A stress tensor eigenvector projection space for the (H2O)5 potential energy surface

A stress tensor eigenvector projection space for the (H2O)5 potential energy surface

Accepted Manuscript Research paper A Stress Tensor Eigenvector Projection Space for the (H2O)5 Potential Energy Surface Tianlv Xu, James Farrell, Roya...

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Accepted Manuscript Research paper A Stress Tensor Eigenvector Projection Space for the (H2O)5 Potential Energy Surface Tianlv Xu, James Farrell, Roya Momen, Alireza Azizi, Steven R. Kirk, Samantha Jenkins, David J. Wales PII: DOI: Reference:

S0009-2614(16)30917-4 http://dx.doi.org/10.1016/j.cplett.2016.11.028 CPLETT 34335

To appear in:

Chemical Physics Letters

Received Date: Revised Date: Accepted Date:

7 October 2016 14 November 2016 15 November 2016

Please cite this article as: T. Xu, J. Farrell, R. Momen, A. Azizi, S.R. Kirk, S. Jenkins, D.J. Wales, A Stress Tensor Eigenvector Projection Space for the (H2O)5 Potential Energy Surface, Chemical Physics Letters (2016), doi: http:// dx.doi.org/10.1016/j.cplett.2016.11.028

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A Stress Tensor Eigenvector Projection Space for the (H2O)5 Potential Energy Surface Tianlv Xu1, James Farrell2, Roya Momen1, Alireza Azizi1, Steven R. Kirk*1, Samantha Jenkins**1 and David J. Wales2 1

Key Laboratory of Chemical Biology and Traditional Chinese Medicine Research and Key Laboratory of

Resource Fine-Processing and Advanced Materials of Hunan Province of MOE, College of Chemistry and Chemical Engineering, Hunan Normal University, Changsha, Hunan 410081, China. 2

Department of Chemistry, Lensfield Road, Cambridge University, UK.

**

email: [email protected] email: [email protected]

*

A Stress tensor eigenvector projection space is created to describe reaction pathways on the (H2O)5 MP2 potential energy surface. Evidence for the stabilizing role of the O---O bonding interactions is found from the length of the recently introduced stress tensor trajectory in the stress tensor eigenvector projection space. The stress tensor trajectories demonstrate coupling behavior of the adjoining covalent (σ) O-H and hydrogen bonds due to sharing of covalent character. Additionally, the stress tensor trajectories can show dynamic coupling effects of pairs of σbonds and of pairs of hydrogen bonds.

The mixed bonding environment in water is responsible for the dynamics of the hydrogen-bonding network rearrangements in water complicate the characterization of these conformational spaces1–3. The effects of the bonding environment include cooperative polarization that occurs between the (σ) O-H and hydrogen-bonding networks, as well as the dipole–dipole interactions within QTAIM, quantified by the O---O interactions, which may stabilize small hydrogen–bonded clusters1–6. In other words, the wide range of stabilization energies found for all types of hydrogen-bonds suggests contributions from a variety of sources, including, among others, electrostatic, induction, electron delocalisation, exchange repulsion, and dispersion. Further effects of the mixed bonding environment include dynamic coupling effects between hydrogen bonds due to the stretching modes of the water molecule, and chemical coupling effects between σ and hydrogen bonding, which accounts for the unusual strength of ice Ih

7,8

. Direct X-ray measurements of Isaacs9 on ice Ih confirmed Pauling’s prediction

that the weak hydrogen-bonds in water obtain part of their bonding character from nearby σ-bonds and therefore are not purely electrostatic in nature. Recently, an attempt to comprehensively map out the (H2O)5 MP2 potential energy surface (PES) resulted in nine transition states and corresponding pairs of energy minima in the forward(f) and reverse(r) directions6. The existence of the O---O bonding interactions was unexpectedly found to be very prevalent, in fact all possible QTAIM topologies were found to have associated O---O bonding interactions. The goal of the present work is to create a methodology to track the chemical and dynamical coupling character of the different types of bonding interaction present in these (H2O)5 clusters and to explain the importance of the O---O bonding interactions evident from the recent (H2O)5 MP2 PES analysis6. This correlation will be achieved by the creation of so-called trajectories that relate the change in position of each of the bond critical points (BCPs) and frame of reference, which is described in terms of the eigenvectors of the Hessian of the total charge density distribution ρ(r), and also the stress tensor within the QTAIM portioning scheme. In this investigation we will use Bader’s definition of the stress tensor 10,11. Using QTAIM we are concerned with the chemical properties obtained from analyzing the properties of the Hessian matrix of the charge density ρ(r) evaluated at each critical point. A diagonalization of the Hessian matrix of ρ(r) gives the set of ordered eigenvalues λ1 < λ2 < λ3, with the Laplacian ∇2ρ(r) of the electron density ρ(r) being the algebraic sum of these eigenvalues evaluated at one of the four types of critical point which are connected by the Poincaré–Hopf relation: n-b+r–c=1.

(1)

These four types of critical point, n, b, r, and c, are given by the numbers of nuclear critical point (NCPs), bond critical points (BCPs), ring critical points (RCPs), and cage critical points (CCPs), respectively. The description

in real space of the nature of the topology of the total electronic charge density distribution ρ(r) is condensed into the molecular graph 12–14, which is defined as the set of critical points and associated bond-paths. The solution sets of equation (1) were mapped out to form a quantum topology phase diagram (QTPD)15– 18

where it was previously noticed that all the solutions Poincaré–Hopf relation (1) corresponded to points on the

PES associated with the existence of O---O interactions6. If the bonding was only examined in structures corresponding to the energy minima on the (H2O)5 PES then the underlying importance of the O---O BCPs would have been overlooked. None of the molecular graphs corresponding to energy minima contain O---O BCPs; however, away from the energy minima all possible QTAIM topologies on the QTPD correspond to molecular graphs containing O---O BCPs. A diagonalization of the Hessian matrix of ρ(r) also gives the set of ordered eigenvectors e1, e2, e3, where the eigenvector e3 indicates the direction of the bond-path at the BCP. The most and least preferred directions of electron accumulation are e2 and e1, respectively,19–21 where the e2 and e1 eigenvectors also define the most and least facile directions of the electronic charge density accumulation. Water consists of shared-shell, O-H (σ) covalent BCPs and closed-shell BCPs; the hydrogen-bond H--O BCPs and O---O BCPs. The closed-shell interactions are characterized by positive values of the Laplacian ∇2ρ(rb) and low ρ(rb) values (< 0.1 atomic units); these types of interaction are dominated by the contraction of charge away from the BCP toward each of the nuclei. Conversely, shared-shell interactions, e.g., O-H BCPs, have both negative ∇2ρ(rb) values and high values of ρ(rb). QTAIM, or the stress tensor eigenvector based descriptors, can be used to track the changing orientation of the (e1, e2, e3) eigenvectors of the (H2O)5 BCPs with respect to either the transition state (‘downhill’) or associated pair of minima (‘uphill’) for the entire reaction-pathway. The eigenvector projection can be performed for each of the QTAIM or stress tensor eigenvectors in two directions; downhill and uphill from the transition state to each minimum in the ‘forward’ and ‘reverse’ directions. The downhill reaction-pathway eigenvector projection for each of the e1, e2 or e3 corresponding to the transition state is formed with each of all the corresponding eigenvectors in sequence, starting from the transition state, along the reaction-pathway. We will focus on the e2 eigenvectors because of the association with the most facile direction of electronic charge density accumulation 22

. We will therefore construct stress tensor eigenvector trajectories Tσ(s) as opposed to QTAIM eigenvector

based trajectories on the basis of the superior performance of the e1σ stress tensor eigenvector compared with the e2 QTAIM eigenvector. The stress tensor eigenvector trajectories Tσ(s) are constructed from the set of shifts dr(s), associated with steps s, of a given BCP in 3-D Cartesian space as an ordered set of vectors dr (s) in the stress tensor eigenvector projection space Uσ, where dr (s) = {dr(s).e1σ, dr(s).e2σ, dr(s).e3σ}. In this investigation, the order of shifts dr(s) used to generate the trajectory Tσ(s) is associated with following the intrinsic reaction coordinate (IRC) from the transition state down to the appropriate minimum. Additionally, for

a given BCP, the stress tensor eigenvectors,{e1σTS, e2σTS, e3σTS} for that BCP at the transition state are used as the projection set for the entire trajectory Tσ(s). The corresponding trajectory length Lσ in the stress tensor eigenvector projection space Uσ, is calculated as the sum: Lσ= .

(2)

The trajectories, Tσ(s) and the associated trajectory length Lσ can then be applied to several of the reactionpathways to highlight different behaviors. Such behaviors include comparing the Tσ(s) and the associated trajectory length Lσ for pairs of coupling and non-coupling hydrogen-bond BCPs and shared-shell BCPs, i.e. H(rb) < 0 or H(rb) > 0, respectively, for the same reaction-pathway. We can then also investigate the Tσ(s) and the associated trajectory length Lσ to understand the role of O---O BCPs existing either at the ends of partial reaction-pathways or in the middle of complete reaction-pathways. Note that when we refer to partial reactionpathways we do so because some of the closed-shell H--O BCPs and O---O BCPs only exist for a portion of the complete reaction-pathway. Referring to a partial reaction-pathway is a shortened notation for ‘the partial reaction-pathway corresponding to the closed-shell BCP or BCPs, which exists for only a portion of the complete reaction-pathway’. In this work we examine five of the nine initial 651 TTM3-F transition state configurations geometry optimized to transition states on the MP2 PES found in a recent investigation6. The hyphenated numbers (e.g. 04_0028) used to label the (H2O)5 clusters have no intrinsic meaning themselves and are simply an artifact of the code that was used to generate a given transition state and the associated reverse (r) and forward (f) minima using the analytic potentials. Single-point wavefunctions, also at the MP2/63-11++G** level of theory, were then calculated for every generated step along all of the full IRC reaction-pathways. All IRCs associated with highly symmetric transition states were screened to detect the failure of the reaction path following algorithm associated with very flat potential energy surfaces. The remaining details of the computational procedure used were previous published6 but are also provided in Supplementary Materials S1. The QTAIM and stress analysis was performed with the AIMAll23 suite on each wave function obtained in the previous step. A sub-set of the five of the molecular graphs corresponding to the nine (H2O)5 transition states and the corresponding minima in the forward (f) and reverse (r) directions along the associated intrinsic reaction coordinates are provided in Figure 1. A detailed specification of the computational details required to generate all nine (H2O)5 transition states for this work is provided in Supplementary Materials S1. A selection of five of the nine plots of the variation of the relative energies ΔE along the (H2O)5 reaction-pathways are provided in Supplementary Materials S2. The stress tensor trajectory Tσ(s) and the length Lσ for selection of the reaction-

paathw wayys aare preesenntedd inn Fiigu ure 2 aand Taablee 1 rresppecctiveely. A sellecttionn off thee pllotss off thee vaariaatioon oof thhe ttotaal loocall ennergy H((rb) allongg thhe reaaction--patthw way foor thhe cloosedd-shhelll annd shaaredd-shhelll B BCP Ps aare prroviidedd inn Supp plemeentaaryy Mat M terials S33. Thhe pprocedduree too ggenneraate thee trrajeectoory leengtth Lσ in thhe sstreess tennsoor uppllem men ntarry M Maaterrialss S44. eiigennveectoor pprojectiionn spacee Uσ is givvenn in Su W ddeteerm We mineed tthe strresss tennsoor trrajeectooriees Tσ(ss) thhat aree caalcuulatted in thee strresss teensoor eeigeenvector pproj ojecctionn sppace Uσ (secctioon 22.2), in thhree ddiffeerennt ssetss off boonddingg ennvirronm mennts. We W w willl deenoote hyddroggenn boondd BC CP Ps w h annd w with withhoutt a deggreee off coovallentt chharaacteer bby H Hx---O Oy aand Hxx--O Oy, resspecctivvelyy.

(aa)

b) (b

(cc)

(d d)

(ee)

ure 11. T Thiss figguree coorreespoondss to a rreceent ppublicaationn6 aandd prooviddes thee atoom num mbeeringg scchem me to aaccoomppany T Table Figu mollecuular graaphss off the m miniima andd trranssitioon sstatee foor thhe 111_00001, 04__00228, 09__00007, 11_00002 and 16_000088 1.. Thhe m (H H2O O)5 cclussterss arre shhow wn in suub-ffiguuress (a--e), resspecctiveely.. Thhe trranssitioon sstatee (T TS) andd thhe reeverrse andd forrwaard m minnima arre inndiccateed bby M M(rr) aand M(f (f) rresppecttively. Thee unndeecorratedd ggreenn, rred, annd bbluee sppherres rreprreseent thee boond criticaal poointts (B BCP Ps),, rinng ccriticcal points (RC CPss), aand cagge crriticcal ppoinnts (CC CPss), rresppectiivelly.

T firsst bbondding enviironnmeent is tthe com The mpletee reeacttionn-paathw wayys oof a hyydroogeen-bbonnd B BCP P annd thee shhareed-sshelll O BC O-H CP thaat sshaare an H NC CP on thee assoociaatedd boondd-paathss. T Therrefoore, w we w willl exxam mine thhe streess tennsoor trrajectooriess Tσ(s)) foor thhe ppressennce or aabssencce oof ccoupplinng, on thee baasiss off H((rb) < 0 or H H(rb) > 0 rresppectiveely,, foor thhe hhyddroggenn-boond BC CP.. Foor 111__00001; wee chhooose thee H H11--O O1, O110-H H111 pair andd thhe H5---O O7,, O4-H H5 paiir annd for 044_000288 we hhavee thhe H H8----O O1, O77-H H8 ppairr. Evviddencce oof ccouplinng of tthe strresss tennsoor trrajeectooriees Tσ(ss) off thhe 11__00001 reaactiion-paathw wayy thhe hhyddroggenn-boondd H11---O1 B BCP P hhas H((rb) < 0, theerefforee w we maay sseekk evvideencce oof ccoupplinng bbehhaviior to com mpare tw wo ppairrs oof B BCP Ps, whheree eaach paair ssharres an H NC CP. A deggree oof cooupplinng oof tthe H111---O11 B BCP P with w thee shharred--sheell O10-H H111 BCP P occcurrs, sinnce H((rb) < 0 aalonng thee enntiree reeacttionn-paathw wayy. T Thiis ccoupplinng is aapppareent froom thee siimilariity of thee paths trraceed outt byy thhe streess tennsoor trrajectooriess Tσ(s)) of thhe 111__00001 reaactioon-patthw wayss asssocciatted witth tthe H111---O11 BCP P annd O O100-H H11 BC CP, seee F ure 2(aa). Thiis ssimiilarrity cann bee coonttrastedd wiith tthe lacck oof ccouplinng eevidennt inn thhe ddisssimilarr Tσ(s)) paathss foor Figu

the H5---O7, O4-H5 pair, where H(rb) > 0 for the H5---O7 BCP along the entire pathway, see Figure 2(b). It can be seen that the values of the stress tensor trajectory length Lσ are more similar for the H11--O1 BCP, O10H11 BCP pair that possess a degree of covalent coupling (H(rb) < 0 for the H11--O1 BCP ) than between the H5---O7 BCP, O4-H5 BCP pair where there is no coupling present (H(rb) > 0 for the H5---O7 BCP), see Table 1 section (I). Table 1. The stress tensor trajectory Tσ(s) length Lσ for the reverse and forward directions, (reverse, forward) in the stress tensor eigenvector projection space Uσ. We list the results of Lσ for the reaction-pathways the three bonding environments; the first bonding environment (I) where σO-H BCP and the H--O BCPs share an H NCP on the associated bond-paths, refer to Figure 1 for the atom numbering scheme. The second bonding environment (II) corresponds to the complete reaction-pathways; H--O→O---O→H--O. The third bonding environment (III) is the partial reaction-pathways H--O→ O--O and O---O→H--O→O---O. See the main text for further details and Figure 2-4 respectively. See the figure caption of Figure 2 for further details. Reaction-pathway (I )

O-H (reverse, forward)

H--O/H---O (reverse, forward)

11_0001

O10-H11 (0.0317, 0.0360)

H11--O1 (0.0270, 0.0387 )

O4-H5 (0.0956, 0.1496)

H5---O7 (0.0511, --- )

O10-H11 (0.0548, 0.0610)

H11---O1 (0.0450, 0.0456 )

O7-H8 (0.0634, 0.0528)

H8---O1 (0.0475, 0.0543)

(II)

H--O

O---O

09_0007

H5--O1 (0.0720, --- )

O1---O4 (

--- ,0.5678)

H6---O1 ( --- ,0.1696)

11_0002

H9--O1 (0.0965, --- )

O1---O7 (

--- ,0.5285)

H8---O1 ( --- ,0.0809)

16_0018

H9--O10 (0.1929,

O10---O7 (0.0521, 0.0913)

H8--O10 ( --- ,0.2872)

(III)

O---O

H---O

O---O

H5---O7 (0.0511, --- )

O4---O7 ( --- ,0.1249)

H5---O1 (0.0217,0.0210)

O1---O4 ( --- ,0.0682)

04_0028

--- )

11_0001 16_0018

O1---O4 (0.0628, --- )

H--O/H---O

None of the hydrogen-bond BCPs for the 04_0028 pathway possess any degree of covalent character. This result is consistent with the findings from inspection of the paths of the stress tensor trajectories Tσ(s), which show that there is no coupling between H8---O1 or H11--O1 hydrogen-bond BCPs and the O7-H8 or O10-H11 shared-shell BCPs that share an H NCP on the associated bond-paths, see Figure 2(c). Consistent behavior is seen in the paths of the stress tensor trajectories Tσ(s) of the reaction-pathways of the two hydrogen-bond BCPs; H8---O1 BCP and H11--O1 BCP and separately between the reaction-pathways of the two shared-shell BCPs; O7-H8 BCP and the O10-H11 BCP. This ‘coherence’ appears to be as a result of a dynamic coupling due a stretching motion of the H3-O1-H2 molecule that is suggested by the form of the molecular graph of the 04_0028 transition state and the two associated minima, see Figure 1(b). The motion seems more likely to originate from a stretching motion of the H3-O1-H2 molecule because the Tσ(s) has a large extent along the e3 direction, i.e. parallel to a bond-path stretching direction of any of the Tσ(s) investigated. The effect of coupling,

either due to hydrogen-bond BCPs possessing H(rb) < 0 or dynamic coupling, that seems to occur for the 04_0028 reaction-pathway, result in more similar values of Lσ for the closed-shell BCPs and shared-shell BCPs, see the first section of Table 1. Figure 2. The x-, y- and z- axes comprise the projections of the real space shifts dr of a given BCP onto the e1σ, e2σ and e3σ unit stress tensor eigenvectors, respectively. The first bonding environment is presented, where the stress tensor Uσ space length Lσ is determined from equation (2), also see (I) in Table 1. The reverse and forward reaction paths are indicated by ‘r’ and ‘f’ respectively. All of the space tensor trajectory paths begin and terminate with a square and cross symbols respectively. Values of the total local energy density H(rb) < 0 and H(rb) > 0 for a given BCP are indicated by “-“ and “---“ respectively. The reaction pathway for 11_0001 with the presence and absence of coupling behavior in stress tensor trajectories Tσ(s) of the O-H BCP and H--O BCP/H---O BCP are shown in sub-figures (a) and (b) respectively. The ‘dynamic coupling’ for the Tσ(s) of the σbonds; O7-H8 BCP and O10-H11 BCP is evident as well as for the Tσ(s) of the H8---O1 BCP and the H11---O1 BCP the Tσ(s) of the 04_0028 (H2O)5 cluster in sub-figure (c).

The second bonding environment is the complete reaction-pathway consisting of an O---O BCP and two different hydrogen bond BCPs, where we will examine the bonding transitions with the stress tensor Tσ(s). For the 09_0007 reaction-pathway we have H5--O1→ O1---O4→ H6---O1, for 11_0002 we choose H9--O1→ O1--O7→ H8---O1 and for 16_0018 we have H9--O10→O10---O7→H8--O10, see Figure S8(b-d). Notice, that the number of steps of the complete reaction-pathways consisting of O---O BCPs is only a small fraction of that compared with that of the H--O BCPs, see Figure 2. The values of the stress tensor trajectory length Lσ, however, do not reflect this property and show that the values of Lσ for the O---O portions of reaction-pathways are either longer or a significant fraction of the Lσ values of that of the H--O reaction-pathways despite the much fewer number of steps occupied by the O---O BCPs, see Table 1.

Figure 3. The tracks of the stress tensor trajectories Tσ(s) for the second bonding environment, see section (II) in Table 1; for the reaction pathway of the 09_0007, 11_0002 and the 16_0018 (H2O)5 clusters are shown in sub-figures (a), (b), and (c) respectively. See the figure caption of Figure 2 for further details.

The third bonding environment considered involves the partial reaction-pathways for a hydrogen-bond BCP terminated at one or both ends by an O---O BCP, where we will again examine the bonding transitions using Tσ(s). For a partial reaction-pathway that terminates at one end with an O---O BCP for the 11_0001 reactionpathway; the H5---O7→O4---O7, O4-H5 pair, for comparison, we include the complete reaction-pathway for

the H11--O1, O10-H11 pair, see Figure S8(a). We choose the 16_0018 reaction-pathway because the H5---O1 terminates at both ends with the O1---O4, see Figure S8(e). It can be seen that there is a smooth transition from the O1---O4→ H5---O1→ O1---O4 for the 16_0018 reaction-pathway and the Tσ(s) extends a much longer distance along the e2, the most facile direction, than for the other Tσ(s) investigated. Conversely, the 16_0018 trajectory Tσ(s) extends a much shorter distance along the e1, the least facile direction, than any other of the investigated Tσ(s). In both cases the values of Lσ for the O---O portions of partial reaction-pathways are longer than the Lσ values of the H--O reaction-pathways, see the third section of Table 1; O---O→ H--O→ O---O.

Figure 4. The tracks of the stress tensor trajectories Tσ(s) for the third bonding environment, see section (III) in Table 1; the reaction pathway of the 11_0001 and the 16_0018 (H2O)5 clusters are shown in sub-figures (a) and (b) respectively. See the figure caption of Figure 2 for further details.

The length Lσ of Tσ(s) in the stress tensor eigenvector projection space Uσ was used to demonstrate the effects of chemical coupling, or lack of it, of the shared-shell O-H BCPs with hydrogen bond H--O BCPs that share an H NCP on the associated bond-paths. The stress tensor trajectories Tσ(s) also demonstrated the evidence of dynamic coupling of hydrogen bonds and separately the covalent bonds caused by the stretching motion of the central H3-O1-H2 centrally located water molecule. In addition, the length Lσ of Tσ(s) was used to demonstrate that there is a significant contribution from the O---O BCPs that is not otherwise visible in conventional QTAIM or energetics analysis. This observation may be the reason for the presence of the O---O BCPs for all points on the quantum topology phase diagram (QTPD) recently observed in a parallel examination of the (H2O)5 PES6. We suggest that in future the stress tensor trajectory Tσ(s) may also be used to analyze other more general nonisomeric reaction-pathways, in particular to better understand the role of the longer closed-shell BCPs. Acknowledgements

The One Hundred Talents Foundation of Hunan Province and the aid program for the Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province are gratefully acknowledged for the support of S.J. and S.R.K. The National Natural Science Foundation of China is also acknowledged, project approval number: 21273069. References 1. Heather A. Harker, Mark R. Viant, Frank N. Keutsch, Ernest A. Michael, Ryan P. McLaughlin, and Richard J. Saykally. Water Pentamer:

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Spectroscopy. J. Phys. Chem. A 109, 6483–6497 (2005). 2. Keutsch, F. N. & Saykally, R. J. Water clusters: Untangling the mysteries of the liquid, one molecule at a time. Proc. Natl. Acad. Sci. 98, 10533–10540 (2001). 3. Brown, M. G., Keutsch, F. N. & Saykally, R. J. The bifurcation rearrangement in cyclic water clusters: Breaking and making hydrogen bonds. J. Chem. Phys. 109, 9645 (1998). 4. David, J., Guerra, D. & Restrepo, A. Structural Characterization of the (Methanol)4 Potential Energy Surface. J. Phys. Chem. A 113, 10167–10173 (2009). 5. Murillo, J., David, J. & Restrepo, A. Insights into the structure and stability of the carbonic acid dimer. Phys. Chem. Chem. Phys. 12, 10963–10970 (2010). 6. Tianlv Xu, James Farrell, Yuning Xu, Roya Momen, Steven R. Kirk, Samantha Jenkins, and David J. Wales. QTAIM and Stress Tensor Interpretation of the (H2O)5 Potential Energy Surface. Accepted. J Comp Chem (2016). 7. Jenkins, S. & Morrison, I. The dependence on structure of the projected vibrational density of states of various phases of ice as calculated by ab initio methods. J. Phys. Condens. Matter 13, 9207 (2001). 8. Li, J. Inelastic neutron scattering studies of hydrogen bonding in ices. J. Chem. Phys. 105, 6733–6755 (1996). 9. Isaacs, E. D. et al. Covalency of the Hydrogen Bond in Ice: A Direct X-Ray Measurement. Phys. Rev. Lett. 82, 600–603 (1999).

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phicaal abstractt Grap



A Stress tensor eigenvector projection Uσ space formalism is created for the reaction-pathways of the (H2O)5 MP2 potential energy surface.



Explained the presence of O---O BCPs for all points on the quantum topology phase diagram found in a parallel examination of the (H2O)5 PES.



Length Lσ of the stress tensor Tσ(s) trajectory indicates the significant contribution of O---O BCPs in (H2O)5 reaction pathways.



Stress tensor trajectories Tσ(s) demonstrate covalent coupling behavior of the adjoining covalent (σ) OH and hydrogen bonds.



Stress tensor trajectories Tσ(s) demonstrate dynamic coupling effects of pairs of σbonds and of pairs of hydrogen bonds.