Preface
Chapter 3: Method for calculating steric strain 3.1 Introduction
3.3 Programming ideology
3.3.3 SterixLB calculations
3.3.3.2 Modified techniques for calculating the dissociating ligand and vacant space sizes
As mentioned earlier, the basis of calculating the size of the dissociating ligand and the size of the vacant space in SterixLB is the modified techniques, namely the modified Tolman, the outer pocket, the inner pocket and the inner-inner pocket techniques.
The calculated dissociating ligand size and the vacant space size are influenced by the type of substituents (X in PX3) of the dissociating ligand and the groups on the carbene carbon.
Figure 3.8 showed a representation of the modified techniques. In Figure 3.8(a), the dissociating ligand (inner-inner pocket technique excluded) and in Figure 3.8(b), the vacant space, is shown. The dissociating ligand has a dummy atom (Figure 3.8(a)) attached to it, which will be discussed in the modified Tolman technique in section 3.3.3.2.1.
In the next sections, an overview of the mathematical approach of the four modified techniques will be provided, as well as the application of each technique for the calculation of the dissociating ligand and the vacant space sizes. Notably, the calculated values for each of the techniques must be multiplied by two since the techniques were based on 2D representations of half of the cone. Consequently, the techniques calculate only half of the cone.
3.3.3.2.1 Modified Tolman technique
As with the Tolman cone angle technique, the concept of a cone around the ligand with A as the apex of the cone is also used in the modified Tolman technique. However, in contrast to Tolman, who built physical models for his calculations in the Tolman cone angle technique,[1]the calculations performed with this modified Tolman technique use quantum mechanical (QM) data obtained from molecular modelling. Molecular modelling mimics the behaviour of molecules, and therefore electronic and structural properties as well as the dynamic process are included in the obtained modelling data. In Figure 3.14, the mathematical representation of the modified Tolman technique is illustrated. SterixLB uses the Cartesian coordinate data and the equations in Appendix C.1 to calculate the modified Tolman values.
Figure 3.14: Mathematical representation of the modified Tolman technique The modified Tolman technique for the dissociating ligand
In the Grubbs 1-type complexes, the ruthenium metal is the apex of the cone, while PH3 is the dissociating ligand, as shown in Figure 3.15(a). However, this is only true while the ruthenium metal is bonded to the dissociating ligand. The P(4)-Ru bond length increases until the bond breaks, enabling the dissociating ligand to dissociate from the Grubbs 1-type complex. After the breaking of the bond, the ruthenium metal cannot be used as the apex to calculate the size of the dissociating ligand accurately, since the increasing P(4)-Ru bond length forces the cone angle smaller than it actually is. Consequently, a dummy atom (D) is used in the place of the ruthenium metal, as shown in Figure 3.15(b). The value of the P(4)-D bond length is fixed at the optimum P(4)-Ru bond length obtained from the geometric optimisation before the PES scan.
(a) (b)
Figure 3.15: The modified Tolman technique for (a) the dissociating ligand before bond breaking and (b) the dissociating ligand with a dummy atom after bond breaking However, the Cartesian coordinates for the dummy atom are needed in order to successfully calculate the size of the dissociating ligand with the modified Tolman technique, as shown in Figure 3.16. The size of the dissociating ligand (θT) is calculated from θ1 and θ2, as shown in Figure 3.16. The equations used to calculate all of the variables shown in Figure 3.16 can be found in Appendix C.2.
Figure 3.16: The mathematical representation of the modified Tolman technique with the use of a dummy atom for a dissociating ligand
SterixLB does not calculate the dissociating ligand size with the modified Tolman technique, because Cartesian coordinate data for the dummy atom are not available.
The modified Tolman technique for the vacant space
The representation of the modified Tolman technique on the vacant space in the Grubbs 1- type complexes, which was adapted from the above-mentioned dissociating ligand calculations, is shown in Figure 3.17. In order to calculate the vacant space left by the dissociated ligand, the apex of the cone is taken as the non-dissociating phosphine ligand (P(5)
atom). As shown in Figure 3.17, the vacant space is calculated from the outer radius of the carbene carbon atom and the two chlorine atoms. The groups on the carbene carbon are not included in the calculations, since they are not directly connected to the vacant space.
Furthermore, the close contacts will detect whether these groups on the carbene carbon will have an effect (repulsive force) on any of the remaining atoms in the Grubbs 1-type complexes. In SterixLB, the size of the vacant space is calculated by calculating all of the variables shown in Figure 3.17 using the equation list found in Appendix C.2.
Figure 3.17: The modified Tolman technique for the vacant space 3.3.3.2.2 The pocket techniques
The inner pocket, the outer pocket and the inner-inner pocket techniques are based on the same concept, where the open volume between four atoms is calculated, as shown in Figure 3.18. The open volume in the dissociating ligand is the space between the substituents where an object such as a ball can fit. On the other hand, the open volume in the vacant space is the vacant coordination site between the carbene carbon and the chlorines.
Figure 3.18: The open volume is indicated with a sphere situated between four atoms The reasoning behind creating three different techniques was to observe how much the van der Waals radii of the substituents influence the size of the dissociating ligand and the size of the vacant space.
3.3.3.2.2.1 The inner pocket technique
The inner pocket technique could be defined by a triangular pyramid as shown in Figure 3.19(a). It is important to note that the inner pocket technique is based on the observation that the D, B, and C atoms are in a different (xyz) plane than A. Therefore, the triangular pyramid is calculated by calculating the radii of D (r1), B (r2) and C(r3) , which are in the same (xyz) plane as the triangular centroid (referred as a midpoint in the rest of the document) (M), shown in Figure 3.19(b). After the radii were calculated for the D, B and C atoms, the apex angle (at A) of the triangular pyramid could be calculated by using the radii (r1, r2, r3), as shown in Figure 3.19(c). This is done by calculating the individual angles of each atom (Figure 3.19(c)) using the Pythagoras theorem. These individual angle must be multiplied by two, since the representation is a 2D image. The average of the above- mentioned calculated angles was taken to obtain the inner pocket angle. SterixLB uses the imported Cartesian coordinate data to compute the inner pocket technique in two parts. In the first part, SterixLB calculates all of the variables shown in Figure 3.19(b). Thereafter, SterixLB uses the calculated variables from the first part to calculate the θ (θB, θC, θD) angles (Figure 3.19(c)), before calculating the inner pocket value from the average of the three atoms. The inner pocket technique uses only the bond length of the atoms (which is determined from the centre of the connected atoms) with no additional bond lengths (van der Waals radii).
(a) (b) (c)
Figure 3.19: The mathematical representation of the inner pocket technique calculations where (a) is a side view, (b) is the top view of the triangle where the radii intersect at the midpoint (M) and (c) is the side view of the triangle with the angles (θ, θB, θC, θD) and the
midpoint (M).
The inner pocket technique for the dissociating ligand
The inner pocket technique for the dissociating ligand is calculated, as shown in Figure 3.20.
SterixLB calculates all of the variables shown in Figure 3.20 from the imported Materials Studio[3]Cartesian coordinate data.
Figure 3.20: Detailed mathematical representation of the inner pocket technique for the dissociating ligand (top view)
Thereafter, the radii (f1, d1, e2) calculated from Figure 3.20 are used to calculate the angle at the apex (θ), which is the average of the θA, θB and θC angles, as shown in Figure 3.19(c).
SterixLB uses the variables calculated in Figure 3.20 to calculate the inner pocket values of each substituent separately (Figure 3.19(c)). Thereafter, the inner pocket value for the dissociating ligand is calculated from the average of the three substituents’ inner pocket values. Each of the substituents’ inner pocket values is multiplied by two (since the representation is a 2D image) before calculating the average.
The inner pocket technique for the vacant space
The calculation of the inner pocket technique for the dissociating ligand and for the vacant space is the same, with the exception of the atoms used in the calculation. The inner pocket technique for the vacant space does not include the ruthenium atom in the calculations. This is because the ruthenium metal is in the same (xyz) plane as the chlorides (D, B) and the carbene carbon atom (C) (as shown in Figure 3.21), therefore another midpoint (M), is used.
Subsequently, the P atom in the non-dissociating phosphine ligand is used as the apex (A) in the inner pocket calculations for the vacant space, as shown in Figure 3.21. With this in mind, SterixLB imports the Cartesian coordinate data from the Materials Studio[3]output file and proceeds to calculate all of the variables shown in Figure 3.21 using the mathematical equations shown in Appendix C. With the radii known (r1, r2 and r3), the inner pocket (θ) values for the Cl1, Cl6 and C2 atoms are calculated separately, as shown in Figure 3.19(c).
Furthermore, the inner pocket values for each atom are multiplied by two. Finally, the inner pocket (θ) for the vacant space is calculated from the average inner pocket values of the Cl1, Cl6 and C2 atoms.
Figure 3.21: Detailed mathematical representation of the calculations used in the inner pocket technique for the vacant space
3.3.3.2.2.2 The outer pocket technique
The outer pocket technique builds on the concept of triangular pyramid calculations, as mentioned in the inner pocket technique. The only difference is the inclusion of the van der Waals radii in the calculations to calculate the outer-most size of the dissociating ligand or vacant space (Figure 3.22). Before the outer pocket can be calculated, the inner pocket should be calculated for the dissociating ligand and the vacant space. The values obtained for the inner pocket technique in the 2D image (as shown in Figure 3.19 (c)) should be used. In Figure 3.22, the van der Waals radius of the substituent (B) is added to the bond length shown as the tangent line (TOP) (green line). The tangent line is calculated with the Pythagoras theorem. Subsequently, the θx angle is calculated from the tangent line. Lastly, the
outer pocket angle (θOP) is calculated by adding the inner pocket angle (θB in Figure 3.22) to the θx angle. The outer pocket angle of the other two substituents (C and D) is calculated in a similar manner. Each of the individual outer pocket angles is then multiplied by two, before the average is calculated for all three substituents to obtain the outer pocket angle for the dissociating ligand or vacant space.
Figure 3.22: Modification of Figure 3.19(c) to represent the outer pocket (θOP) and inner- inner pocket (θIIP) techniques, TIIP = Tangent inner-inner pocket (TOP = Tangent outer pocket,
θy = inner-inner angle, θx = outer angle) The outer pocket technique for the dissociating ligand
SterixLB calculates the size of the dissociating ligand with the outer pocket technique from the values obtained from the inner pocket technique (Figure 3.21). The atom used in the calculation of the dissociating ligand is A (where A = P4), and B, C and D are the substituents (F, Cl, Br, I or H) on the dissociating ligand. The tangent line (TOP) is calculated with the Pythagoras theorem in order to obtain the angle for each substituent (θx where B = F, Cl, Br, I or H), as shown in Figure 3.22. Thereafter, the outer pocket value θOP is obtained by adding the inner pocket angle (θB) to the outer pocket angle (θx) for each of the atoms separately. The outer pocket angle for the dissociating ligand is then calculated from the average of the substituents’ outer pocket angles. Furthermore, each of the substituent angles (θB) must be multiplied by two before calculating the average outer pocket value. As mentioned previously, the value is multiplied by two, since the angle calculated for the technique is only for half of the cone (Figure 3.19(c)).
The outer pocket technique for the vacant space
The outer pocket technique for the vacant space uses the inner pocket value calculated for the vacant space (Figure 3.22). The outer pocket technique for the vacant space is calculated in the same manner as the outer pocket for the dissociating ligand. The only difference between these two techniques is that the atoms used in the vacant space calculation are different than those used in the dissociating ligand calculations. In the case of the vacant space, B, C and D can be Cl1, Cl6 or C2. SterixLB firstly calculates the inner pocket angles for the Cl1, Cl6 and C2 atoms, separately from the imported Cartesian coordinate data. Thereafter, the outer pocket angles are calculated for the Cl1, Cl6 and C2 atoms, separately from the inner pocket angles. The outer pocket values for each of the substituents is multiplied by two before the outer pocket value for the vacant space is calculated from the average (Cl1, Cl6 and C2
atoms).
3.3.3.2.2.3 The inner-inner pocket technique
The inner-inner pocket technique is the opposite of the outer pocket technique in the terms of the θy angle, as shown in Figure 3.22. The inner pocket values should be calculated before the inner-inner pocket technique can be used. The tangent line (TIIP or red line) for the D, C and B atoms is calculated with the Pythagoras theorem using the radii of the atoms and the respective bond lengths (A-B, etc.). Subsequently, the θy value is calculated from the tangent line. Lastly, the inner-inner pocket (θIIP) value is obtained by subtracting the θy value from the calculated inner pocket (θB) value. In the case of the vacant space, A = P5 whereas B, C and D can be Cl1, Cl6 or C2. As mentioned in the previous techniques, the inner-inner pocket for the vacant space is obtained from the average inner-inner pocket values for each of the atoms (Cl1, Cl6 and C2). It should be noted that the inner-inner pocket is only calculated for the vacant space, because the inner size of the dissociating ligand is not of interest for this study.