Theory of quasibound states. Comparison with experiment for HeH+ isotopes

Physica 64 (1973) 93-113 0 North-Holland PubIishing Co.

THEORY COMPARISON

OF QUASIBOUND

WITH EXPERIMENT

STATES.

FOR HeH+ ISOTOPES*

J. M. PEEK Sandia Laboratories, Albuquerque, New Mexico 87115, USA and Joint Znstitute for Laboratory Astrophysics, # University of Colorado, Boulder, Colorado 80302, USA

Received 22 May 1972

syoopsis The energy and widths for all quasibound states of the 4HeH+, 3HeH+, 4HeD+, and 3HeD+ isotopes are calculated by methods that utilize their near-discrete state character. A method for the enumeration of these states is also proposed. Comparisons with the standard phase-shift method indicate these proposed techniques are useful and offer certain advantages. The energies of the quasibound states show at least semiquantitative agreement with sharp features observed in the energy of the H+ or D+ fragments which are produced both by unimolecular and singlecollision dissociation mechanisms. Requirements on the widths (lifetimes) of the states used in this comparison are also satisfied. The agreement is best for the 4HeH+ and 3HeH+ cases. Tests of two potentials for the HeH+ system indicate that part of the discrepancy between theory and experiment is due to errors in the best available potential function. Hence, quasibound states are presumed to be responsible for the sharp features observed in the H+ and D+ dissociation energy. This conclusion is in agreement with the mechanisms proposed by the experimental groups.

1. Introduction. Recent experimental measurements1-3) of the proton production cross section from isotopes of HeH+ show very distinct peaks in the energy distribution of protons produced at zero angle with respect to the initial HeH+ beam direction. These sharp features occur without a target gas and these same features plus additional structure are observed when a target gas is present. The first experiments1*2) considered only 4HeH+, where a quadratic-like sequence in the center-of-mass energies of these features was observed. The accompanying experimental paper3) extends this measurement to include several isotopes of the * This work was supported in part by the U.S. Atomic Energy Commission. # JILA Visiting Fellow 1971-1972. 93

94

J. M. PEEK

HeH+ ion and a careful recheck of the earlier results. The narrow energy distributions of the H+ fragments, observed without a target gas, were attributed1r2) to the decay of quasibound states in the rotational-vibrational continuum of the electronic ground state of HeH+ which were present in the initial beam. Also, the new features observed with a target gas are presumed to be due to the collisional excitation of these quasibound states1s2). If this explanation is to be accepted, the amount by which the energies of the quasibound states exceed the dissociation limit must correspond to the energies of the observed features. The lifetimes or, equivalently, the energy widths of these states must also conform to certain restrictions imposed by the experimental apparatus. These lifetime arguments are presented in the accompanying paper3). Attempts to further interpret the experimental data2) by postulating the existence of rotational series to fit the quadratic patterns in dissociation energy were not entirely satisfactory. To better assess the role of quasibound states in this experiment, this paper reports a detailed investigation of all rotational-vibrational states in the electronic ground state of HeH+ that are metastable to dissociation. The above hypotheses are tested by the calculation of the spectra of states metastable to dissocation for 4HeHf, 3HeH+, 4HeD +, and 3HeD +. This presents the task of locating a large number of such states. A total of 112 states are found for one potential function and another potential is given consideration. To facilitate this survey, a rapid method of locating quasibound states is proposed and shown to be quite useful in this application. The information generated in locating the energy of a quasibound state is shown to provide simultaneously an estimate of the width of the state. The formula for the width is motivated by a well-known result due to Gamow4) but differs in some details from the usual applications. These arguments are given in section 2 except for the derivation of the width formula, which is given in the appendix. The proposed rapid search method can miss a quasibound state if an initial choice of one of the search parameters is incorrect, so it is important to have an enumeration technique for this problem as is available for discrete states in the form of Levinson’s theorem5). A simple and easy to apply enumeration technique for these metastable states is described in section 3. Section 4 contains a numerical test of the development presented in sections 2 and 3. Results for the four HeH+ isotopes are also tabulated in section 4. In addition, two potential functions for the electronic ground state of HeH+ are given consideration6*7). Other HeH+ potentials are availables*g*lO) but they appear to contain no additional information other than to emphasize the existence of 0.001 a.u. or larger differences. Arguments based on the variational principle for energy and from comparisons with experiment, which are presented in section 5, identify the better potentia17). However, the better potential may have errors several times 0.0001 a.u. Because of the sensitivity of quasibound-state energies to small changes in the potential curve over a large range of internuclear separations

QUASIBOUND

STATES FOR HeH+ ISOTOPES

95

and the relatively few experimental data, no attempt is made to derive a quantitative estimate of the changes in the potential curve required by experiment. Section 5 contains the conclusions along with the comparison of experiment with theory. The positions in energy of quasibound states are found to correlate well with the features observed experimentally. An important aspect of this comparison is the range of widths that prove necessary to obtain this correlation. Two ranges are required, one for the unimolecular features and one for the collision-induced features, both of which prove to be consistent with known requirements3). The comparison is best for the 4HeHf and 3HeH+ isotopes where the number of observed features and predicted states are in essential agreement. It is possible to assign rotational-vibrational quantum numbers to the quasibound states; these assignments indicate the existence of near-quadratic sequences in energy as the vibrational quantum number v decreases by unity and the rotational quantum number Jincreases by two. This behavior is quite similar to that observed in H211). (These two cases are not believed to be a hint of some general property. It seems, however, that sequence-like behavior usually will occur but the change in J between members of the sequence is not necessarily two. Preliminary considerationG2) of the 2px, state of Hz indicate that J changes by increments of three for these sequences. The increment in J appears to be controlled by the change in J required to change each successive discrete vibrational state into a quasibound state.) Members from these quadratic-like sequences are just the states that correlate well with the experimental data. Hence, quasibound states for the isotopes of HeH+ seem to have the correct properties to explain most of the features observed in this interesting experiment. 2. Characterization of quasibound states. The use of the complex-energy method to describe quasibound states4) or to treat resonances in elastic scattering13) is well known. In this method, the characterization of the complex energy E = EO - i$r,

(1)

where EO is the (real) energy above the dissociation limit and r describes the width, or lifetime t = fir-l, of the quasibound state, serves the same purpose as finding just the eigenenergy of a discrete state. The argument that the resonance energy, occurring when EO = ER, and r can be found more rapidly by using the “near-discrete” properties4) than by using the scattering properties13) follows. Constraining the energy to the physically accessible real axis, solutions to the equation Y’; (R, k) + [k2 - W(R) - J(J + 1) K2]

YJ (R, k) = 0,

(2)

on 0 I R < co are required subject to the conditions YJ (0, k) = 0

(3)

96

J. M. PEEK

and lim U: (R, k) = k-l

sin (kR - -LfJx+ d,),

(4)

R-+OI

where ~3,is the phase shift. Here k2 = (2,u/fr2) Eo,

(5)

where p is the reduced mass of the two dissociation fragments, fik is the dissociation-momentum magnitude relative to the center of mass of the fragments, and W(R) = (2,G2) W),

(6)

where V(R) is the potential function and R is the internuclear separation. The above equations are obviously based on the Born-Oppenheimer approximation to electronic and nuclear motions14). The parameters ER and F are determined by fitting a relationship liker3) 6J = g(O) + #l’E J .I

0

+ ..,

-

tan-l

[(& - Eo)/+.F]

(7)

to 13,. Here S:“‘, S:“, etc. are constants. Eq. (7) requires S, at least over a range like ER - .F I E. 5 ER + .ZT The calculation of ~3,is a simple and quick task, hence the information required by eq. (7) can easily be obtained provided one has good estimates of both ER and r and provided r has a size convenient for computational purposes. Unfortunately, random study of ~3,does not provide a suitable method for the quick location of ER as 8, is weakly dependent on E, when E. is far from ER; that is ~3:” and higher coefficients are usually small and often are set equal to zero. Also, r may be so small that the interesting range of E, is too small to be explored conveniently on a digital computer. Both of the difficulties mentioned above can be avoided by exploiting the discrete-state character of ‘u, (R, k). A quasibound state can occur when eq. (2) represents a three turning-point problem. A typical example is shown in fig. 1 where the three turning points el, e2 and Q3, and KJ are defined. The idea of a quasibound state15) can qualitatively be represented by the condition max [I% (R,,

k)121,

O
on k. R, is the value of R for which YJ assumes its maximum value under the condition R, < ,02. In other words, for E. = ER, !PJ (R, k) will decrease as much as possible as R increases from &k) to e3(k). This character is similar to that of a discrete state and this property can be assured by prescribing appropriate conditions on !PJ (R, k) and ul; (R, k) at R = ,03(k). These requirements can be expressed in terms of the logarithmic derivative of Y, (R, k) as %

(e3

3 WC

(e3

2 4

=

0~.

(8)

QUASIBOUND

97

STATES FOR HeH+ ISOTOPES

Conditions stated by eqs. (3) and (8) define an eigenvalue problem for eq. (2). If & = E for these conditions, ordinary discrete-state methods14) can be used to rapidly locate E. [Eq. (8) is not intended as a replacement for eq. (4); solutions satisfying eqs. (3) and (8) will also satisfy eq. (4) when integrated from es(k) to large R and appropriately normalized.] The notion of creating an eigenvalue problem in the continuum is of course not new and has been used in a number of contextP). k2A W(R)+

_-------_---;;)

J(J+I)R-2

* P3(‘o

R

Fig. 1. A graph of the effective potential for eq. (2). The constants K: and RJ for a given value of J are shown along with the three turning points el(k), &), and es(&) for a given value of k.

The degree to which E approximates ER depends on the ability to pick do and is somewhat a matter of taste. The choice 01g (k - KJ is the correct value if the repulsive part of V(R) is a sufficiently wide rectangular barrier and 01 = 0 is correct for an infinite barrier of zero widthIs). It seems obvious that E > ER if d = -M, as M becomes infinite, is used and E < ER if LX= 0 is used. Note the requirement on the variation of the logarithmic derivative of !PJ (R, k) as E,, changes17). Since KJ z la,’ is expected for many states in the present study, a = -la;’ seems a plausible value. All these choices were tested for the 4HeH+ problem and .s z ER 2 O.lrforol = - 1, E z ER + Pforol = co, and .s z ER -I’ for OL= 0 were observed for a number of quasibound states. Some different choices of ochave been discussed elsewhere’ 8*1g). The most obvious value of a for this problem is constructed as follows. For this case and most cases encountered in this type of study, (W(R)1 4 J (J + 1) Rw2 will be the case for R x es. Hence the two independent solutions of eq. (2) can be approximately expressed in terms of Bessel functions. The above discussion makes it clear that the behavior required by quasibound character is represented by the irregular solution for R z es. Then Ly=-

d

TR f

dR [( 2k >

YJ+O.S

(W

-)yJ+o.s @ed 19

(9)

98

J. M. PEEK

is the desired condition on 01. Recognizing that ke, z J + 0.5 and using the expansion of the Bessel function Y at the turning pointzO) a g ,oo’ [0.5 - (b/u) (J + 0.5)2’3]

(9a)

can be used as a convenient and reasonably accurate representation of eq. (9). The constants a and b are tabulated elsewhere*O). The present use of the discrete-state method consists in finding the eigenvalues E = fi2k2/(2,u) for eq. (2) subject to the initial values required by eqs. (3) and (9). For the problem under consideration eq. (9a) will be used to replace eq. (9). A test of this definition and a comparison with other discrete-state methods is presented in section 4. The possibility of estimating I’ from the information generated in a single calculation by the discrete-state method is established in the appendix. This result is in the spirit of a treatment due to Gamow4) but relies more on the techniques expounded by Blatt and Weisskopf*l). Ref. 21 reduces the formula for I’ to one equivalent to the study of the phase shift as a function of scattering energy. This is undesirable for the present problem, as discussed above, but their method can be used to obtain an estimate in terms of YJ (R, k) for just one value of fi*k* = 2,~s and 0 5 I? I p3(k). The formula for r obtained by Gamow4) does meet the present requirements but is less convenient, in that it requires the evaluation of an additional integral, and relies on an analogy with a one-dimensional problem. The WKB method lg**r) and another independent method**) also appear well suited to this problem but it is believed that the proposal offered here has some minor advantage in convenience. To adapt the general result, eq. (A.13) of the appendix, to the proposed discretestate method, !& (e3, k) must be specified. The present use of the discrete-state method implies !PJ (es, k) # 0 so the arbitrary normalization !P__(e3, k) = la, is assumed. The estimate for r is then T z l.O6ti*ku; [/A (J + +)1’3 f dR I!i?J (R, k)l*]-‘,

(10)

where the R integral extends over the interior part of the effective potential 0 5 R i e3(k). Eq. (10) can be simplified even further by using, in analogy with the estimate made by Gamow4), the estimate j

dR IYJCR,WI* z Mm Ilu, (R,, W.

The use of this simple formula is adequate for on the error of

approximation in conjunction with eq. (10) provides an extremely for r and should give at least order of magnitude accuracy, which many purposes. A factor of three proved to be an adequate bound this approximation in tests on the present problem.

QUASIBOUND

STATES FOR HeH+ ISOTOPES

99

3. Enumeration of quasibound states. Consideration is now given to the problem of enumerating the number of quasibound states supported by the effective potential in eq. (2). Although arguments based on the behavior of the phase shift, as used by Levinsor?), seem to be natural for this problem in the continuum part of the spectrum, it proved impossible to make much progress in this way. Instead, the nodal behavior of YJ (R, k) will be utilized. To start the discussion, it is necessary to make the following definitions: 1) The number of zeros of !PJ (R, k) for a < R I b is defined as NJ (a, b; k). 2) The number of eigenstates supported by W(R) + J (J + 1) Re2 is &d). 3) The number of quasibound states is r],(q). The following property is also required. If the eigenstates of eq. (2) are distinct and there is no eigenstate at E, = 0, then

v,(d) = Nr (0, ~0 ; 0).

(11)

No proof of this statement will be given since it is a fairly obvious consequence of the comparison theorem of Sturm (p. 101 of ref. 23), the condition for nonoscillatory behavior of !PJ (p. 98 of ref. 23), and the conditions imposed on the motion of zeros of YJ as E0 changes (pp. 100, 101 and 110 of ref. 23, and ref. 17). The enumeration of quasibound states is given by

(12) If the logarithmic derivative of Y, (R, &) at R = RJ is (13) one additional state having near-quasibound character, termed false resonance, may occur. A false resonance represents a state with characteristics intermediate between a quasibound state, one of which the phase shift is well represented by eq. (7) and an ordinary scattering state where S, is much too weakly dependent upon k to conform to eq. (7). As the value found for /3 becomes more negative, the expected false resonance will more resemble a quasibound state. The following discussion indicates that p < 0 represents a good criterion for the existence of a false resonance, but a definite upper limit on /? is not proposed. A dimensionless parameter, such as RJ, would perhaps be less model-dependent. The above enumeration technique is similar to a proposed phase-integral methodz4). Neither technique has been put on a rigorous mathematical basis but intuitive motivation for the use of eq. (12) and the introduction of the falseresonance term will be given. Besides these arguments, a successful test of eqs. (12) and (13) is presented in section 4 and an application to the Hz resonance spectrum’l) has also proven successful.

100

J. M. PEEK

Another problem is defined by replacing W(R) + J(J + 1) Re2 by KJ’ for RJ -< R < co. The number of discrete states fJ(d) supported by the new potential is

which follows from eq. (11). The bars over symbols defined for the actual problem indicate they apply to the problem with the distorted potential. Furthermore, if 1, which is defined by analogy with eq. (13), is greater than or equal to zero, q.,(d) = NJ (0, R.,; KJ>can be established and, if b < 0 the equality rjJ(d) = ii& (0, R,; KJ> + 1 follows. Since U, (R, k) = ~9~ (R, k) for 0 I R 2 R,, where a is some normalization constant, it follows that ,8 = fi and NJ (0, R,;,K,) = fl; (0, R.,; KJ). The quasibound states for eq. (2) are associated with discrete-state character so the present hypothesis is qJ(q) + I = r5,(d) for the case fl = B 2 0. This is just eq. (12). The definition of a new term for the case /? c 0 is motivated by the following considerations. The motion of some zero of YJ (R, I&) for 0 < R c R,, as KJ is replaced by k and is allowed to decrease, must either move to a finite R or approach infinity as k --f 0. If this zero approaches infinity there must be some finite &k) which coincides with this node and it is this circumstance which is associated with the appearance of a quasibound state in section 2. In this instance one expects eq. (7) to fit the phase-shift data near ER quite well. On the other hand, the first node in U: (R, KJ>for R > RJ does not necessarily approach the value ,03(k) as k is changed and its motion will depend a great deal on the detailed shape of W(R) + J(J + 1) RM2. For this reason, eq. (7) may not provide a good fit of S., although rapid changes similar to those required by eq. (7) could occur. The resonance-type behavior associated with the first node of ul, (R, K_,) for R > RJ will be enhanced by the proximity of this node to RJ, a circumstance correlated with increased negativity of j3. It seems probable that false-resonance behavior will be observed when k z KJ and only if fi --x0. The enumeration of the discrete and quasibound spectra for eq. (2) requires two integrations subject to the boundary condition of eq. (3). One integration for k = 0 provides NJ (0, co ; 0) and the other for k = KJ provides NJ (0, RJ; KJ). These data predict the number of quasibound and discrete states as implied by eq. (12). The logarithmic derivative of U, (R, KJ>at R = RJ indicates the likelihood of finding a false resonance for k GZKJ. The latter possibility is enhanced by a large negative value for p of eq. (13). This enumeration device also provides an extremely useful tool for making initial estimates of the E defined in section 2. The number of quasibound states for a given vibrational quantum number v can be deduced from the above method. Perturbation theory shows that the energy difference between successive E’Sas J changes for v fixed is linear in J, provided Jis not too small. Also, the value of @ for the last member of this J sequence gives a good indication of how close this E

QUASIBOUND

STATES FOR ‘HeH+ ISOTOPES

101

is to @KJ2/(2~); this E will approach (fiKJ2/(2p) from below as /I approaches infinity. Since the dissociation limit is known, it is then a simple matter to approximately locate the remaining E in the J sequence. 4. Quasibound states for isotopes of HeH+. The HeH+ potential V(R) is required to predict the quasibound spectra to be compared with experimentlm3). The character of the states to be calculated requires V(R) over the entire range of R, not just near the range where maximum binding occurs. For this reason an analytic form6) defined for all R, V(H), and a numerical potential, V(G), were used. V(G) consists of Wolniewicz’sz5) data plus additional information provided by Green and Michels’). The data from ref. 7 cover an extensive R range and are based on an ab initio calculation which was scaled to connect smoothly to Wolniewicz’s dataz5) and to the correct long-range polarization interaction. Sample data from the two potentials are shown in table I. TABLE I

Sample numerical data for the HeH+ potential. The V(H) potential is from ref. 6 and V(G) is from ref. 7. Both R and V are in atomic units.

R 1.0 1.4 1.6 2.0 3.0 5.0 7.0 9.0

J'(H) 0.00305 - 0.07407 - 0.07223 -0.05164 -0.01389 -0.001227 - 0.000286 -0.000104

V(c) -0.00145 - 0.07426 - 0.07253 - 0.05305 - 0.01476 -0.001350 - 0.000300 -0.000106

The integration of eq. (2) under the conditions specified by eqs. (3) and (8) was carried out by standard techniques14). The integration was started at R x 0 and R = es(k) with some guess for k. A Newton-Raphson technique14) was used to iterate k until a sufficiently continuous Y, (R, k) for 0 < R < e3(k) was generated. This method has been tested extensively on discrete-state problems, including difficult cases extremely close to the dissociation limitz6). Phase shifts were generated with the same integration procedure, either by integrating from R x 0 to very large R for a given value of k or by integrating from e3(k) tb large R using the value of k demanded by the value of cxassigned to eq. (8). The latter method avoids serious error-propagation problems when E0 z ER for a case in which r is very small. Any error in E is believed to be confined to the last figure quoted in the tables.

102

J. M. PEEK

Preliminary to comparing the present calculation with experiment, some numerical examples of the results presented in sections 2 and 3 will be given. This information is included in table II and fig. 2. Phase shifts S, for the 4HeH+ case, using V(G), are shown in fig. 2. Values of J are given periodically along the appropriate curve and the short vertical lines intersecting the 8, curves indicate the maximum energy of the trapping potential, fi’KJ” (2,~)~~. For most cases in fig. 2 no attempt was made to show the shape of the 8, curve for E. x ER and the near-vertical lines only schematically represent the actual shape. The ranges in E. and J were selected for their pertinence to the experimental data. The S, curves show the behavior required by Levinson’s theorem5) as E,, approaches zero and the near-quadratic progressions in ER as J changes by two are apparent for this case as in the H, casell).

0

1

2

3

4

5

8

7

6 E

ld

9

10

11

12

13

i4

(a. u. )

Fig. 2. A graph of the phase shift divided by x, 6.,/x, shown as a function of Eo. These data are based on the V(G) potential and are for the 4HeH+ isotopic species. E. is in (Hartree) atomic units.

Table II contains values of e predicted by the discrete-state method described in section 2, and of r resulting from the evaluation of eq. (10). These data are given for both the V(H) and V(G) potentials. The range of U,which is the number of zeros in Su, (R, k) for 0 c R I &k), and of J are those predicted by the enumeration method described in section 3. The parenthetical entries are false resonances based on the criterion ,6 I -l&l in eq. (13), and the .Zentries for

QUASIBOUND

STATES

FOR

HeH+

103

ISOTOPES

TABLE II Quasibound-state

energies and widths for 4HeH*

was used to calculate

are given in (Hartree) atomic units. Eq. (7)

ER and r*. The quantities E and I’ were determined by the methods

discussed in section 2. The sign and integer following each width indicate the power of ten that multiplies this entry. V(H) Potential

V(G) Potential u

J

r*

ER x lo3

r

E x lo3

u

0

27

2.9 -

5

2.6 -

5

12.847

12.8508

0

(:)

(23) 25

(4.9 8.7 -

4) 5

7.9 -

5

10.099

10.1141

(1)

(7.760)

(8.0401)

J

(Z,

& x 103

r 1.2 -

4

14.2592 (11.181)

0

26

8.9 -

8

8.3 -

8

9.2679

9.2679

0

26

1.3 -

6

10.7020

1

24

3.5 -

7

3.3 -

7

7.0944

7.0944

1

24

4.7 -

6

8.2507

2

22

1.1 -

6

1.0 -‘6

5.3761

5.3762

2

22

1.3 -

5

6.3161

3

20

3.9 -

6

3.7 -

6

4.0525

4.0528

3

20

3.8 -

5

4.8009

(l)

(::)

(7.0 1.6 -

5) 5

1.5 -

5

3.0337

(4)

(18)

3.0323 (2.245)

0

25

1

23

1.1 -

12

2.

21

3

19

8.0 -

12

4

17

8.5 -

5 6

15 13

7

11

(3.7529)

(2.3939) 5.4417

25

2.2 -

10

6.8444

3.7813

23

6.7 -

10

4.9893

2.5747

21

2.1 -9

1.7603

19

1.2 -

8

1.2511

1.2511

17

1.4 -

7

1.9969

9 7

0.9427

0.9427

15

2.8 -

6

1.5303

0.7320

0.7320

13

3.5 -

5

1.1465

8.6 -

6

0.5364

0.5341 2.8047

1.8 -

12

2.5 -

12

3.3 -

12

8.8 -

12

1.7603

11

9.2 -

11

3.7 2.8 -

9 7

3.9 2.9 -

8.6 -

6

5.4417

3.6236 2.6584

0

24

4.6 -

25

1.4689

24

1.4 -

18

1

22

1.8 -

35

0.3072

22

4.0 -

20

1.5218

20

1.7 -

22

0.7084

18

2.6 -

25

0.2785

16

2.5 -

25

0.1361

14

1.8 -

18

0.1724

12

2.4 -

11

0.2671

10

5.3 -

7

0.2996

8

8

4.9 -

13

0.0240

9

6

6.0 -

7

0.0417 10

< 0.0060

these cases are just the energies of the maximum of the appropriate trapping barriers. The quantities labelled ER and F* were obtained from fits of eq. (7) to S, data for selected resonances shown in fig. 2. It was practical to use eq. (7) only for those cases in which .F 2 10-l* a.u. Comparisons of E with Ex for the quasibound states show a bound of E = ER + O.lW. The use of @&)*/(2,u) for the energy of the false resonances is not particularly good but, since r* is typically several percent of ER, precise evaluation of ER for these cases often is not necessary.

J. M. PEEK

If it is, the study of S, and the use of eq. (7) seem most appropriate and do not involve problems with error propagation as do the cases for which F is small. Table II indicates _F = (1 f 0.1) r*, except for the cases u = 0, J = 25. The study of S, for this case proved to be very near the limit of ability to control error propagation in the CDC 6600 digital computer and it is believed that F is the preferred value. The discrete-state method used here differs from the techniques used elsewhere18*lg) only in the choice of LXdefined by eq. (8). It appears that the choice used for this problem, eqs. (9) and (9a), is competitive with the best results obtained to datelg). Another method of locating quasibound states, based on the study of the reaction matrixZ7), appears to be useful although it has not been compared directly with results from eq. (7). This method does provide information on the number of quasibound statesz7) but it has not been used to generate r and it does not seem to offer many advantages for treating states with small F. The formula derived for r, eq. (IO), seems to achieve the accuracy available from the semiclassical method1g32’*28) an d one other proposed technique22). The convenience of one method versus another will depend somewhat on one’s point of view. The one advantage of the proposals made here is that the usual discrete-state programs available for many digital computers can be used to simultaneously generate E and r with very minor modifications. TABLE III Quasibound-state

energies E and widthsrfor

‘HeH+

in (Hartree)

atomic units. These data are based on the V(G) potential. U 0

(:,

J

EX lo3

r

26

2.0 -

5

24

6.5 -

5

12.4603 9.7091 (7.7365)

(22)

8.7359

0

25

4.3 -

8

1

23

1.6 -

7

6.5789

2

21

5.0 -

7

4.9039 3.6430

3

19

2.0 -

6

4

17

l.O-

5

5

15

0

24 22

1 2

20

2.6951 <2.1529

2.7 -

13

4.7672

2.2 1.5 -

13 13

3.1534 2.0213

3 4

18

2.9 -

13

1.2998

16

4.2 -

12

0.8909

5

14 12

4.4 9.1 -

10 8

0.6771

6 7

10

5.6 -

6

0.4059

0

23

3.0 -

32

9

5

0.5454 0.6519 < 0.0427

QUASIBOUND STATES FOR HeH+ ISOTOPES

105

TABLEIV Quasibound-state energies and widths for 4HeD+ are given in (Hartree) atomic units. This calculation is based on the V(G) potential. V

J

r

36

7.8 - 5

E

(34) 35 33 31 29

-

6 6 5 5

12.4243 10.3171 8.5007 6.9551 (5.7425)

3.5 1.7 5.8 2.0 7.8 3.5 1.5

-

9 8 8 7 7 6 5

9.5593 7.7913 6.3056 5.0832 4.0868 3.2726 2.5983 (2.0941)

2.0 5.3 1.1 2.6 1.1 9.8 1.9 5.5 1.3 1.3

-

13 13 12 12 11 11 9 8 6 5

6.5608 5.1024 3.9254 3.0087 2.3189 1.8128 1.4450 1.1650 0.9243 0.6956

(Z) 0 1 2 3 4 5 6 7 8 9

33 31 29 27 25 23 21 19 17 15

15.0944 (12.594)

1.8 8.1 2.2 5.0

(27) 34 32 30 28 26 24

x lo3

V

J

1 2 3 4 5 6 7 8 9 10 11

32 30 28 26 24 22 20 18 16 14 12 10

3.9 3.8 9.8 8.3 4.2 1.2 4.2 7.9 3.0 1.0 3.43.0 -

0

31

1.0-52

12 13

7 5

0

&x 103

r 21 22 24 26 28 29 28 23 16 11 8 6

3.4696 2.3110 1.4374 0.8220 0.4322 0.2255 0.1510 0.1529 0.1818 0.2017 0.1917 0.1449 0.3106 < 0.0471 CO.0135

Table II shows that the enumeration of the quasibound states exactly predicts the resonance behavior exhibited in fig. 2 when ER is less than ti*KJ’ (2~)~~. The behavior of S, for E,, z fi*Kf (2,~)~~ does show the false-resonance character, especially for the cases J = 14, 16, 21 and 23. Table II shows false-resonance entries only for J = 16 and 23. This results from selecting false resonances from the criterion /? I - la; 1 for eq. (13). Had p I 0 been used, the J = 14 and 21 cases would also be counted as false resonances. As discussed in section 3, this choice is somewhat arbitrary. (The values of /3 in atomic units were -7.38 for J = 16, -3.53 for J = 23, -0.328 for J = 21, and -0.092 for J = 14.) In fitting these false-resonance data to eq. (7) extreme care in selecting S, data was necessary and only the J = 16 case was relatively insensitive to choice of input data. Results for ER and r* are shown in table II only for J = 16 and J = 23.

106

J. M. PEEK

The quasibound states and false resonances for the 3HeH+, 4HeD+, and 3HeD+ isotopic species are given in tables III, IV, and V. The discrete-state method, described above for 4HeH+, and the V(G) potential‘were used in this calculation. Tables II-V have the data arranged according to the quadratic-like sequences noted earlier. The reason for this grouping is not because of the quadratic-like energy relationship but is motivated by the similarity in widths r for the members of each sequence. The _P for any given sequence tend to be quite similar for the small-v and large-J entries, tending toward larger J’ as z, increases and J decreases. Here, as in most similar situations, the lifetime rather than energy is the more critical factor. TABLE V

Quasibound-state

energies E and widths I’ for 3HeD+

are given in (Hartree)

atomic

units.

These data are from the V(G) potential. ”

J

r

E x 103

v

14.8113

0

31

3.0 -

14

5.7759

12.280

1

29

5.4 -

14

4.3324

J

EX 103

r

0

34

(1) 0

(32) 33

1.4 -

6

11.9862

2

27

7.3 -

14

3.2004

1

31

6.2 -

6

9.8330

3

25

1.3 -

13

2.3524

7.3 -

5

2

29

1.7 -

5

7.9986

4

23

5.6 -

13

1.7478

3

27

4.0 -

5

6.4566

5

21

7.0 -

12

1.3348

(4)

(25)

6

19

2.6 -

10

1.0573

7

17

1.7 -

8

0.8557

8

15

7.9 -

7

0.6784

9

13

1.1 -

5

0.4994 2.5050

(5.3109)

0

32

1.7 -

9

8.9508

1

7.7 -

9

7.1666

2

30 28

2.5 -

8

5.6939

3

26

8.9 -

8

4.5083

0

30

5.5 -

24

4

24

3.9 -

7

3.5657

1

28

1.2 -

26

1.3936

5

22

2.1 -

6

2.8143

2

26

9.8 -

32

0.5983

6

20

1.2 -

5

2.2025

3

24

1.5 -

47

0.0847

(1.7544)

9

12

1.4 -

14

0.0715

10

10

9.8 -

9

0.0980

11

8

2.4 -

6

0.0754

(7)

(18)

5. Comparison’ with experiment. The experimental data are of two types; dissociation fragments observed without a target gas, presumably due tospontaneous or unimolecular dissociation of ions which are part of the incident beam, and fragments observed after single collisions with a target gaslP3). Both types of measurements were performed on 4HeH+, 3HeHf, and 3HeD+ 1*2*3)while data for 4HeD+ could not be distinctly separated into the two modes of dissociation3). The number of distinct features in the experimental data is exceeded by the number of quasibound states and false resonances found theoretically. Fortunately, many of the theoretically calculated states can be eliminated by arguments based

QUASIBOUND

107

STATES FOR HeH+ ISOTOPES

on lifetimes t = fir-‘. The ranges of lifetimes that can be observed in the experiment are discussed in the accompanying paper3) with the result that, using the atomic units, 2 x lo-l3 < F < 4 x lo-lo is required for the spontaneous case and 10mg c .F < 10m3 is required for the collision case. Applying these criteria to the widths listed in tables II-V requires caution because of our imprecise knowledge of the HeH+ potential and the rather strong dependence of F’ on this function. The V(G) potential is presumed to be less binding than the true TABLE VI

The energies and widths of the quasibound states that most likely correspond to the experimentally observed unimolecular features are given in (Hartree) atomic units for the indicated HeH+ isotopes. The experimental data3) for the observed center-of-mass dissociation energies are given in the same units.

V(H) V

r 0

1 2 3 4

2.2 6.7 2.1 1.2 1.4-

r

EX lo3

10 10 9 8 7

6.8444 4.9893 3.6236 2.6584 1.9969

Ex

EX 103

1.8 2.5 3.3 8.8 9.2

-

12 12 12 12 11

5.4417 3.7813 2.5747 1.7603 1.2511

24 22 20 18 16 14

2.7 2.2 1.5 2.9 4.2 4.4

-

13 13 13 13 12 10

4.7672 3.1534 2.0213 1.2998 0.8909 0.6771

34 33 31 29 27 25 23 14

3.5 2.0 5.3 1.1 2.6 1.1 9.8 1.0

-

9 13 13 12 12 11 11 11

9.5593 6.5608 5.1024 3.9254 3.0087 2.3189 1.8128 0.2017

25 23 21 19 10

1.3 5.6 7.0 2.6 9.8

-

13 13 12 10 9

2.3524 1.7478 1.3348 1.0573 0.0980

25 23 21 19 17

1

3 4 5 6 10

Experiment3)

V(G)

J

a The experimental a.u.

10s

4.96 + O.la

3.40 + 0.09 2.31 +_ 0.05 1.52 f 0.04 1.05

1

6.51 4.60 2.83 1.79 1.15 0.69 9.59

7.24 5.62 4.10 3.16 2.66 1.56

1 4HeD +

J

1.24

3HeD+

0.076

error bars are relative to the location of the peak observed at 1.05 x 10e3

108

J. M. PEEK

adiabatic potential, so it is anticipated that the actual I’ will tend to be smaller than those found here. Hence, the greatest deviations from the above criteria will be in using states having larger calculated r. Table VI lists the only choices consistent with the restriction on F for spontaneous dissociation. Experimental data3) for the spontaneous dissociation of the 4HeH+, 3HeH+, and 3HeD+ isotopes are also shown along with the data available for 4HeD+. The agreement for 4HeH+ is satisfactory for the V(G) results while neither the values for E nor for F from the V(H) are particularly good. Similar agreement is found for 3HeH+ except for the lack of a theoretical state to correspond with the peak observed at 6.51 x 10m3 a.u. The 4HeDf data are in rough agreement with the predicted spontaneous dissociation states. The fact that the experiment was not able to separate the spontaneous from the collisionally produced fragments also makes this comparison more tentative. States that agree with the observed spontaneous fragments from 3HeDf are found, but three unobserved states are also predicted.

THEORY I 1’

3He H+ EXPERIMENT

I

/

I

THEORY 4He H+ EXPERIMENT

/ 0

I

i

1

J 2

3

4

5

6

7

E -IO3

8

9

IO

II

12

13

14

la. u.)

Fig. 3. The observed energies for the collisionally produced features in the 4HeH+ and 3HeH+ proton production measurementj) are shown along with the values of E from tables II and III that have the appropriate widths3). The numbers in parentheses indicate the values of v and J for the theoretical state. The dashed lines suggest the most probable identification of experimental states.

The features observed for the collisional dissociation of 4HeH+ and 3HeH+, which presumably include spontaneous dissociation, are plotted in fig. 3. The predicted states are also shown and are connected to the most likely corresponding

QUASIBOUND

STATES FOR HeH+ ISOTOPES

109

observed feature. The number of observed and predicted states agree, except for a few close-lying states at small energy that are not clearly resolved3) in the 3HeH+ case. Two false resonances are included in the comparisons because they nearly qualify as quasibound states; u = 5, J = 16 for 4HeH+, where p = -7.38 and v = 2, J = 22 for 3HeH+, where /? = -8.69, are the states in question. The agreement in energy is typical of that found for the spontaneous-dissociation cases. As noted above, the 4HeD+ data could not be resolved into the two cases of dissociation3) and the features that were observed correlate fairly well with spontaneously dissociating states. The fact that the series starting with u = 0 and J = 36, 35, and 34 were not observed from collisional processes cannot be explained. The situation is quite similar for the collisional spectrum of 3HeDf, where two features are observed3) while series starting with v = 0 and J = 34, 33, 32 plus the u = 8, J = 15 and v = 9, J = 13 states are predicted. In the 4HeH+ and 3HeH+ cases, where the agreement between experiment and theory is good, the tendency for E to be greater than the energies of the observed features is notable. This again suggests that V(G) is very likely larger than the true potential. Since V(G) is from a variational calculationz5) for R rz 1.4~~) the lowering of V(G) seems plausible as well as demanded if theory is to agree better with experiment. The trend resulting from the V(H) and V(G) data indicates the changes in E to be expected are of the order of - 0.001 a.u. Changes in the potential need not be this large, however, since e is roughly related to an integral of the potential. Any changes in the overall shape of the potential will also be of primary importance and, from the experience gained in this calculation, shape changes for 1.0 < R -c 6a,, would be especially critical. Also the full adiabatic potential is nat available and effects from these additional terms may be important. It is obvious that a potential of higher quality than is yet available is needed. The above discussion shows a reasonable correlation between the theoretical E and the energies of the observed features from the proton-containing isotopes. An important aspect of this comparison is the conformity of the theoretically predicted states to the width requirements3). Quantitative agreement was not observed for the deuterium-containing isotopes. In particular, several sequences expected in the collision-produced spectra were not observed3). This paper has not considered any detailed aspects of the collision mechanism, as has been done elsewhere2g), and it may be necessary to understand this part of the problem better before a complete rationalization is possible. The degree of agreement between theory and experiment does make the assignments of ZIand J plausible, at least for the 4HeH+ and 3HeH+ cases. Accepting this argument, it is possible to make a remark concerning the ion source. The pertinent values of J are quite large. Hence, the reactions which produce HeH+ ions create a good many ions with large angular momentum or the large angular momentum is created upon extraction of the ions. The latter alternative seems less likely. This point is also important in searching for this type of unimolecular

110

J. M. PEEK

dissociation products since large J values are required to produce quasibound states with reasonably large dissociation energies. Acknowledgements. The author would like to recognize the invaluable aid of many colleagues. T.A. Green provided considerable help with the potentialfunction problem. P. B. Bailey gave considerable assistance -with the enumeration problem. D. E. Ramaker provided the program used to fit eq. (7) to the phase-shift data. M. E. Riley and members of each of the experimental groups participated in many stimulating discussions. The numerical computations would have been impossible without the aid of Mrs. M. M. Madsen.

APPENDIX

The derivation of the F estimate is as follows. The equality IX-l

j dR I@ (R, &)I2 = A (2pi)-l

j dS. [@* V@ - cf V@*],,

64.1)

has been established4) where qS is a solution of Schriidinger’s equation for complex E [see eq. (l)]. The dR integral is over a finite volume of radius Q and dS is over the corresponding surface. In the present problem, the function @ can be written as 0 (R, k) = (2/n)’ c i’ Yl,, (2) Y&,,(i) exp ( -iSl) FJ (R, E) R-‘, 1.m

(A-2)

where U, is a solution of eq. (2) for complex E satisfying eq. (3) and unit vectors are signified by ff and $. In the following the symbol & indicates a complex magnitude while k is defined by eq. (5). Substituting eq. (A.2) into eq. (A.l),

F jdR I(21 + 1) lul, 0

(R, @I”

I

=

@P-’ C (21+ 1) I% (e,W Im P-C(e, Q/U:(e, Ql ,

1I

1

(A-3)

results where Im [. . .] indicates that only the imaginary part of the quantity in brackets is to be used. The assumption that only one quasibound state, with I = J, will occur for any given E is made, hence the integral on the left side of eq. (A.3) is dominated by the Jth term. The approximation ;dR c (21 + 1) [u: (R, It)l’ g (2J + 1) a dR I!& (R, @I’-= (2J+ 1) N2 0

1

(A.4) then seems reasonable. The radius Q has been identified with &k).

QUASIBOUND

STATES FOR HeH+ ISOTOPES

111

The existence of a quasibound state for 1= J requires exp (-id,) ul, (R, k) R- ’ to become an outgoing wave when E is equal to the resonance energy13). This is equivalent to a simple pole in the Jth component of the S matrix; the remaining components of this diagonal matrix will lie near the unit circle. The outward flux is then controlled by the Jth term on the right side of eq. (A.3) and hence

results when eq. (A.4) is used. The techniques of Blatt and Weisskopf*l) can be used to evaluate the Im [. . .] term appearing in eq. (A.5). This requires the assumption*‘) that the complex part of E, i.e. T, is small. The complex solution of eq. (2) can be written as U: (R, It) = -!PJ (R, k) + i0, (R, k),

(A.6)

where Y, (R, k) is the function generated by the discrete-state method and 0, is an appropriately normalized independent solution*‘). For R 1 e3(k), [k* - W(R) - J(J + 1) R-*1 z [k* - J(J + 1) R-*1 and the approximation YJ (R, k) = c (xR/2#

[- yJ+O.J (kR) + iJJ+o.s VW,

can be used, where c is some normalization estimate

constant.

Eq. (A.7) provides the

Im [Yi (es, Wul, (es, 61 g 2k (xke3 [JJ”++ (ked +- YJ’,t(kedl)-'. If the assumption ke, z J + $, and

concerning the effective potential

(A-7)

(A.8)

for R 1 e3(k) is correct,

Im [Yi (es, r;>/YJ(e3. r;>] z 2k [47nz*(J + +)1’3]-1,

(A-9)

where a r 0.4473 and J must not be too small*O). Note that eq. (A.7) displays the appropriate outgoing-wave property for R large. In the same spirit, eq. (A.6) can be used to evaluate N* of eq. (A.4). The estimate N* = !dR

I -YJ (R, k) + i0, (R, k)l* g jdR

IY_,(R, k)l*,

(A. 10)

0

will be a good approximation. This follows because 0, is identified with the decreasing solution for e*(k) I R I e3(k) and hence will have the minimal amplitude for R < e3(k) while ul, (R, k) will have the maximal amplitude in the same region of R. The evaluation of lYJ (e3, E)I* is accomplished by requiring the function resulting from the integration of eq. (2) for 0 I R I e3 and the function used in

112

J. M. PEEK

evaluating eq. (A.lO) to be continuous. This conditions results in

IYJ(e3, W = 1y.1(e34 11- i F?I(~23,WC (e3, W1112,

(A.11)

provided YJ (es, k) IS . not zero but otherwise has the value used in constructing 01 of eq. (8). The various approximations leading to eq. (A.9) provide an estimate of 3-* for the ratio of functions appearing in eq. (A. 11). Hence, the result

IY7(e3, @I’ = 4 1%(e3, k)12,

(A. 12)

is obtained provided !P, (es, k) is not zero. Note the identification of Y’, (eJ, k) and 0, with the Bessel functions Y and J, respectively, as justified by the discussion in constructing eq. (9). The estimate of I’ is provided by eqs. (A.5), (A.9) (A.lO), and (A.12) as I’ 2 2+i2kIlu, (es, k)lz [3x,& (J + $)1’3 ;‘dR Ilu, (R, k)12]-‘,

(A. 13)

0

where !PJ is the solution to eq. (2) for real k and provided PJ (e3, k) is not too small. Two points concerning eq. (A.13) should be made. First, the form is such that r is independent of the normalization of YJ, so the function found subject to the conditions of eqs. (3) and (8) need not be normalized to conform to eq. (4) before evaluating N2 and !P_,(es, k). The requirement that u/, (e3, k) not be small guarantees P = 0 will not occur and places a restriction on the construction of OL in eq. (8). This restriction is not severe since YJ (es, k) = la, can be assumed and yet 01 can be made as small or large as one likes since Yj (es, k) may take any value. In fact, the condition that cx be varied until r assumes its minimum value is a good criterion to use in optimizing the discrete-state method described in section 2. REFERENCES 1) Schopman, J. and Los, J., Physica 48 (1970) 190. 2) Houver, J.C., Baudon, J., Abignoli, M., Barat, M., Fournier, P. and Durup, J., Internat. J. Mass Spectrom. Ion Phys. 4 (1970) 137. 3) Schopman, J., Fournier, P.G. and Los, J., Physica 63 (1973) 518. 4) Gamow, G., Structure of Atomic Nuclei and Nuclear Transformations, Clarendon Press (Oxford, 1937), Ch. V. 5) Levinson, N., K. Danske Vidensk. Selsk. mat.-fys. Medd. 25 (1949) No. 9. 6) Helbig, H.F., Millis, D.B. and Todd, L. W., Phys. Rev. A2 (1970) 771. 7) Green, T.A. and Michels, H.H., private communication. 8) Miller, W.H. and Schaefer III, H.F., J. them. Phys. 53 (1970) 1421. 9) Mackrodt, W.C., J. them. Phys. 54 (1971) 2952. 10) Weise, H.P., Mittmann, H.U., Ding, A. and Henglein, A., Z. Naturforsch. 26a (1971) 1122. 11) Buckingham, R. A., Fox, J. W. and Gal, E., Proc. Roy. Sot. (London) A284 (1965) 237.

QUASIBOUND

STATES FOR HeH+ ISOTOPES

113

12) Peek, J. M. (unpublished). 13) Landau, L.D. and Lifshitz, E.M., Quantum Mechanics. Non-Relativistic Theory, Pergamon Press (London, 1958) p. 440. 14) Beckel, C.L., Hansen III, B.D. and Peek, J.M., J. them. Phys. 53 (1970) 3681. 15) Wu, T.Y. and Ohmura, T., Quantum Theory of Scattering, Prentice-Hall (Englewood Cliffs, N. J., 1962) Sec. T. 16) Wigner, E.P. and Eisenbud, L., Phys. Rev. 72 (1947) 29. 17) Messiah, A., Quantum Mechanics, North Holland Publ. Comp. (Amsterdam, 1966) p. 100. 18) Eu, B.C. and Ross, J., J. them. Phys. 44 (1966) 2467. Wasch, T.G. and Bernstein, R.B., J. them. Phys. 46 (1967) 4905. Jackson, J.L. and Wyatt, R.E., Chem. Phys. Letters 4 (1970) 643. 19) LeRoy, R. J. and Bernstein, R.B., J. them. Phys. 54 (1971) 5114. 20) Abramowitz, M. and Stegun, I.A., eds., Handbook of Mathematical Functions, U.S. Govt. Printing Office (Washington, D.C., 1964) p. 368. 21) Blatt, J.M. and Weisskopf, V.F., Theoretical Nuclear Physics, John Wiley (New York, 1952) p. 412. 22) Bain, R.A. and Bardsley, J.N., J. them. Phys. 55 (1971) 4535. 23) Tricomi, F. G., Differential Equations, Blackie (London, 1961). 24) Dickinson, A.S. and Bernstein, R.B., Mol. Phys. 18 (1970) 305. 25) Wolniewicz, L., J. them. Phys. 43 (1965) 1087. 26) Peek, J.M., J. them. Phys. 50 (1969) 4595. 27) Johnson, B.R., Balint-Kurti, G. G. and Levine, R.D. , Chem. Phys. Letters 7 (1970) 268. 28) Dickinson, A.S., Mol. Phys. 18 (1970) 441. 29) Sizun, M. and Dump, J., Mol. Phys. 22 (1971) 459.