ab initio calculations and modeling of three-dimensional adiabatic and diabatic potential energy...

Ab Initio Calculations and Modeling ofThree-Dimensional Adiabatic andDiabatic Potential Energy Surfaces ofF(2P). . .H2(1��) Van der WaalsComplex*

JACEK KŁOS,1,† GRZEGORZ CHAŁASINSKI,1,2 M. M. SZCZESNIAK2

1Faculty of Chemistry, University of Warsaw, Pasteura 1 02-093 Warsaw Poland2Department of Chemistry, Oakland University, Rochester, Michigan, USA

Received 11 September 2001; accepted 7 March 2002Published online 16 August 2002 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.10328

ABSTRACT: Three lowest diabatic potential energy surfaces for the F(2P) � H2 Vander Waals complex are derived from accurate ab initio calculations of the T-shaped(C2v) and collinear geometries (C�v), at the coupled-cluster single, double, andnoniterative triple excitations [CCSD(T)] level of theory with a large basis set. For theintermediate geometries the angular dependence is modeled by a simple Legendre-polynomial interpolation. The nonadiabatic coupling (off-diagonal derivative) matrixelement, and the fourth, off-diagonal, diabatic surface are determined by separatemultireference configuration interaction (MR-CI) calculations with a somewhat smallerbasis set. Three adiabatic potential energy surfaces (PESs) are also obtained, bydiagonalizing the total Hamiltonian in the diabatic basis. Both the nonrelativistic andrelativistic (including spin-orbit coupling) PESs are evaluated. They are found to be invery good agreement with the entrance channel of the ASW PESs (Alexander,Manolopoulos, and Werner, J Chem Phys 2000, 113, 11084). The dependence of the PESson the H2 stretching coordinate is also incorporated and analyzed. © 2002 WileyPeriodicals, Inc. Int J Quantum Chem 90: 1038–1048, 2002

*This contribution is dedicated to the memory of our latecolleague, Staszek Kwiatkowski, renowned scientist, faithfulfriend, and noble, honorable man.

†Current address: Institute of Theoretical Chemistry, NSRIM,University of Nijmegen, Nijmegen, The Netherlands.

Correspondence to: G. Chałasinski; e-mail: [email protected]

Contract grant sponsor: National Science Foundation.Contract grant number: CHE-0078533.Contract grant sponsor: NATO.Contract grant number: CRG.LG 974215.Contract grant sponsor: Polish Committee for Scientific Re-

search KBN.Contract grant number: 3 T09A 112 18.

International Journal of Quantum Chemistry, Vol 90, 1038–1048 (2002)© 2002 Wiley Periodicals, Inc.

Ah, when to the heart of manWas it ever less than a treasonTo go with the drift of things,To yield with a grace to reason,And bow and accept the endOf a love or a season?

——R. Frost, Reluctance

Introduction

B ecause of its experimental accessibility, thereaction of F with H2 and its isotopomers has

become the paradigm for exothermic triatomic re-actions [1, 2]. The high-quality ab initio potentialenergy surface of Stark and Werner (SW) [3] hasbeen used in a number of quantum-scattering cal-culations [4–11], quasiclassical trajectory studies [5,12], and other investigations of the properties of theF-H2 system [13–15]. This theoretical work has suc-cessfully reproduced the major features seen inboth the photodetachment spectrum of the FH2

� ion[4, 13] and the molecular-beam scattering studies ofthe reaction of F with H2 [10], D2 [16], and HD [17].Recently, full framework for the quantum treat-ment of reactions of the fluorine atom with molec-ular hydrogen was developed [2] and involved fourpotential energy surfaces (PESs) and two coordi-nate-dependent ab initio–derived spin-orbit inter-action terms. This pioneering study establishedsmall overall reactivity of the excited (2P) spin-orbitstate of F (which is not allowed adiabatically) andled to the conclusion that the dynamics of the reac-tion will be well described by calculations on asingle, electronically adiabatic PES.

Rapid and remarkable progress in quantifyingthe details of the F � H2 reaction brings about theissues previously deemed of either no or secondaryimportance. One such issue is the role of the Vander Waals interactions in the entrance channel. Thistrend has been explicitly expressed by Skouteris etal.: “The study of chemical reaction dynamics hasnow advanced to the stage where even compara-tively weak Van der Waals interactions can nolonger be neglected in calculations of the potentialenergy surfaces of chemical reactions” [18]. Thisstatement was prompted by the authors’ establish-ing in the reaction of the Cl atom and HD thatstrong preference of production DCl versus HClcan be directly traced to the existence of a shallowVan der Waals well in the entrance channel.

A beneficial role of the H2 rotation in promotingthe reaction was already established on the SW

surface favoring bent transition state [19, 20]. Re-cently, Balakrishnan and Dalgarno [21] showed thatin the low temperature limit its rate coefficient iscontrolled by the attractive Van der Waals interac-tion. In addition, the latter work proved the rate tobe sensitive to the details of the PES in the entrancechannel, especially the heights of the entrance chan-nel barrier and the depth of the Van der Waals well.

To accurately quantify and model the entrancechannel, it is necessary to have accurate ab initiodescription of the Van der Waals interaction thatdetermines PES in this region. For closed-shell sys-tems, such an accurate treatment has been achievedfor many model complexes. However, for open-shell systems, the necessity to consider severalstates at a time presents a serious challenge thatinvolves nonadiabatic and spin-orbit couplings. Aninteresting alternative to modeling of such hyper-surfaces within the empirical approach has beenrecently proposed by Aquilanti and collaborators[22].

At the ab initio level of theory, the multireferenceconfiguration interaction (MR-CI) techniques,which are best suited for multiple-surface problemsincluding transition states, are difficult to apply forVan der Waals interactions because of the issues ofconsistent evaluation of monomer and dimer ener-gies with respect to size, one-electron basis set, andconfiguration selection [23]. In contrast, the cou-pled-cluster (CC) techniques, which easily copewith the above-mentioned problems and are so suc-cessful with closed shells [24], are useless when itcomes to several states of the same symmetry, es-pecially when close to the avoided crossings.

To alleviate these difficulties, we have proposeda combined application of both the coupled-clustersingles, doubles, and noniterative triples excitations[CCSD(T)] and MR-CI approaches to obtain accu-rate PESs in the Van der Waals region. The ap-proach, termed CCSD(T)-model (CC-M), was usedin the entrance channel of the Cl � H2 [25] reactionand has provided reliable PESs for this complex.

In this article, to provide a strong verification forthe CC-M approach, it is used to generate a set ofnew ab initio–based model potentials for threestates of the F � H2 complex in the Van der Waalsregion. Benchmark PESs for all three states werecalculated by Alexander et al. [2], Stark and Werner[3], and Werner and collaborators [13] and thus areavailable to test the quality of our PESs. The threestates arise from the interaction of H2 with thetriply degenerate 2P F atom. The electron configu-ration of F gives rise to 2�� and 2� states in the C�v

POTENTIAL ENERGY SURFACES OF F(2P) � H2 VAN DER WAALS COMPLEX

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1039

configuration, to 2A1, 2B1, 2B2 states in the triangularC2v geometry, and to 12A�, 22A� and 2A� states in Cs

geometries.The essence of the CC-M approach is to calculate

accurate CCSD(T) interaction energies only for twohighly symmetrical configurations, the C2v and C�v

geometries, and then model three lowest diabaticsurfaces of the Cs symmetry by means of a simpleangular interpolation, which proved successful forthe Cl � H2 system. The approach takes advantageof the oblate shape of the H2 molecule, which re-sults in a relatively simple anisotropy of the inter-action. To obtain the fourth diabatic surface (relatedto the nonadiabatic coupling element of the Ham-iltonian matrix), separate MR-CI calculations areperformed over the complete range of geometries.To obtain three lowest nonrelativistic adiabatic hy-persurfaces, the Hamiltonian in the diabatic basis isdiagonalized. The relativistic spin-orbit couplingeffects are included by using the formalism recentlydeveloped by Alexander, Manolopoulos, andWerner [2], assuming empirical value of the split-ting parameters. The dependence of the PESs on theH2 stretching coordinate is also incorporated andanalyzed.

Computational Methods and Results

GEOMETRIES AND BASIS SETS

The F-H2 complex is described in Jacobi coordi-nates (R, � ). The R variable denotes the distancebetween the center of the H2 monomer and the Fatom, and � denotes the angle between the R� vectorand the H2 bond axis. � � 0° corresponds to theF. . .H-H collinear arrangement. The H2 monomerstretch is described by r coordinate. Calculationswere done for r � 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, and 2.0bohr, and distance R ranged from 1.5 Å to 5.5 Å.The origin of the system of coordinates was placedat the center of the H2 molecule. In the calculationsof the diabatic energies, the H2 molecule was lo-cated along the x axis, and the z axis was perpen-dicular to the triatomic plane. Calculations em-ployed the augmented correlation-consistentpolarized basis sets of quadruple zeta quality (aug-cc-pvqz) basis function set of Dunning [26], Kendalland Dunning [27], and Woon and Dunning [28].The variant of CCSD(T), dubbed R-UCCSD(T), wasused, based on restricted open-shell Hartree-Fock(ROHF) orbitals but with spin contamination al-lowed in the linear terms of the wavefunction [29–

31]. The CCSD(T) calculations (but not the MR-CIones) included also bond functions, with the expo-nents: sp 0.9, 0.3, 0.1; d 0.6, 0.2 [32], in the form ofset: [3s3p2d] denoted as (332). Bond functions werecentered in the middle of the vector R� . Bond func-tions have been shown [33] to be both effective andeconomical for a number of Van der Waals com-plexes including those with an open-shell moiety[34–37].

AB INITIO ADIABATIC AND DIABATICPOTENTIAL ENERGY SURFACES

Building of the CC-M potentials is carried out inthree steps:

(1) Accurate CCSD(T) calculations are per-formed for the C2v and C�v geometries, witha large basis set, to obtain benchmark inter-action energies; see below. The model dia-bats for the Cs-symmetry geometries are ob-tained by a simple Legendre-polynomialinterpolation between the C2v and C�v geom-etries.

(2) The MR-CI calculations with smaller basisset are performed to obtain nonadiabaticcoupling (off-diagonal derivative) matrix el-ement and the fourth diabatic (off-diagonal)surface, see below. These calculations do notrequire the self-consistency corrections or thecounterpoise correction.

(3) The adiabatic PESs are obtained by diagonal-izing the Hamiltonian matrix in terms of dia-batic basis set: see below.

All ab initio calculations reported in this articlewere performed using the MOLPRO package [38].The supermolecular method was used in calcula-tion of three adiabatic potential energy surfaces.This method derives the interaction energy as thedifference between the energies of the dimer ABand the monomers A and B

En� � EABn� � EA

n� � EBn� (1)

The superscript (n) denotes the level of ab initiotheory. In the CCSD(T) calculations the use of theabove equation is straightforward, and free fromarbitrary choices, as long as the dimer and mono-mer energies are calculated with the same dimer-centered basis set to counterpoise the basis set ex-tension effect [39]. The CCSD(T) method is well

KŁOS, CHAŁASINSKI, AND SZCZESNIAK

1040 VOL. 90, NO. 3

known to be very effective in recovering electroncorrelation effects in Van der Waals complexes cal-culations [23, 24] and is preferred as long as thesingle-reference approach is valid. If not, one has touse the MRCI approach. The MRCI calculations aremore involved, as they require the size consistencycorrections and counterpoise correction at the dia-batic level. We describe below application of MRCI.

MODEL DIABATIC PESS

Legendre Polynomial Expansion

The CCSD(T) approach can provide us with veryaccurate results, close to saturation with respect tobasis set and correlation effects. It can be used withconfidence for the lowest state of a given symmetrybut also for excited states that can be adequatelyrepresented by a single Slater determinant, for ex-ample, when the excited state is related to a single-electron promotion from one p orbital to another,orthogonal p orbital [40].

The latter feature was successfully exploited inour recent study of the HCl–Cl Van der Waalscomplex [41], where two A� states in the Van derWaals region were well separated in a wide rangeof geometries. However, whereas formally the sit-uation in the H2-F(2P) case is identical, the H2 mol-ecule produces a much smaller splitting of the 2Pstate of F, and a significant nonadiabatic mixing oftwo adiabatic A� states takes place, which culmi-nates in the conical intersection for the collineararrangement.

The benchmark diabatic surfaces of F-H2 reveal asimple and regular shape [1, 2]. For a given R, the�-dependence has been found monotonic betweenthe C2v and C�v geometries. On this basis, we haveproposed the CC-M method, which assumes theCCSD(T) interaction energies for the C2v and C�v

geometries and derives the interaction energies forthe Cs geometries from the Legendre expansiontruncated at L � 2, which is equivalent to the fol-lowing angular dependence:

H11CC-MR, r, � � � VA1

CCSD(T)R, r�sin2�

� V�CCSD(T)R, r�cos2� (2)

H22CC-MR, r, � � � VB2



H33CC-MR, r, � � � VB1



In essence, it takes advantage of the simple elip-soidal symmetry of the electron density of the H2moiety in its ground state. The CC-M diabatic sur-faces were prepared for the full range of geometriesusing ab initio CCSD(T) results for the A1, B1, andB2 representations of the C2v symmetry and �� and� representations for the C�v symmetry.

2D Fitting

In the framework of the CC-M, the r-dependencehas to be incorporated only for C2v and C�v sym-metries. The analytic expression is based on theTaylor expansion in the r coordinate with addi-tional exponential r-dependent terms, and the R-dependence is of the Degli Esposti–Werner [42]type:

VR, r� � �GR�e�a1R�a2��b1��b2�2� TR� �

i�5

9 Ci

Ri�H��

(5)

where

GR� � �j�0

8

gjRj (6)

H�� m�0

2

hm�m, � �r � re

re, re � 0.7408 Å

(7)

and damping function,

TR� �12 1 � tanht1 � t2R�� (8)

The fitting procedure resulted in 2D surfaces, whichrepresent ab initio data with root-mean-squareranging from 1 cm�1 to 8 cm�1. Two-dimensionalfits were applied to model 3D diabatic surfacesusing equations 2, 3, and 4. (FORTRAN codes gen-erating PES are available on request from J. Kłos.)

Contour plots of H11CC-M, H22

CC-M, and H33CC-M for

r � 0.7408 Å are shown in Figures 1, 2, and 3,respectively. In Figures 1 and 2, a bold line indi-cates the crossing of diabats, H11

CC-M � H22CC-M. It

originates at the region of the ��-� conical inter-section, at � � 0°).



MR-CI CALCULATIONS AND FITTING OFFOURTH DIABATIC POTENTIAL

MRCI Calculations

The MR-CI calculations began with the determi-nation of the state-averaged CASSCF orbitals,which included all the F-moiety–related orbitals(except for 1s) as well as the �g and �*u molecularorbitals of hydrogen molecule in the active space.

The state-averaged complete active space self-con-sistent field (CASSCF) orbitals allowed for all threestates, 1A�, 2A�, and 1A� [43, 44]. The subsequentinternally contracted MR-CI calculations [45, 46]included all single and double excitations relativeto the full-valence CASSCF reference wavefunc-tions. Calculations were performed for several val-ues of interhydrogen distance r and in a wide rangeof R and � variables. In the case of two A� states, theexcitations relative to both reference states wereincluded, and both states were optimized simulta-neously. This guarantees a balanced treatment inthe regions of the conical intersections [47].

The diabatic surfaces provide more convenientrepresentations for simulations of the Van derWaals spectra of the system. These potentials con-tain information about couplings between the adi-abatic wavefunctions of the same symmetry. Theadiabatic–diabatic transformation yields diabaticstates for which the nonadiabatic coupling matrixelements approximately vanish. The diabatic statesare obtained by an unitary orthogonal transforma-tion of adiabatic states [48, 49]

� �1d

�2d � � � cos � sin �

�sin � cos � �� 1a

�2a �, (9)

where the transformation angle � depends on thenuclear coordinates. The resulting diabatic wave-functions are no longer eigenstates of the electronicHamiltonian. The Hamiltonian in the diabatic (px,

FIGURE 1. Contour plot of the modeled H11CC-M diabat

(values in cm�1, r � 0.7408 Å).


(values in cm�1, r � 0.7408 Å).


(values in cm�1, r � 0.7408 Å).


1042 VOL. 90, NO. 3

py, pz) basis is not diagonal, and the matrix elementsare modeled as:

H11 � H11CC-M

H22 � H22CC-M

H33 � H33CC-M

H12 � �2E1A�MRCI � E2 A�

MRCI�cos � sin � (10)

The transformation angle �, so-called “mixingangle,” is defined as the angle between the vector ofthe singly occupied p orbital and the R� vector. It isa function of Jacobi coordinates of the system. Con-tour plot of the mixing angle is shown in Figure 4.Within a two-state model the mixing angle canin principle be obtained by numerical integrationof the nonadiabatic coupling matrix elements(NACMEs). We used the method that uses themaximal overlap with orbitals of the reference ge-ometry to calculate mixing angle. The referencegeometry is taken to be collinear one for large in-termolecular distance. This choice of reference ge-ometry assures that diabatic states coincide withadiabatic ones. This method calculates NACME inan approximate way, using two slightly displacedgeometries and method of finite differences. Be-cause it is convenient to perform diabatic transfor-mation in a system of coordinates with one axisalong the R� vector, the actual � was redefined as:

�R � � � � ��

2 (11)

The plot of �R values is shown in Figure 4. Thisfigure clearly shows the region where the A� statesavoid crossing each other and the point where the�� and � states cross. This is the region where themixing is the strongest, and the angle in Eq. (9)reaches 45°. The conical intersection occurs atF. . .H-H, � � 0°, R 2.9 Å. The intersection isrelated to the crossing of �� and �, which switchthere.

Contour plot of the H12 diabatic surface for r �0.7408 Å is shown in Figure 5.

3D Fitting

To fit 3D set of ab initio data for H12 we usedanalytical expression based on the expansion inLegendre functions Pl

1 and the Taylor expansionaround r � re interhydrogen distance:

H12r, R, � � � Vshr, R, � � � Vasr, R, � � (12)

where

Vshr, R, � � � �l�1

6 �i�0

3 �j�0

3

gljciRj�ie�1/ 2a�bR��c��d�2

�4l � 12 P2l

1 cos � � (13)FIGURE 4. Contour plot of the �R mixing angle calcu-lated in aug-cc-pvqz/tz basis set, r � 0.7408 Å.

FIGURE 5. Contour plot of the MRCI � Q H12 � 2�1/2

coupling (values in cm�1, r � 0.7408 Å).



and

Vasr, R, � �

� �n�5

7 �i�0

3

ci�iCn

Rn �2n � 4� � 12 P2n�4�

1 cos � �,

� �r � re

re, re � 0.7408 Å (14)

The fit contains 35 optimized parameters. The qual-ity of the fit can be expressed in root-mean-squarevalue of 9 cm�1.

MODEL NONRELATIVISTIC ADIABATIC PESS

To obtain adiabatic counterparts of the diabaticsurfaces, the diabatic matrix:

Hel � � H11 H12 0H12 H22 00 0 H33

� (15)

was diagonalized. Global minima and stationarypoints of the modeled diabatic and adiabatic sur-faces are shown in Table I.

Effect of H2 Stretch

In Figures 6, 7, and 8, we show R- and r-depen-dent contour plots of V(R, r) for the C�v symmetry(the �� and � states, Figures 6 and 7, respectively),and for the C2v symmetry (for the sake of brevityonly the most interesting A1 state is shown, Figure8). One can see that the A1 and �� potentials, whichdefine the ground adiabatic state, are the most sen-sitive functions of r, whereas the other V(R, r) po-

tentials weakly depend on r. The V� potential re-veals a weak Van der Waals minimum for large R,with r close to re and shorter, and has a large wellfor small R and large r, where the reaction region isreached. The VA1

potential for large R is everywhereattractive and fairly wide and flat with respect to r.On decreasing R and for small r, a repulsive bank isbeing raised, the attractive region is narrowed tolarger r only, where the potential falls steeply intothe reactive region.

TABLE I ______________________________________Stationary points of modeled diabatic surfaces.a

Diabat De/cm�1 Re/Å �e Type

H11CC-M 130.9 (124.0) 2.55 (2.55) 90 (90) Minimum

2.7 (4.7) 3.35 (3.45) 0 (0) Saddle pointH22

CC-M 45.2 (47.6) 3.35 (3.35) 0 (0) Minimum10.8 (9.0) 3.60 (3.60) 90 (90) Saddle point

H33CC-M 45.2 (47.6) 3.35 (3.35) 0 (0) Minimum

18.5 (16.2) 3.45 (3.5) 90 (90) Saddle point

a Numbers in parentheses represent values of ASW [2] po-tential.

FIGURE 6. (R, r)-dependence of energies of � sym-metry (values in cm�1).

FIGURE 7. (R, r)-dependence of energies of � sym-metry (values in cm�1).


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The behavior of V(R, r) determines the behaviorof the diabats and adiabats, according to Eqs. (2),(3), and (4). The changes of r have the most signif-icant impact on the H11 diabatic surface and only amild effect on the other two, H22 and H33. Com-pared with the plot for r � re (Fig. 1), upon com-pressing H2 to r � 0.5292 Å (Fig. 9), the T-shapedgeometry minimum rises from �130 cm�1 to �80cm�1, the pass in the barrier at around 40° disap-pears, while the collinear stationary point is slightly

lowered (by a few cm�1). In other words, the min-imum energy path around F is now shallower andflatter, and the pass across the reaction barrier isclosed. By contrast, stretching H2 to r � 0.8466 Å(cf. Figs. 1 and 10) makes the T-shaped minimumdeeper—from �130 cm�1 to �170 cm�1—whereasthe saddle in the barrier falls to �80 cm�1 andwidens. At the same time, the barrier for the reac-tion is lowered from 600 cm�1 to �100 cm�1 and isshifted from 40° to 70°, which is closer to the T-shaped geometry. The collinear configuration isonly slightly shallower, and at the stationary pointthe interaction practically vanishes. In other words,stretching H2 deepens the entrance well but alsobroadens and lowers the pass across the reactionbarrier.

The effect of stretching and compressing of H2on the H22 and H33 diabats is minor, so we do notshow it in figures. The H22 collinear stationarypoint rises by 10 cm�1 on squeezing and a few cm�1

on stretching. H33 is affected even less.

SPIN-ORBIT COUPLING AND MODELRELATIVISTIC ADIABATIC PESS

Upon allowing for spin-orbit coupling of thehalogen atom, one obtains two atomic terms: 2P3/2and 2P1/2, separated by SO � 404 cm�1. The inter-action with H2 further splits the 2P3/2 state into twostates. To evaluate the resulting adiabatic potentialenergy surfaces we used the procedure described

FIGURE 8. (R, r)-dependence of energies of A1 sym-metry (values in cm�1).


(values in cm�1, r � 0.529177 Å).

FIGURE 10. Contour plot of the modeled H11CC-M dia-

bat (values in cm�1, r � 0.846683 Å).



by Alexander, Manolopoulos, and Werner [2]. Thematrix of the electrostatic interaction plus the spin-orbit Hamiltonian was expressed in the basis setinvariant to time reversal (see Ref. [2] for details).Then the total matrix decouples:

Hel � HSO � � H 00 H† �, (16)

where H is the 3 � 3 Hermitian matrix expressed inthe basis invariant to time reversal:

H � � H11 � V1 � i�2B V1

V� � A V2

V� � A�, (17)

where V1 � H12, V� � (H33 � H22)/2 and V2 �(H33 � H22)/2. A and B are spin-orbit matrix ele-ments:

A � i��y�HSO��x� (18)

and

B � �� x�HSO�� (19)

where, after Alexander et al. [2], we use compactCartesian notation for diabatic states: ��x�, ��y�, and��, related to the projections of the electronic or-bital and spin momenta along the vector R. The barabove the ��x� state in Eq. (19) denotes spin. We

have found in the Van der Waals region that A andB are practically independent of R and �, and wefixed both at the value of 1/3SO (in the limit oflarge R, A equals B). Significant changes in A and Bstart when the F atom approaches the H2 moleculecloser than at 2.0 Å. On diagonalizing the matrix ofEq. (17), three adiabatic potential energy surfaceswere obtained, shown in Figures 11, 12, and 13.They are numbered in order of increasing energy.In the limit of large R, the first two adiabatic PESs

FIGURE 11. Contour plot of the spin-orbit correctedadiabat 1, which correlates to F(2P3/2) (values in cm�1).




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correlate to the 2P3/2 state of F, with the projectionof j upon the R vector equal to 3/2 and 1/2 for theground state and the first excited state, respectively.The third adiabat (Fig. 13) correlates with the 2P1/2term of F atom.

One can see that the relativistic adiabatic sur-faces significantly differ from nonrelativistic ones.The lowest adiabatic surface is now half as deep atthe T-shaped minimum (De � 64 cm�1) and con-siderably flattened, with a 24 cm�1 barrier for theH2 rotation, around 40°, and another local mini-mum for the collinear arrangement (De � 45 cm�1).The other adiabatic surface related to the same 2P3/2asymptotic limit is shallower, with a maximum at 90°and a minimum at 0°. The third state, asymptoticallyseparated by 404 cm�1 SO coupling, resembles thesecond one in shape but is slightly deeper.

The results for the lowest relativistic adiabaticPES may be compared with those of Alexander andcollaborators [2] and Castillo and collaborators [5].Our results of Re � 2.8 Å and De � 64 cm�1 agreevery well with the more accurate values from Ref.[2]: 2.81 Å and 57.9 cm�1.

These results corroborate the finding of Alex-ander, Manolopoulos, and Werner that the “spin-orbit coupling cannot be neglected in the region ofthe van der Waals minimum [. . .] and significantlyalters both the depth and position of the van derWaals well.” The De and Re parameters of the threeadiabatic PESs are listed in Table II.

Summary and Conclusions

Model diabatic potentials CC-M for the firstthree states of the H2-F complex have been derivedfrom ab initio calculations for the T-shaped andcollinear forms at the UCCSD(T) level of theorywith a large basis set. The three adiabatic surfaces

are in very good agreement with those of Alex-ander, Stark, and Werner from Ref. [2], as evi-denced in Table I, where the parameters for allstationary points are compared. In addition, theheight of the barrier predicted by our model in thereaction channel is 1.66 kcal/mole, also very closeto the estimate of Stark and Werner of 1.53 kcal/mole [3] and Alexander et al. [2] of 1.546 kcal/mole.This is gratifying, because these hypersurfaces wereobtained by different ab initio methods, differentscaling was applied, and we did not optimize theH–H stretching coordinate. (For a discussion of cal-culations of accurate barrier heights, see also Peter-son and Dunning [50].)

The 3D CC-M PESs that include functional de-pendence on the r distance have also been pro-posed, fitted, and analyzed. It has been found thatthe ground state is quite sensitive to the changes inr. In particular, the height of the barrier falls withstretching the H2 molecule and rises with com-pressing it.

The ground-state diabatic nonrelativistic 1A�PES of H2-F may be compared with the groundstate 1A� PES of H2-Cl, which was derived recentlyby us within the same CC-M framework. The inter-mediate and long ranges of these surfaces are sim-ilar, with the H2-Cl Van der Waals well beingdeeper, by 30 cm�1 (De of 164 cm�1), but withalmost the same hindrance for the H2 rotation ofapproximately 85 cm�1. A major qualitative differ-ence is in the character and magnitude of the reac-tion barrier. The F-H2 system with H2 at the equi-librium re has a low and narrow pass in the barrierof �750 cm�1 at � � 40–45°. By way of contrast, wedid not find any such pass in the Cl-H2 short-rangerepulsive wall [25]. In addition, Bian and Werner[51] reported that the barrier for the reaction iscollinear and more central, of 7.6 kcal/mole � 2658cm�1, with the H2 considerably stretched.

The characteristics of the PESs are dramaticallychanged upon allowing for the spin-orbit effects.The relativistic ground-state adiabatic PES is half asdeep as the nonrelativistic one, 64 cm�1 versus 131cm�1, and the barrier to the rotation of H2 is shiftedtoward 40° and lowered from 86 cm�1 to 24 cm�1,whereas the collinear saddle point transforms into aminimum. The second and third adiabatic surfacesare also considerably flattened. These results agreewith the finding of Alexander, Manolopoulos, andWerner that the “spin-orbit coupling cannot be ne-glected in the region of the van der Waals minimum[. . .] and significantly alters both the depth andposition of the van der Waals well.” Very good

TABLE II ______________________________________Stationary points of modeled spin-orbit correctedadiabatic surfaces.

Adiabat no. De/cm�1 Re/Å �e Type

1 64 2.80 90 Minimum1 45 3.35 0 Minimum2 19 3.40 37 Minimum2 18 3.35 0 Saddle2 14 3.50 90 Saddle3 30 3.35 0 Minimum3 26 3.30 90 Saddle



quantitative agreement with the results of Ref. [2]should be stressed. We also found that the spin-orbitcoupling raises the reaction barrier by about 200 cm�1

with respect to the F � H2 reactants, again in fairlygood accord with the more accurate result of 131cm�1 of Alexander, Manolopoulos, and Werner [2].

Finally, it is worthwhile to stress that our CC-Mapproach is expected to provide reliable PESs alsofor other atom–H2 complexes, which are in theentry channels of many reactions, such as O � H2,N � H2, and S � H2.

ACKNOWLEDGMENTS

The authors are grateful to Drs. Millard Alex-ander and Ad van der Avoird for reading andcommenting on the manuscript. Two of the authors(M. M. S. and G. C.) are grateful for support fromthe National Science Foundation (grant no. CHE-0078533). International collaboration was sup-ported by NATO through the Linkage GrantCRG.LG 974215. Two of the authors (J. K. and G. C.)acknowledge support by the Polish Committee forScientific Research KBN (grant no. 3 T09A 112 18).

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1048 VOL. 90, NO. 3

ab initio calculations and modeling of three-dimensional adiabatic and diabatic potential energy...

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