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Ab initio study of the long-range interaction between He+ and H2
Volume 2 13, number $6
CHEMICAL PHYSICS LETTERS
15 October 1993
Ab initio study of the long-range interaction between He+ and H2 M.F. Falcetta and P.E. Siska Departmentof Chemistry, Universityofpittsburgh, Pittsburgh,PA 15260, USA
Received 5 August 1993
A new estimate of the anisotropic long-range potential energy surface for the interaction of He+ ( 1s 3) and HZ(X ‘Zl) has been computed using extended atomic basis sets to construct three-configuration self-consistent-field (MCSCF) molecular orbitals for use in a multireference configuration interaction expansion including all single and double excitations (MRCISD) The most stable geometry is Czv, in agreement with earlier studies, but the well depth is found to be 3.15 kcal/mol at an intermolecular distanceof 2.42 A, 57% deeper and 0.2 A smaller, respectively, than the best previous estimate. The ab initio points are fitted to a potential function with correct long-range behavior, suitable for scattering calculations.
1. Introduct ion
Despite the simplicity of the (HeH,) + ion, several aspects of the electronic structure and collision dynamics involving this species continue to invite both experimental and theoretical investigation. Recent theoretical studies have explored intennediate-tohigh energy (100-1000 eV) inelastic collisions of He+ and Hz [ 1,2], with a view to the delineation of the appropriate diabatic states to use in describing curve crossings occurring at small separations R, the distance between He+ and the center-of-mass of HZ. At thermal and subthermal energy, this system can still undergo exothermic charge transfer (AE= - 9.1 eV), although the rate of this process is four to five orders of magnitude below the Langevin capture rate. A recent two-body quantum modeling study [ 3 ] intended to emulate ultra-low temperature rate measurements [4] suggests that the rates of both the dominant dissociative charge transfer channel and radiative charge transfer depend sensitively on the lifetime of the [He+...H,] collision complex. The lifetime in turn is dependent upon the features of the attractive part of the He+ + Hz potential energy surface (PES); this sensitivity is heightened if orbiting resonances are involved, as indicated by the modeling study. Here an adiabatic picture is more appropriate, in order to capture the perturbing effects
other states of like symmetry on the attraction. Our own interest in the He+ + Hz PES stems from its role as the core potential in the Penning ionization systems He*( 2 ‘3%)+Hz; for a recent review, see ref. [ 5 1. The ionic surface well shape has been shown to influence the shape of the low-energy repulsion in the excited neutral systems [ 61, and the dependence of the electronic wavefunction on R may provide clues to the mechanism of Penning ionization in these systems and in general. In each case the state of interest is the lowest-lying excited state of HeHt (for H2 at or near its equilibrium distance r= 1.4 bohr), 2 ZA’in C, symmetry, a classic example of a charge-transfer state. Among the ab initio studies of this state [l-3,7,8], noting particularly the extensive calculation of McLaughlin and Thompson [ 8 1, the only one to report details of the PES for r= 1.4 bohr and R> 5bohr has been that of Hopper [ 7 1. Hopper examined both CZvand C,, geometries, employing a multiconfiguration selfconsistent-field (MCSCF) approach, using three configurations approximately designated as HelsZH20g, HelsH&, and HelsH&, and variationally minimizing the second energy eigenvalue (2 ‘A, in C&, 2 ‘Z’ in C,,). 4s2p contracted Gaussian atomic basis sets were used on both He and H. While these are large basis sets for first-row atoms, Dykstra [9] has shown that an accurate polarizaof
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bility tensor for H2 is not obtainable, at least at the self-consistent-field (SCF) level, unless a d function is present. An accurate polarizability is essential for proper behavior of the ab initio PES at intermediateto-large R, especially as the polarization of HZby He+ is the most important contributor to the interaction energy for Rx34 A. In a recent calculation of the Na+ + H2 PES by our group [ 10 1, it was found that the 5s3pld H atom basis set of Dykstra [9] produces both polarizabilities and quadrupole moment of good accuracy for H2 at the SCF level. This resulted in an apparently highly accurate PES, for which the effects of electron correlation were shown to be nearly negligible. Electron correlation, however, is expected to be more important in the present openshell excited system. We report here MCSCF calculations using extended atomic basis sets, and multi-reference configuration interaction including all single and double excitations (MRCISD) to account for electron correlation effects, on the He+ +HZ excited state PES for CZVand C,, geometries using the COLUMBUS program package [ 111. Basis set superposition error (BSSE) corrections, ignored in all previous ab initio studies of HeH: , are made at each computed point on the surface using the counterpoise method of Boys and Bemardi [ 121. The ab initio points, for r= 1.4 bohr (0.741 A) and 1.6 A
2. Theoretical method Using the 5~3~1d hydrogen atom basis set of Dykstra with the 6s3pld helium basis of Khan and Jordan [ 131, we carried out a three-configuration MCSCF calculation identical, except for the larger basis, to that of Hopper described above. At this level of theory the extended H atom basis yields a larger, more accurate perpendicular polarizability component al, as given in table 1; we therefore expected and found improved long-range behavior of the PES in addition to a deeper well in the most stable CzV geometry, even after correcting for BSSE, without the 532
15 October 1993
Table 1 Computed properties of H2 (r= 1.4 bohr) in atomic units at various levels of theory Basis/level
Ref.
8
ffl
ffJ.
4s2plMCSCF Ss3pld/SCF Ss3pld/CISD accurate
171
0.4735 0.496 0.4506 0.4574
6.533 6.43 6.443 6.381
3.660 4.54 4.507 4.578
1101 this work 1141
inclusion of configuration interaction. The reference list for the CI step consisted of all configurations allowed by symmetry and spin arising from distributing the three electrons among the He 1s, H20g,and H20, orbitals, with the restriction that the He Is orbital be singly or doubly occupied. This resulted in three references in C2,,and six in C,,. All single and double excitations out of these references within the MCSCF orbital space were included. As table 1 indicates, CI improves the quadrupole moment 8 of H2 somewhat while the polarizabilities change little; but the surface, as shown below, alters to a greater degree. Further enlarging either the basis sets or the reference configuration list had little effect ( < 0.05 kcal/mol) on the total energy near the minimum in the surface. All calculated energies were corrected for BSSE by the counterpoise method [ 121. The corrections are small but increasing in magnitude with decreasingR, - 0.023 kcal/mol at the surface minimum and -0.20 kcal at R = 1.6 A in C,, in the CI energy, and are due almost entirely to Hz borrowing the He basis functions. We note that the BSSE vanishes as R-m, is necessarily negative, and must be subtracted from the total energy; therefore surface basins are rendered shallower. The large and balanced basis sets used here serve to keep the corrections relatively small.
3. Results The total energy and BSSE were computed at 26 Rvaluesfrom1.6to12A,andatR=105Atoprovide an energy asymptote, in both CZV(y= 90”, y the angle between R and r) and C,, (y=O”) geometries, for r= 1.4 bohr. Table 2 presents the corrected MRCISD interaction energies, and fig. 1 compares
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CHEMICAL PHYSICS LETTERS
Table 2
I
MRCISD interaction energiescorrected for BSSE‘)
I
I
I
30
R (A)
1.60 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.40 3.60 4.00 4.40 4.80 5.20 5.60 6.00 8.00 12.00 l)
He+ Ha (C,)
Energy (kcal/mol)
+
Cl”
CCaY
17.1941 4.9103 1.5916 -0.5573 - 1.8865 -2.6489 - 3.0270 -3.1511 -3.1133 -2.9777 -2.7878 -2.5729 -2.3516 -2.1358 - 1.9323 - 1.7444 - 1.4200 - 1.1601 -0.7912 -0.5581 -0.4070 -0.3059 -0.2358 -0.1854 -0.0685 -0.0175
44.7499 20.4873 13.4821 8.6508 5.3600 3.1512 1.6962 0.7617 0.1823 -0.1583 -0.3413 -0.4228 -0.4411 -0.4218 -0.3820 -0.3325 - 0.2297 -0.1410 - 0.0240 0.0334 0.0596 0.0701 0.0722 0.0697 0.0432 0.0166
The asymptote is taken as the MRCISD energy at R = 105 A, -3.170032536 hartree. 1 hartreez627.510 kcal/mol.
the CZVMCSCF results of Hopper with the extendedbasis MCSCF and MRCISD energies from this work. In agreement with Hopper’s work, we find the CZV geometry to be most stable in both the MCSCF and MRCISD calculations; this is determined mainly by the anisotropy of the ion-quadrupole interaction. The three CZVpotentials may be concisely compared through the zero-crossing radii RO,minimum positions R, and well depths O,, as given in table 3. We note that Hopper’s calculations have not been corrected for BSSE; his De is therefore expected to be larger and his R, and & smaller than for a corrected calculation at the same level of theory. Even so, improving the basis set at the MCSCF level deepens the well by 17%without substantially affecting the range, while including the effects of electron correlation produces a 57% increase in De and a decrease of 0.2
20
Fig. 1. Comparison of ab initio potential energy curves for CsV approach of He+ to HI. The MCSCF and MRCISD points are the present results; the MRCISD values are given in table 2. The large-R asymptotic energy has been subtracted out in each case. (0) Ref. [7]; (0) MCSCF; (0) MRCISD. Table 3 Comparison of ab initio He ++Hz C,, well depths 0, (kcal/mol) and positions R,, ~0 (A) Ref.
Q
R,
J&l
18,151 t31
1.87 2.05
2.35 2.35
2.155 2.10, 1.97,
[71
2.00
2.60~
Js3pldfMSCSF MRCISD
2.34, 3.150
2.632 2.42s
A
in both R. and R, compared to the earlier findings.
4. Analytic representation of the PES As in the earlier work on Na+ ’ + HZ [ 101, we use the ab initio points to construct the first two terms in the Legendre expansion of the potential energy V(R, y) and tit the radial coefficients V,(R) to continuous
functions
havior.
We employ
tion
having the correct a modified
long-range
Tang-Toennies
befunc-
[10,16,17], 533
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CHEMICAL PHYSICS LETTERS
Table 4 Parameters (kcal/mol and A) for He+ +Hz analytic PES
Vo(R)=Vsw+VL(R),
V,(R)=a,Vs(R)+a,(R)+l/h(R),
(1)
where Vs and I’, are the short-range and long-range potentials, respectively, as and a,_ are anisotropy coefficients, and Vo is the ion-quadrupole interaction. These functions are given by V,(R)& -Bexp(
exp( -bR)
- jbR),
y= :b(R-R,)
i
(bR)k/k!.
are
A B b c d
2.0323 2.5713 2.1362 0.3309
0.2505 -41.90 126.8 14182.1 75.975 3.5587 0.01841 0.00149
(3)
This function is more flexible than that used in refs. [ lo] and [ 17 ] by virtue of the Hulburt-Hirschfelder [ 181 modification of the short-range repulsion, the 1 + cy3 +dy4 factor in eq. (2). (The expression for V,(R) can be rewritten as a HulburtHirschfelder potential.) The long-range coefficients are given in terms of the asymptotic properties of H2 from table 1: a,=2(cu,,-~!,)/((y,,+2~,), C4= d (a,, + 2a, ), and C, = - 8. The parameters A, B, b, c, d, and us are varied in a nonlinear least-squares lit to the ab initio points. Our potential-function subroutine allows us to specify e, the well depth of Vo, R,, the well position, and a, the zero-crossing distance, from which A, B, and b may be obtained algebraically. The rms error in the fit was 0.03 kcal/ mol, with the largest deviation, +O. 16 kcal/mol, occurring on the repulsive wall at R = 1.6 ii, y = 90’. Table 4 collects the potential parameters, while fig. 2 illustrates the fit to the Legendre terms in the well region.
5. Discussion Our results illustrate the need for going beyond SCF or few-configuration MCSCF levels of treatment to obtain accurate results for weak interactions in excited ion-molecule systems. In this system at least, and perhaps more generally, even the accurate representation of the multipole and response prop534
u
as G
,
(2)
ks0
t &
c-3
VQW=-hUW’,R-3, where the damping functions f,(R)
Parameter
aL
(1 +cy3+dy4)
V,(R) = -f,uw4~-4,
j,(R)=l-exp(-bR)
15 October 1993
Fig. 2. Two-term Legendre expansion in the vicinity of the PES minimum. Points are MRCISD ab initio from this work, cu-ves are the analyticfit usingeqs. (1)-( 3). (0 ) VO;(A ) Vz.
et-ties of the neutral molecule does not guarantee an accurate surface at separations where repulsive forces begin to compete. Some light is shed on the effect of CI on the properties of the He+ t HZ PES by comparing it with the Li+ +H, PES [ 19,201. Both systems exhibit Czy surface minima, but that for Li++Hz occurs at &=1.99 A, D,=5.64 kcal/mol, a substantially shorter, stronger “bond” than He+ + Ha. Elementary considerations would suggest that He+ should have a smaller ionic radius than Li+,
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CHEMICAL PHYSICS LETTERS
yet the surface properties indicate the opposite. This reversal is clearly due to the interaction with the charge-transfer ground state, which pushes the eigenvalues apart when the fragment orbitals begin to overlap, in the same region of the surface where the ordinary Pauli repulsion sets in. Obtaining a highly accurate surface for the excited state, therefore, requires that the higher-lying states, which countervail the upward push of the ground state, be accurately located. Thus, either a larger number of configurations in the MCSCF stage or the more efficient MRCISD is required. That charge-transfer effects in this system are significant is also indicated indirectly by the need to augment the analytic potential function beyond what was found suitable for Na+ +Hz [ 101 in order to tit closely the ab initio energies. Without the HulburtHirschfelder modification the rms fitting error was 0.1 kcal/mol (a factor of 10 higher in x2). Hulburt and Hirschfelder [ 181 originally proposed their modification of the Morse potential in part to account for anomalous behavior in the spectroscopic constants of metal hydrides MH, in which there is well-established charge transfer from M to H as the bond is formed. The present results should be useful in the interpretation of low-temperature charge-transfer rates through a generalization to include anisotropy of the complex-potential method employed by Rimura and Lane [ 31 in their two-body modeling of the tunneling process believed responsible for the dominant dissociative charge-transfer channel. Close-coupling methods for treating complex, anisotropic potential scattering are well-established [ 2 1,6], and should naturally yield the orbiting or shape resonances that are apparently playing an important role. In this connection, the recent fit by Aguado et al. [ 151 of the Mclaughlin-Thompson 2 2A’ surface [ 81 using a modification of the Sorbie-Murrell function [22] does not have the appropriate R -” asymptotic behavior, and thus cannot be expected to yield either Langevin-type capture or accurately located resonances. Improved behavior in the attractive limb of the He+ +HZ potential should also aid in the interpretation of unusual repulsive-wall structure found in He*( 2 ‘S) +H2 from nonreactive scattering measurements [ 6 1. One-electron calculations [ 61 using
15 October 1993
a model Hamiltonian based on the Hopper ionic surface yielded only fair agreement with the experimentally derived He*t H2 surface as to the markedness of the repulsive structure; the deeper well and consequent greater curvature of the present ionic PES are bound to improve the comparison. The behavior of the resonance width r(R) for Penning ionization of H2 by He* might also be clarified by noting the extent of charge-transfer state mixing, in light of the charge-transfer model of Penning ionization [ 23 1. We have also calculated the ground-state surface for fixed Hz bond length at the same level of theory, and have developed a more efficient and general, although somewhat less accurate, approach to these states using the orbitals of the neutral system for the CI expansion. These results will be reported in a future paper.
Acknowledgement Support received from the donors of the Petroleum Research Fund, administered by the American Chemical Society, and from the National Science Foundation is gratefully acknowledged. Ab initio calculations were performed on the departmental FPS 500 EA superminicomputer, while fitting and plotting utilized the VAX cluster at the University of Pittsburgh.
References [ 1] C. Kubach, C. Courbin-Gaussorgues and V. Sidis, Chem. Phys. Letters 119 (1985) 523. [2] A. Russek and R.J. Furlan, Phys. Rev. A 39 ( 1989) 5034; R.J. Furlan, G. Bent and A. Russek, J. Chem. Phys. 93 (1990) 6676; R.J. Furlan and G. Russek, Phys. Rev. A 42 (1990) 6436. [3] M. Kimura and N.F. Lane, Phys. Rev. A 44 ( 1991) 259. [4] MM, Schauer, S.R. Jefferts, S.E. Barlow and G.H. Dunn, J. Chem. Phys. 91 (1989) 4593. [5] P.E. Siska, Rev. Mod. Phys. 65 (1993) 337. [6] D.W.MartinandP.E.Siska, J. Chem. Phys. 82 (1985) 2630; 89 (1988) 240. [7] D.G. Hopper, Intern. J. QuantumChem. Symp. 12 (1978) 305; J. Chem. Phys. 73 (1980) 3289; 73 (1980) 4528. [8] D.R. McLaughlin and D.L. Thompson, J. Chem. Phys. 70 (1979) 2748. [9] C.E. Dykstra, J. Chem. Phys. 82 (1985) 4120.
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[lo] M.F. Falcctta, J.L. Pazun, M.J. Dorko, D. Kitchen and P.E. Siska, J. Phys. Chem. 97 (1993) 1011. [ 111R. Shepard, I. Shavitt and J. Simons, J. Chem. Phys. 76 (1982) 543; H. Lischka, R. Shepard, F.B. Brown and I. Shavitt, Intern. J. Quantum Chem. Symp. 15 (1981) 91. [ 121S.F. Boys and F. Bemardi, Mol. Phys. 19 (1970) 553. [ 131A. Khan and K.D. Jordan, Chem. Phys. Letters 128 ( 1986) 368. [ 141W. Kolos and L. Wolniewicz, J. Chem. Phys. 43 (1965) 2429; 46 (1967) 1426. [ 151A. Aguado, C. Sudrez and M. Paniagua, J. Chem. Phys. 98 (1993) 308. [ 161K.T. Tang and J.P. Toe&es, J. Chem. Phys. 80 ( 1984) 3726.
536
I5 October 1993
I171 P.E. Siska, J. Chem. Phys. 85 (1986) 7497. [ 181H.M. Hulburt and J.O. Hirschfelder, J. Chem. Phys. 9 (1941) 61. [ 191W.A.LesterJr., J. Chem.Phys. 53 (1970) 1511;54 (1971) 3171. [20] W. Kutzelnigg, V. Staemmler and C. Hoheisel, Chem. Phys. 1 (1973) 27. [21] A.P. Hickman, A.D. Isaacson and W.H. Miller, J. Chem. Phys. 66 (1977) 1492. [22] A. Agundo and M. Paniagua, J. Chem. Phys. 96 ( 1992) 1265. [23] W.H. Miller and H. Morgner, J. Chem. Phys. 67 (1977) 4923.
CHEMICAL PHYSICS LETTERS
15 October 1993
Ab initio study of the long-range interaction between He+ and H2 M.F. Falcetta and P.E. Siska Departmentof Chemistry, Universityofpittsburgh, Pittsburgh,PA 15260, USA
Received 5 August 1993
A new estimate of the anisotropic long-range potential energy surface for the interaction of He+ ( 1s 3) and HZ(X ‘Zl) has been computed using extended atomic basis sets to construct three-configuration self-consistent-field (MCSCF) molecular orbitals for use in a multireference configuration interaction expansion including all single and double excitations (MRCISD) The most stable geometry is Czv, in agreement with earlier studies, but the well depth is found to be 3.15 kcal/mol at an intermolecular distanceof 2.42 A, 57% deeper and 0.2 A smaller, respectively, than the best previous estimate. The ab initio points are fitted to a potential function with correct long-range behavior, suitable for scattering calculations.
1. Introduct ion
Despite the simplicity of the (HeH,) + ion, several aspects of the electronic structure and collision dynamics involving this species continue to invite both experimental and theoretical investigation. Recent theoretical studies have explored intennediate-tohigh energy (100-1000 eV) inelastic collisions of He+ and Hz [ 1,2], with a view to the delineation of the appropriate diabatic states to use in describing curve crossings occurring at small separations R, the distance between He+ and the center-of-mass of HZ. At thermal and subthermal energy, this system can still undergo exothermic charge transfer (AE= - 9.1 eV), although the rate of this process is four to five orders of magnitude below the Langevin capture rate. A recent two-body quantum modeling study [ 3 ] intended to emulate ultra-low temperature rate measurements [4] suggests that the rates of both the dominant dissociative charge transfer channel and radiative charge transfer depend sensitively on the lifetime of the [He+...H,] collision complex. The lifetime in turn is dependent upon the features of the attractive part of the He+ + Hz potential energy surface (PES); this sensitivity is heightened if orbiting resonances are involved, as indicated by the modeling study. Here an adiabatic picture is more appropriate, in order to capture the perturbing effects
other states of like symmetry on the attraction. Our own interest in the He+ + Hz PES stems from its role as the core potential in the Penning ionization systems He*( 2 ‘3%)+Hz; for a recent review, see ref. [ 5 1. The ionic surface well shape has been shown to influence the shape of the low-energy repulsion in the excited neutral systems [ 61, and the dependence of the electronic wavefunction on R may provide clues to the mechanism of Penning ionization in these systems and in general. In each case the state of interest is the lowest-lying excited state of HeHt (for H2 at or near its equilibrium distance r= 1.4 bohr), 2 ZA’in C, symmetry, a classic example of a charge-transfer state. Among the ab initio studies of this state [l-3,7,8], noting particularly the extensive calculation of McLaughlin and Thompson [ 8 1, the only one to report details of the PES for r= 1.4 bohr and R> 5bohr has been that of Hopper [ 7 1. Hopper examined both CZvand C,, geometries, employing a multiconfiguration selfconsistent-field (MCSCF) approach, using three configurations approximately designated as HelsZH20g, HelsH&, and HelsH&, and variationally minimizing the second energy eigenvalue (2 ‘A, in C&, 2 ‘Z’ in C,,). 4s2p contracted Gaussian atomic basis sets were used on both He and H. While these are large basis sets for first-row atoms, Dykstra [9] has shown that an accurate polarizaof
0009-2614/93/$ 06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
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bility tensor for H2 is not obtainable, at least at the self-consistent-field (SCF) level, unless a d function is present. An accurate polarizability is essential for proper behavior of the ab initio PES at intermediateto-large R, especially as the polarization of HZby He+ is the most important contributor to the interaction energy for Rx34 A. In a recent calculation of the Na+ + H2 PES by our group [ 10 1, it was found that the 5s3pld H atom basis set of Dykstra [9] produces both polarizabilities and quadrupole moment of good accuracy for H2 at the SCF level. This resulted in an apparently highly accurate PES, for which the effects of electron correlation were shown to be nearly negligible. Electron correlation, however, is expected to be more important in the present openshell excited system. We report here MCSCF calculations using extended atomic basis sets, and multi-reference configuration interaction including all single and double excitations (MRCISD) to account for electron correlation effects, on the He+ +HZ excited state PES for CZVand C,, geometries using the COLUMBUS program package [ 111. Basis set superposition error (BSSE) corrections, ignored in all previous ab initio studies of HeH: , are made at each computed point on the surface using the counterpoise method of Boys and Bemardi [ 121. The ab initio points, for r= 1.4 bohr (0.741 A) and 1.6 A
2. Theoretical method Using the 5~3~1d hydrogen atom basis set of Dykstra with the 6s3pld helium basis of Khan and Jordan [ 131, we carried out a three-configuration MCSCF calculation identical, except for the larger basis, to that of Hopper described above. At this level of theory the extended H atom basis yields a larger, more accurate perpendicular polarizability component al, as given in table 1; we therefore expected and found improved long-range behavior of the PES in addition to a deeper well in the most stable CzV geometry, even after correcting for BSSE, without the 532
15 October 1993
Table 1 Computed properties of H2 (r= 1.4 bohr) in atomic units at various levels of theory Basis/level
Ref.
8
ffl
ffJ.
4s2plMCSCF Ss3pld/SCF Ss3pld/CISD accurate
171
0.4735 0.496 0.4506 0.4574
6.533 6.43 6.443 6.381
3.660 4.54 4.507 4.578
1101 this work 1141
inclusion of configuration interaction. The reference list for the CI step consisted of all configurations allowed by symmetry and spin arising from distributing the three electrons among the He 1s, H20g,and H20, orbitals, with the restriction that the He Is orbital be singly or doubly occupied. This resulted in three references in C2,,and six in C,,. All single and double excitations out of these references within the MCSCF orbital space were included. As table 1 indicates, CI improves the quadrupole moment 8 of H2 somewhat while the polarizabilities change little; but the surface, as shown below, alters to a greater degree. Further enlarging either the basis sets or the reference configuration list had little effect ( < 0.05 kcal/mol) on the total energy near the minimum in the surface. All calculated energies were corrected for BSSE by the counterpoise method [ 121. The corrections are small but increasing in magnitude with decreasingR, - 0.023 kcal/mol at the surface minimum and -0.20 kcal at R = 1.6 A in C,, in the CI energy, and are due almost entirely to Hz borrowing the He basis functions. We note that the BSSE vanishes as R-m, is necessarily negative, and must be subtracted from the total energy; therefore surface basins are rendered shallower. The large and balanced basis sets used here serve to keep the corrections relatively small.
3. Results The total energy and BSSE were computed at 26 Rvaluesfrom1.6to12A,andatR=105Atoprovide an energy asymptote, in both CZV(y= 90”, y the angle between R and r) and C,, (y=O”) geometries, for r= 1.4 bohr. Table 2 presents the corrected MRCISD interaction energies, and fig. 1 compares
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I5 October 1993
CHEMICAL PHYSICS LETTERS
Table 2
I
MRCISD interaction energiescorrected for BSSE‘)
I
I
I
30
R (A)
1.60 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.40 3.60 4.00 4.40 4.80 5.20 5.60 6.00 8.00 12.00 l)
He+ Ha (C,)
Energy (kcal/mol)
+
Cl”
CCaY
17.1941 4.9103 1.5916 -0.5573 - 1.8865 -2.6489 - 3.0270 -3.1511 -3.1133 -2.9777 -2.7878 -2.5729 -2.3516 -2.1358 - 1.9323 - 1.7444 - 1.4200 - 1.1601 -0.7912 -0.5581 -0.4070 -0.3059 -0.2358 -0.1854 -0.0685 -0.0175
44.7499 20.4873 13.4821 8.6508 5.3600 3.1512 1.6962 0.7617 0.1823 -0.1583 -0.3413 -0.4228 -0.4411 -0.4218 -0.3820 -0.3325 - 0.2297 -0.1410 - 0.0240 0.0334 0.0596 0.0701 0.0722 0.0697 0.0432 0.0166
The asymptote is taken as the MRCISD energy at R = 105 A, -3.170032536 hartree. 1 hartreez627.510 kcal/mol.
the CZVMCSCF results of Hopper with the extendedbasis MCSCF and MRCISD energies from this work. In agreement with Hopper’s work, we find the CZV geometry to be most stable in both the MCSCF and MRCISD calculations; this is determined mainly by the anisotropy of the ion-quadrupole interaction. The three CZVpotentials may be concisely compared through the zero-crossing radii RO,minimum positions R, and well depths O,, as given in table 3. We note that Hopper’s calculations have not been corrected for BSSE; his De is therefore expected to be larger and his R, and & smaller than for a corrected calculation at the same level of theory. Even so, improving the basis set at the MCSCF level deepens the well by 17%without substantially affecting the range, while including the effects of electron correlation produces a 57% increase in De and a decrease of 0.2
20
Fig. 1. Comparison of ab initio potential energy curves for CsV approach of He+ to HI. The MCSCF and MRCISD points are the present results; the MRCISD values are given in table 2. The large-R asymptotic energy has been subtracted out in each case. (0) Ref. [7]; (0) MCSCF; (0) MRCISD. Table 3 Comparison of ab initio He ++Hz C,, well depths 0, (kcal/mol) and positions R,, ~0 (A) Ref.
Q
R,
J&l
18,151 t31
1.87 2.05
2.35 2.35
2.155 2.10, 1.97,
[71
2.00
2.60~
Js3pldfMSCSF MRCISD
2.34, 3.150
2.632 2.42s
A
in both R. and R, compared to the earlier findings.
4. Analytic representation of the PES As in the earlier work on Na+ ’ + HZ [ 101, we use the ab initio points to construct the first two terms in the Legendre expansion of the potential energy V(R, y) and tit the radial coefficients V,(R) to continuous
functions
havior.
We employ
tion
having the correct a modified
long-range
Tang-Toennies
befunc-
[10,16,17], 533
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Table 4 Parameters (kcal/mol and A) for He+ +Hz analytic PES
Vo(R)=Vsw+VL(R),
V,(R)=a,Vs(R)+a,(R)+l/h(R),
(1)
where Vs and I’, are the short-range and long-range potentials, respectively, as and a,_ are anisotropy coefficients, and Vo is the ion-quadrupole interaction. These functions are given by V,(R)& -Bexp(
exp( -bR)
- jbR),
y= :b(R-R,)
i
(bR)k/k!.
are
A B b c d
2.0323 2.5713 2.1362 0.3309
0.2505 -41.90 126.8 14182.1 75.975 3.5587 0.01841 0.00149
(3)
This function is more flexible than that used in refs. [ lo] and [ 17 ] by virtue of the Hulburt-Hirschfelder [ 181 modification of the short-range repulsion, the 1 + cy3 +dy4 factor in eq. (2). (The expression for V,(R) can be rewritten as a HulburtHirschfelder potential.) The long-range coefficients are given in terms of the asymptotic properties of H2 from table 1: a,=2(cu,,-~!,)/((y,,+2~,), C4= d (a,, + 2a, ), and C, = - 8. The parameters A, B, b, c, d, and us are varied in a nonlinear least-squares lit to the ab initio points. Our potential-function subroutine allows us to specify e, the well depth of Vo, R,, the well position, and a, the zero-crossing distance, from which A, B, and b may be obtained algebraically. The rms error in the fit was 0.03 kcal/ mol, with the largest deviation, +O. 16 kcal/mol, occurring on the repulsive wall at R = 1.6 ii, y = 90’. Table 4 collects the potential parameters, while fig. 2 illustrates the fit to the Legendre terms in the well region.
5. Discussion Our results illustrate the need for going beyond SCF or few-configuration MCSCF levels of treatment to obtain accurate results for weak interactions in excited ion-molecule systems. In this system at least, and perhaps more generally, even the accurate representation of the multipole and response prop534
u
as G
,
(2)
ks0
t &
c-3
VQW=-hUW’,R-3, where the damping functions f,(R)
Parameter
aL
(1 +cy3+dy4)
V,(R) = -f,uw4~-4,
j,(R)=l-exp(-bR)
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Fig. 2. Two-term Legendre expansion in the vicinity of the PES minimum. Points are MRCISD ab initio from this work, cu-ves are the analyticfit usingeqs. (1)-( 3). (0 ) VO;(A ) Vz.
et-ties of the neutral molecule does not guarantee an accurate surface at separations where repulsive forces begin to compete. Some light is shed on the effect of CI on the properties of the He+ t HZ PES by comparing it with the Li+ +H, PES [ 19,201. Both systems exhibit Czy surface minima, but that for Li++Hz occurs at &=1.99 A, D,=5.64 kcal/mol, a substantially shorter, stronger “bond” than He+ + Ha. Elementary considerations would suggest that He+ should have a smaller ionic radius than Li+,
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yet the surface properties indicate the opposite. This reversal is clearly due to the interaction with the charge-transfer ground state, which pushes the eigenvalues apart when the fragment orbitals begin to overlap, in the same region of the surface where the ordinary Pauli repulsion sets in. Obtaining a highly accurate surface for the excited state, therefore, requires that the higher-lying states, which countervail the upward push of the ground state, be accurately located. Thus, either a larger number of configurations in the MCSCF stage or the more efficient MRCISD is required. That charge-transfer effects in this system are significant is also indicated indirectly by the need to augment the analytic potential function beyond what was found suitable for Na+ +Hz [ 101 in order to tit closely the ab initio energies. Without the HulburtHirschfelder modification the rms fitting error was 0.1 kcal/mol (a factor of 10 higher in x2). Hulburt and Hirschfelder [ 181 originally proposed their modification of the Morse potential in part to account for anomalous behavior in the spectroscopic constants of metal hydrides MH, in which there is well-established charge transfer from M to H as the bond is formed. The present results should be useful in the interpretation of low-temperature charge-transfer rates through a generalization to include anisotropy of the complex-potential method employed by Rimura and Lane [ 31 in their two-body modeling of the tunneling process believed responsible for the dominant dissociative charge-transfer channel. Close-coupling methods for treating complex, anisotropic potential scattering are well-established [ 2 1,6], and should naturally yield the orbiting or shape resonances that are apparently playing an important role. In this connection, the recent fit by Aguado et al. [ 151 of the Mclaughlin-Thompson 2 2A’ surface [ 81 using a modification of the Sorbie-Murrell function [22] does not have the appropriate R -” asymptotic behavior, and thus cannot be expected to yield either Langevin-type capture or accurately located resonances. Improved behavior in the attractive limb of the He+ +HZ potential should also aid in the interpretation of unusual repulsive-wall structure found in He*( 2 ‘S) +H2 from nonreactive scattering measurements [ 6 1. One-electron calculations [ 61 using
15 October 1993
a model Hamiltonian based on the Hopper ionic surface yielded only fair agreement with the experimentally derived He*t H2 surface as to the markedness of the repulsive structure; the deeper well and consequent greater curvature of the present ionic PES are bound to improve the comparison. The behavior of the resonance width r(R) for Penning ionization of H2 by He* might also be clarified by noting the extent of charge-transfer state mixing, in light of the charge-transfer model of Penning ionization [ 23 1. We have also calculated the ground-state surface for fixed Hz bond length at the same level of theory, and have developed a more efficient and general, although somewhat less accurate, approach to these states using the orbitals of the neutral system for the CI expansion. These results will be reported in a future paper.
Acknowledgement Support received from the donors of the Petroleum Research Fund, administered by the American Chemical Society, and from the National Science Foundation is gratefully acknowledged. Ab initio calculations were performed on the departmental FPS 500 EA superminicomputer, while fitting and plotting utilized the VAX cluster at the University of Pittsburgh.
References [ 1] C. Kubach, C. Courbin-Gaussorgues and V. Sidis, Chem. Phys. Letters 119 (1985) 523. [2] A. Russek and R.J. Furlan, Phys. Rev. A 39 ( 1989) 5034; R.J. Furlan, G. Bent and A. Russek, J. Chem. Phys. 93 (1990) 6676; R.J. Furlan and G. Russek, Phys. Rev. A 42 (1990) 6436. [3] M. Kimura and N.F. Lane, Phys. Rev. A 44 ( 1991) 259. [4] MM, Schauer, S.R. Jefferts, S.E. Barlow and G.H. Dunn, J. Chem. Phys. 91 (1989) 4593. [5] P.E. Siska, Rev. Mod. Phys. 65 (1993) 337. [6] D.W.MartinandP.E.Siska, J. Chem. Phys. 82 (1985) 2630; 89 (1988) 240. [7] D.G. Hopper, Intern. J. QuantumChem. Symp. 12 (1978) 305; J. Chem. Phys. 73 (1980) 3289; 73 (1980) 4528. [8] D.R. McLaughlin and D.L. Thompson, J. Chem. Phys. 70 (1979) 2748. [9] C.E. Dykstra, J. Chem. Phys. 82 (1985) 4120.
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[lo] M.F. Falcctta, J.L. Pazun, M.J. Dorko, D. Kitchen and P.E. Siska, J. Phys. Chem. 97 (1993) 1011. [ 111R. Shepard, I. Shavitt and J. Simons, J. Chem. Phys. 76 (1982) 543; H. Lischka, R. Shepard, F.B. Brown and I. Shavitt, Intern. J. Quantum Chem. Symp. 15 (1981) 91. [ 121S.F. Boys and F. Bemardi, Mol. Phys. 19 (1970) 553. [ 131A. Khan and K.D. Jordan, Chem. Phys. Letters 128 ( 1986) 368. [ 141W. Kolos and L. Wolniewicz, J. Chem. Phys. 43 (1965) 2429; 46 (1967) 1426. [ 151A. Aguado, C. Sudrez and M. Paniagua, J. Chem. Phys. 98 (1993) 308. [ 161K.T. Tang and J.P. Toe&es, J. Chem. Phys. 80 ( 1984) 3726.
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I171 P.E. Siska, J. Chem. Phys. 85 (1986) 7497. [ 181H.M. Hulburt and J.O. Hirschfelder, J. Chem. Phys. 9 (1941) 61. [ 191W.A.LesterJr., J. Chem.Phys. 53 (1970) 1511;54 (1971) 3171. [20] W. Kutzelnigg, V. Staemmler and C. Hoheisel, Chem. Phys. 1 (1973) 27. [21] A.P. Hickman, A.D. Isaacson and W.H. Miller, J. Chem. Phys. 66 (1977) 1492. [22] A. Agundo and M. Paniagua, J. Chem. Phys. 96 ( 1992) 1265. [23] W.H. Miller and H. Morgner, J. Chem. Phys. 67 (1977) 4923.