J. Phys. Chem. 1995, 99, 1711-1718
1711
Doubly-Excited States in the Spectrum of Molecular Hydrogen Chris H. Greene" Department of Physics and The Joint Institute for Laboratory Astrophysics, University of Colorado, Boulder, Colorado 80309-0440
Byungduk Yo0 Laser Spectroscopy Laboratory, Korea Atomic Energy Research Institute, P.O. Box 105, Yusong, Taejon, South Korea Received: August 10, 1994; In Final Form: October 17, 1994@
An ab initio calculation of the fixed-nuclei scattering information required for multichannel quantum defect applications is presented for several symmetries of molecular hydrogen. This represents a modest step toward the longer-range goal of constructing a unified theoretical description of all fragmentation channels in H2, including ionization and both neutral and ionic dissociation channels. The errors in our calculated fixednuclei quantum defects are generally in the range 0.005-0.05. This is sufficiently accurate to consider a fully ab initio calculation of H2 photoabsorption near the ionization threshold, including rovibrational channel interactions by the usual frame transformation method.
I. Introduction Only one type of theoretical approach to date has been able to successfully describe the complicated rovibronic channel interactions in Hz Rydberg spectra near the ionization threshold, of the type observed by Dehmer and Chupka.' This successful approach is based on multichannel quantum defect theory (MQDT), including a rovibrational frame transformation. The work of Jungen, Dill, Atabek, Raoult, and c o - ~ o r k e r s ~gives -~ several beautiful examples of how this theoretical scheme quantitatively describes perturbed HZ Rydberg spectra of impressive complexity. Moreover, the conceptual basis of this MQDT frame transformation approach is quite simple and intuitive, and the calculations are small in scale compared to many numerical schemes used nowadays to treat complex spectra. The goal of this paper is to show some of the physical considerations needed to implement this MQDT approach in a completely ab initio inanner. We show ab initio calculations of the required fixed-nuclei scattering information and discuss potential difficulties that are likely to arise if they are used in an MQDT frame transformation calculation of the full H2 spectrum. The treatment of H2 ungerade states excited from the ground state is the simplest, because at the relevant internuclear distances p-f mixing is extremely weak and can be entirely neglected for most purposes. This reduces the problem to one of describing the p-wave Rydberg states, which are accurately characterized in the body frame by a quantum defect p~ (R) that depends on the internuclear distance R and on the projection A of the total orbital angular momentum onto the internuclear axis. In a f i s t approximation, this body-frame p-wave quantum defect function can be extracted from accurate ab initio calculations of the H2 potential curves U,,a(R), such as those of Kolos and Wolniewicz.6 This extraction uses the Rydberg formula applied at each internuclear distance, namely (in au),
@Abstractpublished in Advance ACS Abstracts, February 1, 1995.
where U'O"(R) is the ground state (lsa) potential curve of Hz+. Jungen and Atabek2 showed how these quantum defect functions serve as input to a rovibrational frame transformation, leading to a matrix formulation that requires matrix elements such as (v+I~AIv+'). Here the notation Iv+) indicates a vibrational state of the H2+ ground state potential curve. The resulting MQDT calculation based on this matrix generates far more accurate energy levels than if the same information is used within the context of a Born-Oppenheimer approximation calculation, owing to its essentially exact inclusion of a class of nonadiabatic effects. Still more accurate results were obtained by Jungen and Atabek after they made small ad hoc adjustments to their quantum defect functions; these adjustments were chosen simply to give better agreement between the MQDT calculation and the experiment. Ideally, it would be preferable to generate the input information &A(R))needed for these MQDT calculations entirely from ab initio theory, to eliminate all uncertainty about what physics is included in such semiempirical fitting procedures. The gerade singlet states of HZare far more complicated to describe theoretically at small R than the ungerade states, because even at relatively small internuclear distances the 2p02 doubly-excited state comes "crashing down" (as R increases) across the lsanso and lsondo Rydberg manifolds. This doublyexcited state causes an infinite number of avoided crossings and produces a pronounced double minimum in several of the Born-Oppenheimer potential curves, such as the EF state. In fact, each of the doubly-excited Rydberg states 2ponpo causes an additional infinite number of avoided crossings. One consequPnce of these crossings is an enhanced importance of strong nonadiabatic coupling effects. In a series of papers, Ross and J~ngen'-~have recently shown that the complicated physics associated with the gerade Rydberg state manifold can be described within the same MQDT and frame transformation treatment. Their work starts from accurate ab initio BomOppenheimer potential curves calculated by others.l0 In contrast to the ungerade states at small R, the fixed-nuclei physics for 'C, symmetry is intrinsically multichannel in nature, and accordingly the fixed-nuclei information takes the form of a 3 x 3 multichannel reaction matrix K;(R). The indices of this
0022-365419512099-1711$09.0010 0 1995 American Chemical Society
1712 J. Phys. Chem., Vol. 99, No. 6, 1995 matrix, e.g., for the case of l& electronic symetry, refer to the three channels i = { lsacso, lsacda, 2pacpa). (As usual, the channel notation €pa is intended to refer to the entire set of p a Rydberg states and to the adjoining continuum.) Once the body reaction matrix K$(R) has been determined, as in the work of ref^,^-^ one must again calculate vibrational A +’ matrix elements of the type (vi+ IKiitlvi, ) to obtain the laboratory frame reaction matrix. Now the calculation grows in complexity because the ionic vibrational states T ( R ) are different for Rydberg levels attached to the different H2+ l s a and 2pa electronic states. Moreover, the purely repulsive nature of the 2pa potential curve forces the introduction of a finite “box” for the internuclear coordinate, 0 I R I Ro, which discretizes the vibrational spectrum for that electronic channel. (This discretization introduces “unphysical 2po ionization thresholds” into the calculation that would cause unphysical effects in the spectrum, but those artifacts are limited to energies above the dissociation threshold of H2+. This high-energy range is more complicated by the presence of a three-body continuum and is beyond the scope of the present work. Some additional issues relating to the discretization of the vibrational spectrum of a repulsive Born-Oppenheimer potential curve are addressed in ref.”) Nevertheless, after these steps are performed, standard MQDT matrix manipulations convert the laboratory-frame reaction matrix into a prediction of observables such as energy levels, oscillator strengths, etc. The procedure just described has omitted one important detail. The “true” body-frame reaction matrix should be written K: (E,R) to show explicitly that it depends on 6, the body-frame energy. While the body-frame energy dependence can frequently be ignored or else handled indirectly, especially over a limited energy range close to an ionization threshold, its effects are important in some cases.12 The importance of quantum defect energy dependences can be dramatized by considering the example of a l s m d o Rydberg series with a constant (energyindependent) quantum defect ,u(R). The potential curves of these molecular Rydberg states are given (in au) by
These potential curves are nearly parallel to the ionic l s a potential curve for sufficiently high principal quantum numbers n, but not quite parallel owing to the R dependence of the quantum defect. Note that nothing in the quantum defect formula itself limits the range of n values, which suggests that eq 2 seems to imply the existence of a “2dd’ state and also a “ l d d ’ state, which are known not to exist since the lowest possible principal quantum number for a d state should be n = 3. The only way to “terminate” the bottom end of a Rydberg series within the context of quantum defect theory is to include the energy dependence of the quantum defect. Analytical and numerical studies have both shown, in a variety of contexts, that the quantum defect acquires a characteristic (and strong) energy dependence as the energy decreases below the lowest member of a Rydberg series.I3 This energy dependence is precisely that which is needed to eliminate these unphysical roots (such as the l s d d a state) from the Rydberg formula. (By working with the so-called 7 quantum defect, Ross and Jungen eliminate some of these unphysical states, but this procedure will not generally eliminate all spurious roots.) A first study by Ross and Jungen7 ignored all energy dependences, but the most recent work8s9 approximately included some energy dependence of the (7-type) quantum defect parameters. Strong energy dependences of quantum defect parameters are exceedingly difficult to incorporate into theoretical descriptions
Greene and Yo0 based on a frame transformation. This difficulty has been documented both in atomic systems14 and in a range of molecular systems.’* Consequently, it is important to study the full energy dependences of H2 body-frame quantum defects calculated using an ab initio approach. While a fitting procedure as implemented by Ross and Jungen can be expected to give higher quantitative accuracy, given current limitations on ab initio quantum defect calculations, it remains important to test the reliability of the resulting fitted parameters. A large number of papers have appeared in the literature dealing with these bound state potential curves of H2 and with the doubly-excited autoionizing states (for a sampling, see refs 6, 10, and 15-30). To the best of our knowledge, however, only two previous theoretical studies directly calculated the fixed-nuclei quantum defect parameters. Both of these calculations were restricted to the ungerade symmetry. The first was carried out by Raseev15 with the goal of generating the fixednuclei information needed to treat H2 photoionization processes. Doubly-excited states were included, but only through their effects on the single-electron p-wave phase shift of an outermost electron that escapes from the H2+ ground state. These calculations give a p a quantum defect at R = 1.4 au and zero electron energy that is about 0.02 lower than the exact value. Raseev’s treatment of doubly-excited states is different from -~ our approach. Whereas we, like Ross and J ~ n g e n , ~treat doubly-excited states such as 2ponsa as an explicit (weaklyclosed) channel in the usual sense of MQDT, Raseev does not have an explicit MQDT channel refemng to the double excitations. Raseev’s resulting quantum defects are consequently more poorly suited to a description of doubly-excited state effects, as these states induce strong phase variations with E and R. Such strong dependences on body-frame energy and internuclear distance render the rovibrational frame transformation (as currently implemented) virtually useless. The second ab initio calculation of fixed-nuclei quantum defects was carried out by Stephens and McKoy16 using the Schwinger variational principle, reformulated to use a “smooth” Coulomb Green’s function to handle negative body-frame energies as in ref 32. This calculation was carried out at the independent-electron (Hartree-Fock) level and ignored all doubly-excited states. The Schwinger results are qualitatively correct at small internuclear distances, e.g., giving errors of order 0.1 in the quantum defects near R = 2 au, but doubly-excited state effects would have to be incorporated to render such calculations useful at larger distances.
II. Ab Znitio Calculation of Body-Frame Quantum Defects With these motivations in mind, we have carried out a set of fixed-nuclei eigenchannel R-matrix calculations using a prolate spheroidal coordinate system. The streamlined form of the eigenchannel R-matrix method has been described elsewhere33 and will not be repeated in this paper. We summarize briefly the novel aspects of these calculations and focus most of our discussion on the interpretation of our numerical results. Each R-matrix calculation is performed at a fixed value of the internuclear distance R and also at a fixed value of A. In this study we treat only states having A = 0. The eigenchannel R-matrix calculation singles out a set of stationary states qp inside a reaction volume, each of which has a constant logarithmic derivative bp on the surface of this volume. In this study our reaction volume is the interior of the spheroidal region whose surface is a set of points E = 60, and 60 is the value of the R-matrix “box radius”. Here E = ( r ~ 1g)/2R is the “radial-
+
J. Phys. Chem., Vol. 99, No. 6, 1995 1713
Spectrum of Molecular Hydrogen type” spheroidal coordinate of either electron, where rA and ~g are the distances of that electron from nuclei A and B, respectively. For the calculations reported in this paper, the spheroidal box radius was chosen to vary continuously with R, after some experimentation, through the equation 60 = 15/R 1. As normally formulated, the value of bp relates the solution v p and its normal derivative aly,&3n at the surface S of the reaction volume according to
+
but both values of the total spin S = 0, 1 and parity u, g were considered. The basis set included the following angular configuration types for gerade symmetry:
sas’a,pap’a,sad’a,dad’o,faa,paa, p ~ ‘ n , p ~ n , f ~ n , ~ d ‘ n , d s d ’ s(,10) fsrs A total of 201 such configurations were included for ’& symmetry and 187 for 32gsymmetry. The angular configuration types included for ungerade symmetry were
(3) Following a suggestion by R o b i ~ h e a u xwe , ~ ~instead define bp in this study through the modified relation
(4)
This last relation can be expressed more transparently as
This same choice was adopted by S ~ h n e i d e rin~ ~his early R-matrix treatment of e-H2 scattering in spheroidal coordinates. A major advantage of this procedure is that it automatically leads to a symmetric R-matrix in spheroidal coordinates, irrespective of the level of convergence of the variational c a l ~ u l a t i o n .In ~ ~eq 2, the radial coordinate metric coefficient evaluated at the reaction surface is given by
Here 7 = (rA - m)/R is the usual (nonazimuthal) angular spheroidal coordinate, which has the range - 1 5 7 5 1. Inserting a basis set expansion in terms of unknown coefficients ck of superposition into the variational expression for bp gives the following (modified) generalized eigenvalue equation
r C = bAC
(7)
in which b is one of the desired eigenvalues bp Here the matrices r and A are real symmetric matrices, provided the phases of the basis functions are chosen appropriately. They are given explicitly by
(9) Here H is the fixed-nuclei (Bom-Oppenheimer) electronic Hamiltonian. The volume integrals (dV) extend over the interior of the reaction volume, while the surface integrals (dS)include all points in configuration space such that max{51,&} = (0. The two-electron basis functions used are antisymmetrized products of single-electron eigenfunctions of the H2+ Hamiltonian. The angular (a) spheroidal equation was solved by diagonalization in a basis of associated Legendre functions while the radial one-electron equation was integrated numerically. It was found to be important to pay careful attention to the behavior of these solutions near their singular point 6 1. The calculations of this initial investigation treated only Z states,
-
leading to 180 two-electron configurations for both values of total spin, S = 0, 1. A transformation equivalent to the “streamlined formulation” of ref 33 was also performed, to improve the efficiency of solving the generalized eigenvalue equation (6) at many different energies E . The motion of the (outermost) electron beyond the reaction surface was described using quantum defect techniques. Two different methods for handling this part of the calculation were explored. In the first method, generalized quantum defect t h e 0 1 - y ~was ~ ~used ~ ~ to construct regular and irregular two-center Coulomb functions directly in spheroidal coordinates. The construction of these matching solutions is based on the Milne phase-amplitude method and has been described in ref 38. The second method starts from the variational wave functions determined by solving eq 6 within the spheroidal reaction volume, followed by projection of the resulting solutions (and their normal derivatives) onto the surface of a sphere contained wholly within the spheroid. These gave spherical coordinate representations of the solutions, which were next matched to ordinary (single-center) Coulomb functions at the appropriate channel energies. This determined the reaction matrix or the equivalent quantum defect matrix in a form suitable for eventual application of the rovibrational frame transformation, although this last frame transformation step was not carried out in this exploratory study. Because these two methods were found to give almost identical results for the adiabatic potential curves, calculations based only on the second method are shown in this paper.
111. Comparisons with Other Fixed-Nuclei Calculations The most stringent test of the accuracy of the fixed-nuclei calculations is to compare with extremely accurate BomOppenheimer potential curves that have been calculated by other groups. Figure la,b shows the resulting HZ potential curves for 1,32g symmetries, respectively, along with some of the most accurate calculations available for comparison. Of course, only the potential curves below the H2+ Isa ground state are truly bound within the Bom-Oppenheimer approximation. The clearest demonstration of our results can be found in Figure la, which excludes the H2 ground state potential curve to permit us to concentrate on the energy range closer to the ionization threshold. The differences between our calculation and that of Wolniewicz and Dressier" can barely be distinguished in this figure for R < 4 au. At larger R, however, the differences are increasingly noticeable. The potential curves calculated variationally by Wolniewicz and Dressler are generally (but not always) lower than out potential curves, which is expected because their calculation should be more accurate. The Hylleras-Undheim theorem stating that the more accurate calculation should always lie lower in energy is not strictly applicable here, because our calculations are variational only within a finite volume: a nonvariational (quantum defect) approximation was used beyond the reaction surface which could in principle cause
1714 J. Phys. Chem., Vol. 99, No. 6, 1995
,
0.0
Greene and Yo0
TABLE 1: Positions (Er) and Widths (r,)of the Lowest lZg
I,
Resonance in Hz Are Shown as Functions of the Internuclear Distance Ra
NE,or r,) mesent 1.O (Er) 1.2 1.4 1.6 1.8 2.0 2.2 2.5 2.6 1.o (r,) 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.5 2.6
$
Q
Ez w
-Oe4:
w
-0.8 -
0.237 26 0.041 58 -0.107 99 -0.228 86 -0.327 68 -0.409 12 -0.477 31 -0.554 32 -0.573 88 2.29(-2) 2.41(-2) 2.51(-2) 2.59(-2) 2.65(-2) 2.88(-2) 3.35(-2) 5.52(-2) 4.59(-2) 5.25(-2)
Shimamura et aLZ9 Takagi and Nakamuraz8 0.2485 0.0512 -0.0994 -0.2194 -0.3174 -0.3985 -0.4660 -0.5486 -0.5717 1.50(-2) 2.07(-2) 2.76(-2) 3.52(-2) 4.31(-2) 5.09(-2) 5.85(-2) 6.52(-2) 6.76(-2) 6.98(-2)
0.053 43 -0.091 58 -0.217 53 -0.301 65 -0.401 63 -0.475 94
1.9(-2) 1.8(-2) 4.3(-4) 5.9(-2) 6.7(-2)
“All results are given in atomic units. Each energy position displayed is the total energy of the fied-nuclei system, Le., including the proton-proton Coulombic repulsion 1/R. The notation (-n) means lo-”.
L:
1 9 3 2
=!
3 * (3
a W
with the errors obtained in several atomic calculations using the eigenchannel R-matrix method.39 Our resdts could presumably be improved by increasing the number of partial waves in our basis set expansion, but we have not made any systematic attempt to do so. We do not present all of our calculated potential curves because of space limitations, but they are available from the first author upon request, for any state of symmetry. Figure l a also shows two ‘Z, “autoionizing potential curves” of H2 at body-frame energies above the H2+ ground state curve, which are not bound in this fixed-nuclei calculation. These are the lowest two members (2p02 and 2pa3po) of a Rydberg series of electronically autoionizing states that converge (with increasing principal quantum number) to the excited H2+ 2pa potential curve. Being unbound, these are not adiabatic potential curves in the usual sense. These are frequently portrayed in the literature as “diabatic” potential curves which are not eigenvalues of the full fixed-nuclei Hamiltonian H,only of a “portion” of H with the electronic continua projected out. As R increases, these eigenvalues eventually cross below the 1sa potential curve of H2+ and cause complicated avoided crossings and perturbations among the H2 bound state potential curves. For instance, the lowest curve shown in Figure l a is the well-known EF state, whose double-minimum nature derives from the plunging 2 p d doubly-excited state potential curve. Figure l b displays some similar avoided crossings among the 3Z,potential curves, although they are less pronounced for the triplet potential curves than for the singlets. The smoother nature of the triplet potentials is largely due to the fact that the lowest doubly-excited curve is now 2pa3pa, whereas the lowest singlet curve (2pa2) is far lower in energy. Table 1 compares the positions and widths of the lowest ‘Z, doubly-excited autoionizing state with other calculations and shows that a reasonable solid consensus has been reached concerning the resonance positions. On the other hand, the calculated resonance widths show wide variations. These widths were obtained by searching for maxima of the energy derivative of the (fixed-nuclei) eigenphase sum that was calculated through the standard MQDT “elimination” of closed channels. Figure l b compares our calculated 3Zgpotential curves with that of Kolos and Wolniewicz22for the lowest state a13C,. The
-0.4-
3
-0.6 -
-0.8
4 0
2
4
6
1 8 1 0 1 2 1 4
R(a.u.) Figure 1. Calculated gerade symmetry Bom-Oppenheimer potential curves are shown as the total body-frame energy versus internuclear distance, in au. Bound state potential curves obtained in the present calculations are shown as thin solid lines, while the double-excited resonance positions are shown as thick solid lines. The l s a and 2pa ionic Hz+potential curves are drawn as dashed curves. (a) Potential curves for lXg symmetry. The bound state potentials of Wolniewicz and Dressier'* are drawn as dotted lines. (b) Potential curves for 3X, symmetry. Dotted lines show the bound state potentials of Wakefield and Davidson:O while a fine dashed lined shows the potential curve calculated by Kolos and Wolniewic~.’~
small errors in the energy having either sign. The quantitative errors in our potential curves at R = 2 (assuming that the ref 17 calculation can be regarded as “exact”) are (in au) AU = 0.OOO 53,O.OOO 32, and 0.000 16 for the lowest three ‘Egcurves of Figure la, respectively. These translate into (dimensionless) quantum defect errors of -0.0049, -0.0082, and -0.0049. At larger internuclear distances our convergence is clearly poorer, as the same three curves have quantum defect error of -0.024, -0.020, and -0.046 at R = 9. These errors are competitive
J. Phys. Chem., Vol. 99, No. 6,1995 1715
Spectrum of Molecular Hydrogen
TABLE 2: Same as Table 1, Except for the Lowest 'C, Resonance of €Iz R(E, or I?,) present Collins et aL3' Takagi and NakamuraZ8
0.0
-0.2
?
?
J
>
Q
-0.4
3
3 -0.8
-0.8
0
2
4
6
8
1 0 1 2 1 4
R(a.u.) 0
-0.2
A
2
-0.4
Y
>.
xg
-0.8
W
-1
0
2
4
8
8 1 0 1 2 1 4
R ( a.u. 1 Figure 2. Same as Figure 1 except for ungerude symmetry of Hz. (a) Potential curves for 1& symmetry. The bound state potential curves of Kolos and Wolniewicd9 are shown as dotted lines. (b) Potential curves for 3& symmetry. The lowest bound state potential curve of Kolos and Wolniewicz6is drawn as a dotted line, as is the third lowest bound state of Davidson.z' Of these potential curves calculated , only this latter potential curve of Davidson previously for 32,symmetry, can be distinguished from our results, on the scale of this figure.
1.O (Er) 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.5 2.6 3 .O 3.5 1.O (rr) 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.5 2.6 3.0
0.284 10 0.099 24 -0.037 81 -0.143 07 -0.226 77 -0.293 18 -0.346 15 -0.388 05 -0.405 72 -0.421 29 -0.467 58 -0.510 66 1.12(-2) 1.40(-2) 1.61(-2) 1.92(-2) 2.30( -2) 2.70(-2) 3.25(-2) 3.92(-2) 4.25(-2) 4.42(-2) 4.49(-2)
0.290 21 0.0990 -0.033 47 -0.1430 -0.222 65 -0.290 23 -0.342 84
-0.2917 -0.3853
-0.415 83
-0.4185 -0.4659
8.30(-3) 1.29(-2) 1SO( -2)
1.27(-2) 1.95(-2)
2.20(-2) 2.70(-2) 3.10(-2)
2.37(-2) 2.93(-2)
3.50(-2)
3.20(-2) 3.92( -2)
of magnitude to I'= 2.4 x au at R = 4 au. G ~ b e r m a n ~ ~ does not present calculated widths for this state. Figure 2a,b presents our calculated ungerade Bom-Oppenheimer potential curves for l s 3 2 , , symmetry, respectively. The lowest states agree well with the calculations of Kolos and Wolniewicz, although the agreement deteriorates noticeably as the intemuclear distance increases. The third-lowest 3C, potential curve is seen in Figure 2b to agree with that of Davidsonzl for R < 4 au, but at larger values of R Davidson's curve is much higher and we suspect it is not converged. Table 2 shows the positions and widths of the lowest doubly-excited autoionizing resonance of IZ,, symmetry. The calculated widths are in somewhat better agreement here than for the lZgresonance tabulated in Table 1. The variations among the various theoretical calculations for resonance widths are somewhat surprising for this relatively simple and fundamental diatomic molecule. Because fixednuclei resonance widths are not directly measurable experimentally, it is difficult to judge which calculation is likely to be the most accurate. In variational bound-state calculations, the Hylleras-Undeheim theorem guarantees that the lowest calculated levels are the most accurate. By this criterion the present calculations might be expected to be the most accurate, as the present values for the resonance potential curves are generally (with the exception of the R = 1.2 lZ,, resonance) the lowest in Tables 1 and 2. However, it should be remembered that the present calculations are variational only over the finite reaction volume and also that the Hylleras-Undheim theorem is not strictly applicable to autoionizing resonances. Nevertheless, we can see some advantages of the present spheroidal coordinate approach compared to that of Collins et such as the fact that we incorporate the (essentially) exact energy eigenvalues and eigenfunctions of H2+ instead of using a truncated basis set expansion for the ionic states. One aspect of our numerical results that looks suspicious and deserves further study is the nonmonotonic behavior of the '2, resonance widths from R = 2.2 to R = 2.6. The MQDT fits of Ross and Jungen for 'Z, symmetry predict resonance widths Trof 0.024 and 0.060 au for R = 2.0 and R = 2.5, respectively. Inspection of Table 1 shows that Ross and Jungen's value at R = 2.0 is much closer to our value than to either that of Shimamurafetal. or that of Takagi and Nakamura. On the other hand, the calculation of ~
next two higher-lying states are compared with the values of Wakefield and Davidson*O out to R = 6 au. Wakefield and Davidson also calculated the lowest state, but their values for this lowest Bom-Oppenheimer potential curve of 32,symmetry are higher than, and presumably inferior to, those of ref 22 and those of the present study. Figure l b shows only one autoionizing state (2pa3pa). The energies of this autoionizing state lie slightly lower than Guberman's values23for R < 2 au but at larger intemuclear distances the present values are somewhat higher. Our calculated width for this autoionizing state at R = 2 au is I'= 2.8 x au and it increases by nearly an order
1
.
~
~
9
~
~
Greene and Yo0
1716 J. Phys. Chem., Vol. 99, No. 6, 1995 Shimamura et al. is in better agreement with the prediction of Ross and Jungen at R = 2.4 In summary, we have not yet been able to determine whether our results for doubly-excited state resonance properties are better or worse than the results reported by other authors. In view of the importance of the resonance width for calculations of other properties, such as dissociative recombination cross sections,@it will be important to improve the reliability of the calculated widths.
IV. Quantum Defect Parameters versus Energy and Internuclear Distance The results of the preceding section have shown that these eigenchannel R-matrix calculations in prolate spheroidal coordinates can achieve a useful level of accuracy for both the singlyand doubly-excited states of H2. For the lowest states of each symmetry, our results are clearly not as accurate as the impressive calculations of Kolos, Wolniewicz, Dressler, and coworkers. Nevertheless, the errors in our calculated quantum defects are not large, typically IApl 0.05. The small error implies that we have correctly determined the phase of the outermost H2 electron to within a fraction ( < 5 % ) of its halfwavelength: accuracy in this phase information is the most important prerequisite to using the fixed-nuclei quantum defects in a subsequent frame transformation calculation that includes coupling of the electron motion to the rotational and vibrational motions. Nearly all previous frame transformation studies have used energy-independent body-frame quantum defects ~ A ( Ras ) the starting point, or else they have incorporated a very small and smooth energy dependence in a simple fashion. It is consequently of some importance to ascertain whether our calculated body-frame phase information has smooth and simple dependences on both the body-frame energy E and the internuclear distance R. Figure 3 indicates some of these joint c and R dependences for eigenvalues p t ( ~ , R )of the “quantum defect matrix” p i ( ~ , R for ) ‘Z, symmetry and for 1 IR 5 6 au. The zero of the body-frame energy (E) scale has been taken in these plots to coincide with the H2+ l s a ground state potential curve. Thus, the Rydberg state potential curves lying just below the (adiabatic) ionization threshold are determined mainly by the values of these phase parameters at E x 0. An immediate conclusion from Figure 3 is that the fixednuclei scattering parameters p t ( ~ , R )depend strongly on both E and R. This is somewhat disappointing, since this implies that an application of the rovibratinal frame transformation starting from this information would be difficult. At the same time, it should be remembered that Ross and Jungen successfully developed such a frame transformation description for this same symmetry. The connection between our results and those of Ross and Jungen needs to be explored further. A first item which is relevant to a comparison of our ab initio ‘Z, MQDT parameters with those fitted by Ross and Jungen is a specification of the fixed-nuclei electronic channels included. Ross and Jungen fitted a 3 x 3 quantum defect matrix for this symmetry, with the three channels lsocsa, IsaEda, and 2 p a ~ p a .The last of these three channels is responsible for the doubly-excited resonances of the type 2panpa which form a Rydberg series (at each R) converging (with increasing principal quantum number n) to the ionic 2pa potential curve of H2+. Our fixednuclei quantum defect matrix includes one additional channel omitted by Ross and Jungen, namely, the channel 2pacfa. This would be relevant for any processes that involve body-frame energies near the 2pa ionic potential curve, but for a description of the bound state part of the (fixed-nuclei) spectrum well below the ionic l s a potential, it makes sense for Ross and Jungen to
E I2k-li F R=4.0
R=5.0 15.0
-0.4
-0.8
R=6.0
-8.4
I -a.0-
-0.2
0.0
0.2
c(a.u.)
0.4
1
,
0.6-0.2
,
.
.
!
0.0
1
0.2
e(a.u.)
Figure 3. Eigenvalues p: of the ‘2,quantum defect matrix are shown at internuclear distances R = 1, 2, 3, 4, 5 , and 6 au versus the body-
frame energy E , whose zero is referred to the lsa potential curve of H2+.The four channels included are described in the text. omit this channel. Our inclusion of this channel undoubtedly increases the energy dependence of the body-frame quantum defect matrix (or reaction matrix), especially at lower energies E I0. A closer comparison between our results and those of Ross and Jungen could in principle be made if we had both chosen the same set of channels. However, numerical tests showed that our calculated quantum defect matrices still had far greater energy dependences than those fitted by Ross and Jungen, when we used only their three channels. Again, this presents a problem, because the strong E dependence would severely complicate any attempt to use this information in a frame transformation calculation. On the other hand, we believe that this energy dependence obtained in our calculations is quite realistic and even physically important. The fact that we obtain reasonably accurate ab initio bound and resonance states (at the Born-Oppenheimer level) suggests that our more energydependent parameters may not be “unphysical”. In fact, we interpret this stronger body-frame energy dependence as reflecting an intrinsic complication in the channel structure of the HZ ‘Z, symmetry that must be faced by theory. The basic problem of a frame transformation description is the fact that the “appropriate” number of electronic channels changes as the intemuclear distance R changes. The physical issues can be illustrated by referring to the Bom-Oppenheimer potential curves for this symmetry in Figure la. At small internuclear distances R I2 au, the lowest bound state potential curve (the EF state) has a minimum near U = -0.72 au or at E = -0.12 au relative to the H2+ l s a potential curve. At these values of E and R the nature of this potential curve is essentially lsdsa, and only one electronic channel
Spectrum of Molecular Hydrogen (lsoeso) is relevant. The next two higher potential curves at R = 2 are different and can be viewed as two linear combinations of the electronic states Isa3so and lso3du. Thus, in the energy range E > -0.06 au, at R = 2, two electronic channels are apparently appropriate to describe the Rydberg state dynamics, namely, lsacsa and lsmda. The situation changes again as the body-frame energy increases further to E > 0.15 au, and the lowest doubly-excited autoionizing state 2pa2 is encountered at R = 2. At this and higher body-frame energies, it is clearly essential to include the electronic channel 2 p a ~ p ain any MQDT description. By the time R increases to the value R = 4, the relevant channels are quite different at each of these energies. At R = 4, the lowest (EF) state shown in Figure l a is most sensibly labeled as 2pu2, so that the fixed-nuclei physics near the second minimum of the EF curve involves the two electronic channels Ismso and 2 p a ~ p a . It still does not involve the lsacda channel in this energy range because at E < -0.06 au the d channel is still “strongly closed” in the sense that no bound states are present in this channel for such low energies. For higher members of the Rydberg series close to E 0 (Le, near the l s a H2+ potential curve), the d channel “turns on”, and three channels are clearly relevant. (At still higher body-frame energies, as mentioned before, the fourth channel 2 p o ~ f o becomes relevant, but for our discussion of the HZbound states at R I4 this channel is not significant.) The preceding discussion dramatizes how the number of relevant channels changes from 1 to 3 as the intemuclear distance changes and as the body-frame energy range changes. The first reaction to this problem is that one should include all three channels in any MQDT and frame transformation treatment, and this is precisely the viewpoint adopted by Ross and Jungen. However, we stress that this choice of three channels at all values of E and R introduces difficulties at once. If one includes the 2paepa and lsoeda channels at small intemuclear distances and at energies as low as the EF state minimum, the 3 x 3 fixed-nuclei quantum defect matrix must acquire a strong energy dependence. Without this energy dependence unphysical potential curves would emerge from the quantum defect calculation, such as a “ l s d d d ’ potential. Moreover, the actual viability of a rovibrational frame transformation approach has never been established in the presence of such strong bodyframe energy dependences.
J. Phys. Chem., Vol. 99, No. 6, 1995 1717
information in calculations such as those of refs 35 and 2831, etc., could be used to obtain realistic H2 wave functions including nuclear motion, without using frame transformation and multichannel quantum defect techniques. A scheme for including the effects of nuclear motion has been described by Schneider et al.,”l but it has not yet been shown to quantitatively reproduce the Dehmer-Chupka’ photoabsorption spectrum. Consequently, we regard this class of non-MQDT methods41 as untested for describing rovibrational HZchannel interactions in this energy range. On the other hand, the energy dependence of quantum defects is seen from this work to be far from slow (although it is smooth). This rapid dependence reflects the changing number of channels as the internuclear distance varies, for any chosen value of the body-frame energy E. Similar strong energy dependences near the lower end of a Rydberg series are familiar from atomic studies carried out using eigenchannel R-matrix methods. The atomic examples also showed that quantum defect parameters are sensitive to the R-matrix box size in this energy range far below an ionization threshold, although the final observables remain properly independent of box size. While one is free to increase the number of MQDT channels in an atomic calculation as the energy increases, it has not yet proven feasible to change the number of channels as the internuclear distance varies, in the context of rovibrational frame transformation calculations. If a way can be found to do this, it would be possible to use the present fixed-nuclei calculations. If not, it may become necessary to utilize the “nonuniqueness” of the quantum defects at very low channel energies to determine an “effective” reaction matrix that is smooth enough (versus E, R ) to use in frame transformation calculations. We suspect that the smooth parameters fitted by Ross and Jungen amount to one particular choice of this effective reaction matrix, but the extent of its nonuniqueness remains to be delineated in the future.
Acknowledgment. We thank Ch. Jungen and J. Stephens for discussions about the implications of these results and B. Schneider for communicating the results of ref 3 1 prior to their publication. This work was supported in part by the National Science Foundation. References and Notes
V. Conclusions The present work has established that ab initio calculations of the smooth short-range scattering parameters needed to describe the body-frame electronic motion in Hz can now be carried out directly. While the calculations are not yet as accurate for determining the lower-lying Born-Oppenheimer potential curves as other methods, the calculated quantum defects have a precision comparable to that achieved in typical atomic calculations. l4 Moreover, by expressing the results in terms of smoothly energy-dependent quantum defects, the infinity of Rydberg state potential curves are simultaneously determined. The same information obtained here could be extracted in principle from other methods, such as the R-matrix calculations of S ~ h n e i d e r ,provided ~~ one additional step is included in which the R-matrix solutions are matched to Coulomb functions on the reaction surface to obtain fixed-nuclei multichannel quantum defect parameters. This connection with MQDT, which appears to be important for the eventual incorporation of the nonperturbative effects of nuclear rotation and vibration through a frame transformation, has apparently not been accomplished in previous ab initio calculations of HZ doubly-excited states. It is possible that the fixed-nuclei
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