The HF–Ar system is an important benchmark for tdevelopment and understanding of potential energy surfaThere has been much experimental and theoretical interethis system due mostly to the use of hydrogen fluoridelasers. Even though there exists a great deal of experimeand computational data for the system, little is known abthe details of its vibrational energy transfer. This mayattributed to both the experimental and the computatiodifficulties that are inherent to its study. One of the chcomputational problems is the development of an accupotential energy surface~PES!. Here we test several methodfor calculating the potential energy surface for the collisiovibrational energy transfer of this and similar systems. Tsurfaces generated are tested against known spectrosand rotational activation data. The final surfaces obtainedused to investigate rovibrational energy transfer ofHF–Ar collisional system.
Vibrational energy transfer involving small moleculesof interest for a variety of reasons. In the atmosphere, higvibrationally excited molecules~e.g., CO2, O3, OH, H2O,NO! are produced as a result of chemical reactions, quening of excited electronic states, and absorption of light.
a!Present address: Computational Biochemistry, Biophysics, and BioGroup, Environmental Molecular Sciences Laboratory, Pacific NorthwNational Laboratory, Richland, WA 99352.
4570021-9606/2001/115(10)/4573/13/$18.00
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low altitudes, the high collision frequency maintains a Bozmann vibrational energy distribution characterized bylocal translational~kinetic! temperature: ‘‘Local thermody-namic equilibrium’’~LTE!. At high altitudes, ‘‘non-LTE’’ dis-tributions are produced as the result of radiative and checal processes in competition with collisions. Radiatienergy is lost from the local atmosphere, affecting the eciency of heat deposition.1–5 The occurrence of non-LTE caalso compromise the interpretation of atmospheric limb raances measured from satellites~e.g., Solomonet al.6!. Thepresent study was motivated by the desire to investigate nLTE in HF(v), which is likely produced in atmospheric colisions of HF with O2(
1D) or O2(1S).7
The vibrational deactivation of HF in HF–Ar mixturehas been studied in laser fluorescence and shockexperiments,8–17but accurate measurement of HF–Ar enertransfer is difficult because it makes only a minor contribtion compared to the much more efficient HF–HF enertransfer. At 295 K the rate of self-deactivation is reported'2.0310212cm3/molecule•sec, while the upper limit forthe deactivation by argon at the same temperature is torders of magnitude smaller. The experimental rate constwere determined from plots of inverse relaxation time verHF mole fraction extrapolated to infinite dilution. At highetemperature, the HF–Ar rate constants are measurable, blow temperatures, the intercepts fall close to zero and expmental uncertainties prevent evaluation of the HF–Ar r
4574 J. Chem. Phys., Vol. 115, No. 10, 8 September 2001 Shroll, Lohr, and Barker
constant. Thus, the results from the lower temperatureperiments were reported as upper bounds. Computatiprediction of the rate constant is also challenging sincedependence of the intermolecular potential on the HF vibtional coordinate must be assessed and the number ofrovibrational channels may be prohibitive. In contrast toexperiments, the computational procedure is simplifiedlower temperatures because fewer channels are availab
There have been several previous theoretical studiethe collision induced deactivation of HF by Ar.18–23Ovchin-nikova used a quasiclassical approximation with several splifying transformations to calculate the vibrational deactivtion rate constant from the first excited vibrational state wfour different PESs.22 Two of them, which were based opreviously determined self-consistent field~SCF! data,24 pro-duced rate constants within the range of values obtainedexperiments for temperatures between 800 and 4000 K.results are not in close agreement with the present studythey fall within the wide scatter of experimental values. Brend and Thommarson performed a quasiclassical trajecstudy of HF (v51) – Ar vibrational deactivation that provided good results for higher temperatures but overestimthe upper limits set by experimental observations at 294350 K.20 Thompson conducted quasiclassical trajectory cculations using an additive pair potential for several vibtional states of hydrogen fluoride.18,19He provided a detailedpreliminary survey of the state-to-state cross sections, bunot determine the relative translational energy dependencthe cross sections. This dependence is required in ordecalculate rate constants and is also helpful in interpretingqualitative behavior of the collisional system. When this pper was nearing completion, Kremset al.23 released a quantum mechanical study of the vibrational relaxation of tHF–Ar system. Their intermolecular potential is basedthe diatomics-in-molecule approach,25–27which is much dif-ferent than the augmented potentials used here. Their reare in good agreement with existing data.
The predicted rate constant for the vibrational deactition of hydrogen fluoride by argon may be greatly improvby using better computational techniques and new specscopic data for the van der Waals complex. Recent advain computer hardware have made it possible to use higaccurate quantum scattering methods such as the coustates~CS! approximation28 to calculate state-to-state crosections. These same advances have made it possible toout more accurate and computationally demandingab initiocalculations of the PES. Much is known now about thetermolecular potentials of van der Waals complexes and hto useab initio quantum mechanics to assess them reliabl29
There has been great interest in the development oftermolecular potential energy surfaces for prototype systconsisting of an atom and a diatomic molecule.24,30–50Poten-tial surfaces for the intermolecular interaction of hydrogfluoride and argon have been reported numertimes.18–22,24–26,44–51In Jacobi coordinates, the relative postions of the particles are represented by a vectorr whichpoints from the fluorine atom to the hydrogen atom, byvector R which points from the diatomic center of massthe argon atom, and by the angleu between the vectors~the
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zero for u is chosen to be the linear Ar-H-F geometry!. Inorder to obtain a potential energy surface approaching stroscopic accuracy directly fromab initio calculations, accu-rate electron correlation techniques must be employed wlarge basis sets and many different geometries.48,52 For mostsystems it is not yet practical to perform calculations wthis level of accuracy or complexity and there is clearlyneed for effective methods for enhancing the more modab initio methods that are currently accessible.
Various approaches to potential energy surface enhament have been reported.30–32,37,53–61Some decompose thpotential into physically meaningful terms that canmodified.31,37,54,55,61Others scale the potential by a constafactor or scale the coordinates.56–60Recently a technique wadeveloped which scales both simultaneously.32 For an atom-diatom collisional system, spectroscopic data for vanWaals complexes may be used with nonlinear least squalgorithms to determine the modifications.45,62 In the presentstudy we perform bound state calculations for the vanWaals complex and use the results to augment or morppotential energy surface derived from second-order MølPlesset theory~MP2!. We then determine the effects of thpotential energy enhancements on the calculation of vibtional energy transfer.
Once a suitable potential is obtained, it is used to predthe rotational energy transfer cross sections and ratestants. These cross sections provide a test of the intermollar potential that is a prelude to determining the vibrationrate constants. Rotational activation cross sections that hbeen measured63 at a relative translational energy of 35cm21 may be compared to highly accurate close-coupled toretical calculations and are a useful test of the intermolelar potential energy surface anisotropy at the base ofrepulsive wall. The CS calculations are used to predict rotional cross sections over the range of energies studied.is known experimentally about how the individual cross stions vary with the relative translational energy and therefthese dependencies cannot be compared directly to exmental data. However, the calculated energy dependeprovides valuable insight into the qualitative behavior of tcollisional system and is shown for selected cross sectioWe also examine the relationship between the rotatiocross sections corresponding to HF in different vibratiostates and the suitability of the empirical power-gap lawdata modeling and reduction. Representative rotationalergy transfer rate constants are presented and discussed
The vibrational energy transfer rate constants are calated from the individual rovibrational state-to-state rate costants and are shown to be in good agreement with theited experimental measurements. The original MP2 surfdoes well at calculating the vibrational deactivation rate cstants. The surface enhancing techniques discussed herelittle effect on the resulting vibrational energy transfer, hoever they do improve the predicted rotational activation crsections and bound state energies. Analysis of the statestate rate constants shows a strong tendency toward puretransitions. Qualitative structural features are evidentlow-temperature rate constants, due to low energy crosection resonance structures. All of the results are consis
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4575J. Chem. Phys., Vol. 115, No. 10, 8 September 2001 Energy transfer of hydrogen fluoride
with the available experimental observations and the knoqualitative behavior of energy transfer cross sections andconstants.
II. POTENTIAL ENERGY SURFACES
The empirical potentials developed here begin with3-D potentialV(R,u,r ) based on a grid of points determinevia MP2 calculations. The 3-D potential is modified eitherscaling the MP2 potential or by adding terms that enhathe dispersion interaction, which is known to be underemated by the MP2 level of electron correlation. This iscomputationally simple step and could be replaced by onthe other methods of surface augmentation or morphFinding the best parameters is greatly accelerated by theistence of previously determined vibrationally averagedtentials~averaged overr!, which may be used to make smaimprovements to the 3-D potential. By themselves thesepotentials have been used for rotational energy transfer,63 butthey do not provide the necessary information to predictbrational state changes.
The 3-D potential can be vibrationally averaged by ingrating over the diatomic wave functions as shown in SII C. The resulting 2-D potential is used to calculate the bining energy and spectroscopic properties of the van der Wcomplex for comparison to experimental quantities.64–68TheMP2 PES is adjusted until satisfactory agreement is reacwith the experimental spectroscopic properties. This metproduces an improved 3-D potential energy surface forvan der Waals complex without a significant increase incomputational effort. By applying augmentation or morphitechniques to the 3-D surface and comparing the vibtionally averaged surface to experimental data, this technpreserves the dependence of the MP2 potential on the innal coordinater of the molecule.
Potential energy surfaces generated in the aforemtioned manner are augmented based on the properties ofvan der Waals potential wells. However, the collisional eergy transfer cross sections and rate constants are mdependent on the repulsive part of the potential. The mdeficiency of the MP2 surfaces is the underestimation ofattractive dispersion interaction29,69 which is most importantnear the van der Waals minimum. However, when the dpersion interaction is augmented, the repulsive wall ofpotential is also modified. In order to determine the sensiity of the energy transfer results to the form of the augmtation, three different augmented potentials are generfrom an MP2 surface and used in scattering calculations
A. ab initio calculations
The quantum chemistry method chosen for this stuwas second-order Møller-Plesset perturbation theory~MP2!with the aug-cc-pVTZ basis.70–72This method has the advantage of being size consistent, which simplifies the deternation of the intermolecular potential. Mourik and Dunninhave recently performedab initio calculations for the HF-Arsystem using this and other computational technique48
Their results show that the MP2 method is capable of repducing important features of the potential energy surface,the HF-Ar binding energy is underestimated even in the
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timated limit of a complete basis. For the HF-Ar compleMP2 with the aug-cc-pVTZ basis recovers only appromately 80% of the binding energy De . The deficiency isexpected, since an accurate calculation of the attractivepersion interaction requires highly correlated techniques.29,69
All of the MP2 calculations presented here were conducusing NWChem73 software.
The intermolecular potentialV(R,u,r ) is obtained viathe supermolecular approach29,38 and is expressed as
V~R,u,r !5EHF–Ar~R,u,r !2EHF ~`,r !2EAr~`!
1DEcp~R,u,r !2DEsc~r !, ~1!
where EHF–Ar(R,u,r ) is the potential energy of the wholsystem,EHF(`,r ) is the isolated diatom potential energEAr(`) is the isolated Ar atom potential energDEcp(R,u,r ) is the counterpoise correction for the basissuperposition error~BSSE!, and DEsc(r ) is a size consis-tency correction.
The counterpoise correction is expressed as
DEcp~R,u,r !5EHF~`,r !2EHF~R,u,r !
1EAr~`!2EAr~R,u,r !, ~2!
and substituting this result back into Eq.~1! gives
whereEHF(R,u,r ) andEAr(R,u,r ) are fragment energies requiring separate calculations. The size consistent termequal to zero for the MP2 method.74 The dependence oEHF(R,u,r ) on R and u as well as the dependenceEAr(R,u,r ) on R, u, andr is due to the inclusion of the fulset of basis functions that were used for the complex.
In order to obtain an accurate analytical representationthe potentialV(R,u,r ), ab initio calculations were carriedout for many values ofR, u, and r. The range ofR fellbetween 1.5 and 20 Å and included 23 unequally spapoints. This range was chosen to include values smaller tthe lower integration limits of the subsequent scattering cculations~Sec. III A!. The values ofr used were 0.7, 0.96481.1, 1.3, and 1.5 Å. The value of 0.9648 Å is the expectatvalue of r for v51 in isolated HF. The range ofr was de-termined from the HF vibrational wave-function amplitudeThe angleu varied between 0° and 180° with 11 equalspaced values and thus covered the range of symmunique angles.
B. Analytical representation of the interactionpotential
Theab initio calculations provide a discrete set of poinby which a functional representation must be determinThere are numerous ways of accomplishing this step, ewith its own strengths.75–80 Here the potential function isrepresented by a Legendre polynomial expansion with coficients determined using a matrix inversion technique.38,81
The potential may be expressed as
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4576 J. Chem. Phys., Vol. 115, No. 10, 8 September 2001 Shroll, Lohr, and Barker
V~R,u,r !5(k
(l
Vlk~R!Pl~cosu!~r 2r e!
k
5dTV~R!s, ~4!
where
dl5Pl~cosu! and sk5~r 2r e!k ~5!
andPl(cosu) are Legendre polynomials of orderl. The ma-trix V(R) is given by
V~R!5D21B~R!S21, ~6!
where
Dlm5Pl~cosum! and Skn5~r n2r e!k. ~7!
The ab initio calculations are performed for the anglesum
and the diatomic bond lengthsr n , constrained by the conditions l5m andk5n. These conditions are required by thmatrix inversion step in Eq.~6!. The termsBmn(R) are rep-resented by functions of the form
Bmn~R!5exp@2amn~R2Rmn!#F(i 50
3
bmn~ i !Ri G
2tanh~R!F(i 53
5
cmn~2i !R22i G , ~8!
where the coefficientsamn, bmn( i ) and cmn
(2i ) were determinedusing least squares. The hyperbolic tangent function inlast term was used as a damping function for the disperinteraction. Various damping functions have been reviewin the literature.30,69
The potential expansion convergence, with respect tonumber of angular functions, is affected by the locationthe coordinate origin. For example, an isotropic exponenrepulsive potential may be expressed as,
V~R1!5A exp~2bR1!, ~9!
whereR1 is a distance from the first originO1 ~see Fig. 1!.Transforming to a second originO2 gives,
V~R2 ,u!5A expH 2bR2F (l50
`
Pl~cosu!S a
R2D lG21J
~10!
with the constraint
FIG. 1. Two coordinate systems for the expansion of the isotropic potenThe origin O1 corresponds to the center of the isotropic potential andorigin O2 corresponds to an off center expansion.
end
efl
R22.ua222aR2 cosuu, ~11!
wherea is the magnitude of the displacement,u is the anglebetween the position vector and the displacement vectoshown in Fig. 1, anda/R2,1. For a50, only theP0 termremains and the form of the original potential is obtaineFor a.0, higher order Legendre polynomials are requireThe expansion coefficients are proportional to (a/R2)l and itfollows that for larger values ofa ~greater displacement othe coordinate origin from the isotropic potential cente!more terms are required for the expansion to converge.same considerations apply to anisotropic potentials, howethe algebra is much more cumbersome.
The repulsive wall of the HF–Ar potential closely resembles an ellipse~with nearly equal major and minor axes!as show in Fig. 2. The convergence of the angular part of~4! is expected to be best for a coordinate system withorigin located at the center of the ellipse. With the correchoice of origin for the HF–Ar potential, we also expect ththe isotropic term should be the largest term for values oRthat correspond to the potential wall. Since Legendre ponomial expansions of the HF–Ar intermolecular potentV(R,u,r ) are done with the origin located at the center-omass, we expect the convergence of the angular expansibe best when the center of the ellipse is located close tocenter-of-mass. Fortunately, this is the case for the MP2tential.
An ellipse with the center located atRi50.06 Å closelymatches the classical turning points on theV0,0
3 (R,u) poten-tial at a relative translational energy of 1000 cm21. The con-vergence of the expansion may be qualitatively seen byrelative magnitudes of the expansion coefficientsV(R)l
k . For
l.e
FIG. 2. The HF-Ar intermolecular potential classical turning points. T
V0,02 (R,u) matrix element is shown with the H6~4,3,2! potential for the
ground vibrational state of HF. Each ellipse corresponds to a constanergy. From the outside ellipse inward, the energies are 1000, 2000, 4and 8000 cm21. The HF-Ar center-of-mass is taken as the origin of tgraph.
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4577J. Chem. Phys., Vol. 115, No. 10, 8 September 2001 Energy transfer of hydrogen fluoride
HF–Ar the isotropic termV(R)00 is large and the Ar atom ca
only access regions in the scattering calculations whhigher-order (l.4) Legendre polynomial expansions asmall. For R52.12 Å, theV(R)0
0 term is 8.4 times largerthan theV(R)4
0 term. Espostiet al.82 discuss this effect forsimilar systems, as well as provide several graphs ofexpansion coefficients.
The same matrix technique was applied to Thompsopotential. Figure 3 shows the angular expansion coefficiefor five different origin locations. From the graph it is evdent that the best choice of origin is close to a 0.4 Å dplacement from the center of mass toward the hydrogendisplacement of between 0.3 and 0.4 Å is necessary to fitrepulsive wall to an ellipse at a classical turning point1000 cm21. The saw tooth shape of the coefficients~smallerodd terms! indicates that for this choice of origin, the coodinates more correctly represent the symmetry of the potial. For a perfect ellipse, only even terms appear in thepansion. The variation in convergence between these separameters is significant. For the matrix inversion techniql can be interpreted as the number of angles in Eq. 7. Foab initio surface, optimizing the choice of origin could dcrease the number of angles necessary to describe the ptial.
C. Surface augmentation
Once an analytical representation was obtained, two npotential surfaces were generated by adjusting the expancoefficients of Eq.~8!. This allows the augmentation proceto retain the same level of flexibility as the original fittinfunctions. In the present work the coefficientscmn
(2i ) weremodified until the potential reproduced observed bindenergies64–68 for the van der Waals complex to within thdesired tolerance. The resulting surfaces differ in the formEq. ~8! as shown by
FIG. 3. The magnitude of the angular expansion coefficients for the Thoson ~Refs. 18,19! potential. The radius is held constantRn52.11 Å for ndifferent polar coordinate origins. Numbers on the graph represent theplacement of the origin along the bond from the center-of-mass witpositive displacement toward the hydrogen. The coefficients for an origthe center-of-mass have been emphasized. The plot for 0.4 Å shows sture ~smaller odd terms! that is due to the presence of symmetry.
re
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f
Bmn1 ~R!5Bmn~R!2tanh~R!(
i 53
5
dm~2i !R22i ~12!
Bmn2 ~R!5Bmn~R!2tanh~R!(
i 53
5
dmn~2i !R22i , ~13!
where the augmentation ofBmn1 (R) depends only on the
angleu, and the augmentation ofBmn2 (R) depends on both
the r andu. The coefficientsdmn were assumed to vary linearly with r n . Three-dimensional potentials are generafrom Eq. ~12! and Eq.~13! via Eq. ~6!.
The two augmented potentials were constructed in silar ways. TheV1 potential was augmented by adding an agular dependent dispersion interaction. TheV1 potential wasmade to reproduce the geometry and depth of the minbased on the highly accurateab initio calculations of Mourikand Dunning48 while reproducing the experimental spectrscopic bound state energies for the first five HF vibratiostates of the van der Waals complex.64–68,83,84TheV2 poten-tial was augmented by adding a dispersion interaction thaa function of the angleu and of the HF bond length. TheH6~4, 3, 2! potential45 was used to determine the anguldependence of the coefficientsdmn from Eq. ~13!. TheV1,1(R,u) element of the augmented potential was compato the corresponding HF(v51) surface of the H6~4, 3, 2!potential. A least squares algorithm was used to minimizedifference between the potentials over all eleven anglesfrom R53.0 Å to R55.0 Å. In this way information fromthe high quality vibrationally averaged potential H6~4, 3, 2!was included during the construction of the 3-D potentiaThe angular dependence was determined from the H6~4, 3,2! potential first, followed by the bond length dependenThe bond length dependence was assumed to be lineardetermined by fitting the van der Waals complex bound senergies for the first three HF vibrational states. The statiary points of the potential energy surfaces are givenTable I.
D. Potential scaling
A third potential was obtain by scaling the MP2 potentby a constant factor
V351.23VMP21Vdisp~u!. ~14!
The coefficient was chosen to bring the van der Waals coplex well depths into agreement with Mourik and Dunning48
Spectroscopic binding energies for the van der Waals cplex were then fitted by adding an angle-dependent dission interactionVdisp(u) using Eq.~12!. The angular dependence of this interaction was designed to broaden the wThe stationary points of the potential energy surfacegiven in Table I.
E. Bound state calculations
In order to calculate the spectroscopic properties frthe potentials and to perform quantum scattering calcutions, a potential coupling matrix is required. The matrix wobtained from expectation values
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4578 J. Chem. Phys., Vol. 115, No. 10, 8 September 2001 Shroll, Lohr, and Barker
Vv,v8~R,u!5^v,J50uV~R,u,r !uv8,J50&, ~15!
whereuv,J50& are the diatomic vibrational wavefunctionThe vibrational wave-functions needed in Eq.~15! were de-termined from Numerov integration85,86 of RKRpotentials87–89VHF
RKR using both software developed in houand the software programLEVEL.90
The programBOUND91,92 was used to perform closecoupling calculations of the bound state energies and totermine the binding energiesD0 , the energy differencesEJ512EJ50 , and the approximate centrifugal distortioconstant DJ5@2EJ5213EJ5122EJ50#/24. Calculationswere performed in the manner described by Changet al.,who performed bound state calculations on the H6~4, 3, 2!potential.93 Spectroscopic properties were calculated ign
TABLE I. The potential energy surface stationary points for the total pottial Vtot5VHF
RKB1V(R,u,r ). The ab initio data were obtained from Mourikand Dunning.a
ing vibrational coupling, since the off-diagonal elementsthe vibrational matrix do not greatly affect the eigenvalues45
The spectroscopic properties calculated for all of thetentials are shown in Table II along with their experimencounterparts. The unmodified MP2 surface recovered 66%the binding energyD0 , which is even less than the 80%recovery if measured from the bottom of the wellDe .48 Allof the augmented potentials reproduce the spectroscopicreported in Table II significantly better than did the originMP2 surface.
III. ROVIBRATIONAL ENERGY TRANSFER
The HF–Ar collisional energy transfer may be reprsented by the following second-order reaction
HF~v i ,Ji !1Ar ——→k~v i ,Ji ,v f ,Jf ;T!
HF~v f ,Jf !1Ar, ~16!
where v is the vibrational quantum number,J is the rota-tional quantum number,k(v i ,Ji ,v f ,Jf ;T) is the temperaturedependent rate constant,i labels initial values, andf labelsfinal values. The rate constant is expressed in ucm3/molecule•sec and is related to the reaction rate by
d@HF~v f ,Jf !#
dt52
d@HF~v i ,Ji !#
dt
5k~v i ,Ji ,v f ,Jf ;T!@HF~v i ,Ji !#@Ar#,
~17!
where square brackets denote number density in unitsmolecule/cm3. The energy transfer rate constants are relato the state-to-state cross sectionss(v i ,Ji ,v f ,Jf ;Ek) usingthe Maxwell-Boltzmann distribution
-
4654894
TABLE II. Spectroscopic properties of the potential energy surfaces. The binding energiesD0 , state energydifferencesEJ512EJ50 , and a centrifugal distortion constantDJ are presented.a
aFor a description of the quantum numbers characterizing the states of the complex see Hutson~Ref. 45!.bSee Refs. 64–68.
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4579J. Chem. Phys., Vol. 115, No. 10, 8 September 2001 Energy transfer of hydrogen fluoride
k~v i ,Ji ,v f ,Jf ;T!
51/kTS 8
pmkTD 1/2E0
`
Eks~v i ,Ji ,v f ,Jf ;Ek!
3exp~2Ek /kT!dEk , ~18!
whereEk is the initial relative translational energy andm isthe HF–Ar reduced mass. Rate constants for vibrationallaxation are obtained by averaging the state-to-state ratestants over the initial rotational states and summing overfinal rotational states.
The experimental measurements cited herein havebeen carried out on mixtures of HF and Ar in which the Hmolecules undergo multiple collisions before being vibtionally deactivated. Since the cross sections for rotatioenergy transfer are much larger than those for vibratiodeactivation, the rotational states are well-represented bBoltzmann distribution. Therefore, the energy dependcross section for vibrational energy transfer is given by
s~v i ,v f ;Ek!51
q (Ji
open
~2Ji11!
3exp~2EJi/kT!s~v i ,Ji ,v f ;Ek!, ~19!
where
q5(Ji
~2Ji11!exp~2EJi/kT!, ~20!
EJiis the energy of theJi state, and
s~v i ,Ji ,v f ;Ek!5(Jf
open
s~v i ,Ji ,v f ,Jf ;Ek! ~21!
is the cross section summed over all open final states.thermal rate constant for vibrational energy transfer is givby
k~v i ,v f ;T!51
kT S 8
pmkTD 1/2E0
`
Eks~v i ,v f ;Ek!
3exp~2Ek /kT!dEk . ~22!
A. Quantum scattering calculations
All quantum scattering calculations were performeding the HIBRIDON94–96 software package developed by MH. Alexander and co-workers. For details, see the HIBDON distribution literature. Convergence of the calculatcross sections was verified with respect to all relevant pareters and was better than 2%. A hybrid log-derivative/Apropagator was used.97 Log-derivative integration was carried out from 4.0 a0 to 30.0 a0 with an interval of 0.05 a0.Airy integration was used from 30.0 a0 to 100.0 a0 with avariable interval. A step size of 5 was used for the orbangular momentum in the CS calculations. The reduced mof the 1H19F and40Ar system is 13.331 917 a.u. The vibrational coupling matrices were approximated as shown in~15!. This approximation is expected to be least accuraterate constants below room temperature; discussion of itslidity may be found elsewhere.53,98,99
e-n-ll
ll
-alal
at
hen
-
-d
-
lss
q.r
a-
For comparison with experimental results of Chapmet al.,63 rotational cross sections were determined by solvthe close-coupled quantum scattering equations for 35cm21 relative translation energy. TheV0,0(R,u) element ofthe potential coupling matrix was used from Eq.~15!. Thematrix element represents a vibrationally averaged potenthat is appropriate for rotational energy transfer at low retive velocity.
The energy dependent rovibrational energy transfer crsections, necessary for calculating the vibrational rate cstants, were determined using the CS approximation.28,100–103
Using this approximation, the state-to-state cross sections
s~v i ,Ji ,v f ,Jf ;Ek!5(V
sV~v i ,Ji ,v f ,Jf ;Ek!, ~23!
whereV denotes the projection of the total angular mometum on the body-fixed quantization axis and the sum is oall values of the projection quantum number such thatuVu<min(Ji ,Jf). The CS approximation has been shown tohighly accurate for rovibrational energy transfer of atodiatomic molecule collisions, except at very low relative vlocities. For neutral systems, if the relative translationalergy of both the initial and final states is greater than the wdepth then the CS approximation is expected to be valid104
Comparison between the CS approximation and clocoupling results shows no distinguishable differences forvibrational deactivation cross section of He1H2,
105,106a sys-tem less anisotropic than HF–Ar. Kouri provides a detaidescription of the CS approximation with its strengths aweaknesses.104 Rovibrational energy transfer cross sectiowere calculated over a range of total energies from 611716512 cm21 relative to the bottom of the hydrogen fluoridintermolecular potential well.
B. Rotational energy transfer
The close-coupled rotational activation calculationssummarized in Table III for a relative translational energy350 cm21. The experimental and theoretical results of Chaman et al.,63 corresponded to a narrow distribution of reltive translational energies centered at 350 cm21. In theirwork, the width of the distribution opens theJf54 channel,which lies slightly above 350 cm21. Their experimental re-
TABLE III. Relative cross sections for rotational activation of HF by Awith a relative translational energy of 350 cm21. Experimental values are
given by Chapmanet al.a The results reported as H6(4,3,2) were deter-mined from the H6~4,3,2! potential by Chapmanet al. by using a615%distribution of energies about the relative translational energy.
Total cross section~Å2!25~5! 21.3 21.10 16.89 20.24 15.53 17.5
aSee Ref. 63.
ns
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4580 J. Chem. Phys., Vol. 115, No. 10, 8 September 2001 Shroll, Lohr, and Barker
sults and their theoretical calculations based on HustoH6~4,3,2! potential45 are summarized in Table III by columnlabeled Expt. and H6~4,3,2! respectively. All of the otherdata presented in Table III were calculated in the presstudy. A direct comparison with the experimental measuments may be made without taking into account the effecthe energy distribution, which is very narrow.
It is evident from the table that all of the potentials giqualitatively similar results. The results obtained with theV2
potential agree with the experiments to within reported erTheV2 potential is most accurate for the calculation of rotional cross sections and therefore it was used for all ofcalculations presented for rotational energy transfer crsections and rate constants. State-to-state rotational entransfer rate constants were obtained using the CS appmation. An example of these rate constants as a functiotemperature is shown in Fig. 4. The rate constants arevibrationally elastic collisions of Ar with HF(v51) whererotationally elastic terms have been omitted in order to shthe relatively smaller inelastic terms more clearly. The figushows a rapid decrease in the rate constant as a functiouDEHFu5uEHF(v f ,Jf)2EHF(v i ,Ji)u. The exponential character of this decrease is evident for both upward and doward transitions shown in Fig. 5.
There has been much effort to create empirical modand fitting functions for rotational rate constants.107–124Langet al., observed rotational relaxation of HF in HF–Ar mixtures and fitted their data to a simple exponential mode125
Their observations are consistent with ours: At a given teperature, higher J levels have lower probability of rotatiodeactivation, downward transitions are favored over upwones for comparableuDEHFu, and the rate constants decreaas uDEHFu increases.
The power-gap law has been used to model the matriJi→Jf integral cross sections,111 but it is not accurate for thecompleteJf-distribution.126 It has been shown that at leatwo sets of parameters are needed to fit the data:108 One setfor uDEHFu<uDEHFu* and one set foruDEHFu.uDEHFu* ,
FIG. 4. The rotational rate constantsk(v i51,Ji ,v f51,Jf ;T) for vibra-tionally elastic collisions of HF(v51) and Ar as a function of temperaturand the change in internal energyEHF(v f51,Jf)2EHF(v f51,Ji) in wavenumbers. The change in internal energy depends on the initial androtational angular momentum quantum numbersJi andJf . Temperature isvaried from 100 to 1500 K in increments of 100 K. For each temperatlines are plotted representing the rate constants for all open channelsJfÞJi .
’s
nt-f
r.-essrgyxi-ofor
weof
n-
ls
-ld
e
of
where uDEHFu is the energy gap between initial and finrotational levels anduDEHFu* is the intersection of the regressions~indicated by the vertical line in Fig. 6!. Thepower-gap law for the cross sections can be written,
s~v i ,Ji ,v f ,Jf ;Ek!5a~2Jf11!~Ekf /Ek!
1/2uDEHFu2g,~24!
wherea and g are fitting parameters andEkf is the kinetic
energy of the final state. It follows from this equation that fone set of parameters$a,g% a plot of ln@sV(vi ,Ji
50,v f ,Jf ;Ek)(Ek /Ekf )1/2/(2Jf11)# versus lnuDEHF u should
yield a straight line. An example of such a plot is presenin Fig. 6, where two sets of parameters were determinedseparate linear regressions over the regions to the leftright of the vertical line. The plot is for pure-rotational tran
al
,ith
FIG. 5. The rotational rate constantsk(v i51,Ji510,v f51,Jf ;T) for vibra-tionally elastic collisions of HF(v51) and Ar as a function of temperaturand the change in internal energyEHF(v f51,Jf)2EHF(v i51,Ji510) inwave numbers. Numbers located above data points represent the finaltional angular momentum quantum number for the series of temperatData for the rotationally elastic transitionJi5Jf510 have been excluded.
FIG. 6. ln@sV50(vi51,Ji50,v f51,Jf ;Ek)(Ek /Ekf )1/2/(2Jf11)# vs lnuDEHFu
for a relative translational energy of 12 878 cm21 with respect to theHF(v51,J50) state energy. Circles represent final rotational angular mmentum states and the vertical line represents the boundary between ccally allowed and classically forbidden transitions. It is estimated fromintersection of the two power-gap law data fits.
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4581J. Chem. Phys., Vol. 115, No. 10, 8 September 2001 Energy transfer of hydrogen fluoride
sitions with Ji50 andv51. It is clear that a single set oparameters could not fit the data well over the entire ran
It should be emphasized that not all energetically opchannels are classically allowed.107 Some transitions areclassically inaccessible due to momentum conservation cstraints and from limitations placed by the topology of tsurface on angular momentum change. Classically theremaximum allowable value for the torque-arm~effective im-pact parameter! in the impulsive limit. At sufficiently highrelative velocities, angular momentum conservation is mrestrictive than energy conservation, hence a channel ca‘‘open’’ as far as energy is concerned, and yet be classicforbidden.109,121
The angle dependent classical turning point at a relatranslation energy of 12 878 cm21 for the HF–Ar intermo-lecular potential was fitted to an ellipse, as described abThe center of the ellipse was found to be displaced 0.0from the center-of-mass toward the hydrogen and the semajor and semiminor axes were 1.092 and 1.034 Å, resptively. For the ellipsoid model, the maximum classical limof the angular momentum transfer is,110
~DJ!max5A2m~AE1AE8!~A2B! ~25!
whereE andE8 are the initial and final translational kinetienergies,A andB are the semimajor and semiminor axes, am is the reduced mass. Using the ellipsoid model for HF–at this energy, the maximum classically allowed changeangular momentum is (DJ)max516, which corresponds toJf516 in Fig. 6.
Agrawal et al., have shown that the intersection of thregressions in Fig. 6 provides an estimate of the boundbetween the classically allowed and classically forbiddregions.110 The parameters for the power-gap law taken frothese regions areg51.311, lna56.065, andg533.79, lna5290.6, respectively. This yields an intersection at~8.762,25.422! and a classical limit of 6385 cm21 above the inter-nal energy of thev51, Ji50 state. This agrees well with thellipsoid model since it also predictsJf516 to be the lastclassically allowed rotational level. The classically forbiddregion extends 6492 cm21 past the classical limit and includes 8 channels that are open with respect to energy,closed with respect to angular momentum. The cross sectdecrease rapidly with increasinguDEHFu in the classicallyforbidden region.
C. Vibrational energy transfer
In order to determine vibrational energy transfer raconstants, collisional cross sections were calculated betwthe first five vibrational states of hydrogen fluoride usingV2 potential. Cross sectionss(v i ,Ji ,v f ,Jf ;Ek) were calcu-lated for all of the open rotational states of each vibratiolevel. Examples of these cross sections for two relative tralational energies are shown in Fig. 7, where a line indicathe locus of V-R state changes that occur without an assated change in internal energy or relative translational ene(uDEHFu5uDEku50). Transitions close to this line exhibprimarily V-R energy transfer where the change in vibtional energy is accompanied by a change in rotational
.n
n-
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e
e.Åi-c-
drn
ryn
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ene
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ergy of nearly identical magnitude, but of opposite sigthere is virtually no change in translational energy. Tgraphs show that V-R energy transfer is strongly favoredthis system, in agreement with previous findings.18,19For ac-tivation, for deactivation, and for transitions where the vibtional state changes by more than one quantum numberenergy transfer was found to be favored.
To obtain vibrational energy transfer rate constancross sections from an initial rotational stateJi were summedover the set of all open final states@Eq. ~21!#. Calculationswere performed for a range of relative translational energEk resulting in a cross section that depends on the energythe initial rotational state. An example of these cross sectiis shown in Fig. 8 forJi50. For lower energies the crossection exhibits structure that has been associatedresonances.127–133 These resonances invert the slope of tcross section as a function of kinetic energy and some invidual resonances are evident as spikes on plots of this fution. Qualitatively these resonances are due to quasi-bostates as the Ar orbits the HF in the region of the attractwell. However, a full characterization requires analysis127 ofthe S matrix, which has not been attempted here.
The cross sectionss(v i ,Ji ,v f ;Ek) for Ar1HF ~v52and 4;Ji54, 6, 8, and 10! were calculated by Thompson19
for a relative translational energy of 5251 cm21. From hisdata, Thompson concluded that a relatively small increasrotation causes a dramatic increase in the cross sectionboth upward and downward transitions. In a later publicat
FIG. 7. State-to-state vibrational deactivation cross sections of HF(v51)by Ar. The cross sectionss(v i51,Ji ,v f50,Jf ;Ek) are functions of therelative translational energyEk and depend parametrically on the initiarotational angular momentum quantum numberJi and on the final quantumnumberJf . The top graph is for a relative translational energy of 2651cm21 and the bottom is for 6469.70 cm21. A line in thex2y plane of bothgraphs is the locus of V-R state changes that occur withuDEHFu50.
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4582 J. Chem. Phys., Vol. 115, No. 10, 8 September 2001 Shroll, Lohr, and Barker
Thompson showed that the trend was not well-behaved,observed that the cross sectionss(v i ,Ji ,v f ;Ek) are stronglydependent on the initial rotational statesJi .18 The presentresults are in qualitative agreement with Thompson. Tmagnitudes of the cross sections are significantly differfor varying Ji at constant relative translational energy,shown in Fig. 9 for energies in the neighborhood of thinvestigated by Thompson. The ordering of the cross sectoften changes as a function of relative translational ener
The predicted vibrational deactivation rate constantsHF(v51) as a function of temperature are shown in Fig.for all four potentials. From the figure it is apparent thatof the rate constants are very similar. This is in spite ofdifferent modifications made to the repulsive walls of tpotentials and indicates that the calculated results arevery sensitive to the kind of PES enhancing techniquesployed here. The original unmodified MP2ab initio surfaceperforms well and predicts the lowest cross sections.
FIG. 8. The cross sectionsV50(v i51,Ji510,v f50;Ek) as a function of therelative translational energy for the vibrational deactivation of HF(v51) byAr. Cross sections are summed over all finalJf states. The graphical inseshows the low-temperature region of the plot expanded.
FIG. 9. The cross sectionssV50(v i ,Ji ,v f ;Ek) as a function of relativetranslational energy for the vibrational deactivation of HF(v51) by Ar.Cross sections are summed over all finalJf states. Cross sections forJi
.10 and odd cross sections have been excluded to enhance graphicaity.
ut
et
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predicted vibrational deactivation rate constant for HFv51) as a function of temperature is shown in Fig. 11, whthe theoretical rate constant obtained with theV2 potential ispresented along with the available experimental data.
As discussed previously, experimental measuremenHF(v) deactivation by Ar is very difficult at low temperature. Quantitative experimental values only extend down800 K and there are large differences in reports from diffent groups. The reported experiments at 294 and 350 K wonly able to provide upper limits. In the present work, it wpossible to evaluate theoretical rate constants from be100 K to as high as 1500 K. Our calculated results aregood agreement with the available experiments, givenspread of the measurements.
At low temperature, the cross-section resonances cathe rate constant to pass through a maximum located aro
lar-
FIG. 10. A comparison of the vibrational deactivation rate constants forfour potentials.
FIG. 11. The vibrational deactivation rate constant for HF(v51) collisionswith Ar as a function of temperature. Theoretical data is represented
, the experimental data of Blairet al. ~Ref. 15! is represented by ,the experimental data of Vasil’evet al.17 is represented by • • • , andthe experimental data of Bott and Cohen~Ref. 10! is represented by• • • •.The experimental rate constant upper limits of Hancock and Green~Refs.11,16!, Hinchen~Ref. 14!, Airey et al. ~Ref. 9!, and Friedet al. ~Ref. 13! atlower temperature are represented bys. Hinchen reported the same valufor the upper limit at 295 K as did Hancock and Green, and, therefore, tis only one circle for both. The graphical inset expands the low-temperaregion.
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4583J. Chem. Phys., Vol. 115, No. 10, 8 September 2001 Energy transfer of hydrogen fluoride
9.4 K, as shown in an expanded scale~see the graphical insein Fig. 11!. The calculated rate constants are less accuratemperatures below 100 K due to limitations of the couplstates approximation and the influence of the resonanHowever, the description of thev51 to v50 transition inthe 0 to 100 K temperature regime is in good qualitatagreement with close-coupled calculations that we pformed.
IV. CONCLUSIONS
Theoretical vibrational energy transfer studies requireaccurate description of the intermolecular potential enedependence on the diatomic bond length. The inversionexperimental data is often technically challenging andquires spectroscopic data or collisional energy transfer dthat may not exist or may only describe part of the necesspotential surface. The production of theoreticalab initio sur-faces is technically and computationally demanding becahighly correlated quantum chemistry methods must beployed with large basis sets for many nuclear configuratioSpectroscopically accurate surfaces still only exist for a fsystems. The approach taken here was to combine a comtationally modestab initio method with a simple yet effective inclusion of experimental data. The results show thatmethod is capable of reproducing experimental observatiwhile at the same time producing a wealth of new detaiinformation.
All three modified potential energy surfaces perform bter than the original MP2 surface for the calculations psented here. Of the three modified surfaces, none is clesuperior for all applications. TheV2 potential more closelyreproduces the experimental binding energies and rotatiactivation cross sections. However, this potential underemates the depth of the global minimum atu50° ~by ;22cm21! and overestimates the depth of the local minimumu5180° ~by ;10 cm21, see Table I!. TheV1 andV3 poten-tials reproduce the stationary points more accurately anderwise are similar to each other, with only modest differenin the calculated properties.
Accurate determination of the rotational energy transcross sectionss(v i ,Ji ,v f ,Jf ,Ek) and s(v i ,Ji ,v f ,Ek) re-quired inclusion of an angle-dependent augmentation ofPES. All of the augmented potentials were in good agrment with spectroscopic measurements and with experimtal scattering cross sections. The rate constants show mcharacteristics which are consistent over the temperatstudied: HigherJ levels have lower probability of rotationadeactivation, downward transitions are favored over upwones for comparableuDEHFu, and the rate constants decreaasuDEHFu increases. However, the calculated rotational crsections show complicated behavior and even the relamagnitudes of the state-to-state rate constants depentranslational energy.
The magnitudes of the state-to-state rate constantssubject to both energy conservation and momentum convation, and they may be limited by either one. The costraints placed on the rotational cross section by momenaffect the rotational energy transfer rate constants as wethe vibrational energy transfer rate constants for the ene
at-s.
r-
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se-
s.
pu-
iss,d
--rly
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r
e-n-nyes
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range studied. The constraints have the effect of loweringcross sections and therefore the rate constants. Theycomplicate the use of empirical fitting functions like thpower-gap law.
The vibrational energy transfer rate constants showlittle sensitivity to the specific augmented potentials, all throf which give rate constants within a factor of two of thofor the MP2 surface. The calculations reaffirm that enetransfer for the HF–Ar system is primarily V-R with littlechange in internal energyuDEHFu. This is the case for up-ward or downward vibrational transitions, transitions whethe vibrational state changes by more than one quannumber, and for purely rotational transitions as well. Fdownward vibrational transitions this would result in a noBoltzmann increase in the population of the higher rotatiolevels and the opposite trend for the upward transitions. Tvibrational rate constants for HF are in agreement withperimental values and predictions have been made inlow-temperature regime where experimental measuremare lacking.
ACKNOWLEDGMENTS
The authors thank Jeremy M. Hutson for providingwith the H6~4, 3, 2! potential in the form of FORTRAN codeand Millard H. Alexander for assistance in porting his quatum scattering code HIBRIDON to the Linux operating sytem. The authors thank Roman Krems and Alexei A. Buachenko for technical advice and helpful discussconcerning the diatomic-in-molecule potential. Finally tauthors thank M. G. Mlynczak for discussions and NAS~Office of Space Sciences! for funding.
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