mechanically defined charges of atoms, which are more indicative of properties than e.g. Mulliken charges.

A selection of charges and populations derived by different methods is given in table 7. For the silatrane studied here we find a very positive silicon atom and corresponding negative nitrogen and oxygen atoms. This indicates a strong ionic contribution to the bonding.

Table 7: Atomic charges and populations. Columns marked with — were not explicitly calculated due to time limits.

Atom Mull.Pop. Charge Low.Pop. Charge Bader Pop. Charge
1 SI 12.623108 1.376892 13.227044 0.772956 10.65411 3.34588
2 O  8.740145 -0.740145 8.451786 -0.451786  9.5138  -1.5138
3 O  8.732526 -0.732526 8.452240 -0.452240  —  —
4 O  8.767042 -0.767042 8.440570 -0.440570  —  —
5 N  7.793785 -0.793785 7.191056 -0.191056  8.47333  -1.47333
6 C  6.700731 -0.700731 6.725870 -0.725870  —  —
7 C  5.919583 0.080417 6.138055 -0.138055  —  —

3http://www.chemistry.mcmaster.ca/faculty/bader/aim/aim 0.html

3.5 Computational details

The calculations where performed with GAMESS^{4} [17].

The geometry chosen for this study is the same as determined by x-ray structure [7], as our main interest is in the solid state properties. For the RHF calculations an 6-31G* basis set was used (358 basis functions, 692 primitives). The results are summarised in table 8.

Table 8: Results of the RHF calculation

final energy  -1151.4256118559
total potential energy -2307.2144953598
total kinetic energy  1155.7888835039
virial ratio (V/T)  1.9962248541
dipole moment μ  7.302916 D

The wavefunction table of the calculation was used as input for Bader’s AIMPAC package.

All calculations were performed under Linux 2.0.32^{5} on a Pentium 90 PC with 96 MB of RAM and about 3 GB free disk space. A typical RHF calculation of this size took about 25–30 h wall clock time. An integration over an atomic basin took about 1 week.

4 Conclusion

We have demonstrated that chiral silatranes can be prepared from terpene epoxides in very few steps, and have analysed the electron density distribution by the Bader technique. The weak Si-N bond is mainly ionic (bond order 0.270) and responsible for the enhanced hydrolytic stability compared with most other derivatives of silicic acid. Silicon ends up with a positive (3.34588) and nitrogen with a negative charge (-1.47333), which is somewhat counter-intuitive (the Si-N bond is normally formulated as a dative bond involving the nitrogen lone pair). The results show that the topological analysis of the electron density is possible even for such quite large systems. Calculations of the hyperpolarizability (for the non-linear optical properties) should therefore be feasible as well.

References

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[7] Details on the crystal structure will be published elsewhere. We wish to thank Dr. Wolfgang Scherer, Anorganisch-chemisches Institut der TU München, for the crystal structure analysis.

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4http://www.msg.ameslab.gov/GAMESS/GAMESS.html

5http://www.linux.org

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