8364
J. Phys. Chem. 1991,95, 8364-8370
A Diatomlcs-in-Molecules Model for the 'A' Potential Energy Surfaces of H2CI+ P.J. Kuntz,* Hahn- Meitner Institut (Berlin) GmbH, Glienicker Strasse 100, D- IO00 Berlin 39, Germany
A. C . Roach, Department of Chemistry, Paisley College of Technology, High Street, Paisley, Renfrewshire PA1 ZBE, Scot land
and D.M. Hirst Department of Chemistry, University of Warwick, Coventry CV4 7AL, England (Received: February 6, 1991)
-
A model Hamiltonian for the 'A' states of the H2CP molecule is constructed from the diatomics-in-moleculesmethod using 10 basis functions. It is then applied to the charge-transferreaction H+ + HCI(X*Z+) H + HCl+(A21;+).The suitability of the basis set is assessed by means of valence bond calculations on the diatomic fragments as well as by comparison with molecular orbital calculations, enabling the introduction of empirical data into the diatomic fragment input. The otherwise unadjusted model agrees very well with recent ab initio calculations, and is shown to reproduce the shapes of the lowest three potential energy surfaces. Contour diagrams are presented for collinear and sideways approaches of H+ to HCl, with emphasis on the role of curve crossings between the surfaces. The locus of a conical intersection between surfaces 2 and 3 is computed and its implications for the reaction dynamics are discussed.
Introduction Dick Bernstein was fond of pointing out that much of research arises from a 'need to know".1*2 Often, very interesting experimental observations of complex systems expose a surprising lack of any knowledge relevant to those particular observations. So it was with the topic of the present paper, which was suggested to us some time ago by Ottinger and co-workers, who observed visible light from the X211 A2Z+transition of H C P produced in the reaction H+ + HCl(X'C+) H + HCl+(A2Z+) AE = +2.7 eV (1) There is certainly a great deal one needs to know about this reaction, for there are many low-lying potential energy surfaces that exhibit any number of curve crossings or avoided crossings. This is evident from the extensive ab initio calculations published recently by Glenewinkel-Meyer et who investigated many of the surfaces in all the relevant symmetries, and who give an up-to-date summary of the work pertaining to H2C1+. Our interest in this system arises out of the possibility of performing dynamical calculations on such systems, as reported recently for Hz0.4 A diatomics-in-molecules (DIM) approach5 is adopted in order to describe the various potential energy surfaces and their derivatives, which are needed in very many configurations by the dynamics program. The DIM model is applied directly in these calculations, the Hamiltonian matrix being diagonalized every time a point on the surface is needed. Recently, a new procedure has been developed: which combines a classical path approach with the DIM Hamiltonian matrix in order to treat systems where many electronicstates lie close together, amplifying the effects of nonadiabatic coupling. A modification of this method' has recently been applied to ion-molecule reactions and to the dynamics of small argon cluster ions.* It is essential in such applications to have a good model of the electronic Hamiltonian matrix for those states that are relevant to the dynamics investigation. This is the main reason for our investigating the 'A' states of the H2CP molecule by the DIM method. A second reason is the chance to compare two systems with the same valence shell structure: H 2 0 and H2C1+. One expects some similarities but also many differences, for although the two are similar electronically, the presence of low-lying charge-transfer states in
-
-
'This work was sponsored by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 337.
0022-3654191/2095-8364S02.50/0
H2C1+leads to a completely different relative disposition of the surfaces. This in turn should produce changes in the shapes of at least some of the surfaces interpolating between the reactant and product states. Our goal here is to develop the simplest DIM model compatible with a realistic description of the lowest three or four surfaces needed to describe the path for reaction 1. This model should also, of course, be applicable to the reaction CI+('DJ
+ H2
-
H
+ HCI+(X211)
AE = -1.5 eV
(2)
which is analogous to the reaction extensively studied for H20a9 Although there are ab initio points available for this system? we make no attempt to fit the DIM model to these. Our objective is to develop a 'zeroth-order" small model suitable for later dynamical applications. Such a model can always be improved upon. Before attempting to do this, however, it is important first to demonstrate that such an unadjusted model is capable of reproducing the shapes of the surfaces in a semiquantitative fashion. Prior to doing this, we give a short exposition of the model.
DIM Model The structure of a DIM model is specified by a list of basis functions for all of the atomic fragments considered necessary for an adequate description of the system. The direct products of these generate all of the basis functions for the polyatomic Hamiltonian matrix as well as all of the basis functions for the diatomic fragment Hamiltonian matrices that are required as input. The relationship between the polyatomic and fragment matrix elements is implicit in the list of atomic functions but the model can be considered fully defined only after the fragment matrix elements have been specified as a function of the internuclear distances of each diatomic pair. This step requires a (1) Bemstein, R. B. In Atom-Molecule Collision Theory: A Guidefor the Experimentalist; Bernstein, R. B., Ed.;Plenum: New York, 1979; Chapter 1, pp 1-43. (2) Levine, R. D.; Bernstein, R. B. Molecular Reaction Dynamics and Chemical Reactiuity; Oxford University Press: Oxford, U.K., 1987. (3) Glenewinkel-Meyer,Th.; Ottinger, Ch.; Rosmus, P.; Werner, H. J. Chem. Phys. 1991, 152, 409. (4) Kuntz, P. J.; Niefer, B. 1.; Sloan, J. J. J . Chem. Phys. 1988,88,3629. ( 5 ) Ellison, F.0.J . Am. Chem. Soc. 1%3,85,3540. (6) Gad& F. X.; Amarouche. M. Chem. Phys. 1990, 140, 385. (7) Kuntz, P. J. J . Chem. Phys., in press. (8) Kuntz, P. J.; Hogreve, J. J. J . Chem. Phys., in press. (9) Aker, P. M.; Sloan, J. J . J . Chem. Phys. 1986, 85, 1412.
0 1991 American Chemical Society
DIM Model for 'A' Potential Energy Surfaces of H2CP TABLE I: VB Structurcs for tbe DIM Model
structure CI+(3P,)HH CI+(ID )HH CI+(IS~HH
C1(2P,)HH+ C1(2P,)H+H CI-H+H+
43 12
3 I 2 2
30 11
66 45
1
59
34
TABLE II: Matomic Fnmnent Matrices no. of basis
fragment functions HCI+(X211) 3 HCI+(A~Z+) 3 HCI+(~A)
H2('Z,+) H2(XiZ,+)
TABLE 111: Valence Bond Atomic Term Values
contribution of structure to wave function (5%) funcHCP- HCP'Ai tions HCI (XzII) (A2Z+) (min) l Z c 1
fragment HCI(XiZ+) HCl(lII) Hz+(XzZ+) Hz+('ZU+! H?+
1 1 1
The Journal of Physical Chemistry, Vol. 95, No.21, 1991 8365
term CI+('P,) CI+('D ) CI+('SB)
25 8 0 26 26 I5
IO 5 28 28 29
no. of basis functions 2 1 1 1 1
valence bond (VB) calculation followed by some semiempirical manipulation, so that experimental information or the results of a b initio calculations can be introduced into the model. The assessment of the suitability of a model must consider two aspects: (i) the ability of the implied diatomic fragment basis set to describe qualitatively the bonding in those diatomic states that are considered important for the system under investigation (e.g., the states correlating with reagents and products) and (ii) the ability of the polyatomic model to describe the potential energy surfaces that interpolate between the reactants and products. The first part of this assessment goes hand in hand with the development of the model, to which we now turn our attention. Composite basis functions for H2CPare constructed by taking the direct products of the atomic states Cl(*P,), Cl+(lDB),Cl+('S ), Cl+('P,), C1-('Sg), H(2S,), and H+ in such a way as to yidd spin-coupled basis functions of A' symmetry. The resulting 10 triatomic VB structures and the number of basis functions arising from each of them are listed in Table I. The diatomic fragment manifolds corresponding to these are listed in Table I1 along with the size (Le., the number of basis functions) of the corresponding fragment Hamiltonian matrices that are needed as input to the model. These tables implicitly specify the structure of the model. Using the above bases as a guide, we carried out valence bond calculations for each fragment, using the program MuLTieoND A.IO Our objectives here were (i) to obtain the best possible diatomic "mixing coefficients" (Le., the coefficients of the various (orthogonalized) VB structures) as a function of the internuclear distances, (ii) to determine which of the diatomic states could safely by regarded as "pure* states (Le., those that could later be replaced by more accurate interaction energies), and (iii) to estimate the interactions of the remaining, mixed states. The first step in achieving these objectives was to choose VB basis functions with a view to giving a uniform level of accuracy to the atomic states for C1, CP, and Cl-. It turned out that the basis of Veillard" was adequate for Cl(2P,) and for the C1+ states, provided that the configuration IsxffyJzZI was included in the description of the IS, term for CP. For C1-, an additional single gaussian, p', was added, choosing the exponent (a = 0.087 bohr-') to optimize the energy for C1-. The computed energies are listed in Table 111. For the investigation of the diatomic states, two bases were employed: (i) the restricted basis, consisting of composites formed from the orbitals for the atoms and ions listed above, and (ii) an extended basis, which included a function describing the s'p6 configuration of C1, a scaled 1s orbital on H (the 5-Gaussian 1s function with the exponents scaled uniformly), and a singleGaussian x'orbital on H. The basis was chosen to avoid upsetting the good relative values already achieved for the CI, C1+, and C1-
'
(10) Baht-Kurti, G. G.; Yardley, R. N.MULTIBOND A. Quantum Chemistry Program Exchange, program 335. (11) Veillard. A. Theor. Chim. Acto 1968, 12, 405.
C1(2P,) CI-('SJ WS,)
H+
relative energy, eV exactzi O.oo00 O.Oo0
energy, hartrcc
this work
-458.997931 19 -458.93896306 -458.88563342 -459.474009 40 -459.58965226 -0.499 809 34 0.00000000
1.6046 3.0558 -12.9540 -16.1020 13.6010 13.6010
1.403 3.419 -13.030 -16.645 13.606 13.606
states. The exponents of the added orbitals were optimized separately for the HCl and HCl+ molecules at distances close to the equilibrium separation. In assessing these basis sets, a b initio molecular orbital calc u l a t i o n ~ ' ~of* 'selected ~ states were compared with the results of the VB calculations. The restricted basis gave qualitatively good descriptions of the lowest state of each of the manifolds for which we had data for comparison. The extended basis yielded substantial improvements in the interactions for HCl(X'Z+), HCl+(X211),and HC1+(A2Z+),which followed the ab initio results very well, furnishing a surprisingly uniform approximation to the required states. The HC1+(22Z+)state proved to be an exception, for the MO calculations exhibit a short-range interference from a Rydberg state, leading to a second minimum in this curve, which cannot be produced with the DIM basis-this state must definitely be treated as "mixed". According to the analysis of the VB calculations, the following states could, if desired, be replaced by more accurate interactions: HCl X'Z', All& 'n, 3Z+
HCI' X 2 n , A22+, 2A, 2ZNot all of these states are needed for the 'A' surfaces of H2C1+; we included them because of future applications to surfaces of other symmetries. Armed with the above information, it is possible to introduce experimental data or the results of better calculations into the fragment matrices. The first step is to shift the diagonal elements by constant amounts at all distance^,'^ so that the correct limits are approached as r m. For HCl, the element corresponding to the limit H+ C1- was lowered by 0.4994 eV. For the H C1+ limit, the corrections were as follows: Cl+(3P8),+0.07603 eV; Cl+('D,), -0.1313 eV; and Cl+('S,),. +0.4331 eV. The resulting matrices served as the starting point for all further modifications. The interaction energies for H2(XiZ8+)and H2(%,+) were taken from Kobs and W~lniewicz;'~ those for H:(X2Z +) and H:(2Z,+), from Bates and Reid.I6 The HCl and HClO interactions for the pure states were obtained from the a b initio calculations of Glenewinkel et aL3 and were then slightly adjusted to agree with experimental information. The ground state of HCl was shifted to increase the interaction at the equilibrium distance of 2.439 bohr from -3.8925 eV to -4.6200 eV. The 'II interaction was left unaltered. The HCl+(X211) state was shifted to go from -4.1703 eV to -4.819 eV at r(H-CI) = 2.4939 bohr. The HCl+(A2Z+)curve was left unaltered, producing only a very small discrepancy (0.04 eV) from the experimental value. (This was inadvertent and can be corrected when we improve the model later.) The 2A interaction was also left unaltered. No further adjustments were made to the model. In particular, no information about the triatomic molecule was incorporated into this first model.
+
-
+
Results Throughout the remainder of this paper, all energies are referred to a zero corresponding to the infinitely separated atoms Cl+(3Pg)
Hirst, D. M.; Guest, M. F. Mol. Phys. 1980, 41, 1483. (13) Bettendorff, M.;Peyerimhoff, S.;Buenker, R. J. Chem. Phys. 1982, (12)
66, 261.
P. J.; Chang, C. C. Chem. Phys. 1983, 75, 79. (15) KO&, W.;Wolniewicz, L. J . Chem. Phys. 1965, 43, 2429. (16) Bates, D.R.; Reid, R. H. G. Adu. Ai. Mol. Phys. 1968, 4, 13. (14) Kuntz,
Kuntz et al.
8366 The Journal of Physical Chemistry, Vol. 95, No. 21, 1991
TABLE I V DIM Energim (cV) at sckctcd Cadigumtiom configuration Cl+(3P,) + H + H Eri'Dj H H 'S H H C1(2P,) H + H+ CI- H+ H+ Cl+(3P,) + H2 CI+('D ) + H2 CI+('S3 H2 C1(2P,) H2+(X2Z;) H+ HCI(X'Z+) H + HCI+(X211) H + HCI+(A2Z+) H2CI+('A,) +
+ + + + + + +
+
+ +
(HCI-H)+('ZI+)
(HCI-H)+('II, minimum)
5
5
4
4
3
3
2
2
F1
1
p"
0
ab initio or exptF
DIM O.oo00 1.4030 3.4186 0.5754 10.5259 -4.7470 -3.3440 -1.3284 -2.2175
L
-4.0443 -4.8189 -1.3140' -9.7703 ( r = 2.51 bohr, 8 = 95.52O) -6.7600 -3.8568
-1.2696' -10.11' ( r = 2.46 bohr, 8 = 94.24') -7.45e -3.5 to -4.w
'A blank entry means that this energy was used as input to the DIM model. bReported relative to H+ HCI(XIZ+);here reexpressed relative to the Cl+(3P,) + H H energy. dAb initio.18 *Ab initio.3
+
+
+ H + H.
Table IV collects the energies of some of the most important configurations for convenient r e f e r e n e t h e first three blocks in the table contain data for the channels CI H + H, C1 H2, and H HCl respectively. The fourth block contains a few energies for the triatomic. The channel energies are part of the input to the model, so no comparison with ab initio calculations or experiment is given, except for the H HCI+(A2Z+) entry, for which there is a small discrepancy between the experimental value and the ab initio value that we used for input. All energies are in units of *electronvolts", the product of the Faraday constant and the unit volt;" 1 eV = 96.487 kJ/mol. All distances are in atomic units: 1 bohr = 5.291 77 X lo-" m = 0.529 177 A. From the dissociation limits in Table IV and the diatomic interactions in the HCl fragment, cuts through the HCl asymptotic channel can easily be computed. Such curves, shown in Figure 1, are very useful in displaying the relative positions of the surfaces in the asymptotic region as well as presenting the complete input to the DIM model in a compact way. In this figure, one H atom is infinitely removed from the HCl fragment, and the energy change corresponding to each value of the r(H-Cl) coordinate is plotted. The asymptotic limits A, B,C, and D, correspond to the atomic states CI+(lSg) + H2, Cl+('D ) H2. C1(2P,) HI(XZE,+), and CI+(3Pp,) H2, respectivefy. The curves most relevant to reactions 1 and 2 are the three lowest ones. The H HCI+ states are both bound and straddle the H+ HCI state in the region of the equilibrium distances; hence, reaction 1 clearly involves a passage from surface 2 to surface 3, most likely via an avoided crossing region. States 2 and 3 both approach the same asymptotic limit, C, which is 4-fold degenerate. This behavior is quite different from that for the H 2 0 system, where the curve OH- H+ (analogous to the H+ HCl curve here) lies oboue the OH(A2Z+) H curve and does not cross or meet it, since the second and third curves go to dvferent asymptotic limits. The C1 H2 channel, displayed in Figure 2, is more complicated, because of the degeneracy associated with the states of the C1 fragment at infinity. This is brought out by the lower panel of the figure, which shows cuts through the same channel for a distance R(C1-H2) (the distance from the atom to the center of the diatomic molecule) of 6.5 bohr for an angle 4 (the angle that the line joining the atom fo the center of the molecule makes with the bond axis of the diatomic molecule) of 80 degrees. Here, the degeneracy is lifted, clearly revealing the presence of many surfaces. This suggests that nonadiabatic effects could play an essential role in this channel.
+
+
+
+
+
+
+
+
-1
-2
-2
-3
-3
-4
-4
-5
-5 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 8.5
r(H-CI) [bohr] Figure 1. HCI fragment interaction energies for the first 9 eigenvalues of the DIM Hamiltonian matrix relative to the separated atom limits (R(HCI-H) = 03) (channel H + HCI): (A) CI+('S ) + H + H; (B) CI+('D,) + H + H; (C) C1(2P,) + H + H+; (D) C!+(3P,) + H + H (energy zero), 4
4
3
3
z2
2
-1
1
p
0
W-1
-1
-2
-2
%
-3
+
+
+
-1
+
+
(17) McGlashan, M.L. Physicochemical Quantities and Units, 2nd ed.; The Royal Institute of Chemistry: London, 1971.
4
4
3
3
2
2
Z1
I
5 0
0
0)
5-1
-1
-2
-2 R(HH 43) = 6 5 bohr
-3
-3
-4 1
1 .o
,
/
3.0
1
,
1
1
1
5.0
1
1
1
I
l
1
7.0
r(H-H) [bohr] Figure 2. H2 fragment interaction energies for the first 9 eigenvalues of the DIM Hamiltonian matrix (channel CI + H2): (Top) asymptotic region (R(CI-H2) = m), separated atom limits as in Figure 1; (Bottom) long-range region (R(C1-H2) = 6.5 bohr).
A good picture of the behavior of the ground state surface is obtained by considering the approach of the Cl fragment to the H2 fragment. Since we are here considering only singlet surfaces, this corresponds to the electronic configuration CI+(lDB)+ H2 in the approach channel. In C, symmetry, the configurations
DIM Model for 'A' Potential Energy Surfaces of H2CP
The Journal of Physical Chemistry, Vol. 95, No. 21, 1991 8367 10
t
\
chrsurface1
I 9
a 7 CI
$ 6 Y
is
b
I 4
I
3
2
1
-
CI HH Distance [bohr] Figure 3. Energy contours (eV) for the lowest eigenvalue in C, configurations H-CI-H.The minimum energy, -9.7703 eV, corresponds to the 'Al state; the saddle point at energy -6.7600 eV, to the '2,' state in the D.& configuration. The curve marked X is a crossing between the 'A, and 'B2surfaces. of lowest energy belong to the 'Al irreducible representation. Energy contours are shown in Figure 3 in the plane described by the distance r(H-H) and the atom-molecule distance, R(CI-HI). The minimum energy is -9.770 eV at an H-CI-H configuration having a bending angle 8 = 95.524' for r(H-H) = 3.7172 bohr, R(CI-HI) = 1.6875 bohr, and r(H-CI) = 2.5105 bohr. Accurate ab initio calculations'* yield a bond angle of 94.244O at r(H-Cl) = 1.3030 A 2.4623 bohr; the calculated proton affinity of HCl, when subtracted from the energy for H+ HCl(X'Z+) yields a minimum energy of -10.1 125 eV. This agreement is very gratifying for an unadjusted model employing only 10 basis functions. The saddle point in the Dmhconfiguration (R= 0) lies at -6.7600 eV for r(H-Cl) = 2.627 bohr, which is also somewhat higher than the ab initio value3 of -7.45 eV. At large values of R(C1-HI), the surfaces become almost degenerate, resulting in a crossing of the 'A, surface with the lowest 'B2 surface, as indicated by the crossing seam marked with an X. The weights of the VB structures in the wave functions for the two extrema on this surface are listed in Table I. In both cases, most of the positive charge resides on the H atoms. The second eigenvalue of the model Hamiltonian, shown in Figure 4, exhibits two minima. The one in the D-h configuration corresponds to a Ill, state with r(H-CI) = 2.895 bohr and E = -3.8568 eV. The other minimum, with energy -3.5838 eV, lies in a bent configuration at R = 3.8440 bohr and r(H-H) = 1.5547 bohr. The crossing between the states of different symmetry is shown by the dotted lines. The first two adiabatic surfaces for the collinear approach of CI+ to H2are shown in Figure 5. The lowest surface corresponds to a I Z ' state in the Cl+('DJ + H2 channel, switching over to a III state in the HCl H channel. The crossing seam shifts its position in accordance with the relative importance of the curve crossings in Figure 2 as r(H-Cl) increases. The minimum energy of -5.8838 eV corresponds to r(H-H) = 1.6519 bohr and r(H-Cl)
-
+
+
(18)
Botschwina, P. J. Chcm. Soc., Faraday Tram. 2
1988, 84, 1263.
1
2
--
3
4
5
6
7
CI HH Distance [bohr] Figure 4. Energy contours (eV) for the second eigenvalue in C, configurations. The minima are for the In, (-3.8568 eV) and 'B2(-3.5838 eV) states. The dotted lines show the crossings of these two states (eigenvalues 2 and 3).
= 2.5149 bohr. The second state exhibits a saddle point on the IZ+surface in the Cl+('S ) + H2 channel. Its energy is -3.6292 eV at r(H-H) = 4.9784bhr and r(H-Cl) = 2.4421 bohr. In the other channel, there is a crossing with the 2A state; i.e., eigenvalues 2 and 3 intersect each other. In this region, the 'II, I Z ' , and 'A surfaces are nearly degenerate with each other-it is unlikely that the DIM model can correctly predict the ordering of these states in this channel. Of particular relevance to reaction 1 are those regions corresponding to the approach of the H fragment to the HCI molecule. The collinear approach of H+ to HCl is shown in Figure 6 for the arrangement H-CI-H and in Figure 7 for H-H-Cl. In both cases there is a crossing between the I l l and 'Z+states, which could provide a pathway for the direct production of HCl+(X211): H+ HCl(XIZ+) H HCl+(X211) PE -0.8 eV (3) This is also seen in the contour plots of Figure 8. The minimum (-6.76 eV) on the '2' surface and the saddle (-3.86 eV) on the I l l surface correspond to the D,, extrema on the Ch surfaces in Figures 3 and 4. The lateral approach of H+ to HCl exhibits surfaces which are somewhat sensitive to the position of approach: going toward the middle of the bond produces an interaction quite different from a direct attack on the Cl end of the molecule. Figure 9 shows the perpendicular approach along a line leading to the midpoint of the HCl molecule: there is an avoided crossing between eigenvalues 3 and 4 at R(H-HC1) = 3.25 bohr. Contrast this with the direct attack pictured in Figure 10, where the H-CI-H bond angle is held fixed at 97'. Here there is an additional avoided crossing between states 2 and 3, furnishing a pathway for reaction 2 provided that the H+ has sufficient translational energy to reach the crossing region. The corresponding contour diagrams for the three lowest surfaces at a bond angle of 97O are shown in Figure 1 1. At the C, configuration, surface 3 has a small local minimum (E = 0.17 eV), which is separated from the asymptotic channel (H + HC1+(A2Z+)) by a saddle point (E = 0.4190 eV).
+
-
+
-
8368 The Journal of Physical Chemistry, Vol. 95, No. 21, 1991 5
-
Kuntz et al.
El
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5
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5
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2
.
3
l
,
4
l
,
5
l
.
8
l
.
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7
8
r(HCI) {bohr]
Figure 5. Energy contours (eV) for the first and second surfaces interpolating between the channelsCI + H2and H + HCI for collinear ClHH configurations. The broken lines show the crossing between the lZ+and I l l surfaces, eigenvalues 1 and 2 (both panels), and between the second and third surfaces, In,IA (top). The IZ+state exhibits a minimum (-5.8838 eV) and a saddle point (-3.6292 eV). The asymptotic limits are Cl+(ID ) + H2 (both su;faccs) and H + HCI+(X211)(El) and H + HCI+(A~Z$( E ~ ) .
4.0
5.0
6.0
7.0
R(H - HCI) [bohr] Figure 7. Energy (eV) for the collinear approach of H to the H end of HCI held fixed at its equilibrium position (r(HC1) = 2.439 bohr). 8
-
5
f
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1
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.
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Figure 8. Energy contours (ev) for the first two eigenvalues of H2CI+ in collinear HC1-H configurations. The minimum (-6.86 eV) on the
IZ+surface and the saddle (-3.86 eV) on the In surface correspond to the minima of the 'E and In, statcs rcspoctively for D,h symmetry (see Figures 3 and 4). Tbe broken line is the intersection of the IZ' and I l l surfaces. +
30
40
5.0
60
70
80
R(H - CIH) [bohr] Figure 6. Energy (eV) for the collinear approach of H to the CI end of the HCI molecule near its equilibrium position (r(CI-H) = 2.439 bohr).
The sideways and collinear approaches are combined in the 'fixed molecule" contour diagrams in Figure 12. Here, the HCl fragment is held fixed on the X axis, while the other H atom takes
up arbitrary positions in the X-Y plane. Also shown are contours of the difference in energy between states 2 and 3. These are spaced at 0.1 eV. The position of closest approach of these states moves outward as the distance of the fixed HCI molecule is increased from 2.439 bohr to 2.8614 bohr. This behavior is reflected
DIM Model for 'A' Potential Energy Surfaces of H2C1+
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The Journal of Physical Chemistry, Vol. 95, No. 21, 1991 8369
I
-10
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 6.0
R(H - midpoint HCI) [bohr] Figure 9. Energy (eV) for the perpendicular approach of H to the midpoint of HCI held fixed at its equilibrium configuration ( r ( H 4 ) = 2.439 bohr). .............. .- 7 6
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........ .......... ...............................................................
4
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...........................................................................
11
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45
:
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2
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4 5 6 r(HGI) [bohr]
7
8
Figure 11. Energy contours (eV) for the first 3 eigenvaluesof H2CI+ in bent configurations for a bond angle H C I - H of 97'. The contour E, - E2= 0.1 eV is represented by the long dashed lines in the middle panel.
a circle of small radius around the position of the suspected intersection. Indeed, the eigenvector changed sign! We also assured ourselves that there was no sign change when the suspected point of intersection was nor included in the circle. This is evidence of a bona fide inter~ection.'~" On changing the distance of the fmed HCl molecule, the point of intersection wanders to a different position in the X-Y plane. The locus of this point is shown in Figure 13 as a function of the fixed HCI distance. The inset in the figure displays the energy of the second eigenvalue to which the point corresponds. Also shown is a table of the H-Cl-H angles.
Discussion On the whole, the DIM model does very well, considering that it employs only 10 basis functions and that it is not empirically adjusted with any information about the triatomic molecule. The surfaces behave in much the same way as the ab initio calculations indicate they should, although the absolute energy of any given surface may show discrepancies of up to 0.5 eV with the corresponding ab initio value. Of course, it should be borne in mind that the asymptotic channels were chosen, for all cases but one, to go to the experimental limits and not the ab initio ones, so that a comparison of the absolute energy is difficult. In any case, we can expect that, as the model is capable of further empirical adjustment, it could serve as the basis for a dynamical investigation of this system. The ground-state surface bears a remarkable qualitative similarity to the corresponding H20surface, especially the 'Al surface in close C, configurations. Differences with H 2 0 appear in the asymptotic regions, especially C1 + H2where the avoided crossing (19) Berry, M. V. Phys. Today 1990,13 (12), 34. (20) Mead, C. A,; Truhlar, D. G. 1.Chem. Phys. 1979, 70, 2284. (21) Moore, C. E. Natl. Bur. Srand. Circ. (US.)1949, no. 467. (22) Huber, K. P.; Herzberg, G. Molecular Spectra and Molecular Structure, Constants of Diatomic Molecules; Van Nostrand: New York, 1979.
Kuntz et al.
8370 The Journal of Physical Chemistry, Vol. 95, No. 21, 1991 6
- H-CLH
RBC(CI-H
R(HCI)=2.44 bohi
2.439 2.861
1
5
3 2
-2
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7
4
4
4 2.439
Energ)
k !
We01 99.94
-
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3
2
R(HCI):Fbted Molecule [bohr] 1
2
,
.
,
1
- 3 - 2 - 1 0
,
,
1
,
I
2
,
l
3
,
4
4
,
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X [bohr] 1
F w 13. Locus of the conical intersection in the X-Y plane of Figure 12 as the HCI distancc is varied from 2.439 to 4.5 bohr, along with the HCI-H bond angle and energy at the intersection.
X [bohr]
Figure 12. Energy contours (eV) for eigenvalue 2 for H approaching HCI held fixed on the X axis at dietan- r(H-Cl) = 2.439 bohr (top) and 2.86 bohr (bottom). Also shown arc some conical intersection contours E3 E2in steps of 0.1 eV. The intersection contours from the lower panel are reproduced on the upper one for ease of comparison.
-
region occurs at a smaller H2 distance and where there is increased degeneracy of the accessible surfaces. The excited states of the two molecules show greater differences from one another. The conical intersection betwem surfaces 2 and 3 deserves closer attention. We believe it may simply be due to the extreme simplicity of the DIM model. On the other hand, a cursory inspection of the DIM Hamiltonian matrix does not reveal any obvious reason for there being an intersection: the matrix is not block diagonal. There should, of course, be some transformation of the basis that would make manifest the intersection. In any case, the behavior of the DIM model in the neighborhood of the intersection parallels that of the avoided crossings in the ab initio calculations. It would be easy to test for the existence of the intersection by means of an ab initio calculation of the behavior of the wave function on transporting it around the suspected intersection point of our calculations. Whether or not there is a bonufide intersection in real life, the behavior of the minimum in E' - E2allows us to infer something interesting about this system. Reaction 1, which is endoergic by -2.7 eV, can only proceed via the avoided crossing. The inset
in Figure 13 shows that this has its lowest energy, -1.1 eV, at r(H-Cl) =3.1 bohr, which corresponds to a vibrational energy in the HCI reactant molecule of between 1.O and 1.5 eV. If the HCl were excited by this amount, reaction 1 could proceed at a lower translational energy: 3.0 eV instead of the 3.5 eV for HCI in its ground vibrational state. (We caution again about taking the absolute energy values too seriously--only the relative values should be considered in this qualitative discussion!) This is also illustrated in Figure 11 by the long dashed lines showing the locus of points where the difference in energy between surfaces 2 and 3 is 0.1 eV: the reaction threshold is lowest when the reagents have a combination of vibrational and translational energy. It would therefore be interesting to perform experiments varying the amounts of translational and vibrational energy while keeping the sum of these constant. The cross section might also be augmented by increasing vibration at the expense of translation, since the avoided crossing region shifts to a point corresponding to larger impact parameters. These remarks are in agreement with the conclusion of the ab initio work' that reaction is favored by increasing the HCl distance.
Acknowledgment. We are particularly thankful to Th. Glenewinkel-Meyer for sending us the results of his ab initio calculations on HCI, HCl+, and H2CI+prior to their publication. We also thank the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 337, for supporting this work. R d ~ t r NO. y HZCI', 36658-55-6; , ' H 12408-02-5; HCI, 7647-01-0.
