denis tarasov et al- optimal tooltip trajectories in a hydrogen abstraction tool recharge reaction...

8/3/2019 Denis Tarasov et al- Optimal Tooltip Trajectories in a Hydrogen Abstraction Tool Recharge Reaction Sequence for Po

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Copyright 2010 American Scientic PublishersAll rights reservedPrinted in the United States of America

Journal of Computational and Theoretical Nanoscience

Vol. 7, 129, 2010

Optimal Tooltip Trajectories in a Hydrogen AbstractionTool Recharge Reaction Sequence for Positionally

Controlled Diamond MechanosynthesisDenis Tarasov 1, Natalia Akberova2 , Ekaterina Izotova 2 , Diana Alisheva2,

Maksim Astaev 1 , and Robert A. Freitas, Jr. 3 1Kazan Scientic Center, Russian Academy of Science, Russia

2Department of Biology, Kazan State University, Russia 3Institute for Molecular Manufacturing, Palo Alto, CA 94301, USA

The use of precisely applied mechanical forces to induce site-specic chemical transformationsis called positional mechanosynthesis, and diamond is an important early target for achievingmechanosynthesis experimentally. A key step in diamond mechanosynthesis (DMS) employs anethynyl-based hydrogen abstraction tool (HAbst) for the site-specic mechanical dehydrogenationof H-passivated diamond surfaces, creating an isolated radical site that can accept adatoms viaradicalradical coupling in a subsequent positionally controlled reaction step. The abstraction tool,once used (HAbstH), must be recharged by removing the abstracted hydrogen atom from thetooltip, before the tool can be used again. This paper presents the rst theoretical study of DMStool-workpiece operating envelopes and optimal tooltip trajectories for any positionally controlledreaction sequenceand more specically, one that may be used to recharge a spent hydrogenabstraction toolduring scanning-probe based ultrahigh-vacuum diamond mechanosynthesis. Tra- jectories were analyzed using Density Functional Theory (DFT) in PC-GAMESS at the B3LYP/6-311G(d,p)//B3LYP/3-21G(2d,p) level of theory. The results of this study help to dene equipmentand tooltip motion requirements that may be needed to execute the proposed reaction sequenceexperimentally and provide support for early developmental targets as part of a comprehensivenear-term DMS implementation program.

Keywords: Abstraction, Carbon, Diamond, DMS, Germanium, Hydrogen, Mechanosynthesis,Nanotechnology, Pathology, Positional Control, Reaction Sequence, Tooltip,Trajectory.

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . 43. Reaction I: Join GeRad Tool to Apical Ethynyl C Atom

of HAbstH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.1. Tooltip Geometry and Coordinate System . . . . . . . . . . . 53.2. PES as a Function of Positional Anglesand . . . . . . 63.3. Trajectory Pathologies of Reaction I . . . . . . . . . . . . . . 83.4. PES as a Function of Rotational Angle. . . . . . . . . . . 93.5. Lateral Displacement Error Tolerance . . . . . . . . . . . . . 103.6. Optimal Tooltip Trajectories . . . . . . . . . . . . . . . . . . . 11

4. Reaction II: Abstract Apical H from HAbstH UsingGeRad2 Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.1. Tooltip Geometry and Coordinate System . . . . . . . . . . . 154.2. PES as a Function of Positional AnglesH and H . . . . . 164.3. Trajectory Pathologies of Reaction II . . . . . . . . . . . . . . 184.4. PES as a Function of Rotational AngleH . . . . . . . . . . 20

Author to whom correspondence should be addressed.

4.5. Lateral Displacement Error Tolerance . . . . . . . . . . . . . 204.6. Optimal Tooltip Trajectories . . . . . . . . . . . . . . . . . . . 21

5. Reaction III: Detach GeRad Tool from Recharged HAbst Tool . 215.1. Tooltip Geometry and Coordinate System . . . . . . . . . . . 215.2. PES as a Function of Rotational AngleD . . . . . . . . . . 225.3. PES as a Function of Positional AnglesD and D . . . . . 235.4. Trajectory Pathologies of Reaction III . . . . . . . . . . . . . 235.5. Optimal Tooltip Trajectories . . . . . . . . . . . . . . . . . . . 27

6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1. INTRODUCTION

Arranging atoms in most of the ways permitted by phys-ical law is a fundamental objective of molecular manu-facturing. A more modest and specic objective is theability to synthesize atomically precise diamondoid struc-tures using positionally controlled molecular tools. Such

J. Comput. Theor. Nanosci. 2010, Vol. 7, No. 2 1546-1955/2010/7/001/029 doi:10.1166/jctn.2010.1365 1


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Optimal Tooltip Trajectories in a Hydrogen Abstraction Tool Recharge Reaction Sequence Tarasov et al.

Denis Tarasov is Senior Reasearch Scientist at the Supercomputer Center of Kazan Scentic Center of Russian Academy of Sciences, in Kazan, Russia. He graduated from Department of Biochemistry of Kazan State University in Kazan (Russian Federation) received an M.S. in biochemistry. He received his Ph.D. in Biology from Kazan State Uversity in 2004. His research interests include computer modeling in biology and chemishigh performance computing and molecular machine systems. He has published 15 refer

journal publications.

Natalia Akberova is Assistant Professor of the Kazan State University biology department. She has a Ph.D. in Biology, with specialization in biochemistry (1999) and lectuof courses for biology department students in informatics, bioinformatics, and modelin biology. Her scientic interests include systems biology, bioinformatics, and molecumodeling.

Ekaterina Izotova is a Ph.D. candidate in Kazan State University with reseach interests imolecular mechanics and quantum chemistry methods. She was born in Kazan (Russia) received her M.S. in biochemistry from Kazan State University.

Diana Alisheva is a Ph.D. candidate at the Kazan State University with reseach interesin molecular mechanics and quantum chemistry methods. She was born in Kazan (Rusand received her M.S. in biochemistry from Kazan State University.

Maksim Astaev is Senior Research Scientist at the Kazan Scientic Center of RussiaAcademy of Sciences in Kazan, Russia. Since 2000 he has devoted himself to the creaof high performance computers and organization of computing services for science institof the Kazan Scientic Center of the Russian Academy of Sciences. His research interinclude: high performance computers, information technology and telecommunications research, computational mathematics. He is co-organizer of Public Supercomputing Ceat KSC RAS. He is author of a number of reports for international and Russian conferedevoted to the latest developments of information support for research on high performacomputing, and co-author of publications in various Russian journals.

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Tarasov et al. Optimal Tooltip Trajectories in a Hydrogen Abstraction Tool Recharge Reaction Sequen

Robert A. Freitas, Jr. is Senior Research Fellow at the Institute for Molecular Manu-facturing (IMM) in Palo Alto, California, and was a Research Scientist at Zyvex Co(Richardson, Texas), the rst molecular nanotechnology company, during 20002004. received B.S. degrees in Physics and Psychology from Harvey Mudd College in 1974 a J.D. from University of Santa Clara in 1979. Freitas co-edited the 1980 NASA feasibianalysis of self-replicating space factories and in 1996 authored the rst detailed techn

design study of a medical nanorobot ever published in a peer-reviewed mainstream biomcal journal. Freitas is the author of Nanomedicine, the rst book-length technical discussof the potential medical applications of molecular nanotechnology; the initial two volumof this 4-volume series were published in 1999 and 2003 by Landes Bioscience. His reseainterests include: nanomedicine, medical nanorobotics design, molecular machine syste

diamondoid mechanosynthesis (theory and experimental pathways), molecular assemblers and nanofactories, atomprecise manufacturing, and self-replication in machine and factory systems. He has published 49 refereed journalcations and contributed book chapters, co-authored Kinematic Self-Replicating Machines (Landes Bioscience, 200in 2006 co-founded the Nanofactory Collaboration. His home page is at www.rfreitas.com.

positional control might be achieved using an instrumentlike a Scanning Probe Microscope (SPM). The landmarkexperimental demonstration of positional atomic assemblyoccurred in 1989 when Eigler and Schweizer1 employedan SPM to spell out the IBM logo using 35 xenon atomsarranged on nickel surface, though no covalent bonds wereformed.

The use of precisely applied mechanical forces to inducesite-specic chemical transformations is called positionalmechanosynthesis. In 2003, Oyabu et al.2 achieved the rstexperimental demonstration of atomically precise purelymechanical positional chemical synthesis on a heavy atomusing only mechanical forces to make and break covalentbonds, rst abstracting and then rebonding a single siliconatom to a silicon surface with SPM positional control invacuum at low temperature. Using an atomic force micro-scope the same group similarly manipulated individual Geatoms in 20043 and Si/Sn atoms in 2008.4

The assumption of positionally controlled highly reac-tive tools operating in vacuum permits the use of noveland relatively simple reaction pathways. Following earlygeneral proposals by Drexler5 and Merkle6 for possi-ble diamond mechanosynthesis (DMS) tools and sketchesof conceptual approaches to a few reaction pathways,a comprehensive three-year DFT-based (Density Func-tional Theory) study by Freitas and Merkle7 computation-ally analyzed for the rst time a complete set of DMSreactions and an associated minimal set of nine specicDMS tooltips that could be used to build basic diamond,graphene (e.g., carbon nanotubes), and all of the toolsthemselves including all necessary tooltip recharging reac-tions. Their work dened 65 foundational DMS reactionsequences incorporating 328 reaction steps.

A key step in the process of atomically precisemechanosynthetic fabrication of diamond is to remove ahydrogen atom from a specic lattice location on the dia-mond surface, leaving behind a reactive dangling bond

able to accept adatoms via radicalradical coupling ina subsequent positionally controlled reaction step. Thisremoval could be done using a hydrogen abstraction tool8that has a high chemical afnity for hydrogen at one endbut is elsewhere inert. The tools unreactive region servesas a handle or handle attachment point. The tool wouldbe held by a high-precision nanoscale positioning devicesuch as an SPM tip that is moved directly over particularhydrogen atoms on the surface. One suitable molecule fora hydrogen abstraction tooltip is the acetylene or ethynylradical, comprised of two triple-bonded carbon atoms. Onecarbon of the two serves as the handle connection andwould bond to a nanoscale positioning device through alarger handle structure. The other carbon of the two hasa dangling bond where a hydrogen atom would normallybe present in a molecule of ordinary acetylene (C2H2 ,forming the reactive tip. The environment around the toolwould be inert (e.g., vacuum or a noble gas such asxenon). This tool has received substantial theoretical studyand computational validation,514 and site-specic hydro-gen abstraction from crystal surfaces, though not purelymechanical abstraction, has also been achieved experimen-tally via scanning probe microscopy. For example, Lydinget al.1517 demonstrated the ability to abstract an individ-ual hydrogen atom from a specic atomic position in acovalently-bound hydrogen monolayer on a at Si(100)

surface, using an electrically-pulsed SPM tip in ultrahighvacuum. Hos group18 has similarly demonstrated position-ally controlled single-atom hydrogen abstraction experi-mentally using an SPM.

After each use, the spent hydrogen abstraction tool hasan unwanted H atom bonded to its tip that must be removedbefore the tool can be used again for hydrogen abstractionin a molecular manufacturing system. Freitas and Merkle7describe a mechanosynthetic reaction sequence (Fig. 1)labeled RS5 involving three reaction steps that could beused to recharge a spent ethynyl-type hydrogen abstrac-tion tool (hereinafter HAbst after recharge, HAbstH

J. Comput. Theor. Nanosci. 7, 129, 2010 3


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Reaction I Reaction II Reaction III

Fig. 1. Schematic of reaction sequence 5 from Freitas and Merkle,7 showing recharge reaction for 1-ethynyladamantane hydrogen abstraction tool(end product of Reaction III, structure at top) using sequence of three positionally controlled reactions involving two 1-germanoadamantantooltips. (C= black, H= white, Ge= yellow).

when hydrogenated and requiring recharge), modeled asan ethynyl radical mounted on an adamantane base (i.e.,1-ethynyladamantane). This base would be part of a largerdiamond lattice handle structure in an actual DMS tool.

The recharge reaction requires the use of a second posi-tionally controlled tool, a germanium radical (hereinafterGeRad), which is modeled as an adamantane base withan unterminated germanium atom substituted for carbonat a bridgehead position (i.e., 1-germanoadamantane). Theunterminated Ge atom is the active tip of the GeRad tool.During the recharge reaction RS5, the Ge radical on aGeRad tool is rst bonded to the distal C atom of theethynyl group on the spent HAbstH tool. A second GeRad(hereinafter GeRad2) is brought up to the transactionalH atom on the HAbstH tool and abstracts the H atom; thisH atom is readily donated to a clean diamond surface (notillustrated), recovering the original GeRad2 tool. GeRadis then mechanically debonded from the distal C atom,yielding a recharged HAbst tool.

While the proposed recharge reaction sequence appearsenergetically favorable, to date the positional and rota-tional operating envelopes of specic DMS tools act-ing on specic surfaces, workpieces, or other tools havebeen examined theoretically in only one prior study19that considered a C2 dimer placement tool (DCB6Ge)20interacting with a clean C(110) diamond surface in a sin-gle mechanosynthetic reaction. No such studies have yetbeen attempted for any other DMS tool or depositionsurface, nor have any yet been attempted for completereaction sequences. This paper presents the rst theoret-ical study of DMS tool-workpiece operating envelopesand optimal tooltip trajectories for any positionally con-trolled reaction sequencein particular, one that maybe used in a complete three-reaction mechanosyntheticsequence to recharge a spent hydrogen abstraction toolduring scanning-probe based ultrahigh-vacuum diamondmechanosynthesis. The results of this study help to deneequipment and tooltip motion requirements that may beneeded to execute the proposed reaction sequence exper-imentally and provide guidance on early development

targets as part of a comprehensive near-term DMS imple-mentation program.

2. COMPUTATIONAL METHODS

All studies were conducted using Density Functional The-ory (DFT) in the PC-GAMESS version21 of the GAMESS(US) QC package21A running on several clusters con-suming188,320 CPU-hours of runtime at 1 GHz, andincluded 5,284 separate valid-structure calculations. Dur-ing this four-year course of work (in 20052009) thehardware was upgraded several times, and has includeda cluster of 3 GHz Pentium4 HT (5 nodes), a cluster of 2 GHz AMD Opteron machines (16 nodes) and 1.8 GHzIntel Xeon machines (8 nodes) at the Supercomputing Cen-ter at the Kazan Scientic Center of the Russian Academyof Sciences (KSC RAS), and a cluster of 1.6 GHz AMDOpteron 240 machines (8 nodes, 2 CPU per node).

A comparison (Table I) of the reaction energy (energy of optimized product (HAbstH-GeRad complex) less energyof optimized reactants) for Reaction I shows that com-puted exoergicity is dependent on basis set choice,

Table I. Comparison of basis sets for calculating bond lengths,bond angles, and reaction energies (without zero-point correction) inReaction I.

Reaction CGe bond GeCC bondBasis set energy (eV) length () angle (deg)

AM1 0.708 1.974 122.7STO-3 2.109 1.918 120.7MIDI 1.135 1.953 123.83-21G(d) 0.917 1.980 121.23-21G(2d,p) 0.997 1.967 122.33-21G+ (2d,2p) 1.047 1.966 123.46-31G(d) 1.083 1.953 123.46-31G+ (2d,2p) 0.964 1.958 123.1DZV 1.234 1.953 125.66-311G 0.695 1.969 123.86-311G(d,p) 0.694 1.971 123.56-311G(d,p)//3-21G(2d,p) 0.696 1.966a 123.4aaug-cc-VTZ//6-311G(d,p) 0.634 1.971b 123.5b

a Geometry from 3-21G(2d,p) basis set.bGeometry from 6-311G(d,p) basis set.



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but that DFT energies computed using 6-311G(d,p)//3-21G(2d,p) differ little from 6-311G(d,p)//6-311G(d,p) andare comparable to the highest-accuracy cc-aug-VTZ//6-311G(d,p) energy, with semi-empirical AM1 giving asurprisingly good performance. Optimized reactant andproduct structures are also very similar to the 6-311G(d,p)

results except for STO-3G. Based on these results, weselected the 6-311G(d,p)//3-21G(2d,p) basis set for therest of this study, using B3LYP which is a hybridHartree-Fock/DFT method using Beckes three-parametergradient-corrected exchange functional (B3)23 with theLee-Yang-Parr correlation functional (LYP).24 The meanabsolute deviation from experiment (MAD) for B3LYP/6-311+ G(2d,p)//B3LYP/3-21G(d) energies is estimated25 as0.14 eV for carbon-rich molecules, which should be ade-quate for the purposes of this analysis while conservingcomputational resources, and which appears slightly supe-rior to the MAD of 0.34 eV estimated25 for the more com-monly reported B3LYP/6-31G(d)//B3LYP/6-31G(d) basis

set. In conventional positionally uncontrolled chemistry,errors on the order of 0.14 eV might well inuence reac-tion rates and also the dominant reaction pathway takenwhen multiple alternative reaction pathways are present.However, in the context of the present analysis this shouldnot be an issue because alternative reaction pathways arelimited by using positional control. Zero-point correctionsare not made to the energy data because:(1) the differences between energies with and withouta correction are small (0.008 eV for Reaction I, andsimilarly for Reaction II),(2) the number of points to be evaluated is large (manythousands in this study), and(3) the computational expense is huge (e.g., analyticalsecond derivatives for spin-unrestricted calculations arenot implemented in PC-GAMESS and the calculation of numerical frequencies requires about twice as much CPUtime as geometry optimizations).

For each of the three positionally controlled mechano-synthetic reactions examined, the HAbst, HAbstH andGeRad tooltips are constrained by xing the positionsof the three sidewall carbon atoms (CH2 groups) in thetooltip adamantane base that are located on the side of the cage farthest from the active radical site of the tool.Each tool was thus xed precisely in space by constraining

just 3 toolbase atoms from the start of each run, then thesystem was allowed to relax to its equilibrium geometryduring the run. This method provides the best model foranticipated actual laboratory conditions in which an exper-imentalist will control tooltip position by applying forcesthrough a larger diamond lattice handle structure afxedbehind the base structures of the respective tooltips. Foreach run, each tooltip base was positionally constrained toa specic spherical coordinate(in XY plane) and (inZ direction) in xed increments, and to several xed radialdistancesR, and the incoming tooltip was also constrainedto a specic axial rotational angle.

3. REACTION I: JOIN GeRad TOOL TOAPICAL ETHYNYL C ATOM OF HAbstH

In Reaction I of the HAbst recharge reaction sequenceRS5 (Fig. 1), a GeRad tool is brought up to the distalethynyl carbon atom to which the abstracted hydrogenatom is bonded on the HAbstH tool (which now becomesthe workpiece), and is then bonded. This decreases thecarboncarbon bond order from 3 to 2, causing the car-bon dimer to become nonlinear relative to the adamantanehandle central axis and creating an open radical site on thecarbon atom proximal to the HAbstH base structure.

After dening the tooltip geometry and coordinate sys-tem for Reaction I (Section 3.1) we calculate the reactionPES (potential energy surface) as a function of positionalangles and (Section 3.2), describe some trajectory-related pathologies (Section 3.3), calculate the reactionPES as a function of rotational angleof the incom-ing GeRad tool (Section 3.4), estimate the tolerance forlateral displacement error in tooltip positioning for Reac-tion I (Section 3.5), then recommend some optimal GeRadtooltip trajectories for this reaction (Section 3.6).

3.1. Tooltip Geometry and Coordinate System

The tooltip geometry and coordinate system for Reac-tion I is shown in Figure 2. The rst coordinate origin Ois dened as the point equidistant from the xed carbonatoms C4, C6, and C10 in the HAbstH adamantane base,and lying in the plane containing those atoms. TheX axislies perpendicular to the C4/C6/C10 plane and points from

origin O to a second origin Owhich is initially coinci-dent with atom C12 in HAbstH,5.262 away from O.This axis passes from origin O through atoms C1, C11,C12, and H28, whose equilibrium positions are collinearin HAbstH. TheY axis originates at O, lies perpendicu-lar to theX axis, and runs parallel to a vector pointingfrom atom C10 to origin O. TheZ axis also originatesat O and lies perpendicular toX and Y axes followingthe right-hand rule, running parallel to a vector pointingfrom atom C6 to atom C4. The approaching GeRad toolis initially oriented relative to the HAbstH workpiece suchthat a line extending backwards through atoms C12 (originO) and Ge35 perpendicularly penetrates the plane denedby the 3 xed GeRad tool base carbon atoms (C30, C32,and C34), intersecting a third origin Owhich lies in theC30/C32/C34 plane and is equidistant from C30, C32 andC34, analogous to origin O.

In this spherical coordinate system,is dened as theangle from theX axis to theY axis of the projection of thevector pointing from Oto O onto theXY plane. Note that+ is dened as rotation toward the Y axis in the arrowdirection, since the target hydrogen atom (H28) movesaway from GeRad in the+ Y direction ( direction)upon GeRad bonding to atom C12 at the completion of



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GeRad2

Fig. 2. Coordinate system for Reaction I: Positionally controlled reactant tooltips prior to reaction (top left), and following completion of r(top right), dening phi (and theta ( , including atom labels (bottom, left), and denition of tooltip axial rotation angle rho ((bottom, right).(C= yellow, H= blue, Ge= white).

Reaction I. is dened as the angle from theXY plane tothe vector pointing from Oto O , with 90 + 90and 180 + 180 . Radial distanceR is dened asthe distance between origins Oand O. Although atomC12 moves away from Oas the reaction proceeds, theproper experimental protocol is nevertheless to aim the O-to-Ge35 vector directly toward the xed origin O(not toC12, which moves) to execute Reaction I. The rotationalstate of HAbstH is completely specied after labeling thebase atoms C4, C6, and C10 and dening the positiveY -axis as the C10-to-O direction. The rotational state of GeRad, specied by, is measured as the angle taken fromthe O-to-C10 vector to the O-to-C34 vector when GeRadis virtually repositioned to (, = 180 , 0 making Ocoincident with O, with+ taken in the clockwise direc-tion as viewed from O. Thus, rotation to+ becomesequivalent to rotation to at = + 90 , or to+ at = 90 . This coordinate system was chosen because originsO and O experience negligible reaction-mediated nonther-mal displacement during the course of the reaction andthus may be most directly controlled in an experimentalapparatus.

3.2. PES as a Function of Positional Angles and

The PES as a function of positional anglesand isshown in Figure 3, holding GeRad at the rotational angle

= + 40 , for four different tooltip-workpiece displace-ments (R = 5 60 , 5.35 , 4.85 , and 4.35 ) as theGeRad tool approaches the HAbstH workpiece.

At R = 5 60 , with Ge35 roughly 3.20 away fromO (originally at C12), the PES in a 90 90 angularregion centered on (, = 0 , 0 is largely featurelessand at, with strongly endoergic mountains centered on

( , = 180 , 0 due to rapidly rising steric repulsionbetween approaching HAbstH and GeRad tool handles atthose angles. AtR = 5 35 , with Ge35 roughly 2.95 away from O, a weakly endoergic hill about 30in radiushas appeared at the center of the PES due to rising repul-sion between the Ge35 radical on the GeRad tool andthe target hydrogen atom H28 on the HAbstH workpiece,since H28 lies between Ge35 and C12 at these angles ofapproach. AtR = 4 85 , with Ge35 roughly 2.45 awayfrom O, the hydrogen hill at the center of the PES hasbecome more repulsive, dening an annular region sur-rounding the hydrogen hill within which the GeRad tool



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R = 5.35

R = 4.85 R = 4.35

R = 5.60

Fig. 3. PES for Reaction I (endoergic= blue, exoergic= red, energy in eV) as a function of tooltip positional angles= 180 to + 180 , = 90to + 90 , at tooltip separation distancesR = 5 60 ,R = 5 35 ,R = 4 85 andR = 4 35 , and GeRad rotational angle= + 40 , with labeledisoergic contours. Energy minima are marked with solid dots at (, = 46 , + 74 ( 0.506 eV) and (+ 34 , + 74 ( 0.504 eV) atR = 4 85 ,( , = 23 , + 26 ( 0.486 eV) and (+ 24 , + 26 ( 0.482 eV) atR = 4 35 .



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may approach the distal C12 atom along a maximally exo-ergic trajectory. ByR = 4 35 , with Ge35 roughly 1.95 away from O(the approximate equilibrium GeC bondlength), H28 has moved off to the side and the PES in an80 80 region centered on (, = 0 , 0 is uniformlyexoergic, mostly in the 0.40 to 0.50 eV range, for the

completed Ge35C12 bonding reaction.The spherical representations of the PES (Fig. 4) forR = 4 85 show regions of favorable (red, exoergic) andunfavorable (blue, endoergic) net reaction energies fromthe viewpoint of a GeRad tool as it approaches theHAbstH toolhere represented by the hydrogenatedethynyl (CCH) group, the HAbstH handle bridgehead Catom to which the ethynyl is attached and the three side-wall C atoms attached to the bridgehead. The central endo-ergic (blue) hydrogen hill is surrounded by an exoergic(red) annular region, with another (blue) annular regionbehind it indicating unfavorable energetics, with a yellowregion at negativeX-axis values that is excluded due tohandle collision.

A comparison of the succession of charts in Figure 3indicates the presence of a favorable annular region withinwhich many dozens of useful approach trajectories exist,such as the (, , = + 40 , + 70 , + 40 example.A closer examination of the progression of the Ge35C12 bonding reaction for this exemplar trajectory (Fig. 5)shows the kinking of the bridgehead-ethynyl group andthe60 shift in CCH28 bond angle (left image), alongwith the0.13 increase in CC bond length as bondorder changes from 3 (C C) to 2 (C C) and the changein Ge35C12 equilibrium separation as the GeC bond

forms (center image). The 1D PES (right image) shows aslight uplift in energy peaking atR = 5 65 as GeRadapproaches, prior to consummation of the bonding reac-tion, revealing a small+ 0.09 eV reaction barrier along thistrajectory. The0.18 excess separation (above diagonalline on top curve in center image) due to repulsion betweenGe35 and C12 at theR = 5 65 barrier peak implies thatonly a rather modest mechanical force of 0.80 nN mustbe applied by the incoming GeRad tool to overcome this

Z Z

Y X

Z

X

Fig. 4. Three views of spherical representation of PES for reaction I (endoergic= blue, exoergic= red) as a function of tooltip positional anglesand (see Fig. 2) expressed on the cartesian XYZ coordinate system at tooltip separation distanceR = 4 85 and GeRad rotational angle= + 40 .Yellow region is excluded due to handle collision.

repulsive barrier, when following this particular exemplarapproach trajectory.

3.3. Trajectory Pathologies of Reaction I

Aside from the known possible reaction pathologies of

Reaction I that are energetically disfavored,7

two addi-tional classes of undesired trajectory-associated patholo-gies have been identied in the present work (Fig. 6).

The rst trajectory pathology involves the bonding of Ge35 and C11 (the proximal ethynyl carbon atom onHAbstH to which H28 is not directly attached) in thecase of high- trajectories that allow the Ge35 radicalsite to pass too close to C11 while moving toward C12,enabling an unwanted Ge35C11 bond to form before thedesired Ge35C12 bond can form (Fig. 6(A)). Nonexhaus-tive investigations of numerous relevant (, trajectoriesat = + 40 and R = 4 85 revealed that the Ge35C11misbonding pathology does not occur for 90 , andoccurs only at low for some higher values of , e.g.,at ( , = 100 , + 20 and ( 100 , + 30 , and at(+ 120 , 0 and (+ 120 , 40 , and is not observed insome high- high- trajectories such as (, = + 120 , 70 . Apparently the pathology is absent when the clos-est initial Ge35C11 distance dGe35C11 is 2.74 , withpathology permitted but not mandatory for 2.73 >dGe35C11 2.57 .

The second class of trajectory pathologies involvesunwanted high-energy covalent bond formation betweenHAbstH and GeRad tool handles during very high-trajectories where the handles are computationally forced

into experimentally unrealistic proximity at the start of an energy minimization. Five distinct structural rearrange-ments were recorded, of two general subclasses: twoinstances (Figs. 6(D and F)) in which a handle-cageshoulder-bond to Ge35 is broken, resulting in diradicaliza-tion of the Ge atom, and three instances (Figs. 6(B, C andE)) in which multiple rebonding events between handle-cage atoms are accompanied by the release of an H2molecule, a likely spurious result. These rearrangements



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Fig. 5. Progression of Ge35C12 bonding reaction in Reaction I as a function of R () for a xed (, , = + 40 , + 70 , + 40 approach trajectoryat 0.05 increments. Left: C1C11C12 (black) and C11C12H28 (red) bond angles (deg). Center: Ge35C12 (black) and C11C12 (redistances (). Right: PES (eV).

have not been investigated in further detail since they occuronly well outside of the recommended (, operatingrange of 80 , 80 for Reaction I, and are reportedhere for completeness.

3.4. PES as a Function of Rotational Angle

Figure 7 shows the effects of GeRad tooltip rotationalangle on the 1-dimensional PES for Reaction I. At (,

= 0 , 0 forR = 4 85 , the PES has negligible varia-tion with changingbecause the HAbstH and GeRad han-dles are maximally distant, thus minimizing handlehandlesteric interactions. Using the viable (, = + 40 , + 70trajectory, varying has greater effectthere is a smallervariation of about 0.03 eV at a more distant separation

(A)

(D) (E) (F)

(B) (C)

Fig. 6. Six instances of handlehandle rearrangements observed during experimentally unrealistic close apposition of tooltip handles for Reaat R = 4 85 and = + 40 , for ( , settings: (A) (+ 100 , 30 , ( 100 , + 30 , ( 100 , + 20 , (+ 120 , 40 , and (+ 120 , 0 ; (B) ( 110 , 10 ; (C) ( 120 , 0 ; (D) ( 120 , + 20 ; (E) (+ 150 , 40 ; and (F) ( 150 , 50 . (C= yellow, H= blue, Ge= white).

distance of R = 4 85 and a somewhat larger variation of 0.14 eV as nal Ge35C12 bond formation is reached atR = 4 35 , with reaction energy minimized in this caseat = + 40 , + 160 , and + 280 . There are three peaksand valleys due to the threefold symmetry of the GeRadtooltip: the GeC bonds of the three sidewall CH2 groupsbonded to Ge35 are spaced 120apart, as are the threegaps between those shoulder groups, producing an asym-metric steric interaction when GeRad is brought into closerproximity to the HAbstH handle and rotated to variousvalues of . At the most extreme values of = 90 , thechoice of ( has greatest effect because of the stronghandlehandle interactions as GeRad rotations inmovetool handle sidewall CH2 groups into and out of juxtapo-sition on the apposed tool handles. In particular, there is a



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Fig. 7. PES for Reaction I as a function of GeRad tooltip rotational angle, for trajectory (, = 0 , 0 at R = 4 85 and trajectory (,= + 40 , + 70 at R = 4 85 and 4.35 (at left), and trajectories= 90 at R = 4 85 (at right) where displacements are equivalent to.

variation of 0.38 eV depending on the choice of , andwith the reaction energy for Reaction I minimized at( = 0 , 120 in this case. Note that if the GeRad tool

was held xed in space and instead the HAbstH workpiecewas rotated around its central axis, we would expect to seethe same trimodal pattern in net energy due to the similarthreefold symmetry of the HAbstH handle structure.

Figure 7 shows that the impact of on net reactionenergy can sometimes be signicant, especially near theperimeter of the recommended (, operating range of 80 , 80 for Reaction I. Since the choice of forwhich Reaction I net energy will be minimized is a func-tion of the selected approach trajectory (, but is not afunction of R, Figure 8 explores the space of minimum-and maximum-energy values as a function of andin the two representative (+ , quadrants to 20reso-lution atR = 4 85 . The results conrm that:(a) the impact of on net reaction energy increases withlarger or (Figs. 8(A and B)) and may vary signicantlydepending upon trajectory (Fig. 8(D)),(b) the increase in net reaction energy caused by choos-ing the most detrimental versus the most optimalrangesfrom 0.0 eV to+ 0.7 eV (Fig. 8(B)), and(c) the choice of optimal(Fig. 8(C)) can vary from 0to 100 depending upon choice of (, .

The apparent patternlessness of optimalsettings inFigure 8(C) is partly due to the occurrence in somecases of multiple minima having small energy differences

within the computational error and partly artifactual dueto wraparound (i.e.,= 0 = 120 .For trajectories where the most optimal and most detri-

mental values give reaction energies that differ onlyslightly (i.e., white areas in Fig. 8(B)), the precise valueof is inconsequential in determining reaction reliabil-ity and thus the ability to positionally controlduringthe experiment may not be crucial. For other trajectorieswhere this energy differential is large (i.e., dark blue areasin Fig. 8(B)), can be a signicant factor in determiningreaction reliability in which case the ability to positionallycontrol during the experiment would be very important.

3.5. Lateral Displacement Error Tolerance

To specify a useful experimental protocol it is alsonecessary to determine the maximum tolerable lateralmisplacement error of GeRad that will still result in a suc-cessful consummation of Reaction I. We start by den-ing theU -axis as a vector pointing from Oto C34 andthe V -axis as a vector originating at Oand perpendic-ular to U that points parallel and codirectional with avector from C30 to C32 (Fig. 9). We can then exam-ine whether the Ge35C12 bond still forms when GeRadis translationally displaced within the UV plane awayfrom its intended position at any point within a particu-lar approach trajectory. A comprehensive analysis of allpossible displacements from every point along all possibleapproach trajectories is beyond the scope of this paper, so

we analyzed a representative reaction point (R = 4 35 )along a single exemplar approach trajectory: (, , =+ 40 , + 70 , + 50 . The lateral displacement of GeRad

from its intended trajectory is reported as a displacementangle measured fromU and a displacement radial dis-tance r measured from O.

Simulations began by examining smallr , moving tolarger r , simulating progressively larger displacementerror circles. At somer we would expect to nd thatGe35 and C12 are too far apart to form the desiredbond. However, before that point is reached a compet-ing Ge35C11 bonding pathology is encountered, initially

only at = 90 where the GeRad error displacementmoves Ge35 exactly in the C11 direction. This= 90pathology is observed atr = 0 6 but not at r =0 5 . Thus the maximum tolerable GeRad tooltip dis-placement error for successfully completing Reaction Ialong the exemplar trajectory at= 90 is 0.5 . Withincreasing r beyond 0.6 , the range of over whichthe Ge35C11 bonding pathology occurs increasese.g.,by r = 1 0 the pathology is found over= 50 130although the desired Ge35C12 bonding is still obtained atall other ; at r = 1 5 , Ge35C12 bonding is obtainedonly at = 180 ; and at r = 1 6 the desired Ge35C12



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(A)

(B)

(D)

eV eV deg(C)

Fig. 8. Effects on reaction energy for various choices of GeRad tooltip rotational angleas a function of (, trajectory forR = 4 85 inReaction I: (A) minimum net energyEmin (eV) produced by choosing the optimumvalue for each trajectory; (B) increase in net reaction energy(eV) over minimum (Emin produced by choosing the most detrimentalvalue that yields the highest reaction energyEmax, or EmaxEmin ; (C) choiceof optimum value (deg) that produces the minimum net energy shown in (A) for each trajectory; and (D) energy variations as a function offortrajectories (, = + 80 , + 60 , (0 , + 60 and (+ 60 , + 40 . Charts (A)(C):E or E < 0 (red),= 0 (white),> 0(blue); optimum angles inyellow; excluded data points in gray.

bonding cannot be obtained at any(tested full circle at20 increments). These results are broadly consistent withsimilar results from previous analyses of tolerable mispo-sitioning errors reported elsewhere.19 26

3.6. Optimal Tooltip Trajectories

3.6.1. Single-Setting Optimal Trajectories

Perhaps the simplest method for selecting an optimalapproach trajectory for Reaction I is to choose a single( , trajectory that continuously maintains an acceptablylow net reaction energy throughout the entire approach,starting from innite separation between tool and work-piece and concluding with Ge35C12 bond formation atR = 4 35 . The objective is to allow tool and work-piece to be positioned in the energy landscape on a val-ley oor of acceptable depth but maximum width which

will permit successful consummation of Reaction I evenin situations where tooltip placement error may be signif-icant, e.g., the (, , = + 40 , + 70 , + 40 approachtrajectory described in Section 3.2 (Fig. 5). This methodprovides a safe and adequate approach trajectory that min-imizes demands on the reliability of an experimental posi-

tional control system, since the controls can be set onceat the start of the reaction and held constant through itsconclusion.

A more systematic method for generating a generalrange of workable single-setting trajectories starts with thetwo PES charts forR = 4 85 andR = 4 35 fromFigure 3, which represent the before and after cases ofH28 and C11C12 movement during tooltip approachthe key mechanical events of Reaction I. The intersec-tion of all exoergic points simultaneously present on bothPES surfaces yields a large subset of acceptable approachtrajectories (Fig. 10). The 10 best (, trajectories (as



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Fig. 9. Geometry used for displacement error tolerance, with HAbstHshown in foreground and GeRad shown in background in position toconsummate Reaction I;r and are dened as the radial and angulardisplacement of the GeRad tool relative to an exemplar (, , = + 40 ,+ 70 , + 50 approach trajectory, shown displaced up and to the righttowards 1 oclock in the drawing. (C= yellow, H= blue, Ge= white).

indicated) have net energies within 0.01 eV of the globalminimum. An experimentalist can be reasonably condentthat any trajectory selected from within the annular dis-tribution of (, points (after adjusting for proper;see Fig. 8) should enable Reaction I to successfullyproceed.

3.6.2. Compound Optimal Trajectories

It is also possible to construct a specic compound opti-mal trajectory comprised of two or more single-settingtrajectories that continuously maintains the lowest possi-ble net reaction energy at every separation distance, start-ing from innite separation between tool and workpieceand concluding with Ge35C12 bond formation atR =4 35 . This method maximizes the probability of a cor-rect reaction based on energetics, but assumes that tooltip

Fig. 10. Single-trajectory analysis from Boolean intersection of R =4 85 andR = 4 35 PES exoergic (red) zones, for= + 40 inReaction I. Trajectories within0.01 eV of global minimum are soliddots.

placement error is small enough to allow hopping betweendesignated neighboring PES surfaces (a signicant chal-lenge, experimentally). In this method, during each 0.05 step inR the experimentalist is permitted to alter one orboth of the or settings by at most one increment of minimum resolution (= 10 in either or for this

study) to ensure smooth and continuous motion, shiftingthe tool from one (, trajectory to another immedi-ately adjacent ( , trajectory in order to main-tain minimum energy at each step. Future studies couldemploy a smaller to provide an even smoother com-pound trajectory.

To construct our representative optimized compound trajectory, we start by recomputing absolute energy min-ima for ( , , to 10 resolution at bothR = 4 85 and R = 4 35 , obtaining the following results afterone iteration: ForR = 4 85 , ( , , = + 50 , + 70 ,+ 20 , Emin = 0 508 eV; and (, , = 50 , + 70 ,+ 60 , Emin = 0 505 eV. ForR = 4 35 , ( , , =+ 30 , + 10 , + 90 , Emin = 0 504 eV; and (, , = 20 , + 30 , + 110 , Emin = 0 504 eV. In each case,

an optimum (yielding lowest energy) was determinedfrom a xed (, , after which the new was heldxed and (, was reoptimized. The rst iteration pro-duced only minor 0.002 eV/ 0.001 eV (R = 4 85 )and 0.045 eV/ 0.009 eV (R = 4 35 ) improvements inminimum energy, though a more thorough (computation-ally expensive) analysis might iterate until zero change inminimum energy was attained. Trajectory charts (E vs. R)for constant (, , in all four cases (Fig. 11(A)) reveallarger barrier heights for theR = 4 35 trajectories, so

we choose (, , = + 50 , + 70 , + 20 at R = 4 85 (which also has the lowestEmin of the four choices) as thestarting point for our representative optimized compoundtrajectory.

Starting from this base trajectory,E versusR charts arecomputed for a bracketing 3 3 grid of nearest-neighbor( , trajectories in 10 increments (Fig. 11(B)). Forexample, the (, trajectory yielding the lowest energy atR = 4 85 + 0.05 = 4 90 is selected as the next pointin the compound trajectory. The process is iterated in 0.05 increments outward toR = 5 90 and inward toR =4 35 , yielding the representative locally optimized com-

pound trajectory shown in Figure 11(C) (data in Table II).In the present work,was held constant at+ 20 for com-putational convenience; a more thorough (computationallyexpensive) analysis would reoptimize (, , at everystep to nd the best. Calculating a globally optimizedcompound trajectory would require calculating all possi-ble E versusR curves to create a comprehensive collec-tion from which the absolute energy-minimized trajectorycould denitively be assembled to whatever resolution wasrequired. Trajectory analysis software could be developedto calculate the optimum compound trajectory automati-cally. However, it is not clear that optimized compound



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(A)

(B)

(C)

Fig. 11. Construction of a representative locally optimized compound trajectory for Reaction I: (A) Four-optimized (, , trajectories exhibitingminimum net reaction energy atR = 4 85 ((+ 50 , + 70 , + 20 and ( 50 , + 70 , + 60 and atR = 4 35 ((+ 30 , + 10 , + 90 and ( 20 , + 30 ,+ 110 ; (B) nearest-neighbor trajectories to (, = + 50 , + 70 at = + 20 and R = 4 85 ; and (C) a locally optimized low-barrier compoundGeRad tooltip approach trajectory for= + 20 and allowing 10 angle changes at each 0.05 step (see Table II).

trajectories represent a sufcient improvement over well-chosen single-setting trajectories to justify their additionalcomputational and experimental complexities.

3.6.3. Low-Barrier Optimal Trajectories

Another important criterion for successful completion of a positionally controlled mechanosynthetic reaction is theminimization of energy barriers. If the activation energyfor a selected approach trajectory is too high, the toolsmay not be able to deliver sufcient force to compel thereaction to go forward, or the tool or workpiece might

suffer unacceptably large exures during their attempt toovercome a high barrier, potentially opening the door toadditional trajectory or reaction pathologies. For example,the highest reaction barrier within the recommended (,operating range of 80 , 80 for Reaction I (E + 0 48 eV) occurs near (, , = 0 , 0 , + 40 (Fig. 12).

An energy barrier map for two representative (+ , quadrants of the (, space at xed = 50 , comprisedof a uniform distribution of 102 data points at 10intervalsin the range = + 30 to + 80 and = 80 to + 80 ,shows the maximum endoergicity encountered along each



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Table II. Representative locally-optimized compound trajectory for Reaction I, with= + 20 and allowing 10 angle changes at each 0.05 step.

R () E (eV) R () E (eV) R () E (eV)

5.90 + 40 + 60 + 0.041 5.35 + 60 + 70 0.134 4.80 + 50 + 70 0.5075.85 + 40 + 60 + 0.046 5.30 + 50 + 80 0.453 4.75 + 40 + 70 0.5065.80 + 50 + 60 + 0.052 5.25 + 50 + 80 0.468 4.70 + 40 + 70 0.5035.75 + 40 + 60 + 0.058 5.20 + 50 + 80 0.480 4.65 + 50 + 60 0.500

5.70 + 40 + 60 + 0.065 5.15 + 50 + 80 0.488 4.60 + 50 + 60 0.4975.65 + 40 + 60 + 0.072 5.10 + 40 + 80 0.493 4.55 + 40 + 60 0.4915.60 + 40 + 60 + 0.079 5.05 + 40 + 80 0.498 4.50 + 40 + 60 0.4875.55 + 40 + 60 + 0.087 5.00 + 40 + 80 0.500 4.45 + 40 + 60 0.4795.50 + 40 + 60 + 0.096 4.95 + 50 + 70 0.500 4.40 + 40 + 60 0.4695.45 + 50 + 60 + 0.110 4.90 + 50 + 70 0.505 4.35 + 40 + 60 0.4565.40 + 50 + 70 + 0.112 4.85 + 50 + 70 0.508

approach trajectory (Fig. 13). The peak energy barriersoccur at a variety of separation distancesR (not shown),and range from a minimum of + 0.102 eV at (, =+ 50 , + 60 to a high of + 0.30 eV at (, = + 30 , 0

approaching the hydrogen hill at the leftmost edge of thechart, with the maximum barrier of + 0.48 eV at (, =0 , 0 not directly visible on the chart. The single-setting

( , , = + 50 , + 70 , + 20 approach trajectory liesin a favorable region of the energy barrier map, and otheroptimal trajectories computed using methods described inSections 3.6.1 and 3.6.2 might be enhanced by selectingneighboring paths possessing reduced barrier heights.

The barrier-driven exure through an angleof thetip of the CC C group away from linear congu-ration as the GeRad tooltip approaches C12 can becrudely estimated from the standard bond-bending term inmolecular mechanics:Ebend= 1/ 2kbend ( eq 2 (1+ ksextic

( eq4

, wherekbend is the angle-bending force constant,eq is the equilibrium bond angle, andksextic= 0 754 rad 4for CC bonds in MM2. Data from a published8 bendingpotential chart for the adamantane-handled (C9H15 CC Csystem (the HAbst tooltip) giveskbend370 zJ/rad2, hencesurmounting a0.14 eV barrier on the (, , = + 50 ,+ 70 , + 20 approach trajectory should produce a max-imum exure (immediately prior to Ge35C12 bond for-mation) of 20 , or 0.9 deection at the endof a 2.66 CC C lever arm requiring a very modest0.25 nN of mechanical force to overcome. This estimate

Fig. 12. PES for Reaction I as a function of tooltip separation distancesR using the highest-barrier (E + 0.48 eV) approach trajectory at (, ,

= 0 , 0 , + 40 .

is consistent with the0.4 prebonding ethynyl repul-sion displacement computationally predicted for the O-C12 distance as GeRad approaches (shaded area, Fig. 14),given that no prebonding compression is evident in O-Ge35 but that some portion of the total compression energyis likely partitioned into each of the two tools.

3.6.4. Other Optimal Trajectories

Figure 8(B) indicates another possible optimal trajec-tory criterionminimumangle impact on net reactionenergythat yields specic approach trajectories atR =4 85 such as (, = + 80 , 0 which is both very exo-ergic and maximally insensitive toangle setting, perhaps

Fig. 13. Energy barrier map for Reaction I (endoergic= blue,exoergic= red, energy in eV) as a function of tooltip positional anglesand . Minimum barrier height occurs at various tooltip separation dis-tancesR depending on value of and , taking GeRad rotational angle

= + 50 , with lowest energy value+ 0.102 eV at (, = + 50 , + 60where the guidelines intersect.



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O-C12 distance O-Ge35 distance

Fig. 14. Deection of C12 at ethynyl tip of HAbstH and partial extrusion of Ge35 during Ge35C12 bonding event along (, , = + 50 , + 70 ,+ 20 approach trajectory in Reaction I, as GeRad travels fromR = 6 60 to 4.35 .

ideal in situations where the experimentalist ndsdif-cult or impossible to control. Other specialized trajectorycriteria may be investigated in future work.

4. REACTION II: ABSTRACT APICAL HFROM HAbstH USING GeRad2 TOOL

In Reaction II of the HAbst recharge reaction sequenceRS5 (Fig. 1), a second GeRad tool (GeRad2) is broughtup to the transactional hydrogen atom (H28), and at someseparation distance H28 jumps to the incoming GeRad2tool, having been abstracted away from the tip of theHAbstH tool. This allows the C11C12 bond order to

increase from 2 to 3, causing the carbon dimer to seek toresume linearity relative to the adamantane central axis,and eliminates the open radical site on the carbon atomproximal to the HAbstH base structure that was createdduring Reaction I. The hydrogenated GeRad2-H tool maythen be withdrawn from the system along any noncolli-sional trajectory, the choice of which is noncritical (and isnot examined further in the present work) because thereare no remaining open radicals in the system and so alltooltips are in an inert condition. The GeRad2-H tooltip issubsequently restored to an active GeRad2 tooltip using aseparate recharge reaction sequence7 specic to that tool,e.g., contacting the GeRad2-H to a depassivated at bulkdiamond surface, causing the H to donate to the surface.

After dening the tooltip geometry and coordinate sys-tem for Reaction II (Section 4.1) we calculate the reactionPES (potential energy surface) as a function of positionalangles H and H (Section 4.2), describe the trajectory-related pathologies (Section 4.3), calculate the reactionPES as a function of rotational angleH of the incom-ing GeRad2 tool (Section 4.4), estimate the tolerancefor lateral displacement error in tooltip positioning forReaction II (Section 4.5), then recommend some optimalGeRad2 tooltip trajectories for this reaction (Section 4.6).


A positionally unconstrained single-cage-handle HAbstH-GeRad complex (as created in Reaction I) has two sta-ble minimum-energy congurationsa cis form withthe H28 atom and the radical site on the same sideof the ethynyl C C dimer, and a trans form withH28 and the radical site on opposite sides of the C Cdimer (Fig. 15). We conrm earlier work7 reporting thatthe unconstrained trans form is 0.12 eV lower inenergy than the unconstrained cis form. However, in amechanosynthetic apparatus the two handles comprisingthis structure are not unconstrained, but rather are contin-uously positionally controlled. Using positional control, a

mechanical barrier can be imposed that forces the struc-ture to adopt or to retain either conguration since bothare stable stationary states with no imaginary frequencies.In the present work, as in previous work,7 we presume thatpositional control is employed to establish and maintainthe system in the slightly higher energy cis state becausethis conguration provides greater steric accessibility toH28 by the incoming GeRad2 tooltip.

We start by assuming that the displacement distanceR producing the lowest energy on the optimal trajectoryrepresents the endpoint of Reaction I, and hence consti-tutes the starting point for Reaction II. Hence the tooltipgeometry and coordinate system for Reaction II is basedon the minimum energy conguration for HAbstH-GeRadin cis geometry at (, , = + 50 , + 70 , + 20 andR = 4 85 (Fig. 16). Structural data for this initial geom-etry are as follows: A(C1C11C12)= + 151059 deg,C1C11C12H28 dihedral= + 179775 deg, A(Ge35C12C11) = + 129257 deg, A(Ge35C12H28)=+ 112258 deg, and R(Ge35C12)= 1 984 . The rstcoordinate origin OH is dened as the equilibrium startingposition of H28 with GeRad2 removed to innite distance.The positiveXH axis originates at OH and points in thedirection from the starting position of C12 to OH. The



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Cis ConfigurationA (Ge35C12C11) = +127.820 degA (C1C11C12) = +141.661 degA (Ge35C12H28) = +112.514 degA (H28C12C11) = +119.664 degR (Ge35C12) = 1.96636 E = 0.45 eV

Trans ConfigurationA (Ge35C12C11) = +123.158 degA (C1C11C12) = +140.580 degA (Ge35C12H28) = +116.034 degA (H28C12C11) = +120.808 degR (Ge35C12) = 1.95947 E = 0.57 eV

Fig. 15. Unconstrained single-cage-handle HAbstH-GeRad structure at Reaction I completion has two stable minimum-energy conguration=yellow, H= blue, Ge= white).

Y H axis originates at OH, lies perpendicular to theXHaxis and in the plane containing the starting positions of atoms C1, C11, C12 and Ge35, and points away fromthe GeRad tooltip containing Ge35. TheZH axis alsooriginates at OH and lies perpendicular toXH and Y H axesfollowing the right-hand rule. The approaching GeRad2tool is initially oriented relative to the HAbstH-GeRadworkpiece such that a line extending backwards throughOH and Ge65 perpendicularly penetrates the plane denedby the 3 xed GeRad2 tool base carbon atoms (C59, C66,and C74), intersecting a second origin OH which lies inthe C59/C66/C74 plane and is equidistant from C59, C66and C74, analogous to origins O and O.

In this spherical coordinate system,H is dened as theangle from theXH axis to theY H axis of the projection of the vector pointing from OH to OH onto theXHY H plane.Note that+ H is dened as rotation toward the Y H axisin the arrow direction for consistency with prior usage inReaction I. H is dened as the angle from theXHY H planeto the vector pointing from OH to OH, with 90 H + 90 and 180 H + 180 . Radial distanceRH isdened as the distance between origins OH and OH. The

rotational state of HAbstH-GeRad is completely speciedby ( , , = + 50 , + 70 , + 20 . The rotational state of GeRad2, specied byH, is measured as the angle takenfrom the OH-to-C59 vector to theY H-axis vector whenGeRad2 is virtually repositioned to (H, H = 0 , 0placing both OH and Ge65 on theXH axis, with+ H takenin the counterclockwise direction as viewed from OH look-ing toward OH. Thus, rotation to+ H becomes equivalentto rotation to H at H = + 90 , or to+ H at H = 90 ,consistent with prior usage in Reaction I. The positioningof GeRad2 is controlled by constraining atoms C59, C66,and C74 in the GeRad2 handle base.

4.2. PES as a Function of Positional AnglesH and H

Since well-separated HAbstH-GeRad and GeRad2 eachhave one unpaired electron, the system of workpiece+tooltip can have singlet or triplet multiplicity. The PES asa function of workpiece-tooltip separation distanceRH forthe ideal (H, H, H = 0 , 0 , 0 approach trajectory(Fig. 17) shows that the system remains in the unbondedlower-energy triplet state down toRH = 5 90 (with Ge65roughly 3.05 away from H28), but atRH = 5 80 (orcloser) the singlet state has lower energy and the Ge65H28 bond has formed, indicating a successful abstractionof the hydrogen atom. Figure 17 indicates that the abstrac-tion reaction may be barrierless, consistent with similarresults previously reported elsewhere.7 8

The PES for the Ge65H28 abstraction reaction as afunction of positional anglesH and H is shown inFigure 18 for displacement distanceRH = 5 80 and

H = 0 , compiled from a total of 343 underlying datapoints at 10 intervals. The PES is uniformly at andhighly exoergic across a 40 80 region centered on

( H, H = 0 , 0 , with net reaction energies approximat-ing 1.50 eV over most of this space but rising no higherthan 0.75 eV even at the periphery. (The computed abso-lute minimum energy of 1.576 eV actually occurs at( H, H = 0 , + 10 but this is only 0.004 eV belowthe value at the assumed ideal (H, H = 0 , 0 set-ting, well below the accuracy of the computational methodused.) High endoergic mountains are beginning to appearat H + 50 on the right side of the chart (approach-ing Ge35) and also atH 90 on the left side of thechart (approaching C11C1) due to rapidly rising stericrepulsion between closely proximated HAbstH-GeRad and



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Fig. 16. Coordinate systems for Reaction II: Positionally controlled reactant tooltips prior to reaction, dening phiH ( H and thetaH ( H (top image),and denition of tooltip axial rotation angle rhoH ( H (bottom image). (C= yellow, H= blue, Ge= white).

Fig. 17. Singlet and triplet PES as a function of RH for the ideal (H,H, H = 0 , 0 , 0 approach trajectory for Reaction II.

Fig. 18. Singlet PES for Reaction II (endoergic= blue, exoergic= red,excluded= gray) at tooltip separation distanceRH = 5 80 for approachtrajectories in the range (H, H, H = 180 , 90 , 0 .



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GeRad2 tool handles at those high angles. Spherical repre-sentations of the smoothed PES (Fig. 19) show regions of favorable (exoergic) and unfavorable (endoergic) net reac-tion energies from the viewpoint of a GeRad2 tool as itapproaches the H28 atom on the HAbstH-GeRad complex,with two yellow regions excluded due to handle collisions

on either side. Note that the PES is expected to be slightlyasymmetrical between+ H and H hemispheres becauseGeRad is rotated to= + 20 in the HAbstH-GeRad com-plex, causing that complex to lie asymmetrically across theXHY H plane.

4.3. Trajectory Pathologies of Reaction II

The rst trajectory pathology of Reaction II occurs alonga few mid-range H trajectories that allow the Ge65radical site to pass too close to C11 as GeRad2 approachesH28, enabling the unwanted bonding of Ge65 and C11(Fig. 20) in preference to the desired Ge65H28 bondingto achieve a successful abstraction. On the H side of therecommended (H, H operating range of 40 , 80for the normal abstraction reaction, there is a second low-energy valley about 20wide in H where the Ge65C11pathology occurs, visible on the PES in Figure 18, in therange ( H, H = 60 to 80 , 80 . Interestingly, theGe65C11 pathology also occurs in the entire range (H,

H = 90 , 80 to + 70 , but only the trajectories atH = 60 to 80 and + 50 to + 70 are overall exoer-

gic atRH = 5 80 . The other trajectories are endoergicoverall, but the Ge65C11 bond forms anyway because thebonded structure has lower energy (net exoergic) than theunbonded structure, since at these angles the tools havebeen forced into close proximity and xed in a positionwhere steric repulsion between handles is high. The Ge65C11 pathology also occurs in a similar mixed-energy fash-ion at H = 100 in the ranges H = 50 to 80 and+ 50 to + 80 , but only the H = 50 , 60 and 80trajectories are exoergic atRH = 5 80 . The extent of theGe65C11 pathology at trajectories withH < 100 hasnot been further investigated in the present work because

Z

Y

X

Z

X

Z

YY

X

Fig. 19. Three views of spherical representation of smoothed singlet PES for Reaction II (endoergic= blue, exoergic= red) as a function of tooltippositional anglesH and H expressed on the CartesianXHY HZH coordinate system at tooltip separation distanceRH = 5 80 and GeRad2 rotationalangle H = 0 . The HAbstH-GeRad complex is represented by a hydrogenated ethynyl (CCH) group attached to the Ge35 atom of GeRad. The region is excluded due to handle collision.

Fig. 20. Crossbonding Ge65C11 trajectory pathology observed duringclose apposition of tooltip handles for Reaction II atRH = 5 80 and

H = 0 . (C= yellow, H= blue, Ge= white).

these trajectories, while of general theoretical interest, arenot relevant to dening an experimental protocol involvingthe recommended (H, H operating range of 40 , 80 for Reaction II. The above examples all assume

H = 0 .A question arises as to whether a narrow low-positive-

energy barrier separates the two valleys in the range (H,H, H = 50 , 60 to + 50 , 0 , as is visible in

Figures 18 and 19 as a thin blue vertical reef. Steppingacross the putative barrier in H = 1 increments alongsuccessive (H, H, H = 40 to 71 , 0 , 0 approachtrajectories atRH = 5 80 (Fig. 21) to improve resolutionof this feature reveals a at exoergic net reaction energynear 1.52 eV through H = 45 , followed at H = 46 by singlet energy jumping to a highly endoergic



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Fig. 21. Singlet and triplet PES as a function of H for a ( H, H = 0 ,0 approach trajectory atRH = 5 80 for Reaction II (left ordinate),and Ge65C11 distance (right ordinate), for 71 H 40 in thesinglet case; singletH = 45 approach trajectory is also shown.

+ 0.55 eV, well above the+ 0.01 eV for triplet, coincidingwith failure of the normal Ge65H28 abstraction reac-

tion. Singlet energy again retreats below triplet only atH 54 , indicating the potential onset of Ge65C11pathological bonding as previously discussed: in the sin-glet mode, signicant prebonding movement of atomsGe65 and C11 occurs atH = 46 , but their interatomicseparation stays well above the equilibrium GeC bondlength (according to MM2 parameter set)27 of r 0 = 1 95 until H 61 when the separation falls to2.32 ,the approximate maximum length of a stretched GeCbondand below which distance a stable Ge65C11 bondcan be said to have formed via an apparently barrier-less radicalradical coupling misreaction. (Calculating theMorse parameter = ks/ 2De 1/ 2 = 1 86 1010 m 1 bytaking GeC bond stiffness27 ks = 270 N/m and GeCpotential well depth28 De = 2 44 eV, a GeC bond becomes

Singlet Triplet

Fig. 22. Possible Ge65Ge35 bonding pathology observed for trajectory (H, H, H = + 90 , 90 , 0 at RH = 5 80 for Reaction II. (C= yellow,H = blue, Ge= white).

mechanically unstable5 when stretched past the inectionpoint atr = r 0 + ln 2 / = 2 32 .) However, the sys-tem must lie in the near-equiergic triplet state between

H = 46 to 53 because the singlet energy is muchhigher, hence there is apparently no signicant energy bar-rier blocking entrance to the Ge65C11 pathology from

this direction. Rotating the incoming tool toH = 45 doesnot signicantly alter the endoergic reef structure.Minor features evident in the second valley visible in

Figure 18 are mostly attributable to the atomic granular-ity of approaching handle atoms. For example, at (H, H,

H = 80 , 30 , 0 atom C55 on the GeRad2 han-dle is forced to within 2.59 of atom C8 on the HAbsthandle, producing a large endoergicity for the Ge65C11pathology, whereas at (H, H, H = 80 , + 30 , 0handle atoms C55 and C8 remain 3.45 apart, yieldingan exoergic Ge65C11 pathology.

A second possible trajectory pathology of Reaction IIwas predicted for the

H= 90 case in which structure

optimization in the singlet state leads to partial Ge65Ge35 bonding (Fig. 22) with a computed separation dis-tance of 2.77 (vs.3.93 in the triplet state). Thiswould represent a highly stretched but fully formed GeGe bond, since the Morse parameter= ks/ 2De 1/ 2 =1 6 1010 m 1 taking GeGe bond stiffnessks160 N/m(est.)27 and GeC potential well depthDe = 1 95 eV,29 thenr = r 0 + ln 2 / = 2 84 given an equilibrium GeGebond length of r 0 = 2 41 .29 While typical singlet reac-tion energies are about+ 0.3 eV (slightly endoergic) andtriplet energies are lower, the competing desired Ge65H28 abstraction reaction apparently fails to occur in thesecases. These results may be artifactual due to failure of theoptimization algorithm to locate the actual minimum of the



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H28 abstraction reaction so that the optimization falls intoa local minimum. Further details of this possible pathol-ogy have not been investigated in the present work becausethese trajectories lie outside the recommended (H, Hoperating range of 40 , 80 for Reaction II.

A third class of trajectory pathologies involving

unwanted high-energy covalent bond formation betweenHAbstH-GeRad and GeRad2 tool handles during trajecto-ries involving close handlehandle proximity is possiblebut was not investigated in the present work because thesetrajectories also lie well outside the recommended (H, Hoperating range of 40 , 80 for Reaction II.

4.4. PES as a Function of Rotational Angle H

Figure 23 shows the effects of GeRad2 tooltip rotationalangle H on the 1-dimensional PES for Reaction II. At( H, H = 0 , 0 for RH = 5 80 , the PES has negli-gible variation (0.0042 eV) with changingH and there isno apparent effect on this PES if H is increased to 60at H = 0 , + 30 , or 30 . Due to the threefold symmetryof the GeRad2 tooltip, there are three critical values for

H (0 , 120 and 240 that produce minimum net reac-tion energy. These critical values, as well as the shape of the PES, apparently do not vary signicantly withH atworkable values of H. Within the recommended (H, Hrange of 40 , 80 for Reaction II, the variation innew reaction energy as a function of H increases veryslowly with H, reaching only E H = 0 20 eV by (H,

H = + 30 , 0 , 0.005 eV by (H, H = + 30 , + 60 ,and 0.032 eV by (H, H = 30 , 60 near the edges

of the operating envelope. At higherH, energy variationincreases rapidly toE H = 2 30 eV by (H, H = + 40 ,0 at the outermost edge of the recommended range, withthe desired H28 abstraction reaction now endoergic andfailing to occur at most values of H. Besides becom-ing taller, the PES grows more symmetrical and wider atthe higher-energy peaks with the lower-energy valleys get-ting more sharply dened and narrower at largerH. Most

Fig. 23. PES for Reaction II as a function of tooltip rotational angleH, forRH = 5 80 at various combinations of H and H.

importantly, the three critical values forH that produceminimum net reaction energy and ensure desired reactionexoergicity apparently do not vary withH at workablevalues of H. Thus by operating at or near these three crit-ical values (H = 0 , 120, and 240 , the experimentalistcan be assured of optimal reaction exoergicity through-

out the entire recommended (H, H working range of 40 , 80 for Reaction II.

4.5. Lateral Displacement Error Tolerance

To determine the maximum tolerable lateral misplacementerror of GeRad2 that will still result in a successful con-summation of Reaction II, we start by dening theU H-axisas a vector pointing from OH to C59 and theV H-axis (notshown) as a vector originating at OH and perpendicularto U H that points parallel and codirectional with a vectorfrom C66 to C74 (Fig. 24). We can then examine whetherthe Ge65H28 bond still forms when GeRad2 is transla-tionally displaced within theU HV H plane away from itsintended position at any point within a particular approachtrajectory. The present work analyzes a representative reac-tion point (RH = 5 80 ) along a single exemplar approachtrajectory: (H, H, H = 0 , 0 , 0 . The lateral displace-ment of GeRad2 from its intended trajectory is reportedas a displacement angleH measured fromU H and a dis-placement radial distancer H measured from OH.

Simulations began by examining smallr H, moving tobigger r H, simulating progressively larger displacementerror circles. At somer H we would expect to nd thatGe65 and H28 are too far apart to form the desired bond

to consummate the abstraction. With GeRad2 positionedat RH = 5 80 , where tooltip and workpiece are just nearenough for the H28 abstraction to begin to occur, reac-tion failure rst occurs atr H = 0 5 but only forH =150 210. At r H = 0 6 the failure occurs at a greaterrange of H; by r H 0 7 , failure occurs at all angles

H = 0 360 tested full circle at 30increments. Thiserror tolerance is slightly less generous than the 0.5



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Fig. 24. Geometry used for displacement error tolerance with HAbstH-GeRad complex shown at left and GeRad2 shown at right, in positionto consummate Reaction II;r H and H are dened as the radial and

angular displacement of GeRad2 relative to its ideal entry position atH = 0 , H = 0 . (C= yellow, H= blue, Ge= white).

tolerance found in Reaction I (Section 3.5), in part becausethe GeH bondlength (1.53 ) is shorter than the GeCbondlength (1.95 ) but mainly because the error tol-erance is sensitively a function of RH. Moving GeRad2slightly closer to H28 dramatically increases the lateraldisplacement error tolerance, producing a much largerr cutoff value for achieving a successful H28 abstraction.For example, the maximum tolerable displacement errorincreases fromr H = 0 4 atRH = 5 80 to r H = 1 4

at RH = 5 60 for all anglesH = 0 360 tested fullcircle at 30increments. At higher temperatures the prac-tical lateral displacement error tolerance might be some-what higher than predicted by 0 K simulations because thehydrogen transfer barrier can be more easily surmountedby thermal uctuations. Note that the G65C11 cross-bonding pathology (Section 4.3) is not observed becauseon a ( H, H, H = 0 , 0 , 0 trajectory with GeRad2positioned atRH = 5 80 and an error displacement of

r H = 0 5 with H = 0 (angle of closest approach) theGe65C11 distance is 4.36 , far beyond both the GeCequilibrium bondlength of 1.95 and the GeC Morsebond instability distance of 2.32 (Section 4.3).

4.6. Optimal Tooltip Trajectories

The optimal GeRad2 tooltip trajectory for Reaction IIappears to be (H, H, H = 0 , 0 , 0 , although anyapproach trajectory within the recommended (H, Hoperating range of 40 , 80 should sufce almostequally well if H = 0 . Reaction II is likely to be bar-rierless within the recommended operating range. Afterthe abstraction of H28 is complete, the nonreactive hydro-genated GeRad2-H tool may then be withdrawn from the

system along any noncollisional trajectory that is conve-nient, preferably (H, H, H = 0 , 0 , 0 .

A GeRad2 tool started at (H, H, H = 0 , 0 , 0and rotated in H will encounter a nonreactive increasinglyrepulsive barrier (rising to> 3 eV) blocking further travel at

H > + 40 nearRH = 5 80 , whereas atH < 40 near

RH = 5 80 the tool will encounter no signicant repul-sive barrier and will proceed to pathological Ge65C11bonding, immobilizing both tool and workpiece withoutabstracting H28. Because the GeC bond is the weakest inthe system, tool and workpiece could then be pulled apart,mechanically reversing the Ge65C11 pathology withoutpermanent damage to either structure, but the exit trajec-tory would have to be carefully controlled to avoid unpre-dictably altering the state of H28 bonding during the exit.Thus the operation of GeRad2 nearRH = 5 80 may besaid to be repulsive forH > + 40 (toward GeRad) butsticky for H < 60 (toward HAbstH).

5. REACTION III: DETACH GeRad TOOLFROM RECHARGED HAbst TOOL

In Reaction III of the HAbst recharge reaction sequenceRS5 (Fig. 1), mechanical energy is applied to the rstGeRad tool to pull it away from the tip-dehydrogenatedHAbst tool, breaking the Ge35C12 bond (required ten-sile force3.64 nN)7 to recover the original GeRad toolwith a radical site on atom Ge35 while leaving behind arecharged active HAbst tool with a radical site on the distalethynyl carbon atom C12. The carboncarbon bond orderin the C2 ethynyl group remains at 3.

After dening the tooltip geometry and coordinatesystem for Reaction III (Section 5.1) we calculate thereaction PES (potential energy surface) as a functionof rotational angle D of the departing GeRad tool(Section 5.2) and as a function of departure positionalangles D and D (Section 5.3), describe trajectory-relatedpathologies (Section 5.4), then recommend some opti-mal GeRad tooltip departure trajectories for this reaction(Section 5.5).


The tooltip geometry and coordinate system for Reac-tion III is shown in Figure 25. This coordinate system isdened by xing positions of atoms that are most closelyunder the control of the experimentalist. The leftmostimage in Figure 25 denes the HAbst-GeRad tooltip ori-entation angle beta (, which is the angular displacementof the O-normal vector (pointing from Oto O ) fromthe X-axisor more specically, the angle between theO-normal vector (aka.X-axis) and the O-normal vector.Illustrated here is the= 79 04 case which obtains at thecompletion of Reaction II. The same geometry is foundwhen H28 is deleted from the HAbstH-GeRad complex at



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Fig. 25. Coordinate system for Reaction III: The image at left denes the HAbst-GeRad tooltip orientation angle beta (, which is the angulardisplacement of the O-normal vector (pointing from Oto O ) from theX-axis ( = 0 upon alignment withX-axis, = 7904 in image at left).In the image at right, the (X, Y , Z) coordinate system from Reaction I is translated to O, dening the congruent (XD, Y D, ZD coordinate system.Displacement of the positionally controlled GeRad tooltip from Oto OD by a distanceRD during Reaction III is described by the similarly-dened

D, D, and D displacement angles. (C= yellow, H= blue, Ge= white).

the end of Reaction I (= 50 , = 70 , = 20 , R =4 85 ) and the resulting structure is optimized with the3 control atoms on each handle xed. Changingaltersthe energy of the HAbst-GeRad complex via some com-bination of bond-bending strain and handlehandle steric

interaction. Figure 26 shows that system energy rises withand that the collinear (GeRad aligned withX-axis)conguration at = 0 has the lowest energy (taken asEmin = 0 eV). The minimum system energyEmin at eachbend angle occurs at a different OO separation (Rmin

Fig. 26. Minimum system energyEmin at each HAbst-GeRad bendangle occurs at a particular OO separation (= Rmin which also pro-gressively rises with in Reaction III.Emin and Rmin for each wereidentied using structure optimizations at a series of separations atD =20 . Inset chart shows energy versus tool displacement distanceRD forthe collinear = 0 trajectory.

which also progressively rises with. Most values of >90 are excluded due to tool handle proximity.

In the rightmost image of Figure 25, the (X, Y , Z) coor-dinate system from Reaction I is translated to O, deningthe congruent (XD, Y D, ZD coordinate system for GeRad

departure. Movement of the positionally controlled GeRadtooltip from Oto the displaced origin OD by a distanceRD during Reaction III is described by theD, D, and Ddisplacement angles that are dened similarly as for, ,and in Reaction I (Fig. 2, Section 3.1). Note that at thestart of Reaction III, C1 no longer lies exactly at Oonthe X-axis line but has moved a distance 0.0205 fromthe X-axis line. Similarly, Ge35 has moved 0.067 awayfrom the O-normal vectorthe vector that passes throughO and lies perpendicular to the C30/C32/C34 plane. AfterReaction III is completed, C12 is displaced 0.71 fromorigin Oin the nal H28-removed conguration.

Only four of the six possible degrees of freedom inGeRad departure trajectories are considered here. Pitchand yaw rotations during GeRad retraction are beyond thescope of this present work because they would greatlyincrease the complexity and computational cost of theanalysis but are of uncertain utility because signicantmotion in these dimensions is less likely to be readilyaccessible experimentally for Reaction III.

5.2. PES as a Function of Rotational Angle D

The PES as a function of rotational angleD (i.e., twirlingGeRad around its central axis, the O-normal vector) is



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Fig. 27. Energy as a function of D at R = 4 85 , for = 0 using( , = 0 , 0 and for = 79 04 using ( , = + 50 , + 70 inReaction III.

shown in Figure 27. In the collinear conguration of HAbst-GeRad at = 0 , the energy variation for an entirerepeating cycle of rotation of the threefold-symmetricGeRad tooltip fromD = 0 120 is less than 0.01 eV,rising at = 79 04 to only 0.06 eV (similar to resultsin Fig. 7), both below the 0.14 eV limit of computationalaccuracy (Section 2) for this method. The PES minimumfor the = 79 04 conguration occurs nearD = 20 , thesetting mostly used in the following analyses, but choiceof D has only limited impact on reaction energetics.

Note that in Figure 27 the= 0 PES curve lies about0.4 eV higher in energy than the= 79 04 curveeven though the = 0 PES curve is0.6 eV lower inFigure 26the = 0 curve in Figure 27 is taken at sub-optimalR = 4 85 whose energy lies1 eV higher thanthe 4.15 optimalR for = 0 .

5.3. PES as a Function of Positional Angles D and D

The peak PES as a function of positional anglesD andD for = 79 04 is shown in Figure 28. (For conve-

nience, D = 180 points are replotted at+ 180 .) Toobtain each point, representing a unique departure trajectory, the tools were locked at the indicated (D, Drotational position withD = 20 , thenRD was increasedstepwise from 0 to as much as 7.1 using variable step

increments of between 0.30.5 until either breakage of the Ge35C12 bond (blue squares) or a reaction pathology(yellow squares; Section 5.4) occurred during the restrictedgeometry optimization at each step. Both singlet and tripletPES were calculated, but singlettriplet transition proba-bilities at singlettriplet PES approach points are difcultto estimate, so for convenience the endpoint of Reaction IIIis presumed to occur when the computed spin density of Ge35 reaches a signicant (> 0.5) level. Figure 28 showsthe maximum energyEmax of the system at the givenDand D (likely occurring near the last step yielding Ge35C12 bond scission), along with the separation distance

Rmax at whichEmax occurs.Emax varies from 4.89.7 eV,indicating signicant differences among useful trajectoriesthat well exceed the expected DFT computational error.Rmax spans a relatively large range in part because of the mobility and extensibility of the Ge35C12C11C1chain. Figure 29 shows a representative trajectory at (D,

D, D = 0 , 0 , 20 and = 79 04 for Reaction III, atrajectory having the lowestEmax in Figure 28.The relatively high energies reported near the borderline

of pathological trajectories result in part from the forc-ing of large ethynyl deection angles prior to Ge35C12bond scission but may also include an artifactual compo-nent due to computational limitations, as follows. Reac-tion III imposes rotational forces on the HAbst tooltip,but full rotation is prevented by the xed atoms in thebase, resulting in overall handle deformation and signif-icant bond stretching near the xed atoms. For example,at ( D, D = + 150 , + 30 for RD = 6 6 just priorto bond scission the C6C7 bond (1.83 ,+ 19% strain)and the C6C5 bond (1.80 ,+ 17% strain) in the HAbsthandle are highly stretched, just below the Morse potentialinection point at 1.87 (+ 21% strain) where the CCbond becomes mechanically unstable, driving up apparenttooltip system energy. However, in an actual DMS tool thetooltip base atoms are not xed but are free to move andthus can evenly redistribute the bond strain from the front-line cage back into deeper cages, reducing both tooltipenergy and the risk of handle bond breakage in an actualtool during normal use.

The probability of reaction pathology is minimizedwhen the mechanical dissociation of the Ge35C12 bond

compels the remaining bonds to dissipate the least possi-ble energy. The lowestEmax andRmax are found at (D, D,D = 0 , 0 , 20 , with Ge35C12 bondbreaking occur-

ring at a separation distance of RD 3 1 with a bondscission energy of 4.8 eV. This trajectory runs parallel tothe O-normal vector (i.e., to both+ X and+ XD coordinateaxes).

The peak PES as a function of positional anglesD andD for = 0 is shown in Figure 30, obtained and prepared

similarly to Figure 28.Emax varies only moderately from4.96.2 eV across a broad range of useful trajectories (D,

D, D = 90 , 60 , 20 . These energies are generallyhigher than the 4.8 eV values for the best trajectories at

= 79 04 because at = 0 the system energy is reducedby0.6 eV (much of it removed from the now-unstrainedGe35C12 bond; Fig. 26), hence extra energy must bemechanically applied to the collinear Ge35C12 bond toinduce this bond to break as compared to the= 7904case.

5.4. Trajectory Pathologies of Reaction III

At = 79 04 (Fig. 28) there are 14 unusable trajec-tories (Table III) among the 60 unique trajectories at



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Separation at maximum energy, Rmax()

Maximum system energy, Emax(eV)

Fig. 28. Peak system energy prior to bondbreakingEmax (top) and the separation distance at whichEmax occursRmax (bottom) as a function of (D, Dfor = 79 04 and D = 20 in Reaction III. Blue and green values keyed to scale bar below each chart; yellow areas indicate pathological outcogray areas indicate geometrically unfavorable congurations that were excluded.

Fig. 29. Energy versus separation distanceRD for singlet energy of geometry optimized singlet (Singlet) and for triplet energy of geom-etry optimized singlet (Triplet), for (D, D, D = 0 , 0 , 20 at

= 79 04 in Reaction III.

30 angular resolution for the range (D, D, D = 180 , 60 , 20 . In this coordinate system there is

no reason to expect pathologies to be symmetrical aroundD = 0 ; the relative position of the tools makes HAbst

more likely to interfere with GeRad motion atD < 0 .In 3 instances (gray squares, Fig. 28), the trajectory isexcluded because the motion of Ge35 carries it towardC12 (such that C12 or othe

denis tarasov et al- optimal tooltip trajectories in a hydrogen abstraction tool recharge reaction...

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