New ab initio potential curve and quasibound states of HeH+

Chemiml Physics 12 (1976) 381-386 0 North-Holland Publishing Company

NEW AB INITIO POTENTIAL CURVE AND QUASIBOUND STATES OF HeH+ * W. KotOS

.

Max-P&k-Itlnitute fx Physicsand Astrophysics.SMunich 40, Federal Republic of Germany and Quantum ChemistryLaboratory, University of Warsaw,1X-093 Warsaw, Poland**

and J.M. PEEK SandiaLaboratories,Albuquerque, New Mexico 87112, fJSA Received 14 October 1975 The ground state energy of HeH+has been calculated in the Born-Oppenheimer

approximation for 0.9 G R 4 4.5 au by

using an 83-term variational wavefunction in elliptic coordinates and including the interelectronic tential curve has been used to compute the quasibound state spectrum of the ion.

distance.

The resulting po-

1. Introduction

2. Potential energy curve

Experimental measurements [l-3] of the proton production cross section from isotopes of HeH+ have stimulated theoretical work [4-S] on the quasibound states of the ion. The observed resonances have been identified, and since the theoretical and experimental energies were not in satisfactory agreement it seemed natural to attribute the discrepancy to an insufficiently accurate potential curve for the electronic ground state of He&. To clarify this point a more accurate potential has been computed for a wide range of internuclear distances in HeH+ and the results are reported in section 2 of the present paper, where the correlation energy of the ion is also presented. Section 3 contains the ener ‘es and widths of all quasibound states of 3 He& and BHep obtained with the new potential, their discussion and comparison with the experimental data.

One of the present authors has already calculated [6] the potential energy curve for the ground state of HeH+ for internuclear distances 5 G R G 9 au. The wavefunction was assumed to be in the form of an expansion in elliptic coordinates and depended explicitly on the interelectronic distance. Using the same wavefunction we have extended the calculation to 0.9 Q R G 4.5 au. The 83 terms of the wavefunction used in the final computation had been selected in numerous test runs made at R = l-46,3.0 and 6.0 au. The exponents in the wavefunction were optimized in a 45term expansion at R = 0.9, 1.46, 1.8,3.0,4.5,6.0 and these values were used in the final runs with the 83-term wavefnction. For intermediate internuclear distances the exponents were interpolated. Only the /3* exponent was optimized for some additional R values. The results of the computa: tion are shown in table 1 where in addition to the total energy, E, we list the binding energy, f?, and the derivative dEldR obtained from the virial theorem, I%r 1.1 f R G 1.8 ati. the present results overlap with. those bf Woleiewicz [7] obtained using a &I-term-wavefunction of the sanie form. In this regidn our total energies are 7-9 cm -I lower than his values. It is difficult. to assess the accuracy of the present results. Ho&ever,

*

Supported in part by the Polish &ademy.of Scienceswithin

tt the project PAN-3, and by ERDA, USA. Permanent address.

-.

W.Kohs, J.M. Peek/&w ab initiopotentialcum ofHe&

382

Table 1 &t&ks of He& in the X’Z’ state crrlculated in the BornOppenheimer approximation using an 83-term wavefunction u) R

E ---

i?b)

WdR

0.9

-2.84562504 -2.90478633 -2.9409 1999 -2.96203002

-0.74384593 -0.45989130 -0.27566525 -0.15481689 -0.07524032 -0.02304952 -0.00101162

-2.97848894 -2.97627233 -2.97238658 -2.9675449 I -2.96223694 -2.95679360 -2.94631420 -2.93708703 -2.92542701

-L2748.90 235.50 8165.92 12799.04 15263.97 16302.84 16455.27 16455.64 16455.64 16455.64 16411.35 15924.86 15072.04 14009.41 12844.45 I1649.77 9349.81 7324.68 5643.50

2.8 3.0 3.2

-2.9233 1226 -2.91856551 -2.91495455

4301.47 3259.68 2467.16

o.o27o1sss 0.02074083 0.01565559

3.4 3.5 3.6 3.8 4.0 4.5

-2.91224263 -2.911157S4 -2.91022210 -2.90872167 -2.90760688 -2.90589760

1871.97 1633.82 1428.51 1099.21 854.54 479.39

1.0 1s 1.2 s1.3 1.4 I.46 I.4631 1.4632 I.4633 1.5 1.6 I.7 1.8 1.9 2.0 2.2 2.4 2.6

-2.97326109 -2.97799453 -2.97868906 -2.97869074 -2.97869074 -2.97869074

-0.00002487 O.OOoo~78 0.00003842 0.01071387 0.03195637 0.04463694 0.05145261 o.os425191 0.05433991 0.04977668 0.04235013 0.03438358

0.01169218 o.oiooao28

Table 2 Correlation energy of HeHi (in au) R

E

I.0 1.2 1.3 1.4 1.5 1.6 1.7 1.8

0.04375 0.04464 0.04506 o.o4540 0.04566 0.04582 0.04589 0.04586

con

R

E

2.0 2.2 2.8 3.5

0.04559 0.0451f 0.04353 0.04254 0.04227 0.04215 0.04203

4.0 4.5 m

corr

The latter were obtained from the multipole expansion of the interaction energy va!id in the region where the overiap is negligibIe. The curve in fig. I shows how an approaching proton affects the correiatian energy of the helium atom. The most characteristic feature of the above correlation energy is a m~imum at large R and a very pronounced maximum in the vicinity of the equilibrium internuclear distance. At 1argeR the electron correlation decreases the van der Waals attraction between He and Hi which is understandable since the correlation decreases the induced dipole moment of ‘the helium atom. Flowever, at smallerR the correlation energy increases. Thus the SCF method seems to describe better an atom than a two-center molecular sys-

0.00868343 O.oO644081 o.oo47a957 0.00234968

s) AU results in atomic units, on.& D in cm”. &) Relative to the energy of the heIium atom E = -2.90371332 rnr obtained with the same winefunction [6j.

since the same wavefunction applied to the helium atom [6] gives an energy which is about 2 cm-” higher than the exact vahre, it is likely that for HeHI‘ the error is roughly of the same order of magnitude. Using the present results and the SCF e&g&s computed by Peyerimhoff [S] one gets the correlation energy of the Hep ion listed in table 2 for several internuclear separations. The correlation energy is a!30 shown grap~~Iy in fig_ 1 where the present results are sup plemented with the-asymptotic correlation energies [63.

Fig_ I. Correlation energy of HeH* as function of the internuclear distance. The right hand side has been obtained ns E corr = Ecorr (R = 0) + AEcorr where dEcCrr represents the difference of the induction energies calcuhrted with and without electron correlation using the multipole expansion 161. Note the change of scale.

W. KoTos. J.M. Peek/New ob initio potential curve of HeHf

tern. The maximum change of the helium atom correlation energy due to the interaction with the proton amounts to 9%.

3. Quasibound states The availability of the improved data for the ground state of Hep, presented in the previous Section, makes it possible to remove a serious question concerned with the location of certain quasibound rotation-vibration (QB) states in this potential. A remarkable series of experiments [l-3] succeeded in measuring dissociation fragments from certain HeHf isotopes which have these QB states as their origin. The energy of the resulting fragments defines the energy above the dissociation limit, E, to an accuracy comparable to that obtained in optical spectroscopy and various experimental parameters restrict the range of lifetimes, r, for these states in the continuum. A theoretical study [4] of the QB states for several isotopes of He& was reasonably successful in characterizing the experimental spectrum. Using the restraints imposed by the experiment upon the acceptable range of 7, rotation-vibration indices were assigned to all but one observed state for 4HeH+ and 3HeH’ and all states predicted to be observable were indeed observed for these cases. The deuterium containing isotopes were more difficult to study experimentally and the theoretically predicted spectrum did not correlate as well with experiment for these cases [4]. The theoretical predictions used a ground state HeHf potential constructed from the best information available at that time [9] _As the energies of the assigned states were, almost without exception, higher than those observed, it was stated that this potential could be as much as +O.OOi hartree higher than the true potential if the differences were due to errors in the potential. Subsequently, by using a selected number of these assignments and ignoring the lifetime restrictions, the experimental energies were shown to be accurately represented by standard spectroscopic formulae [S] . It was concluded that some single potential would better represent the various isotopic spectra although a corrected potential was not suggested [5]. The X12+ He& potential presented in section 2 is sufficiently accurate to remove any questions conceming the predicted properties of the quasibound states. Both the energies and widths resulting from this poten-

383

tial should not have errorswhich exceed those of the experimental study. These data, presented below, will resolve the question about the remaining differences between experiment and theory, while satisfying the requirement that a single adiabatic potential correctly predicts the spectral for all Hew isotopic species. The QB spectra for 4HeH’ and 3HeH’ have been recalculated using the a priori ground state potential function presented here. The techniques used in this calculation were substantially those described in ref. [4] _The Born-Oppenheimer equation for nuciear motion \k”t;R, e) +p(R, #‘(R,

E) = 0,

(1)

where ~(R,E) = (2&1*)[e - V(R)] - J(Jf

IF-*,

was solved by standard numerical techniques subject to the conditions rk(O,e) = 0

(2)

and *‘@,,dI\I’(p3

8) =

-0.72901

j-J&13,e)] 1’3.

(3)

Here, I* is the reduced mass (the proton to electron mass ratio was taken as 1836.109), E is the energy of QB state above the dissociation limit, J is the rotational quantum number, andR is the internuclear separation. Primes indicate derivatives with respect to R evaluated at the assigned value of the independent variable. The potential V(R) is defined by a cubic spline fit to the X’ C+ HeH’ data given above and in ref. [6] along with an extension to large R by the formula V(R) = -

[0.6848 R4 + 1.220 15 R-6 + 5.007 R-8] _

The data from table 1 and ref. [6] were normalized to this form by the addition of the He eigenenergy, see footnote b, of table 1. Eq. (10) of ref. 4 was used for the definition of width I’, which is related to T by rr=/i. Eqs. (l)--(3) define an eigenvalue problem and eq. (3) has a restricted applicability. A QB state will occur only when&Z) has three first order zeros if V(R) has the usual properties of a bonding molecular potential. Assume E and J are such that this requirement is satisfied, then&, ,e) = 0 defines p3, where p3 is the largest value of R for whichp(R,e) is zero. It is presumed that an appropriate boundary condition on 9 will provide

.__

,__.-

-.

-.

“.

1

v

I

.. :

“g gdod $~r~x’&$tiqn’t~ E2; fo&ro& th$usual -_ ., : -~~eit~~~~~~-par~~~~~a~~n,of the scatte@gpha%e _.shift [4].~The c&di{ionu.sed here is stated by e&.-(3)..,It is based u~th~~~r.uni~~~~ asymp!otic’approxir+tio{s $the iwo_independent s’{@tionsof eq. (1) :~IJt&e n&ighbo$ood R ltIp3i. The de&redbehavipr.of q ‘forR Slightly.Iesr’thti 6, is asstimed (41 to. be b,&t _de$cribed%y_thes&itioi~ that increases as A decreases for F G@i 1@lso~no~~that in t&r&ion p’(R, e) > 0.t_ -The rightliand side ofeq,,(3) is then th&logarithmic : -derivativeof this choice of tht two Langer apprdxima: tions at R 3 pg. A different condition, eq. (9a) of ref. [A]j was-usedin the previous study of He@ QB states.. Bat+ conditions are &r&n to give good firedictions for thF_QBenergies f4, IO] with the use of&q. (3) being bette! t&n eq. (9a) of ref. 4 [ 111. Eq. (3) fails when a second order zero oFp(R) occurs fl l] . An important exatiple of this problem occurs when E approaches the maximum energy of the trapping barrier. Altema~m-tjve techniques 8repreferred for thi$situ;rtion 141, althou&a ~atisfa~to~ replacement for eq. (3) is known [II]. : -. _Preliminary to finding e and r, it is usual practice to enumerate the number of discrete arr;l$&bound Xv states. Usingthe methods devefoped in ref. [4] to traat the QB part of the spectra, the po:antial presented here differs for #e~t~ton conta~ng isotopes in only one case from the results found for the older Vlp) ‘f4.91. This-difference occurs for the J= 7 and vibr& .: tional index u= 8; (8,7), iti the ‘HeH’ isotope case. Thepoje@ial discussedin ref. [?] supports a bound St&ewith-thislabeling while i&isa quasibauhd state in tfie present poGntiaI. The s&ilarity of tl-,ee~~erat~on resvlts re%&@the ~$1 differences-between these twb -potentia&.The single difference implies the present p* teitial i6 more repulsive th& the pdtential used in the earlier Work,This is indeed the case for 4.0 r$ G R =Z -6.0goan! @e$umabl$tkis_rangeof R isimp&&t --en+@ to @dunt Fo~~~p~orn~tio~ of the {8,7) dis- _ Cretestate.& a QB &te in the cbnti+m. ‘,.-In&ftiality, the %enuni!raiioriproced&s have been im&oved in‘iitie re&ci [l f ] _Some bJoi)e&s-of thk

“-

-_

._

-. -.

‘_

:.. I.

_ < that sqrne $at&pr$&y designated & false ies& tiances now bicome Qti stat&. Th&new d@&ioti does -. : i&pose a mild rest&iol; bn p(R) but it is an impCovement in that, a$ least ib;r the PjKeF case, all re~o&celike behavior which would prerjict an e less than the maxjmum energy of the trapping barrier now is a QB state. -If one uses the most inclusivedefinition OFa false res& ntice discussedin ref. [4], most of ~y-~~ona~~-l~e behavior of the phase shift occurring for energies greater th%n.that of the trapping barrier will now-be associated tiith a false resonance. An example of the preceding discussionis demonstra-ted in table 3, where the complete QB spectrum for 4He@ is listed. The fblse resonances are indicated in _ table 3 as ~ren~etical entries. The energies are assumed to be equal to the maximum ene&y of the trapping barrier and, although their widths were not calculated, they can be expected to have widths as large or larger than a’ few percent of their energy. The (2,23) and (5,16) QB states shown i;l table 3 for the 4HeH’ case were listed as false resonances in ref. f4]. This change in designation is due to the change in enumeration techniques [I l] and not caused by differences in the two potentials. The appearance of false resonances in-table 3 that were not included in table II of ref. [4] is a result of using the relaxed deftitiop disctissedabove and again is n&t a result of the differences between the two potentials. As discussedin-ref. [4] ,-the redassification cif the QB s’tatesand false resonances shown in table ? is-not unexpected and it demonstrates the correlation with the behavior of the phase @Fts shown cy fig. 2 of ref. [4] claimed in the preceding pa&graph. Although th@ phase shift charactei was not :ecaIc!.datedfor the new X’C+ potential, tlit: geq?ral result m&&be quite like‘ &tshowninfig.Zofref. [4]. ‘.. The QB spectrum far 31$K”, based on the enumera- tion sche& diicu&ed in tie prec.$ing paragraphs, is also shown_in table 3. T@ new QB si#e (8,7) is ,seen . when these da+ are cqmpared with table III of wf- [4]. : The reason For‘thisis’d$zutid in a prece&ng paragraph. _. One-falseresonance shownin ref. 141 fqr this caSe .(2,22); is-now-aQB stat&d four neti false re;onances are shown in table 3.. Th&e ., --latter’hi$ereticwF jgpin, tire__

..

.%ble-3.

Table-4

The cbmpletkQm&oun~ spectia for X’ Z’ 4 HeH+&d 3HeH+ are.tabulated. The parenthetical entries are ihe false rkonances discussed in the text The e&rgies, E, and widths, I’, are in Hart&e units ofenergy. The symbols u and J are the usual vibration and rotation quantum nutibers if one associates u with the number of radial nodes “trapped” inside the angular momentum barrier.

Theoretical

J

EX 103

0

28

(15.86)

0

27 25 23 21

12.7772 10.0237 7.1713 (5.94)

3.5-5 1.0-4 2.0-4

26 24 22 20 18 16 14

9.2097 7.0185 5.3053

8.2-S 2.8-7 8.3-7 2.9-6 1.3-S 6.1-S

12.3891 9.6235 7.3881 (5.64)

2.6-S 7.6-5 1.6-4

5.3918 3.7106 25019 1.6941 1.2010 0.9233 0.7479 0.5631

1.5-12 l-7-12 1.8-12 4.2-12 4.5-l 1 2.8-9 4.1-7 1.8-5

8.5786 6.5040 4.8332 3.5853 2.6615 1.9542 (1.34)

4.1-B 1.3-7 3.9-7 1.5-6 8.1-6 4.7-5

1.4247 0.2436

2.2-25 9.1-38

4.7179

2.2-13 1.3-13 7.5-14

3

4 5 6

r

cx

LO3

are also shown

0 25

3.9948 2.9984 2.2225 (1.54)

1 23 2 21 3 19 4 17

Expcrimcnt

5.4417 3.7813 2.5747 1.7603 1.2511

U-12. 2.5-12 3.3-12. 8.8~12 9.2-11

5.3918 3.7106. 2.5019 1.6941 1.2019

-‘1:5-12 1.7-12 1.8-12 4.2-12 4.5-11

4.96 3.40 2.31 a) 1.52 1.05 i 6.51

25 23 21 19 17 15

0 1 2 3 4 5 6 7 8 9 10

24 22 20 I8 16 14 12 10 8 6 4

0,

23

0.6083

5.7-33

8 9 IO

7 5 ‘3

0.0127 d.0210 (0.0007)

3;7-13 2.9-7.

0.0611 0.0487 (0.047)

This work

ref. [4]

r.

0 1 2 3 4 5 6 7

13 11

l an4 t’ data for the QB states predicted to be obscrvabel in the experimental background spectra of 4H$l’ and 3HeH+are-tabulated from reF. [41 and from this work. The .’ corresponding experimental back-&&d spectra from ref. [3j

I31

u

0 1 2

._

3HcH+

4HeQ+

i 2 3

--

,

1.4-9

3.0834 1.9489 I.2346 0.8437 0.6638 OS681 0.4357 (0.235)

1.1-13 1.4-12 2.7-10 1.5-7 1.3-5

21

20 22

2.0213 3.1534

2.2-13 1.5-13

3.0834 1.9489

7.5-14 1.3-13

2.83 1.79

3 4 0 5

18 16 24’ 14

1.2998 0.8909 4.7672 0.6771

2.9-13 4.2-12 2.7-13 4.4-10

12346 0.8437 4.7179 0.6638

l-!-13 LA-I2 2.2-13 2.7-10

1.15 0.69 1 4.60

a) These d&a are for the 4HeH’ isotope. b, These data are for the ‘HeH+ isotope. states satisfying the exp&imentaI constrhts [3]. In fact, although the trends are in the correct direction, the comparison between experiment and theory is substantially identical to that presented in ref. [A]. For this r&son further cotipaiisons will not be presented. As stated above, this new potential.is too accurate to expect-further refinements to produce changes of -any magnitude in the QB spectra. Since Bernstein was able [S] to obtain a rather godd fit to the experimental spectra by using spectr+copic forn&lae: an anotialous situation would seem to exist. However, a resolution appears to exist by drawing an ~&a&y

1.5-6

b)

to the recerit

study of the mass spectrum oF_Hi_aad its deut&iuN containing isotopes 112,131. ne,,H: quasibound ipectrum is measuted innthis‘experiment [ 121 and the t&o_ ieticaliy predicted Cl31 spectrum is btied on a very accurate potential function. Differences not unlike‘those displayed in- table 4 afe_founrl_ Hdwcver, if a .&eoret- , . lcarZy predicted QB statk is cotivolutedwith a fiytion-,.‘T

bonding chara&er gf.the-potential for R G 4.0u. by

+at

beMngsomewhat lower than thg data presented $ ref. -J4] TThe new widt?C!.s a!ze!.I?& G.s?CW, but by &W&I& -too sn-&t~~ch&g+ t&z number and/or indices of@_~ _.:

featiresppea& witi

represeh

the expeiimental at-an enerw~iy

apparatus, Satisf&tory

the resultjngagreemerit

- 1

eXpeshe.nt

t&&&d,-~titi -, 1- ._’

[32,1-31. AlfhWgh k-G y& ro’bk & -. r diffeIencesbitw&efi e$i&+~ts~i$kd :- _’_ :, . _ :-. _. 2. .-I., __ ._-...

-. r

Y-

..

-._:_

_,.

theory shown in table 4 will YL?& likely be removed if the effects of the various experbetital parameters are ~c~uded in an appropriate manner. If this is the case, the Q3 spectra presented here should be the most accurate-data avtiable. This also suggeststhat the potential function presented here is not questioned by the exper~ent~ data [l-33 and that the potential. suggested by the fit of these data to spectroscopic formufae [5] will not represent an improvement.

Acknowiedgement One of the authors (W.K.) wishes to acknowledge the very kind haspita~ty of Dr. G.H.F. Diercksen extended to him at the Max Planck institute for Physics and Astrophysics in Munich.

References [if 1. Schopmanand J. Los,Physica (1970)190. [ 21 J.C. Houver, J. Baudcm, M. Abiinoli, AI. Barat, P. Fournier and J. Durup, Intern&. I. hiss. Spectrum. Ion Phys. 4 (1970) 137. [ 31 J. Schopman, P.G. Fournier and 5. Los, Physica 63 (1973) 518. [4] J.M. Peek, Physica (1973) 93. {S] R.B. Bernstein, Chem. P&s. Letters 25 (1974) 1. [6] W. Kotos, internat. J. Quantum Chem. (in publication}. 171 L. ~~o~~ewic~ J. Chem. Phys. 43 (1965) 1087. IS] S. Peyerimhoff, Y.Chem. Phys. 43 (1965) 998. [Y] T.A. Green, H.H. Michels, J.C. Browne and MM. Madsen, J. C&m. Phys. 61(19?4) 5186. ii01 R.J. LeRoy and R.B. Bernstein, J. Chem. P&s. 54 (1971) 5114. [ 111 J.M. Peek, unpublished. [ LZl J-G. Haaas-, N.P.F.B. van Asselt and 1. Los, Chem. Phys. 8 (1475) 37. (131 J-M. Peek, J.G. Maasand J. Los, Chem.Phys.8 (1975) 46.