Ab initio potential energy surface for HeF2 in its ground electronic state

Chemical Physics 308 (2005) 277–284 www.elsevier.com/locate/chemphys

Ab initio potential energy surface for HeF2 in its ground electronic state U. Lourderaj, N. Sathyamurthy

*

Department of Chemistry, Indian Institute of Technology, Kanpur 208 016, India Received 6 March 2004; accepted 5 May 2004 Available online 2 July 2004 Dedicated to the memory of Gert D. Billing

Abstract The ground state potential energy surface for He–F2 has been generated using the coupled-cluster singles and doubles excitation approach with perturbative treatment of triple excitations [CCSD(T)] and multi-reference configuration interaction (MRCI) methodologies, with augmented correlation consistent quadruple zeta basis set and diffused functions. Both the CCSD(T) and MRCI surfaces are compared and the results analyzed. The CCSD(T) surface exhibits van der Waals minima at different distances for different orientations of He approaching F2 and is adequate to describe accurately only in the region around the equilibrium bond distance of F2 . The MRCI surface, on the other hand, yields reliable results for a wider range of F–F bond distances leading to the correct asymptote. Davidson correction to the MRCI surface makes it purely repulsive over the regions investigated. Ó 2004 Elsevier B.V. All rights reserved. Keywords: He–F2 ; Potential energy surface; Coupled-cluster; Configuration interaction

1. Introduction Rare gas halides form an important class of compounds because of their application in excimer lasers. The role of He and F2 in these lasers is well known. They are often the source for the generation of excimer lasers, in which He is the buffer gas and F2 , the halogen donor. The rare gas–halogen interactions at large intermolecular distances are of the weak van der Waals (vdW) nature and there have been extensive experimental as well as theoretical studies of some of these complexes [1,2]. Although there have been quite a few experimental studies of the van der Waals complexes of rare gases(Rg) with Cl2 , Br2 and I2 , studies on the Rg– F2 complexes have been rather limited, mainly due to the corrosive nature of F2 . Some of the shock wave studies have examined the vibrational relaxation time of F2 [3–6]. *

Corresponding author. Tel.: +91-512-597390; fax: +91-512-597436/ 414. E-mail address: [email protected] (N. Sathyamurthy). 0301-0104/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2004.05.031

Most of the theoretical studies till date on rare gas dihalides have focused on Cl2 containing complexes. There have been detailed studies on the potential energy surface (PES) of He–Cl2 [7,8], Ne–Cl2 [9] and Ar–Cl2 [10,11] using Møller–Plesset perturbation theory of order four (MP4). Naumkim and Knowles [12] modeled the potential energy functions of rare gas halides ArX2 (X ¼ F, Cl, Br, I) using empirical potentials for the diatomic fragments. Naumkin and McCourt generated the PES and predicted the microwave spectrum for He–Cl2 [13], Ne–Cl2 [14] and Ar–Cl2 [15] following the model proposed by Naumkin and Knowles. Cybulski and Holt [16] used the coupled-cluster singles and doubles approach with perturbative triple excitations [CCSD(T)] in their study of the PES for Rg( ¼ He, Ne, Ar)–Cl2 . The use of highly correlated methods such as the CCSD(T) is necessary to bring theory close to experiment. Surprisingly, there have not been many detailed studies of He– F2 interaction. Chan et al. [17] generated the PESs near the minimum for the rare gas difluorides RgF2 (Rg ¼ He, Ne, Ar) at the CCSD(T) level of theory. They used triple zeta augmented correlation consistent basis

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U. Lourderaj, N. Sathyamurthy / Chemical Physics 308 (2005) 277–284

sets of Dunning et al. [18–20] along with a set of bond functions in their study. They generated a surface of nearly 200 points corresponding to different orientations of He with F2 , for three different F2 bond lengths near its equilibrium bond distance. Their study revealed that for the HeF2 complex, the linear geometry is lower in energy than the T-shaped. Since their PES covers only the region near the equilibrium geometry of F2 , its usefulness in carrying out dynamical studies on the surface is limited. Very recently, Stoecklin et al. [21] have studied the vibrational deactivation of F2 by 3 He, using an ab initio surface generated near the minimum at the coupled-cluster [BCCD(T)] level of theory with Brueckner orbitals. Their surface shows similar trends as the one computed by Chan et al. [17] and the data points for 447 geometries were confined to three F–F distances near its equilibrium bond distance. We have been interested in studying He–F2 interaction over the entire range of geometries including the dissociation limit. As a first step, we investigated the ground and excited state(s) of F2 using time-dependent density functional theory [22]. As a second step, we have undertaken the investigation of the ground state PES for He–F2 . In Section 2, we discuss the methodology involved in the generation of the PES. The results obtained are presented and discussed in Section 3. Summary and conclusions follow in Section 4.

2. Methodology All the ab initio calculations for F2 interacting with He were performed using Dunning’s augmented correlation-consistent polarized quadruple zeta basis set with diffused functions (d-aug-cc-pVQZ). The PES was generated using three different approaches: (i) coupledcluster singles and doubles excitation, with noniterative perturbative treatment of triple excitations [CCSD(T)], (ii) multi-reference configuration interaction(MRCI) and (iii) Davidson corrected MRCI (MRCI + Q). Potential energy values were calculated in Jacobi coordinates ðR; r; hÞ, where R is the intermolecular distance from the center of mass of F2 to He, r is the F2 bond distance and h is the angle between R and r, describing the orientation of He with respect to the F–F bond axis. ˚ The CCSD(T) surface was generated for r < 2:0 A, because of the inadequacy in the coupled-cluster approach in describing the dissociation limits (see below). The energies were calculated for a total of 612 geometries in a three-dimensional (16
9
4) grid of
r ¼ 1:012ð0:1Þ1:812 A
and h ¼ 0 ð30Þ R ¼ 1:0ð0:2Þ4:0 A,
an additional set of calculations 90°. For R ¼ 3:38 A were carried out as the vdW minimum occurs at
R ¼ 3.38 A
and h ¼ 0°. r ¼ 1:412 A,

The He–F2 interaction energyðDEÞ was computed using the supermolecule approach DE ¼ EHeF2  EHe  EF2 ðre Þ ;

ð1Þ

where EHeF2 is the energy of HeF2 in an arbitrary geometry and EHe that of He and EF2 ðre Þ the energy of F2 in its equilibrium geometry. The MRCI calculations using the same basis set as mentioned above were carried out for R and h as indi
The reference cated above, but for r ¼ 1:0ð0:2Þ4:0 A. wavefunctions for the CI calculations were obtained from a complete active space self-consistent field (CASSCF) calculation with an active space of 9 orbitals, with 16 electrons distributed among those nine. The CI calculations included all the single and double excitations from the CASSCF space to recover the dynamic correlation energy. In order to correct for the sizeconsistency problem arising from an incomplete CI approach, Langhoff and Davidson [23] suggested the inclusion of a correction factor that accounts for the quadruple (DEquad ) excitation: DEquad ¼ ð1  C02 ÞDEDE ;

ð2Þ

where C0 is the coefficient of the SCF wave function in the CI expansion and DEDE is the correction arising from the double excitation. The Davidson corrected MRCI surface was constructed to see if there was any qualitative change in the PES. All the ab initio calculations reported in this paper were carried out using the MOLPRO suite of programs [24].

3. Results and discussion 3.1. CCSD(T) surface There have been extensive studies on the ground and excited states of F2 . For an overview of the studies till date, the reader may see [22]. The equilibrium bond length (re ) for F2 was found to be 1.41
in close agreement with the experimental value of A,
The variation of the F2 potential as a 1.412 A. function of the bond distance (r), reproduced in Fig. 1(a) shows a smooth behavior around the equilibrium distance but shows an artificial decline for
leading to a wrong dissociation limit. Alr > 2:0 A, though for many molecules this perturbative method [CCSD(T)] has been shown to yield good description of the ground state, it does not work for large values of r indicating the importance of dynamic correlation energy for an accurate description of the PES. Hence the PES using the CCSD(T) was generated for
r < 2:0 A. The diatomic potential of HeF obtained is reproduced in Fig. 1(b). It is repulsive in nature and it leads to the right asymptote.

U. Lourderaj, N. Sathyamurthy / Chemical Physics 308 (2005) 277–284

279

CCSD(T) 6

6

(a)

5 4

4

3

∆ E / eV

∆ E / eV

(b)

5

2

3 2

1 1

0

0

–1 –2

1

1.5

2

r

2.5

3

3.5

–1

1

1.5

2

2.5

3

3.5

4

rHe − / Ao F

o

F−F /A

Fig. 1. Potential energy curve for (a) F–F and (b) He–F, as computed using the CCSD(T) method.

The interaction energy for the He–F2 complex was found to be a minimum ()7.59 meV) at r ¼ re , R ¼ 3:38
and h ¼ 0°. The basis set superposition error (BSSE) A corrected interaction energy was found to be )4.25 meV, in agreement with the result obtained by Chan et al. ()4.42 meV) [17].

Contours of the interaction energy for HeF2 for different h values are plotted as a function of R and r in Fig. 2. For the T-shaped configuration (h ¼ 90°), DE
was found to be a minimum ()6.23 meV) at R ¼ 3:00 A,
Table 1 presents the DE values that show the r ¼ 1:41 A.

CCSD(T) 1.9

1.9

1

o

r/Α

1.8

θ=0

1.7

1.7

1.6

1.6

1.5

1.5

0

1.4

1.3

1.2

1.2

1.1

1.1 1

1.5

2

2.5

3

3.5

4

4.5

1.9

o

r/Α

0

1

1

1.5

2

2.5

3

3.5

4

4.5

1.9

1

1.8

θ = 60

o

1

1.8

1.7

1.7

1.6

1.6

1.5

θ = 90

o

1.5

0

1.4

1.3

1.2

1.2

1.1

1.1 1

1.5

2

2.5

3 o

R/ Α

3.5

4

0

1.4

1.3

1

θ = 30˚

1.4

1.3

1

1

1.8

o

4.5

1

1

1.5

2

2.5

3

3.5

4

4.5

o

R/ Α

Fig. 2. Potential energy contours (eV) for He–F2 interaction in ðR; rÞ space, as computed using CCSD(T) level of theory. Contours are spaced by 0.5 eV. Two of the contours are labeled for quick reference.

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Table 1 Interaction energies (meV) for He–F2 at the CCSD(T) level for dif
ferent values of R and h at r ¼ re (1.412 A)
R (A)

h 0°

30°

60°

90°

9581.48 4602.52 2059.29 863.30 338.18 119.82 34.20 3.35 )6.05 )7.59 )7.58 )6.63 )5.23 )4.01

7350.01 3605.63 1629.76 697.87 283.08 106.22 34.20 6.69 )2.66 )4.89 )4.97 )4.83 )4.06 )3.24

2497.99 1156.88 513.87 217.04 85.02 28.78 6.19 )1.96 )4.23 )4.35 )4.32 )3.74 )3.05 )2.44

835.60 355.61 140.81 48.75 11.52 )2.09 )5.97 )6.23 )5.38 )4.44 )4.34 )3.42 )2.70 )2.17

1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.38 3.40 3.60 3.80 4.00

shift in the location of the minimum as h is varied from 0° to 90°. Since the inclusion of BSSE correction in the PES for all the computed geometries is a fairly big task at the CCSD(T)/d-aug-cc-pVQZ level, the same has been calculated for a limited range of geometries: h ¼ 0°,
and r ¼ 1:212ð0:1Þ1:612 A.
The reR ¼ 1:0ð0:2Þ4:0 A sults are shown in the form of contours in ðR; rÞ space in Fig. 3. It is clear that when the helium atom is away from F2 , the BSSE is almost zero. As the helium atom gets closer to F2 , the correction factor becomes larger. For many geometries, the BSSE is found to be of 1%. Hence it was not computed for the other geometries.

1.65 0.002

0.006

1.6 1.55

r/ A

1.5 o

1.45 1.4 1.35 1.3 1.25 1.2 1

1.5

2

2.5

3

3.5

4

o

R/A Fig. 3. Contours of BSSE in ðR; rÞ space for h ¼ 0°. The values increase from right to left by 0.002 eV as indicated. Two of the contours are labeled for quick reference.

Only the BSSE-uncorrected results are reported in this paper. He and F2 are closed shell species. Therefore, their short range interaction is expected to be exponential (Pauli repulsion). Plots of ln V vs R (not shown) do show straight line behavior for small R, for r ¼ re and h ¼ 0°, 30°, 60°, 90°. It becomes clear from the plots that the interaction is the softest for the sideways approach of He to F2 (h ¼ 90°).

3.2. MRCI surface MRCI methods are known to describe the ground state properties of F2 accurately [22,25]. They give a good estimate of the dissociation energy and lead to the correct dissociation limit. Therefore, we have computed the PES for He–F2 using the MRCI method with the valence active space. The resulting potential energy curve for F2 for varying bond distances is shown in Fig. 4(a). The disso
ciation energy of F2 ðDe ¼ VF2 ðr ¼ 100:0 AÞ VF2 ðr ¼ re ÞÞ was found to be 1.51 eV, as compared to the experimental value of 1.66 eV [26]. The potential energy curve for HeF was also generated using the MRCI method and valence active space orbitals. The results plotted in Fig. 4(b) show that the HeF interaction is repulsive over the entire range of r. Contour plots of the potential energy surface in ðR; rÞ space as obtained from MRCI calculations for four different orientations (h ¼ 0°, 30°, 60° and 90°) are shown in Fig. 5. The PES is purely repulsive in nature. There are some spurious maxima arising at some points on the PES, particularly for h ¼ 30° and their origin is not clear. The dissociation energy of F2 , if estimated from the triatomic potential ðDe ¼ VHe...F...F ðr ¼
 VHe...F ðr¼r Þ Þ, was found to be only 0.96 eV. 100:0 AÞ 2 e This is an outcome of the size-consistency problem of the MRCI methodology. To rectify such discrepancies, a correction factor to account for higher order excitation effects (beyond singles and doubles) in the MRCI wavefunction was introduced by Langhoff and Davidson [23]. The influence of the higher excitations on the diatomic potentials and on the full surface was investigated by us. The Davidson corrected MRCI potential(hereinafter referred to as MRCI + Q) for F2 is reproduced in Fig. 6(a). The potential energy curve shows a smooth behavior leading to the correct dissociation limit. The dissociation energy is found to be 1.62 eV, which is only 0.04 eV less than the experimental value of 1.66 eV. Fig. 6(b) shows the MRCI + Q potential energy curve for HeF. It is repulsive as was found with the CCSD(T) and the MRCI methodologies. The PES generated by the MRCI + Q approach is also repulsive in nature and some of the spurious features persist (not shown). The dissociation energy of F2

U. Lourderaj, N. Sathyamurthy / Chemical Physics 308 (2005) 277–284

281

MRCI 6

2

(b)

(a)

5

1.5 4 1

∆E / eV

∆ E / eV

3 2 1

0.5

0 0 –1 –2

1

1.5

2

2.5

–0.5

3

0.5

1

1.5

rF- F / A

2

2.5

3

3.5

4

rHe-F / Ao

o

Fig. 4. Potential energy curve for (a) F–F and (b) He–F, as computed using the MRCI method.

MRCI 4.5

4.5 o

θ =30

4

3.5

3.5

3

3

1

o

r /A

o

θ =0

4

1

2.5

2.5 2

2

0

1.5 1

1

1.5

2

2.5

3

3.5

4

0

1.5

4.5

4.5

1

1

1.5

2

2.5

3

3.5

4.5

4.5 o

θ =60

1

4

θ =90

4

3.5

3.5

3

3

2.5

2.5

2

2

o

1

o

r /A

4

1.5 1

1

1.5

2

2.5

3

3.5

4

o

R /A

0

1.5

0 4.5

1

1

1.5

2

2.5

3

3.5

4

4.5

o

R /A

Fig. 5. Potential energy contours of the MRCI surface for He–F2 interaction in Jacobi coordinates. Contours are spaced by 0.2 eV. Two of the contours are labeled for quick reference.

computed as HeF2 breaks into He + F + F is 1.49 eV, which is far more reasonable than the uncorrected MRCI estimate (see above). Table 2 gives the interaction energies of the He–F2
complex for different values of R and h at r ¼ 1:40 A.

We can see that there is no minimum in the PES, in contrast to what was predicted by the CCSD(T) surface. The anisotropy of the He  F2 interaction is illustrated in Fig. 7 by plotting the contours of the interaction
(close to energy as a function of R and h for r ¼ 1:40 A

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U. Lourderaj, N. Sathyamurthy / Chemical Physics 308 (2005) 277–284

MRCI+Q 5

2

(a)

(b)

4 1.5

2

∆ E / eV

∆E / eV

3

1

1

0.5

0 0 –1 –2

1

1.5

2

2.5

3

3.5

4

–0.5

0.5

1

1.5

rF - F / A ˚

2

r He

2.5

3

3.5

4

˚ -F / A

Fig. 6. Potential energy curve for (a) F–F and (b) He–F, as computed using the MRCI method including Davidson correction.

4 (a) CCSD(T)

R

3

o

y /A

re ). The potential varies smoothly from h ¼ 0° to 180°. At h ¼ 90°, there is a dip in the potential for small values of R indicating the ease with which He approaches F2 in a perpendicular geometry.

2

3.3. Analytic fit

1

F –4

–1

F 0

60°

90°

9481.33 4496.27 1954.97 781.92 844.60 605.58 452.88 406.57 388.24 381.39 379.08 378.44 378.35

7383.18 3521.17 1747.52 1221.66 741.94 532.73 442.63 404.44 388.59 382.25 379.83 378.97 378.72

3117.18 1681.23 980.60 648.05 493.83 423.78 392.57 378.96 373.22 370.88 369.99 369.71 369.66

1373.35 830.75 576.84 460.58 408.39 385.45 375.70 371.70 370.14 369.61 369.50 369.52 369.57

4

3

0

o

2 1

–3

–2

–1

0

1

2

3

4

0 4

He

(c) MRCI+Q

y R

y /A

30°

2

3

o

1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00

0°

1

(b) Analytic fit of CCSD(T)

Table 2 Interaction energies (meV) for He–F2 at the MRCI + Q level for dif
ferent values of R and h at r ¼ 1:40 A h

–2

4

–4


R (A)

–3

y /A

Any PES calculation of the type we have undertaken has to be followed by an analytic/numerical fit so that the dynamics on the PES could be investigated. Obtaining an accurate fit of the surface is not an easy task [27]. We have tried to use the many body expansion fitting procedure developed by Aguado et al. [28] to fit the CCSD(T) PES. The potential energy for a triatomic system ABC (A ¼ He, B ¼ F, C ¼ F) is expressed as a sum of three ð1Þ ð1Þ ð1Þ monoatomic terms ðVA ; VB ; VC Þ, three 2-body terms ð2Þ ð2Þ ð2Þ ð3Þ ðVAB ; VBC ; VCA Þ and a 3-body term ðVABC Þ:

θ

F

3 F

x

2 1

–4

–3

–2

–1

0

1

2

3

4

0

o

X /A Fig. 7. Potential energy contours as obtained from (a) CCSD(T) calculations, (b) the analytic fit of the CCSD(T) surface and (c) MRCI + Q calculations, plotted in ðR; rÞ space for different h
(b) values and in ðR; hÞ space for (a) r ¼ re ¼ 1:412 A,
and (c) r ¼ 1:4 A,
respectively. Contours are r ¼ re ¼ 1:412 A spaced by 0.5 eV.

U. Lourderaj, N. Sathyamurthy / Chemical Physics 308 (2005) 277–284

283

Table 3 Parameters (all in atomic units) obtained for the analytic fit of the CCSD(T) surface Two-body terms ð2Þ

ð2Þ

VHeF a ¼ 0:1000000E þ 01 b ¼ 0:5000000E þ 00

VFF a ¼ 0:1000000E þ 01 b ¼ 0:8600000E þ 00

i

ci

i

ci

0 1 2 3 4 5 6 7 8 9

)0.157678561773E-01 0.161350991103E-02 )0.210978923011E + 00 0.666444201143E + 01 )0.146549354900E + 02 0.217940461601E + 03 )0.153830589335E + 04 0.718648751921E + 04 )0.183123841271E + 05 0.220052625309E + 05

0 1 2 3 4 5

0.272441003676E + 02 )0.182378130011E + 01 )0.120586127180E + 02 )0.102571233334E + 03 0.403867006905E + 03 )0.748388535624E + 03

ijk

dijk

ijk

dijk

110 101 111 210 201 021 211 121 220 202 310 301 031 221 212 311 131 320 302 032 410 401 041 222 321 312 132 330 303 411 141 420 402 042 510 501 051 322 232

)0.5504749154206993E + 02 )0.4817959242152649E + 03 0.2337779966742512E + 05 0.8496093166151882E + 03 )0.4063661200850774E + 04 0.1635299317358956E + 04 0.1395776470406929E + 06 )0.3918148777211476E + 06 )0.1215242693089760E + 05 )0.7373371162674634E + 06 )0.1244262067127491E + 05 0.1959159215458189E + 06 )0.2045276617130156E + 05 )0.5459683383417798E + 06 )0.2303402466998508E + 07 )0.3856850387292801E + 06 0.3045759834746764E + 07 0.9321284244409615E + 05 0.1121258949410357E + 08 0.7385792326406456E + 05 0.4675429843559697E + 05 )0.6094609981019444E + 07 0.1382302222762481E + 06 0.2191703498557255E + 08 )0.9904858929574540E + 06 )0.3972312893503803E + 08 0.7056015120021741E + 05 )0.2925850950262873E + 06 0.2542563859334254E + 09 0.7047279503913300E + 07 )0.1237752925215042E + 08 0.7698196386598240E + 06 )0.1229358763424459E + 09 )0.2848825169272864E + 06 )0.2053988413462734E + 05 0.6690931629575076E + 08 )0.5238579818901329E + 06 )0.6173728926423030E + 09 0.1052794383055277E + 09

331 313 421 412 142 430 403 043 511 151 520 502 052 610 601 061 332 323 422 242 431 413 143 440 404 521 512 152 530 503 053 611 161 620 602 062 710 701 071

0.3405176643348099E + 08 0.4012868170461608E + 10 0.6992393902600347E + 08 0.3182806405367031E + 09 )0.7668972648049348E + 07 )0.3024075685038025E + 07 )0.5455669629792028E + 10 )0.1872790994334146E + 06 )0.8058792670971338E + 08 0.2649973931143199E + 08 )0.1269338968926766E + 08 0.1180549102394787E + 10 0.9068688554058558E + 06 0.1725047191607673E + 07 )0.3163648529851070E + 09 0.1032746225274528E + 07 )0.3795567503891069E + 09 0.8534789845594817E + 10 0.3470237614837435E + 10 )0.1607362132670536E + 09 )0.1174542234201069E + 09 )0.3340734990132598E + 11 )0.3261284759784712E + 08 )0.5759365697973461E + 06 0.9496738999546915E + 11 )0.6550351510368170E + 09 0.2506504632967104E + 10 0.2179797861681966E + 08 0.3561545569987375E + 08 0.7512653966067584E + 10 0.1339274856120024E + 07 0.2394859774568579E + 09 )0.2412295854796834E + 08 0.3155626851951985E + 08 )0.3266292431477140E + 10 )0.1374169537222445E + 07 )0.7033933029129469E + 07 0.5369904288442296E + 09 )0.8132270916476222E + 06

Three-body terms ð3Þ

VHeFF with M ¼ 8 bHeF ¼ 0:1479979287865189E þ 01 bFF ¼ 0:1009997092404095E þ 01

284

U. Lourderaj, N. Sathyamurthy / Chemical Physics 308 (2005) 277–284 ð1Þ

ð1Þ

ð1Þ

ð2Þ

VABC ðRAB ; RBC ; RCA Þ ¼ VA þ VB þ VC þ VAB ðRAB Þ ð2Þ

ð2Þ

ð3Þ

þ VBC ðRBC Þ þ VCA ðRCA Þ þ VABC ðRAB ; RBC ; RCA Þ; ð3Þ where RAB ; RBC and RCA refer to the A–B, B–C and C–A bond distances. The monoatomic energy is taken to be zero for each atom: ð1Þ VA

¼

ð1Þ VB

¼

ð1Þ VC

¼ 0:

ð4Þ

The diatomic potential is written as a sum of two terms, one representing the short-range and the other the long range potential: ð2Þ VAB

L c0  expðaAB  RAB Þ X ¼ þ ci qiAB ; RAB i¼1

ð5Þ

where qAB ¼ RAB  expðbAB  RAB Þ. ð2Þ ð2Þ A similar definition applies for VBC and VCA . The three-body term is written as a polynomial: ð3Þ

VABC ðRAB ; RBC ; RCA Þ ¼

M X

dijk qiAB qjBC qkCA ;

ð6Þ

ijk

where definitions of qBC and qCA are similar to that of qAB . The indices i; j and k vary from zero to a maximum value such that i þ j þ k 6¼ i 6¼ j 6¼ k;

ð7Þ

i þ j þ k 6 M:

ð8Þ

Parameters resulting from an analytic fit of the CCSD(T) surface are listed in Table 3. The fitted surface showed a root mean square deviation of 36.9 meV from the ab initio potential energy values. The maximum deviation of the fitted surface from the ab initio values was 143.6 meV. It becomes clear from plots of PE contours of the fitted surface (not shown) that they are well behaved for all the orientations in the interaction region for which the ab initio data are available. However, for regions where the ab initio data are not available, the fitted surface does not behave properly. There are artificial dips in the surface for h ¼ 30°, 60° and 90°
Efforts are under way to improve the for r > 2:0 A. analytic fit so that a dynamical investigation of collisioninduced dissociation in He–F2 could be undertaken. Fig. 7 depicts the perspective diagrams of the potential energy contours as He approaches F2 in its equilibrium geometry for (a) CCSD(T) PES, (b) analytic fit of the CCSD(T) surface and (c) MRCI + Q surface.

4. Summary and conclusion We have generated the ab initio PES for He–F2 in three dimensions, over an extended range of geometries, using CCSD(T), MRCI and MRCI + Q methods. It can

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