It is now widely acknowledged that the electrostatic contribution to intermolecular forces is of great significance, anmay often be the most important term in determining thgeometries of van der Waals complexes.1 To provide an ac-curate description of the electrostatic energy requires distruted multipole analysis2–4 ~DMA ! which overcomes the con-vergence problems associated with a central multipoexpansion.5 Numerous applications of the method have beemade,6,7 the best known being Buckingham and Fowler’study which showed that the geometries of numerous hydgen bonded complexes could be reproduced remarkably wusing DMA combined with hard sphere potentials. Howevedespite such successes and the incorporation of a routinecalculate distributed multipoles within theCADPAC package,8
distributed multipole analysis has not been very widely useThis is probably due to the complexity of the approachwhich produces some quite forbidding formulas.2 However,we have now developed a computer code which can calclate the analytic first and second derivatives of the zeroorder electrostatic energy for distributed multipoles up to anincluding rank four,9 which should be sufficient for mostpurposes.
In this paper we will show how these results can be pto use in exploring the potential energy surfaces of ten vder Waals complexes that have featured in this journal ovthe last year or so, namely, eight involving wate~propane–H2O, methane–H2O, CO–H2O, H2CO–H2O,HCl–H2O, ethene–H2O, CO2–H2O, methanol–H2O! and thetwo well-studied dimers~HCl!2 and ~HF!2. A preliminaryreport has appeared elsewhere.10 These first results combinedistributed multipole analysis with simple Lennard-Joneatom–atom terms for the dispersion and repulsion energiand so they are not expected to be of high accuracy. Nevtheless, the ability to search a complex potential energy sface, albeit an approximate one, in a tiny fraction of the timtaken by anab initio calculation, should be of some interestThe other element missing from the present work is the ab
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ity to differentiate the iterated induction energy, as distinfrom the zeroth order electrostatic energy. This is a difficproblem in its own right, and hence we will avoid systemsuch as aromatic–rare gas complexes, where the inducenergy is likely to be important because there is no zorder electrostatic term.
The availability of analytic first and second energy drivatives enables us to employ eigenvector-following11 tosearch routinely for transition states, minima, and to callate reaction pathways. The particular implementation of tmethod has been described in large part elsewhere.10 Here,we simply note that knowledge of rearrangement mecnisms and barrier heights enables one to generate the e
TABLE I. Lennard-Jones parameterse/h ands/a0 .
Atom 1 Atom 2 e s
N N 0.000 118 1 6.255
H H 0.000 027 2 5.310
C C 0.000 162 2 6.331
O O 0.000 195 1 5.575
Cl Cl 0.000 549 6 6.331
H Ar 0.000 205 0a 6.070a
Ar N 0.000 250 6a 6.198a
N C 0.000 138 3 6.293
N H 0.000 056 7 5.783
C H 0.000 066 5 5.820
C O 0.000 177 7 5.953
C S 0.000 306 6 6.491
H O 0.000 072 6 5.442
H S 0.000 125 7 5.981
Cl O 0.000 327 5 5.953
Cl H 0.000 122 4 5.820
aThese values are taken from P. Parneix, N. Halberstadt, Ph. Bre´chignac, F.G. Amar, A. Van der Avoird, and J. W. I. Van Gladel, J. Chem. Phys.98,270 ~1993!.
5552 Wales, Popelier, and Stone: PES’s of van der Waals complexes
TABLE II. Propane–water: Energies/cm21, point groups~PG!, nonzero normal mode frequencies/cm21, rota-tional constants/cm21, and components of the dipole moment,m i /Debye, along the inertial axes~in the sameorder as the rotational constants!.
Label Energy PG Frequencies Rotational constants m i
aThese are the values for the ortho state. For the para state experiment gives 0.82, 0.38, and 0.00 in the sameorder.
ysate
hasningdleis
tive molecular symmetry group12 and hence deduce the pat-tern of tunneling splittings.13
Formulas for the energy derivatives have been presentand discussed elsewhere.9,10 Throughout this paper we willemploy a fairly consistent framework, always using distrib-uted multipoles calculated from basis sets of 6–31G** 14 orDZP quality up to rank 4 with one site on every atom. Theeffects of using different combination rules for the LennardJones parameters and of including correlation effects at thsecond order Mo” ller–Plesset level of theory15 in calculatingthe multipoles have been examined in a few cases. Howeveour main objective is to give some idea of how the generiintermolecular potential performs for a diverse range of complexes. Hence, most of this paper consists of results, prceded by an account of certain aspects of the optimizatioprocedure that were not covered in the previoupublication.10 We envisage that future papers will exploreimproved descriptions of dispersion and repulsion, so thathis work provides a reference point against which the bettedescriptions may be judged.
II. MINIMA, TRANSITION STATES, ANDREARRANGEMENTS
All the geometry optimizations and reaction path calculations in this study were performed with the eigenvectorfollowing ~EF! approach, employing analytic first and sec-ond derivatives of the energy at each step. This providesway to find transition states by minimizing the energy for aldegrees of freedom except one, for which it is maximized.11
Here, we take a transition state to be a stationary point wit
TABLE III. Rearrangement pathways of propane–water. Energies arecm21 andS andD ~defined in Sec. IV! are in bohr.a
E1 D1 Ets D2 E2 S D N
2350.18 5.78 2344.40 2.44 2346.84 2.6 1.9 4.5
2350.22 11.56 2338.66 11.63 2350.29 2.3 2.1 7.4
2343.22 21.11 2322.11 10.54 2332.65 3.5 2.3 6.7
aThe energies2350.18,2350.22, and2350.29 correspond to the samestructure; the differences are due to round-off errors in the geometries amultipoles.
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precisely one negative Hessian eigenvalue.16 Minima arefound by minimizing in all directions and reaction pathwaby minimizing the energy after displacing the transition stgeometry in both senses along the transition vector.10 Forclusters bound by empirical potentials, this approachbeen successfully applied to a variety of systems contaiup to hundreds of degrees of freedom to find both sadpoints and minima.17 The form of the steps taken in thstudy has been described before10,18–20 and a fairly self-
FIG. 1. Low energy stationary points for propane–H2O calculated withORIENT3.
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5553Wales, Popelier, and Stone: PES’s of van der Waals complexes
TABLE IV. Methane–water: Energies/cm21, point groups~PG!, nonzero normal mode frequencies/cm21, rota-tional constants/cm21, and components of the dipole moment,m i /Debye, along the inertial axes~in the sameorder as the rotational constants!.
Label Energy PG Frequencies Rotational constants m i
contained account has also been published recently.21 In thissection we will outline the complications which result whedealing with systems that may contain an arbitrary selectiof atoms and both linear and nonlinear rigid molecules. Wemphasize that the theory presented in this section andone following are independent of the particular form adoptefor the intermolecular potential, and hence will be used umodified in future work involving more sophisticateddispersion–repulsion terms.
When expressed in terms of the eigenvectors and eigvalues of the Hessian the eigenvector-following step in thdirection of a given eigenvector employed here21 involvesthe reciprocal of the eigenvalue. At any stationary point foan isolated complex there are six zero Hessian eigenvalcorresponding to overall translations and rotations; one leif the whole system is itself linear. The energy gradient hano component in these directions because they do not affthe Hamiltonian. Hence, if the directions correspondingzero eigenvalues were evident at a general point we cosimply ignore the steps in such directions. However, the rtational modes are coupled to the vibrational normal modby terms which are linear in the gradient.22 Hence, a generalnonstationary point has three zero Hessian eigenvalues c
TABLE V. Rearrangement pathways of methane–water. Energies arecm21 andS andD ~defined in Sec. IV! are in bohr.
E1 D1 Ets D2 E2 S D N
Results for 6–31G** geometries and multipoles
2215.93 8.02 2207.91 8.02 2215.93 2.1 1.9 4.7
2215.93 12.30 2203.63 12.30 2215.93 3.1 2.7 4.2
2215.93 38.03 2177.90 38.03 2215.93 5.3 2.2 2.4
Results for 6–311G** geometries and multipoles
2218.56 0.39 2218.17 0.39 2218.56 1.0 1.0 4.7
2218.56 1.75 2216.81 1.75 2218.56 2.1 1.9 4.0
2218.56 7.23 2211.33 7.23 2218.56 1.9 1.8 5.0
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responding to translations, and a number of small but nonzero eigenvalues which must be removed before we can ueigenvector-following to take steps. One way to do this is temploy internal coordinates. Nguyen and Case,23 Thomasand Emerson,24 and Taylor and Simons25 have all proposedmethods which enable optimisations to be conducted in Catesian coordinates. However, here we follow Baker anHehre26 and employ projection operators, since previouwork has shown that this provides a good solution to thproblem.20 Details are given in the Appendix.
III. NORMAL MODE ANALYSIS
In this section we describe the calculation of normamode frequencies in more detail. Li and Bernstein27 havepreviously provided details of how to proceed using the FGformalism. However, we prefer the conceptually simpler approach of Pohorilleet al.28 in terms of a double transforma-tion. The first transformation diagonalizes the kinetic energmatrix and must also be applied to the Hessian before ititself diagonalized to find the normal modes and their frequencies. In the case of an atomic system the first transfomation is trivial and results in the familiar reciprocal massweighting of the Hessian. For rigid molecules the kineticenergy is obtained asT5xTKx /2, wherex is the vector ofcenter-of-mass and orientational coordinates, superscriptTdenotes the transpose andK is block diagonal with a 636block for each molecule of the form
FIG. 2. Low energy stationary points for methane–H2O calculated withORIENT3.
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5554 Wales, Popelier, and Stone: PES’s of van der Waals complexes
S M 0 0 0 0 0
0 M 0 0 0 0
0 0 M 0 0 0
0 0 0 I xx I xy I xz
0 0 0 I yx I yy I yz
0 0 0 I zx I zy I zz
D ,
with I the inertia tensor andM the molecular mass. For acomplex containing atoms there are 333 blocks for eachatom with diagonal elements corresponding to the atommass. IfU is the orthogonal matrix which diagonalizesK ,i.e.,
T51
2xTKx5
1
2xTUTUKUTUx5
1
2 (i
yi2ki ,
thenK has eigenvalueski , andy5Ux. Of course, sinceK isblock diagonal we need only diagonalize the individuablocks. The harmonic normal mode frequencies are then otained by diagonalizingLUHU TL , whereLi j 5 d i j /Aki .
For systems containing linear molecules this procedumust be modified because if we use 636 blocks in the ki-netic energy matrix for such species they are singular. In th
TABLE VI. Ab initio results for methane–water.
Structure Point group Hessian index DZP Energy 631GE Ene
Transition state Cs 1 2116.255 17 2116.237 99
Minimum A C1 0 2116.255 18 2116.238 09
Minimum B C1 0 2116.255 20 2116.237 98
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FIG. 3. Ab initio stationary points of methane–H2O with water to methaneC distances marked~DZP/631GE!. ~a! Cs transition state,~b! C1 minimumA for which the transition state in part~a! provides a degenerate rearrange-ment, ~c! C1 minimum B found from the highest energy minimum calcu-lated for 6–311G** geometries and multipoles withORIENT3.
rgy
TABLE VII. Water–carbon monoxide: Energies/cm21, point groups ~PG!, nonzero normal modefrequencies/cm21, rotational constants/cm21, and components of the dipole moment,m i /Debye, along the iner-tial axes~in the same order as the rotational constants!.
Label Energy PG Frequencies Rotational constants m i
aThese are the rotational constants and dipole moment components given for symmetric tunneling states; thevalues for antisymmetric states are very similar~Refs. 40 and 41!.
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5555Wales, Popelier, and Stone: PES’s of van der Waals complexes
case one of theki is zero for each linear molecule~corre-sponding to rotation about the molecular axis! and the row ofU corresponding to this eigenvalue is simply deleted. Tprocedure may be contrasted with that described in the Apendix where the projection operator is modified to cowith linear molecules.
IV. RESULTS
We have chosen a new set of examples to complemthose of our preliminary report.10 From the vast array ofpossible systems we decided to concentrate upon compleinvolving water. For comparison we also consider two hdrogen halide dimers for which rearrangement mechanishave been extensively studied before, due to the importaof tunneling splitting in these systems. In each casepresent a reasonably thorough, but by no means exhaussurvey of the potential energy surface, giving the energipoint groups, and intermolecular harmonic frequencies ofminima and transition states located. We also characterizethe rearrangements identified in terms of the stationpoints involved, the barrier heights and three indices whprovide additional information about the path. The firstS5* ds, the integrated arc length in 3N-dimensionalnuclear configuration space, whereN is the number of atoms.S was calculated as a sum over the eigenvector-followsteps
TABLE VIII. Rearrangement pathways of water–carbon monoxide. Engies are in cm21 andS andD ~defined in Sec. IV! are in bohr.
E1 D1 Ets D2 E2 S D N
Results for 6–31G** geometries and multipoles
2496.15 9.03 2487.12 9.03 2496.15 1.1 1.0 1.2
2496.15 48.56 2447.59 48.56 2496.15 2.2 1.8 2.8
2496.15 128.17 2367.97 128.17 2496.15 3.6 2.6 3.3
Results for 6–311G** /MP2 geometries and multipoles
2404.01 4.66 2399.35 86.90 2486.25 3.6 2.3 2.9
2404.01 22.37 2381.64 22.37 2404.01 1.7 1.5 2.9
2404.01 263.15 2140.86 263.15 2404.01 8.9 2.0 2.5
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g
S5 (stepsA(
i
DQi2, ~1!
whereDQi is the step for nuclear Cartesian coordinateQi
and the outer sum is over all the eigenvector-following stepThe moment ratio of displacement,g, is defined as29
g5N( i@Qi~s!2Qi~ t !#
4
~( i@Qi~s!2Qi~ t !#2!2
, ~2!
whereQi(s) is the value of the nuclear Cartesian coordinateQi for minimums, etc. We prefer to use the quantityN5N/g, which is a measure of the number of atoms involved in threarrangement. If a single atom moves thenN51 and therearrangement is localized, while if all atoms move througthe same distance theng51 and the process is entirely co-operative, withN5N. Finally, the distance between minimain nuclear configuration space30 is
D5A( i~Qi~s!2Qi~ t !!2. ~3!
From these definitions it clearly follows thatD<S.Even for the simplest systems it is unlikely that we have
located all the minima and transition states. Transition statewere located for each minimum by following the softestmodes uphill, and the two minima connected by each transtion state were determined by reaction path calculations a
FIG. 4. Low energy stationary points for water–CO calculated withORIENT3. The distances are all in bohr.
r-
TABLE IX. Water–formaldehyde: Energies/cm21, point groups~PG!, nonzero normal mode frequencies/cm21,rotational constants/cm21, and components of the dipole moment,m i /Debye, along the inertial axes~in the sameorder as the rotational constants!.
Label Energy PG Frequencies Rotational constants m i
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5556 Wales, Popelier, and Stone: PES’s of van der Waals complexes
described above. For every new minimum thus locatedsame procedure was followed until no new minima or trasition states were found.
In order to fix one source of variation in the results thDMA calculations have been carried out for 6–31G** basissets, unless stated otherwise. The effects of electron corrtion have been considered in a few test cases using theond order Møller–Plesset correction.15 In each case the rigidmolecule geometry is that obtained by optimization at tsame level of theory. In every case we truncate the distuted multipoles after the hexadecapole (L54). The Lennard-Jones parameters employed are given in Table I; many wobtained from Allen and Tildesley31 with mixed parametersfrom the standard Lorentz–Berthelot combination rules
sab512~sa1sb! and eab5Aeaeb.
It is known that these rules are not very satisfactory, andparticular that the Berthelot rule foreab tends to overestimateit.32 The alternative Slater–Kirkwood combination rule33 hasbeen employed to obtain parameters for nonbonded intetions in polypeptides.34 However, we found that using thisrule made little qualitative difference to the results. In facfor two of the systems we explored the effect of reducingthe heteroatom well depth parameters by a half, and evenhad little qualitative effect. For each system we will ontabulate results for the three lowest minima and transitstates to save space.
V. PROPANE–WATER
Steyert and co-workers have recently published analyof both microwave35 and far-infrared vibration–rotation–
TABLE X. Rearrangement pathways of water–formaldehyde. Energiesin cm21 andS andD ~defined in Sec. IV! are in bohr.
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tunneling spectra36 of propane–H2O which they viewed as aprototypical water–hydrophobe system. The results of thestudies together indicated that all three atoms of the watmolecule lie in the CCC plane of the propane and that thbarrier to internal rotation is less than about 5 cm21. We havelocated ten minima and ten different transition states~TableII !, with the two lowest minima both matching the experimental rotational constants quite well. The large numberminima of relatively low energy is in agreement with theexperimental conclusion that there is unlikely to be a singlow energy structure. The presence of several low enerrearrangements~Table III! is also in agreement with thefloppy nature of this system.
The experimental results are all consistent with a tunneing motion for the water molecule via aC2v transition statewhich links minima where the three water atoms lie in th
FIG. 5. Low energy stationary points for water–H2CO calculated withORI-ENT3. The distances are marked in bohr.
are
TABLE XI. Water–hydrogen chloride: Energies/cm21, point groups ~PG!, nonzero normal modefrequencies/cm21, rotational constants/cm21, and components of the dipole moment,m i /Debye, along the iner-tial axes~in the same order as the rotational constants!.
Label Energy PG Frequencies Rotational constants m i
aValues calculated by Latajka and Scheiner~Ref. 64!.
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5557Wales, Popelier, and Stone: PES’s of van der Waals complexes
CCC plane. No such mechanism is found in the preswork, but there are some similar processes correspondinthe three lowest transition states. In min1 the water oxygatom is furthest from the carbon atom marked and in mione of the water hydrogen atoms is the furthest~Fig. 1!. Thewater molecule rotates through roughly 90° about itsC2 axisin the process mediated by ts1.
The pathway corresponding to the second lowest trantion state is a degenerate rearrangement of min1 wherewater molecule moves from one side of the CCC plane toother via ts2. TheCs transition state is the closest that whave found to theC2v symmetry postulated experimentally.
36
Min3 is similar to min2 but the relative orientation of thwater and propane molecules is slightly different. Min4 hCs symmetry with the water molecule lying across the prpane mirror plane but at some distance above the CCC plThe mirror plane arises when min4 is produced from mivia ts3. Note that the two hydrogens are again closest topropane molecule in min4. Clearly our results indicate ththis surface is likely to be especially complicated, with seeral low energy minima that are able to interconvert amonthemselves. This would certainly be a fruitful system ffurtherab initio and experimental study.
VI. METHANE–WATER
Szczesniak et al.have recently published37 a detailedabinitio study of methane–H2O and compared their results withprevious theoretical and experimental results~for which werefer readers to their paper!. Their results include the effectsof electron correlation up to fourth order Mo” ller–Plessettheory as well as corrections for basis set superposition erTheir global minimum involves a carbon to H–O hydrogebond in which the H–O group points into the middle of aHHH triangular face.37 They also find a higher energy mini
TABLE XII. Rearrangement pathways of water–hydrogen chloride. Engies are in cm21 andS andD ~defined in Sec. IV! are in bohr.
6–311G** /MP2 or DZP/MP2 geometries and multipoles
21122.59 2.65 21119.94 2.65 21122.59 0.7 0.7 2.9
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mum with a C–H to O hydrogen bond which has a longeC–O distance. We found only one minimum for this systemof Cs symmetry in which the two hydrogen atoms are equidistant to the carbon atom and about 0.6 bohr closer than toxygen atom. Details of this minimum and of the four tran-sition states corresponding to degenerate rearrangementsgiven in Table IV, while the rearrangements themselves aoutlined in Table V. The C–O distance is 6.9 bohr in theminimum.
The lowest energy rearrangement involves reorientatioof the water molecule on the same HHH face of the methanmolecule and proceeds via ts1 withCs symmetry in whichthe oxygen atom lies further from the methane molecul~Fig. 2!. The next lowest energy rearrangement may roughlbe described as internal rotation of methane, so that the wamolecule moves between HHH faces but maintains its relative orientation. If both these mechanisms, which have lowbarriers, are feasible then every possible permutation of thsingle minimum that does not involve breaking covalenbonds is accessible. In the higher energy rearrangementsfind that ts3 is similar to ts1 and ts4 again involves reorientation of the water molecule on the same HHH face. Unfortunately, theab initio investigation of Szcze¸sniak et al. didnot consider any of these unsymmetrical configurations because of the computational expense.
We decided to investigate this system in more detail focomparison with the previousab initio studies. The geom-etries were reoptimized and DMA’s repeated for the standaCADPAC8 6–311G** basis14 sets. We then found two lowenergy and one higher energy minima, along with eight transition states; a summary is given in Tables X and XI. Thelowest energy minimum looks like the previous min1 butwith the water molecule rotated so as to break the mirro
FIG. 6. Low energy stationary points for H2O–HCl calculated withORIENT3.The hydrogen bond lengths are indicated in bohr.
r-
TABLE XIII. Water–ethene: Energies/cm21, point groups~PG!, nonzero normal mode frequencies/cm21, rota-tional constants/cm21, and components of the dipole moment,m i /Debye, along the inertial axes~in the sameorder as the rotational constants!.
Label Energy PG Frequencies Rotational constants m i
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5558 Wales, Popelier, and Stone: PES’s of van der Waals complexes
plane. The new ts1 in fact looks like the old min1 and coresponds to a degenerate rearrangement of the new mwith a very low barrier. The new ts2 is analogous to the oone and the new ts3 is also analogous to the old ts3. The nmin2 looks very similar to min1, whereas in the new minthe water molecule is closer to lying over an H–H edge thaan HHH face. We will not illustrate any of these processebut simply note that these data give us some idea of howpresent results can vary with basis set for a system withrather ‘‘flat’’ potential energy surface.
The above minima were then used as starting geometrfor unrestrictedab initio geometry optimizations. The lowsymmetry made it impossible to use the MP2 correlatiocorrection, however. Geometry optimization was achieveusing the eigenvector-following algorithm described in detaelsewhere;21 the largest ‘‘zero’’ frequencies were convergeto less than 3 cm21 in all cases. The first basis set considerewas DZP employing the Dunning double zeta functions38
with polarization functions having exponents of 0.9, 1.0, an0.8 for O, H, and C, respectively. The two low energminima found from the 6–311G** multipoles converge tothe same minimum,A, while min1 found from 6–31G**multipoles converges to a transition state of practically thsame energy. Starting from the high energy minimum founwith the 6–311G** multipoles leads to a slightly lower en-ergy minimum,B. The energies are all reported in Table VIThe calculations were repeated starting from the DZP opmized geometries using theCADPAC 631GE extended basissets which are between DZP and TZ2P in quality8 and en-tailed 102 basis functions in total. The Hessian indexes rmain the same, but the energy ordering changes~Table VI!;the structures are shown in Fig. 3. For the larger basisminimum B becomes slightly higher in energy than minimumA.
Unfortunately, these results are not directly comparabto those of Szcze¸sniak et al. because they do not includecorrelation corrections and the predicted stationary pointsnot lie particularly close to the idealized geometries consiered in that study. However, they do serve to illustrate thfact that the geometries produced byORIENT3 can provideuseful starting configurations, especially for unsymmetricstationary points. The methane–H2O potential energy surfaceis clearly extraordinarily flat, perhaps reflecting the higsymmetry of the methane molecule and the anticipated ‘‘hdrophobic’’ interaction with water, i.e., the methane molecuis relatively isotropic and the interaction itself relativelyweak.
TABLE XIV. Rearrangement pathways of water–ethene. Energies arecm21 andS andD ~defined in Sec. IV! are in bohr.
E1 D1 Ets D2 E2 S D N
2650.30 6.52 2643.77 6.52 2650.30 5.3 4.7 5.3
2650.30 8.92 2641.38 8.92 2650.30 2.3 1.7 2.1
2650.30 284.70 2365.60 284.70 2650.30 8.8 4.0 5.3
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VII. WATER–CARBON MONOXIDE
This complex was the subject of a recentab initio studyby Sadlej and Buch39 who employed large basis sets andcorrelation corrections up to MP4. Their study was promptedby experimental results due to Yaronet al.,40 using molecularbeam resonance and microwave techniques, and by fainfrared measurements by Bumgarneret al.41 which probethe tunneling dynamics more extensively. Yaronet al.40 de-duced an equilibrium structure for the complex with a non-linear hydrogen bond to the CO carbon and a barrier to exchange of the water protons of about 210 cm21. They alsonote that their data are consistent with a nearly linear arrangement of the three heavier atoms with a CO carbon twater oxygen distance of 3.36 Å. Calculations of the electrostatic energy using a DMA approach were not found to account for the observed nonlinearity in the hydrogen bond.40
The subsequent far-infrared measurements41 confirmed thegeometry of the equilibrium structure and provide rotationaconstants. The lowest minimum located by Sadlej and Bucin their study39 is basically in agreement with experiment.Two other minima were found in theab initiowork, one witha hydrogen bond to the CO oxygen, the other a T-shapestructure with the water oxygen closest to the CO center omass. They also confirmed the presence of aC2v transitionstate for proton exchange with a barrier of 283 cm21. Thepredicted binding energies of the three minima were 651.6301.3, and 256.5 cm21, respectively.
For 6–31G** geometries and multipoles we find twominima and four transition states usingORIENT3 ~Table VII!.The lowest minimum most closely resembles the T-shapeminimum found in theab initio study, with the water oxygenalmost equidistant from the CO atoms. The other minimumhasC2v symmetry and corresponds to the transition state
FIG. 7. Low energy stationary points for water–ethene calculated withORI-
ENT3. The H•••C distances are given in bohr.
FIG. 8. Low energy stationary points for H2O–CO2 calculated withORIENT3.The distances are marked in bohr.
in
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5559Wales, Popelier, and Stone: PES’s of van der Waals complexes
TABLE XV. Water–carbon dioxide: Energies/cm21, point groups ~PG!, nonzero normal modefrequencies/cm21, rotational constants/cm21, and components of the dipole moment,m i /Debye, along the iner-tial axes~in the same order as the rotational constants!.
Label Energy PG Frequencies Rotational constants m i
aThese are the harmonic frequencies calculated by Makarewiczet al. ~Ref. 55!.
o
t
t
h
lry
b
e
t
ahe
-she
d a-
a
ytt-
e
e
in
tee
olldd
e
postulated in the previous studies for proton exchange. Thetwo minima are linked via ts4~see Table VIII and Fig. 4!. Wecould not find a minimum corresponding to the structurethe previous studies. However, CO is a notoriously difficumolecule to treat theoretically because of the small dipomoment, so we reoptimized the geometries and calculaDMA’s for both molecules using a 6–311G** basis14 and anMP2 correction for electron correlation. The two minima tharesulted~Table VII! are similar to those found before, buwith the carbon and oxygen atoms of the CO molecule intechanged. Hence, theC2v minimum, which is now the globalminimum, has the two water hydrogens at 5.62 bohr from tCO carbon atom. The corresponding structure with the rolof the CO atoms reversed is now found to be a transitiostate.
Clearly in this case our model is quite unreliable, athough the stationary points do correspond to a higher eneminimum and the transition state for tunneling. This is likeldue to the particular problems caused by the CO molecuand the representation of dispersion and repulsionLennard-Jones terms.
VIII. WATER–FORMALDEHYDE
This complex has recently been the subject of a detailab initio investigation by Ramelotet al.42 who employedlarge basis sets and various correlated methods. One mmum and two transition states are reported, with the minmum having an unusually bent hydrogen bond. The calclated harmonic frequencies for this structure were foundbe in reasonable agreement with experiment. UsingORIENT3
we find four minima and seven transition states for this sytem, as summarized in Tables IX and X. The lowest minmum corresponds closely to the minimum found in theabinitio studies, except that it is slightly nonplanar. In fac
TABLE XVI. Rearrangement pathways of water–carbon dioxide. Energiare in cm21 andS andD ~defined in Sec. IV! are in bohr.
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there is a degenerate rearrangement of this minimum withvery low barrier where the transition state corresponds to tplanar geometry~Fig. 5!. The two hydrogen bonds are sig-nificantly longer than theab initio predictions.
The next two lowest minima both have the two molecules stacked on top of one another with their planeroughly parallel. There is facile rearrangement between ttwo mediated by ts2. In the more symmetricalCs min3 thewater molecule acts as a double hydrogen bond donor andouble acceptor. This structure is similar to one of the transition states found by Ramelotet al.42 where water is adouble donor but a single acceptor; the two are related byrotation through 90° of water about the formaldehydeC2axis. In the highest energyC2v minimum formaldehyde actsas a double acceptor and water as a double donor with O•••Hdistances of 5.09 bohr.
IX. WATER–HYDROGEN CHLORIDE
Legon and Willoughby43 deduced the presence of a lin-ear O•••H–Cl hydrogen bond and an overall planar geometrfor H2O–HCl from microwave spectroscopy. Subsequencalculations are in good agreement with this result, predicing OClH angles from 0° to 3.4°.44 However, many calcula-tions also predict that the complex is not planar, with thangle between the OCl axis and the waterC2 axis rangingfrom 0° to 46.8°.44,45–48There have been three calculationsof the inversion barrier giving best estimates of 115,49 144,50
and 147 cm21.44 Calculations of the dissociation energyrange from 1871~Ref. 47! to 1894 cm21,51 and both experi-ment and theory consistently yield around 3.2 Å for thO•••Cl distance and 1.8 Å for the O•••H distance. In additionseveral studies have reported frequencies, as summarizedTable XI.
We initially obtained oneC2v minimum and one transi-tion state corresponding to a very high energy degenerarearrangement, as summarized in Tables XI and XII. ThO•••Cl distance in the minimum is 3.75 Å and the O•••Hhydrogen bond length is 2.49 Å, i.e., both are much tolarge. We chose this system to test the effect of dividing athe heteroatomic well depth parameters by two, but founthat it made no significant difference. We then reoptimizethe geometries and performed new DMA’s for 6–311G**basis sets14 with MP2 correlation corrections throughout.This particular basis is not available in theCADPAC library
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5560 Wales, Popelier, and Stone: PES’s of van der Waals complexes
D
TABLE XVII. Water–methanol: Energies/cm21, point groups~PG!, nonzero normal mode frequencies/cm21,rotational constants/cm21, and components of the dipole moment,m i /Debye, along the inertial axes~in the sameorder as the rotational constants!.
Label Energy PG Frequencies Rotational constants m i
for chlorine and so a DZP basis was used employing douzeta functions due to Dunning38 and a set of Cartesiandfunctions with exponent 0.75. With the new geometries aDMA’s we now find a single minimum withCs symmetrycorresponding to a tilt of 146° between the waterC2 axis andthe O•••Cl axis, in better agreement with the more sophiscatedab initio calculations. TheC2v geometry is now a tran-sition state for the motion of the water through a planar cofiguration and the barrier is only 2.6 cm21. These results arealso summarized in Tables XI and XII, and the stationapoints are shown in Fig. 6. This rearrangement is little afected if the heteroatom well depths are halved, althoughdo find an additional high energy rearrangement mechanin this case. The fact that reasonable qualitative agreemwith previous calculations is obtained when the more acrate multipoles are used strongly suggests that the remainerrors~especially the oversize hydrogen bond length! are dueto the particular Lennard-Jones parameters employed.
X. WATER–ETHENE
The water–ethene complex has recently beenexamined by Andrews and Kuczkowski52 using microwavespectroscopy. Their spectra are consistent with a high bartunneling pathway that exchanges the water protons anlow barrier internal rotation which is required to explaianomalous dipole measurements. In previousab initio calcu-lations Del Bene53 found the global minimum to have anO–H bond directed at the ethenep system roughly bisectingthe carbon–carbon double bond. The center-of-mass distawas calculated to be 3.65 Å and the system was found toinsensitive to rotation of the water molecule about the hydgen bond. In the same study the bifurcated structure withwater hydrogens equidistant from the middle of the carbocarbon bond was found to lie only 45 cm21 higher in energy,and it was suggested that tunneling of the water protons
TABLE XVIII. Rearrangement pathways of water–methanol. Energiesin cm21 andS andD ~defined in Sec. IV! are in bohr.
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this route was likely. Molecular beam electric resonance eperiments by Peterson and Klemperer54 produced results inbroad agreement with the previous calculations withcenter-of-mass distance of 3.413 Å, an angle of 60° betwethe waterC2 axis and the etheneC2 axis perpendicular to themolecular plane and the free water hydrogen lying in tmirror plane that bisects the carbon–carbon bond. Sevtransitions in this experiment were split into doublets, andtunneling barrier of 353 cm21 was estimated. The new results of Andrews and Kuczkowski52 are consistent withnearly free rotation about the O–H•••p hydrogen bond andhigh-barrier inversion-type doubling involving a bifurcate‘‘intermediate’’ with two equivalent hydrogen bonds.
We located one minimum and five transition states fthis system as summarized in Tables XIII and XIV. Thminimum agrees with Peterson and Klemperer54 in that thefree water hydrogen lies in the plane that bisects the carbocarbon bond. There are two facile degenerate rearranments, the one with the lowest barrier corresponding to
FIG. 9. Low energy stationary points for water–methanol found wiORIENT3. The distances are marked in bohr.
re
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5561Wales, Popelier, and Stone: PES’s of van der Waals complexes
D
TABLE XIX. Hydrogen chloride dimer: Energies/cm21, point groups ~PG!, nonzero normal modefrequencies/cm21, rotational constants/cm21, and components of the dipole moment,m i /Debye, along the iner-tial axes~in the same order as the rotational constants!.
Label Energy PG Frequencies Rotational constants m i
6–31G** geometries and multipoles48, 156, 179, 314a
aCalculated by Latajka and Scheiner for the linear structure~Ref. 64!.bCalculated by Latajka and Scheiner for the cyclic structure~Ref. 64!.
i
hnr
f
ts
snr
e
bhb
n
e
tialre-
czyby
nein
In-
x-x-eror,-
d–
aher a. 9;thatin
gygled am
e
tation about the O–H•••p bond and the other to interchangeof the two water protons~Fig. 7!. TheC2v transition statecorresponds to a mechanism where a mirror plane is matained throughout. The structure where the water moleculies in the plane through the two carbons and has two equivlent H•••C contacts is found to be an index 2 saddle whicgives the actual tunneling transition state on perturbation areoptimization. Del Bene found very similar energies fothese twoC2v structures.53 In the present calculations thetunneling pathway has a barrier almost as small as thatrotation about the H•••p bond. However, these barriers areprobably not trustworthy in the light of the present resultaken as a whole.
XI. WATER–CARBON DIOXIDE
Makarewiczet al.55 have recently presented calculationincluding basis sets of TZ2P quality and MP2 correlatiocorrections for this system. Their results build upon numeous previous theoretical and experimental studies which agenerally in good agreement with one another—we refreaders to the paper of Makarewiczet al. for a review of theprevious results. There is general agreement that the glominimum of H2O–CO2 has a T-shaped planar structure wita closest contact between the water oxygen and the cardioxide carbon to giveC2v symmetry. In fact, Makarewiczet al.55 find that this structure is actually a transition state foa very facile degenerate rearrangement of the global mimum ~barrier 3 cm21! which is slightly nonplanar. They alsoobtain a barrier to rotation about the C•••O axis of 330 cm21,in good agreement with Peterson and Klemperer’s estimat56
of 300 cm21 and the values obtained by Blocket al.57 fromtheir high resolution infrared spectra~305 cm21! and fromab
TABLE XX. Rearrangement pathways of hydrogen chloride dimer. Energiare in cm21 andS andD ~defined in Sec. IV! are in bohr.
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initio work ~338 cm21!. The latter authors also found ahigher energy hydrogen-bonded minimum on the potenenergy surface, and a similar structure was previouslyported in the calculations of Damewoodet al.58 The bindingenergy of the complex is calculated to be 1190 cm21 byMakarewiczet al.
Our results are in very good agreement with Makarewiet al. in that we also find the global minimum to be slightlnonplanar, and that the barrier to rearrangement mediatedthe C2v transition state is only about 1 cm21 ~Fig. 8!. Welocated a much higher energyC2v minimum where wateracts as a symmetrical double hydrogen bond donor to oend of the CO2 molecule. The results are summarizedTables XV and XVI. The alternativeC2v structure obtainedby rotating one molecule through 90° about the C•••O axisdoes not correspond to a transition state in our model.stead, we find a less symmetrical transition state~ts2! whichis obtained from the hypotheticalC2v structure by formingtwo equivalent hydrogen bonds to one of the CO2 oxygenatoms.
XII. WATER–METHANOL
Bakkaset al.59 have recently presented a combined eperimental and theoretical investigation of this species. Eperimentally they found evidence for complexes with watacting as a hydrogen bond acceptor but not as a donwhereas theirab initio calculations, in agreement with previous work,60 suggest similar stability for two structures inwhich water plays both roles. In the present work we finthree low lying and one high energy minimum for watermethanol along with seven transition states~Tables XVII andXVIII !. Our global minimum has the water molecule asdouble donor with a relatively short distance between twater oxygen and a methyl hydrogen too. The pathway fodegenerate rearrangement of this structure is shown in Figthe transition state, ts4, is probably the closest geometrywe have found to the water single donor minimum foundtheab initio studies.59
A rearrangement between the other two low enerminima is mediated by ts3. In min2 water acts as a sinacceptor, and in min3 it acts as both a single acceptor ansingle donor. Min2 closely resembles the other minimu
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5562 Wales, Popelier, and Stone: PES’s of van der Waals complexes
D
TABLE XXI. Hydrogen fluoride dimer: Energies/cm21, point groups ~PG!, nonzero normal modefrequencies/cm21, rotational constants/cm21, and components of the dipole moment,m i /Debye, along the iner-tial axes~in the same order as the rotational constants!.
Label Energy PG Frequencies Rotational constants m i
aCalculated by Latajka and Scheiner for the linear structure~Ref. 64!.bCalculated by Latajka and Scheiner for the cyclic structure~Ref. 64!.
tion
rkture.
edesn toderestasesex-
ul-theen-lotonter–aryareose
thedund
aningmvenra-
e-o ei-en-orown are-
asnale orent
found in theab initio calculations.59 It is tempting to suggestthat our failure to find a minimum where water acts assingle donor is in agreement with the experimental failurefind such a structure. However, taking into account all thresults in the present study this conclusion would be rathoptimistic.
XIII. HYDROGEN CHLORIDE DIMER
The interconversion tunneling of~HCl!2 has recentlybeen investigated by Schuderet al.61 Following Ohashi andPine62 the global minimum is known to have a Cl•••H hydro-gen bond with a Cl•••H•••Cl angle of around 90°. The bind-ing energy given by Pine and Howard63 is 431 cm21 andabinitio work by Latajka and Scheiner64 confirms the assump-tion of a cyclicC2h transition state for interchange of theroles of the two molecules. Our results give the global minmum as aC2h cyclic structure, with a high energy degeneratrearrangement pathway available via aC2h transition state~Tables XIX and XX!. The H•••H distance in theC2h mini-mum is actually 0.05 bohr shorter than the nonbonded Cl•••Hdistance. This is a very interesting pathway because the sting and finishing geometries are exactly the same. Repeatthe calculation using DZP/MP2 geometries and multipoledoes not change the geometry of the global minimum signicantly, although the high energyC2h minimum with a rela-tively short Cl•••Cl contact now becomes a minimum~seeTables XIX and XX!. For either set of multipoles employingheteroatom well depth parameters of half the size maklittle difference.
XIV. HYDROGEN FLUORIDE DIMER
The HF dimer has been the subject of many more studthan the HCl dimer, the most recent work that we know obeing that of Chang and Klemperer.65 They report a moreacute HFF angle in theC2h transition state than found inprevious calculations.66,67 Pine and Howard63 give the bind-ing energy as 1038 cm21 and Latajka and Scheiner64 reportab initio harmonic frequencies for the intermolecular vibrations. In this case we find a uniqueC2h minimum and a
TABLE XXII. Rearrangement pathways of hydrogen fluoride dimer. Enegies are in cm21 andS andD ~defined in Sec. IV! are in bohr.
E1 D1 Ets D2 E2 S D N
2700.42 602.96 297.46 602.96 2700.42 11.5 0.0 `
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moderately high rearrangement barrier via a linear transistate~Tables XXI and XXII!. Hence, as for~HCl!2 the mostfavorable structure obtained within our current framewoseems to be more appropriate to the transition state struc
XV. CONCLUSIONS
We have applied our basic framework of distributmultipole analysis in combination with Lennard-Jonatom–atom terms to represent dispersion and repulsiostudy the potential energy surfaces of a number of vanWaals complexes. In particular, we have presented the lowrearrangement pathways in each case, and in many cthese have the status of predictions. The agreement withperiment and previousab initiowork ranges from good~e.g.,H2O–CO2! to [email protected].,~HCl!2#. For ~HCl!2, we found thatthis situation was not improved by using more accurate mtipoles or monomer geometries, and it seems likely thatproblem lies in the simplistic dispersion–repulsion represtation used in the present work. The use of the Berthecombination rule is known to give too strong an attractibetween unlike atoms, but alternatives, such as the SlaKirkwood rule, do not seem to help. It is probably necessto introduce anisotropic repulsion and dispersion, whichknown to be important in some systems, especially thinvolving the halogens Cl, Br, and I.68 In terms of predictivevalue, the present model may be less useful thanBuckingham–Fowler prescription of DMA with harspheres, although the stationary points that we have foshould serve as sensible starting points forab initiowork andcan be obtained in a small fraction of the time needed forab initio calculation. This should serve as a severe warnabout the reliability of potential surfaces constructed froatom–atom Lennard-Jones terms only, i.e., without eelectrostatic terms, which are prevalent throughout the liteture.
The degree of cooperativity involved in the various rarrangements does not seem to be strongly correlated tther the barrier heights or the path length. Hence, highergy mechanisms may be localized on a few atomsdelocalized over the majority, and the same is true for lenergy mechanisms. This is in line with results obtained isystematic study of atomic Lennard-Jones clusters, asported elsewhere.21 As the molecules are forced to moverigid bodies in the present work, and there are no interfree rotors, we might expect mechanisms localized on ontwo atoms to be very rare. In fact, for most of the pres
r-
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5563Wales, Popelier, and Stone: PES’s of van der Waals complexes
systems the indexN is just over half the total number ofatoms for many mechanisms.
Previous experience suggests that potential energy sfaces calculated with model intermolecular potentials proably contain more minima than they should, and the presework is in keeping with this observation. However, it is possible that some of the structures we have found that havebeen considered before might be physically meaningful.
For future work we intend to build up a library of mono-mer DMA’s and more accurate dispersion–repulsion reprsentations. DMA’s calculated at the DZP/MP2 level shoube good enough for most purposes, but the dispersiorepulsion terms are considerably harder to calculate reliab
ACKNOWLEDGMENTS
D.J.W. is a Royal Society Research Fellow and P.L.A.is an HCM Fellow of the EU.
APPENDIX
Projection operators have previously been employedstudies of reaction paths69,70 and constrained geometryoptimization,71 among others. First we note that the energyinvariant to overall rotations and translations, so that we cwrite the Taylor expansion for the energy in terms of a projected step, gradient and Hessian20 corresponding to thetransformationy5Px, whereP is the projection matrix,x isa vector in nuclear configuration space andy is the corre-sponding projected vector. The appropriate form forP is70,71
Pab5dab2(i
e~ i !ae~ i !b ,
where the$e( i )% are a set of orthonormal vectors which arobtained from displacements corresponding to the motionsbe projected out, as described below. The action ofP clearlyremoves any linear combination of thee vectors from a gen-eral vector. In the present case we wish to remove compnents of overall rotation and translation, and here we folloPage and McIver’s construction70 of P. However, we differfrom their presentation, and that of Baker and Hehre,26 inthat we do not use mass-weighted coordinates. Throughthis paper we use only non-mass-weighted Cartesian coonates, gradients and Hessians, and hence the approximsteepest descent paths calculated are properties only ofpotential energy surface in question, and not the massesthe general case, if we have a set of vectors$b( i )% each ofwhich is parallel to a geometry change corresponding tomotion that we wish to project out, then the final form forPis70
Pab5dab2(i , j
b~ i !aSi j21b~ j !b ,
where
Si j5(g
b~ i !gb~ j !g .
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utdi-atetheIn
a
For a system consisting of a set ofn atoms an appropriate setof b vectors is given by
~b~1!,b~2!,b~3!,b~4!,b~5!,b~6!!
5S 1 0 0 0 z1 2y1
0 1 0 2z1 0 x1
0 0 1 y1 2x1 0
A A A A A A
1 0 0 0 zn 2yn
0 1 0 2zn 0 xn
0 0 1 yn 2xn 0
D .
The resulting projected Hessian then has six zero eigenvues and steps are only taken in the other 3n–6 directions. Infact, there will be three zero eigenvalues for the three tranlations anyway,22 and so it is not really necessary to projecout these motions.
It only remains to find the appropriateb vectors for rigidlinear and nonlinear molecules. In our current implementtion all rigid molecules are described by three center-of-macoordinates and three angles which specify the orientationa previous study of water clusters Euler angles were usedthe orientational variables,72 and it was then necessary to findthe appropriate Euler angle displacements correspondingoverall infinitesimal rotations numerically. However, in thepresent case the orientation of a given molecule is specifiby rotations aboutx, y, andz axes which have a local originat the molecular center of mass and are parallel to the gloCartesian axis system. Infinitesimal rotations about theaxes commute and the appropriateb vectors for a set ofnrigid nonlinear molecules are simply
~b~1!,b~2!,b~3!,b~4!,b~5!,b~6!!
511 0 0 0 z1 2y1
0 1 0 2z1 0 x1
0 0 1 y1 2x1 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
A A A A A A
1 0 0 0 zn 2yn
0 1 0 2zn 0 xn
0 0 1 yn 2xn 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
2 ,
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lye-
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esallngldo
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5564 Wales, Popelier, and Stone: PES’s of van der Waals complexes
where the order of the variables has the three centre-of-mcoordinates of molecule 1 first followed by its orientationparameters, then the same for molecule 2, etc. Note thatb vectors do not need to be normalized. Furthermore, theof local rotations rather than Euler angles not only eliminathe tedious singularities associated with the latter variabbut also makes the projection process much easier. Satoms carry only three positional coordinates the approprb vectors for a mixture of atoms and molecules are givencombining the above results in an obvious fashion.
For linear molecules only two orientational parameteare required, since the energy is invariant to rotation abthe figure axis. However, this axis will generally not corrspond to a local coordinate axis. We found it most convnient to treat this problem by calculating derivatives wirespect to rotations about the three local axes, as for nonear molecules, so that there is a redundant coordinateeach linear molecule. The projection operator was therefmodified to remove motion corresponding to rotation abothe figure axis of each linear molecule, giving a projectHamiltonian with one extra additional zero eigenvalue feach such molecule. Steps were then taken only along Hsian eigenvectors corresponding to nonzero eigenvaluesusual. To find the appropriate vectorb corresponding to aninfinitesimal rotation about an axis with direction vecto( l ,m,n) we need to express the rotation in terms of infintesimal rotations about the localx, y, andz axes. The figureaxis will not generally pass through the global origin, but thdoes not matter because the local rotations are with respelocal axes that are parallel to the global axes but translatethe molecular centre of mass. Infinitesimal rotations ofa, b,andg about the localx, y, andz axes commute to first ordeand correspond to a transformation matrix
S 1 2g b
g 1 2a
2b a 1D .
Equating this with the transformation matrix correspondito an infinitesimal rotation throughd about the (l ,m,n) axis,
S 1 nd/r 2md/r
2nd/r 1 ld/r
md/r 2 ld/r 1D ,
where r 5 Al 21m21n2, gives a:b:g5 l :m:n. Hence, aset ofb vectors appropriate to a system consisting ofp linearmolecules is given by
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assaltheusetesles,inceiateby
rsoute-e-thlin-fororeutedores-, as
ri-
isct tod to
r
ng
~b~1!,b~2!,b~3!,b~4!,b~5!,b~6!,b~7!,...,b~p16!!
511 0 0 0 z1 2y1 0 ... 0
0 1 0 2z1 0 x1 0 ... 0
0 0 1 y1 2x1 0 0 ... 0
0 0 0 1 0 0 l 1 ... 0
0 0 0 0 1 0 m1 ... 0
0 0 0 0 0 1 n1 ... 0
A A A A A A
1 0 0 0 zp 2yp 0 ... 0
0 1 0 2zp 0 xp 0 ... 0
0 0 1 yp 2xp 0 0 ... 0
0 0 0 1 0 0 0 ...l p
0 0 0 0 1 0 0 ...mp
0 0 0 0 0 1 0 ...np
2 .
Once again, the appropriateb vectors for a mixture of atomsand linear and nonlinear molecules are obtained by simpcombining the above results for the required degrees of fredom.
We experienced no convergence problems with tabove framework, employing a maximum step criterion adescribed elsewhere.10 However, one further complicationarises if the whole system becomes linear when we havcomplex containing no nonlinear molecules. In this case tmatrix S defined above becomes singular, and we must aallow for cases where linear and nonlinear geometries intconvert. To do this a simple scheme was adopted wheredeterminant ofSwas calculated before the matrix inversionand if detS,10210 the dimension ofS was reduced by onebefore continuing. The number of zero Hessian eigenvaluis then also reduced by one. This method works becausethe constraint vectors are already mixed together in formiS, and so the resulting projection matrix will still remove althe required components corresponding to motions thatnot change the energy.
1C. E. Dykstra, Chem. Rev.93, 2339~1993!.2A. J. Stone, Chem. Phys. Lett.83, 233 ~1981!; A. J. Stone and M. Alder-ton, Mol. Phys.56, 1047~1985!.
3W. A. Sokalski and A. Sarawyn, J. Chem. Phys.87, 526 ~1987!; W. A.Sokalski and A. Sawaryn, J. Mol. Struct.256, 91 ~1992!.
4F. Vigne-Maeder and P. Claverie, J. Chem. Phys.88, 4934~1988!.5P. W. Fowler and A. D. Buckingham, Chem. Phys. Lett.176, 11 ~1991!.6A. D. Buckingham and P. Fowler, Can. J. Chem.63, 2018~1985!.7M. A. Spackman, J. Chem. Phys.85, 6587~1986!; J. D. Augspurger andC. E. Dykstra, Mol. Phys.76, 229 ~1992!; J. B. O. Mitchell, C. L. Nandi,J. M. Thornton, S. L. Price, J. Singh, and M. Snarey, J. Chem. SoFaraday Trans.89, 2619~1993!.
8The Cambridge Analytic Derivatives Package Issue 5, Cambridge, 19R. D. Amos with contributions from I. L. Alberts, J. S. Andrews, S. MColwell, N. C. Handy, D. Jayatilaka, P. J. Knowles, R. Kobayashi, NKoga, K. E. Laidig, P. E. Maslen, C. W. Murray, J. E. Rice, J. Sanz, E. DSimandiras, A. J. Stone, and M. D. Su.
9P. L. A. Popelier and A. J. Stone, Mol. Phys.97, 411 ~1994!.10P. L. A. Popelier, A. J. Stone, and D. J. Wales, Faraday Discuss.97, 243
~1995!.11C. J. Cerjan and W. H. Miller, J. Chem. Phys.75, 2800~1981!; J. Simons,
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m
D
.
r,
.
.
.
.
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