Wolfram Language Example Repository
Ready-to-use examples of the Wolfram Language
Molecular Gears
Molecular modeling of dynamic gearing and gear-slippage in bis(9-triptycyl)methane
The bis(9-triptycyl)methane molecule has a central methylene group () and two 9-triptycyl groups. Each 9-triptycyl group is effectively a three-bladed propellor or a three-toothed gear.
CH
2
The three views below (side, top, bottom), with the central methylene colored yellow for reference, show the spectacular interleaving of the "blades" or "gear-cogs" of the triptycyl groups.
Despite the tight meshing of the triptycyl groups, they can rotate freely, although in a highly correlated fashion. That is, they act like well-lubricated bevel gears. The functions in the ConsistentForceField paclet can be used to model the structural, dynamic, and energetic aspects of the gearing and gear-slippage.
Install the paclet:
In[1]:=
PacletInstall["RobertNachbar/ConsistentForceField"];
Load the paclet context:
In[2]:=
Needs["RobertNachbar`ConsistentForceField`"]
There are three stationary (optimized geometry with a 0 gradient vector) structures for bis(9-triptycyl)methane. The gauche-gauche conformation is an energy minimum and the ground state. It has symmetry and is chiral. The syn-anti conformation is a transition state and has symmetry. It is only 0.66 kcal/mol higher in energy than the ground state, and it links two mirror image conformations of the gauche-gauche conformation along the gearing pathway. The anti-anti-Cs conformation is also a transition state, and is 33.74 kcal/mol above the ground state. Its symmetry is also and it links two mirror image conformations of the ground state along the gear-slippage pathway. It has the "Cs" suffix to distinguish it from another anti-anti conformation that has symmetry that is approximately 65 kcal/mol above the ground state.
C
2
C
s
C
s
C
2v
Load the stationary structures of bis(9-triptycyl)methane:
In[3]:=
b9tmData=ResourceData["Example: Molecular Gears","Molecules"];
Out[4]=
 |  |  |
gauche-gauche | syn-anti | anti-anti |
Compute the symmetry, energy, gradient vector norm, and number of imaginary frequencies of the three stationary structures:
In[5]:=
PrependComapResourceFunction["PointGroupSymbol"],CFFEnergy,(Chop[#,]&)@*Norm@*CFFGradient,Count[CFFNormalModes[#,"FrequenciesOnly"True]["Frequencies"],λ_?(Positive@*Im)]&,b9tmData[#],#&/@{"Gauche-Gauche","Syn-Anti","Anti-Anti-Cs"}//TableForm#,
&
-6
10
Out[5]//TableForm=
Conformation | Symmetry | Energy | Gradient norm | # imaginary frequencies |
Gauche-Gauche | C 2 | 44.0736 kcal th | 0 kcal th | 0 |
Syn-Anti | C s | 44.7293 kcal th | 0 kcal th | 1 |
Anti-Anti-Cs | C s | 77.8161 kcal th | 0 kcal th | 1 |
The conformational reaction paths are found by starting at a transition state geometry, perturbing it along the transition vector, and then following the geometry as the the energy is minimized. It's not quite the same as Fukui's intrinsic reaction coordinate (IRC) method, but is nonetheless informative. The transition vector is the eigenvector of the Hessian matrix of second derivatives computed at the transition state that has a negative eigenvalue, and indicates the direction of energy descent.
We'll demonstrate the process for the gearing mode, and then just present all the results.
Compute the transition vector for the syn-anti transition state, echoing the negative eigenvalue in the process:
In[6]:=
vec=Eigensystem[QuantityMagnitude@CFFHessian[b9tmData["Syn-Anti"]]]//Apply[Rule]//Thread//SelectFirstNegative@*(Chop[#,]&)@*Keys//Echo[#,"",Keys]&//Values;
-8
10
»
-0.408183
Generate the starting structure by perturbing transition state coordinates with the transition vector:
In[7]:=
startPosMol=Molecule[b9tmData["Syn-Anti"],"AtomCoordinates"QuantityMagnitude@b9tmData["Syn-Anti"]["AtomCoordinates"]+0.01Partition[vec,3]];
Optimize the geometry and sow the energy and molecule coordinates in the process and reap them at the end:
In[8]:=
{stopPosMol,itinerarySynAntiPos}=Reap@CFFOptimizeGeometry[startPosMol,Method"Newton",MaxIterations500,StepMonitorSow[{CFFEnergy[],CFFAtomCoordinates[]}]];//EchoTiming
⌚
188.651
Check that the optimization arrived at the ground state:
In[9]:=
Comap[{ResourceFunction["PointGroupSymbol"],CFFEnergy,Norm@*CFFGradient},stopPosMol]
Out[9]=
,,
C
2
44.0736/mol
kcal
th
2.68055×/(Åmol)
-6
10
kcal
th
Extract the pairs of energy and coordinates from reaped data:
In[10]:=
itinerarySynAntiPos=First@itinerarySynAntiPos//Echo[#,"",Dimensions]&;
»
{72,2}
Prepend the initial energy and coordinates to the itinerary:
In[11]:=
PrependTo[itinerarySynAntiPos,QuantityMagnitude@{CFFEnergy[b9tmData["Syn-Anti"]],CFFAtomCoordinates[b9tmData["Syn-Anti"]]}];
Now, repeat the process for a negative perturbation of the coordinates.
Perturb the transition state geometry in the negative direction:
In[12]:=
startNegMol=Molecule[b9tmData["Syn-Anti"],"AtomCoordinates"QuantityMagnitude@b9tmData["Syn-Anti"]["AtomCoordinates"]-0.01Partition[vec,3]];
In[13]:=
{stopNegMol,itinerarySynAntiNeg}=Reap@CFFOptimizeGeometry[startNegMol,Method"Newton",MaxIterations500,StepMonitorSow[{CFFEnergy[],CFFAtomCoordinates[]}]];//EchoTiming
⌚
258.374
In[14]:=
Comap[{ResourceFunction["PointGroupSymbol"],CFFEnergy,Norm@*CFFGradient},stopNegMol]
Out[14]=
,,
C
2
44.0736/mol
kcal
th
6.42415×/(Åmol)
-9
10
kcal
th
In[15]:=
itinerarySynAntiNeg=First@itinerarySynAntiNeg//Echo[#,"",Dimensions]&;
»
{93,2}
In[16]:=
PrependTo[itinerarySynAntiNeg,QuantityMagnitude@{CFFEnergy[b9tmData["Syn-Anti"]],CFFAtomCoordinates[b9tmData["Syn-Anti"]]}];
Even though the spacing of conformations along the itineraries is not uniform, we can still get a good approximation of the energy profile from one minimum to the other.
Plot the energy profile from one minimum to the other:
In[17]:=
ListPlot[Join[First/@Reverse@itinerarySynAntiNeg,First/@itinerarySynAntiPos],FrameTrue,FrameLabel{"Energy (kcal/mol)","optimization step"},PlotHighlightingNone]
Out[17]=
Creating the animation of the traversal of the conformational reaction path involves making a new molecule at each step using the atom coordinates that were sown along the optimization itinerary. From there the video is generated from a list of images with .
Generate new molecules from the atom and bond lists of the transition state molecule and the list of atom coordinates from the optimization itinerary:
In[18]:=
traversalMolecules=With[{atoms=b9tmData["Syn-Anti"]["AtomList"],bonds=b9tmData["Syn-Anti"]["BondList"],coords=Join[Reverse@itinerarySynAntiNeg〚All,-1〛,itinerarySynAntiPos〚All,-1〛]},Molecule[atoms,bonds,"AtomCoordinates"#]&/@coords];
To minimize jumps due to possible shifting of the center of mass during the optimization, the and are computed for each frame.
Create a video of side, top, and bottom views of bis(9-triptycyl)methane along the gearing pathway:
In[19]:=
frames=Module{xyz=#["AtomCoordinates"]//QuantityMagnitude,dist=50,sphere=Sphere[Automatic,16],size=200,cameraSide,verticalSide,cameraTop,verticalTop,cameraBottom,verticalBottom},
;RowMoleculePlot3D#,{{1,42,43}},
,MoleculePlot3D#,{{1,42,43}},
,MoleculePlot3D#,{{1,42,43}},
,Spacer[0]//Rasterize&/@traversalMolecules//FrameListVideo[#,FrameRate20]&
views |
Out[19]=
As mentioned above, a transition state on the gearing pathway connects two mirror image ground state conformations. Each triptycyl group rotates only 60°. To see the full 120° rotation one needs to stitch together the traversal over two adjacent transition states along the gearing pathway, and the video for that is shown below.
Heating a solution of bis(9-triptycyl)methane to about 150°C provides sufficient energy for the molecule to surmount the gear-slippage barrier over anti-anti-Cs transition state structure, and has been confirmed experimentally. The transition state is quite remarkable by having two benzene rings of one triptycyl group nested between two benzene ring of the other triptycyl group, as shown below.
Display the anti-anti-Cs transition state from below:
In[20]:=
Module[{xyz=b9tmData["Anti-Anti-Cs"]["AtomCoordinates"]//QuantityMagnitude,dist=50,sphere=Sphere[Automatic,16],size=Medium,cameraBottom,verticalBottom},cameraBottom=xyz〚1〛-distMean[Normalize/@{xyz〚42〛-xyz〚1〛,xyz〚43〛-xyz〚1〛}];verticalBottom=xyz〚43〛-xyz〚42〛;MoleculePlot3D[b9tmData["Anti-Anti-Cs"],{{1,42,43}},PlotTheme"Spacefilling",ViewVector{cameraBottom,xyz〚1〛-cameraBottom},ViewVerticalverticalBottom,SphericalRegionsphere,ImageSizesize]]
Out[20]=
The following video, made in the same manner as the previous two, shows how this happens. Beginning with the ground state conformation, the central C—C—C angle widens and two benzene rings of the left-hand triptycyl group spread apart to admit from above a second benzene ring from the right-hand triptycyl group. As the process continues, the benzene ring in the right-hand triptycyl group that was originally present then departs toward below, and the molecule relaxes to a mirror image ground state.
Source Metadata
-
Citation:
- Buergi, H. B., Hounshell, W. D., Nachbar Jr, R. B., & Mislow, K. (1983). Conformational dynamics of propane, di-tert-butylmethane, and bis (9-triptycyl) methane. An analysis of the symmetry of two threefold rotors on a rigid frame in terms of nonrigid molecular structure and energy hypersurfaces. Journal of the American Chemical Society, 105(6), 1427-1438.
- Guenzi, A., Johnson, C. A., Cozzi, F., & Mislow, K. (1983). Dynamic gearing and residual stereoisomerism in labeled bis (9-triptycyl) methane and related molecules. Synthesis and stereochemistry of bis (2, 3-dimethyl-9-triptycyl) methane, bis (2, 3-dimethyl-9-triptycyl) carbinol, and bis (1, 4-dimethyl-9-triptycyl) methane. Journal of the American Chemical Society, 105(6), 1438-1448.
See Also
Related Symbols
Publisher Information
- Contributed by: Robert Nachbar