J . Phys. Chem. 1988, 92, 1816-1821
1816
I
I
I
1
Experiment - This work
__-Preuninger
-
8 Farrar
Phase Space Theory
= 0 03 eV = 0 33eV
,,,,E, ,,,,E,
, \
\
1
I
I
0' 0
I
0.1
0.2
0.3
0.4
0.5
I
I
0.6
0.7
PRODUCT c m KINETIC ENERGY, eV
Figure 5. Comparison of C3H3+/CH3product kinetic energy distributions from various sources: (-) experimental results of this work; ( 0 , m) phase space theory results of this work: (---) experimental results of Preuninger and Farrar, ref 1 1.
energy. Neither theoretical curve fits the experimental distribution exactly, but both nicely mimic the location of the peak in the distribution and the shape of the falloff in the distribution to higher energy. It is, in fact, reasonable that these distributions do not fit exactly. Under our experimental conditions we have C4H6+ ions formed both by chemical ionization from NO+ (reaction 2) and from electron impact (as evidenced by the fact we still observe a small metastable signal). Hence the experimental photoinduced kinetic energy distribution corresponds to C4H6+ions with a variety
of internal energies. This fact probably accounts for the discrepancy between experiment and theory. Also shown in Figure 5 is the kinetic energy release distribution reported by Preuninger and Farrar." Their distribution should be very similar to ours since the experiments are so similar. Yet, their distribution peaks at much higher energy (>0.2 eV as compared to 0.01 eV) and has a much different shape. Since the same excited state is undoubtedly accessed in both experiments it appears clear that there must be instrumental reasons for the discrepancy. Our data are in very good agreement with those published by Bunn and Baer given the differences in the experimental techniques. In addition, we have now applied our method to more than 30 systems and have had no experimental difficulties. We maintain our count rates well within the ability of the electronics to cope with them and consequently do not suffer from potential saturation effects. We thus conclude, in agreement with Bunn and Baer, that the data of Preuninger and Farrar are most likely flawed, probably due to difficulties in accurately subtracting the huge metastable signal they had from the photoinduced plus metastable signal. Acknowledgment. We gratefully acknowledge the support of the National Science Foundation under Grant CHE85-12711 and the Air Force Office of Scientific Research under Grant AFOSR86-0059. We also gratefully acknowledge advice and assistance from Ms. H-S. Kim and Dr. C-H. Kuo. Registry No. C4H6*,34488-62-5; C,H,, 21540-27-2; CH,, 2229-07-4; C4Hs+,64235-83-2; H, 12385-13-6.
Electronic Structure from Semiclassical Dimenslonal Expansions: Symmetry Breaking and Bound States of the Hydride Ion D. J. Dorent and D. R. Herschbach* Department of Chemistry, Harvard University, Cambridge, Massachusetts 021 38 (Received: September 28, 1987)
-
For a large spatial dimension D, semiclassical techniques become applicableto electronicstructure calculations. The zeroth-order term, corresponding to D m, can be evaluated exactly by finding the minimum of an effective electrostatic potential. This defines a rigid configuration of the electrons (the "Lewis structure"). The first-order term, proportional to 1/D, can likewise be calculated exactly and corresponds to harmonic vibrations of the electrons (the "Langmuir vibrations") about the fixed m limit. For a two-electron atom with nuclear charge Z > 1.237, the effective potential has positions attained in the D a single minimum with the electrons equidistant from the nucleus. For smaller Z, this symmetry is broken. When Z < 1.2279 a double minimum obtains, with one electron much closer to the nucleus than the other. For the short range of intermediate Z , a triple minimum occurs; at Z 1.2334 the symmetric and asymmetric electron configurations become isoenergetic. Convergence of the perturbation expansion in powers of 1 / D is improved by a resummation technique which removes contributions from first- and second-order poles that occur for the singular D 1 limit. The hydride ion, with Z = 1, provides an extreme test since it has only two weakly bound states; in the restricted Hartree-Fock approximation both are unbound. Dimensional perturbation has the virtue of including electron correlation at all orders. The very simple first-order resummed dimensional expansion (Lewis and pole terms only, evaluated at D = 3 and 5 ) gives both H- bound states in a single stroke. The accuracy obtained with optimal truncation is a few tenths of a percent for both states.
-
+
I. Introduction The most notorious failure of the old quantum theory was its inability to determine the spectrum of two-electron atoms.) This stems from the rarity of stable orbits in such systems. Before applying any quantization conditions, one must find families of orbits for which the electrons never collapse into the nucleus or undergo autoionization. Despite modern developments in semiclassical mechanics2 and a long history of attempt^,^ there are still no known quantizing trajectories for the helium atom ground state. Quantum mechanics avoided that issue, yet the early variational calculations posed in its stead the quandary of finding simple 'Present address: AT&T Bell Laboratories, Murray Hill, NJ 07971
0022-3654/88/2092-l816$01.50/0
pictures to describe multielectron systems. This need was largely met by the Hartree-Fock theory of orbitals, but that fails as a zero-order model when electronic motions are highly correlated. Generally, such is the situation for open-shell systems, multiply (1) Born, M. Mechanics o f t h e Atom; Ungar: New York, 1960. (2) For excellent surveys of current semiclassical methods, see several papers in: J . Phys. Chem. 1986, 90, 3453-3862 (a special issue dedicated to
R . A. Marcus). (3) Van Vleck, J. H. Philos. Magn. 1922,44, 842. For recent semiclassical treatments of two-electron atoms, see: Leopold, J . G.; Percival, I. C. J . Phys. B 1980, 13, 1037. Coveney, P. V.; Child, M. S. Ibid. 1984, 17, 319. Wesenberg, G. E.; Noid, D. W.; Delos, J . B. Chem. Phys. Lett. 1985, 118, 72. Solov'ev, E. A. Sou. Phys.-JETP (Engl. Transl ) 1985, 62, 1148. Klar, H. Phys. Rea. Lett. 1986, 57, 66.
0 1988 American Chemical Society
Symmetry Breaking and Bound States of the Hydride Ion excited electronic states, or negative ions. The prototype example of the inadequacy of restricted Hartree-Fwk theory is its failure to predict4 the stability of the ground state of the hydride ion, H-. More general variational methods do attain positive binding energy, although trial wave functions with both radial and angular correlation are required for high a c c ~ r a c y . ~Comparison of accurate wave functions with the restricted HartreeFock trial function shows that the two electrons must lie in different orbitals and the interelectronic separation is highly correlated with radial motion of the electrons. Of course, H- is an extreme case: only the ground state and the doubly excited 2p2 3Peare bound, while for an infinitesimally larger nuclear charge, there are infinitely many bound states.6 For all other excited states of two-electron atoms, Z = 1 appears to lie on the circle of convergence for the 1 / Z ~ e r i e s . ~ Recent work has developed a novel semiclassical approach to electronic This involves generalizing the Schriidinger equation to an arbitrary number of spatial dimensions, D, and scaling the coordinates in proportion to D2. A radial equation is thus obtained in which 1/D appears only as a parameter, with a role similar to Planck’s constant, h, in the familiar version. The limit, tantamont to h 0, the novelty is that in the D effective potential has a definite minimum corresponding to a rigid configuration of the electrons. This is termed the “Lewis structure”; it provides a rigorous prescription for the qualitative electron-dot formulas introduced 70 years ago.I5 When D is large but finite, the electrons can oscillate about the limiting rigid configuration. The first-order contribution, proportional to 1/D, corresponds to harmonic vibrations. These are termed “Langmuir vibrations” to acknowledge another significant prequantum insight.16 In this approach, traditional semiclassical methods become applicable and practicable, at least for few-electron systems. The computational limitations are not in the inherently intractable classical dynamics of such systems. Rather, the method suffers from familiar ailments met in other semiclassical treatments of multiple oscillator problems: quantizing trajectories become difficult to find when there are several degrees of freedom, and tunneling effects are neglected. However, if accurate predictions at D = 3 can be obtained from the semiclassical limit at large D, then the same methods used to determine motion on a potential surface can be applied t o calculate the surface itself. Here we evaluate the leading terms of a semiclassical dimensional perturbation expansion for the hydride ion. In section I1 we consider the classical limit, D m. In section I11 we examine a major feature, the symmetry breaking that occurs as the nuclear charge Z is decreased. The Lewis structure then becomes unsymmetrical, with one electron much closer to the nucleus than the other, as in the actual hydride ion. In section IV, we exploit the dominant role in the dimensional perturbation expansion of singularities at D 1, a hyperquantum limit tantamount to h m . A resummation technique permits the contributions from these singularities to be included to all orders in 1/D and thus markedly improves the semiclassical dimensional expansion. A
-
simple interpolation formula is obtained which involves just three quantities computed at the D = and D = 1 limits. This gives fairly accurate energies for both the H- ground state (at D = 3) and the only bound excited state (isomorphic with D = 5 ) . In section V we comment on some prospects and problems anticipated in computing electronic structures by semiclassical dimensional continuation methods. 11. Large Dimension Limit The Schrodinger equation for S states of a D-dimensional two-electron atom with nuclear charge Z can be cast in the formlo
where
K
= Il2(D - 1) and
-
-
-
The Journal of Physical Chemistry, Vol. 92, No. 7, 1988 1817
-
(4) Ruskai, M. B.; Stillinger, F. H. J . Marh. Phys. (N.Y.)1984,25,2099. (5) Rehmus, P.; Roothaan, C. C. J.; Berry, R. S . Chem. Phys. Lerr. 1978, 58, 321. ( 6 ) Kato, T. Trans. Am. Marh. Soc. 1951, 70, 212. Hunziker, W. Helv. Phys. Acta 1975, 48, 145. (7) Baker, J.; Freund, D. E.; Morgan, J. D. Phys. Rev. A , in press. (8) Mlodinow, L. D.; Papanicolaou, N. Ann. Phys. (N.Y.)1981, 131, 1. (9) Goscinski, 0.; Mujica, V. Inr. J . Quantum Chem. 1986, 29, 897. (10) Herschbach, D. R. J . Chem. Phys. 1986, 84, 838. (1 1) Loeser, J. G.; Herschbach, D. R. J. Phys. Chem. 1985,89, 3444. J . Chem. Phys. 1986, 84, 3882, 3893. 1987,86, 2114, 3512. (12) Doren, D. J.; Herschbach, D. R. Chem. Phys. Lett. 1985,118, 115. Phys. Rev. A 1986,34,2654, 2665. J . Chem. Phys. 1986.85, 4557. 1987, 87, 433. (13) Goodson, D. Z.; Herschbach, D. R. Phys. Rev. Lett. 1987,58, 1628. J . Chem. Phys. 1987, 86, 4997. (14) Loeser, J. G. J . Chem. Phys. 1987, 86, 5635. (15) Lewis, G. N. J . Am. Chem. SOC.1916, 38, 762. (16) Langmuir, I. J. Am. Chem. SOC.1919.41, 868. See also: Van Vleck, J. H. Pure Appl. Chem. 1970, 24, 235.
V = - - - 1- + 1 R I R2
X
(R12
+ R2’ - 2RlR2 COS
Here 8 is the angle between the electron-nucleus radii R , and R2 and X = 1/Z. The energy eigenvalue ED is in hartree units and 0 (with K distances in bohr radii. In the hydrogenic limit X > 0 but otherwise arbitrary), the ground-state energy becomes - Z 2 / ~ 2the , radial distribution function becomes independent of 8, and R1 and R2 appear in separate, equivalent factors that each reach a maximum at a radius of K ~ / ZThe . large dimension limit K (with X > 0 but otherwise arbitrary) can be obtained readily by a simple scaling transformation; R, (K2/Z)ri.The rescaled Schrodinger equation then becomes
-
-
-
[ ( l / m ) T ( r 1 7 2 @+) W(r!,r2,@- ~ D ] h ( r l > r 2 , = 8 )0
-
(2)
as K m, where W = U + Vis now an effective potential, m = K~ is an effective mass, and tD is the rescaled energy defined by ED =
-
=
( z / K ) 2 t ~[2Z/(D
- 1)]’€~
(3)
Thus, in the limit D the effective mass becomes infinite and dD becomes a 6-function located at the minimum of the effective potential, so eD e , = W(rlm,r2m,8m). Calculation of the Lewis structure and its energy thereby reduces to finding this minimum, an exactly solvable problem in classical electrostatics. This classical limit, which emerges from dimensional scaling, is not the same as the conventional limit obtained by h 0 for a fixed dimension.l’ Since the centrifugal term U is proportional to h2,it does not contribute to the conventional semiclassical limit. With dimensional scaling, however, the centrifugal term introduces barriers which prevent the electrons from falling into the nucleus, colliding with each other, or finding the unstable trajectories]* that exist when they are opposed by 180’. As long as the well near the minimum is sufficiently deep, the effective potential W can support stable classical trajectories which can be used for semiclassical quantization. A more typical procedure has been to employ perturbation expansions for the energy or wave function in powers of 1ID. These are readily obtained by expanding the potential about its minimum.’b12 Since the Langmuir vibrations are harmonic, they involve just curvatures of the potential. Higher anharmonic contributions can be evaluated efficiently by exploiting recursion relations among moments of the vibrational displacement coordinates.I 3
-
-
111. Symmetry Breaking
-
If the two electrons are assumed to be equidistant from the nucleus (rlm= rZm)at the minimum of the D potential, (17) Yaffe, L. G. Rev. Mod. Phys. 1982, 54, 407. (1 8) Wannier, G. H. Phys. Rev. 1953, 90, 8 17.
1818 The Journal of Physical Chemistry, Vol. 92, No. 7, 1988 -
T
1
-r--
B
7
I
Doren and Herschbach
----ib1
1
1
l l I
2
0
3
1
0
2
2
1
-
6
B
2
350k
Figure 1. Contour maps of effective potential in the D m limit for a two-electron atom. The coordinates rl and r2 specify the distance of each electron to the nucleus; the interelectron angle is fixed at Om, the value it takes at the minimum of the well. Contours are spaced 0.05 au apart. The heavy lines indicate the dissociation limit beyond which only one electron can be bound; the cross marks the position of the energy minimum. (a) For the helium atom, with Z = 2 and 8, = 95.3O; (b) for Z = 1.5 and Om = 97.2’.
I
11
1
1
1;
Nuclear Charge, Z
Figure 3. Variation with nuclear charge Z of energy at potential minima, W,,, = e,, and corresponding electron-nucleus distance rm and intere, lectron angle 8, for D m limit. For Z < Z, = 1.2279..., the potential OL-1 ohp--pL , 0 2 3 4 0 8 2 16 has two equivalent minima, with rlm r2,,,,respectively.
-
Figure 2. Contour maps similar to Figure 1, but for Z = 1.2, in region of symmetry breaking. Contours are spaced 0.005 au apart. (a) Near the local maximum of the potential, where rlm = r2, and 0, = 93.0°; (b) near the minimum of the well, where rlm = 1.02, rZm= 3 64, and 8, = 93.0’. Not shown is an equivalent minimum related by electron exchange, with r , , = 3.64 and r2, = 1.02
analytic formulas for the Lewis structure (rm,8,) and its reduced energy t, can be This symmetrical structure is indeed the sole minimum for helium and all two-electron cations ( Z > 2). Figure l a shows a contour map of the effective potential W(rl,rt,8,) for helium; at the minimum rm = 1.21, 8, = 95.3O, and E, = -0.684. Comparable quantities for the D = 3 atomi9 are ( r 2 ) I l 2= 1.1, cos-l (cos 8) = 93.1°, and q = -0.726; thus, the reduced energy for the D = m limit differs by only about 5.8% from that for the real atom. Previous studies*-” have noted that the coefficient of l / D in the dimensional expansion for the energy becomes complex when Z is decreased below a critical value Zo = 1/X, given by A, = 25/2[(4/3)1/4- (3/4)’14]
(4)
Hence Zo = 1.22791378.... This occurs because for Z < Z , the force constant for the antisymmetric stretching vibrational mode becomes negative. The symmetric configuration thus corresponds to a saddle point of W(rl,r2,8)rather than a minimum. There are then two equivalent unsymmetrical minima that differ by interchange of rlmand r2,. Figure 1b shows a contour map for Z = 1.5, where the potential still has only a single, symmetrical minimum but clearly exhibits a “bow tie” about the minimum which anticipates the transition to the double-minimum regime. Figure 2 displays contour maps for Z = 1.2, where the double-minimum structure is quite pronounced, showing both (a) the saddle point region about the symmetrical maximum and (b) the markedly unsymmetrical minimum region. In our scaled units, one electron has now moved almost as close to the nucleus as in the hydrogen atom and the other is more than 3 times as far away. The Lewis structure thus corresponds to putting the electrons into (19) Pekeris. C . L. Phys. Reu. 1959, 115, 1216
different radial orbitals. In effect, this simple classical limit predicts that the restricted Hartree-Fock approximation will become increasingly poor when Z < Zo. As Z decreases further below Z,, the twin minima in the D m effective potential become shallower, the inner electron becomes still more hydrogenic, the outer electron retreats further from the nucleus, and the interelectron angle decreases steeply toward 90°. For the hydride ion, with Z = 1, the effective potential is no longer able to bind the outer electron. Figure 3 displays these trends in the electron radii, angle and the reduced energy. In numerical searches, we found no additional minima in the 1 < Z < Zoregion. However, in a short range immediately above Zo we found a pair of unsymmetrical minima, previously unsuspected harbingers of the double minima below 2,. These develop from the “bow tie” contour illustrated in Figure lb. Thus, the symmetry-breaking transition begins slightly above Zo (at approximately Z = 1.235) and involves triple minima. At first the central, symmetric minimum is the most stable, but the three soon become isoenergetic (at approximately Z = 1.2334), after which the pair of equivalent unsymmetric minima become the most stable. Figures 4 and 5 show the changes in electron geometry and energy in this transition region.
-
IV. Dimensional Interpolation In the large-D limit, as exhibited in Figure 3, the effective potential for the hydride ion allows the outer electron to escape and so predicts the same energy as a hydrogen atom: em To obtain binding of both electrons requires a higher order approximation. The customary form of dimensional perturbation expansion,*-” which determines a Taylor series in powers of 1/D, is not feasible for the hydride ion. That expansion involves evaluating higher derivatives of the effective potential at its D 03 minimum, which fails because the minimum disappears as Z 1. Here we employ a different technique which incorporates information from the D 1 limit and was found to be quite successful when tested on helium and two-electron cations.I2 Our method stems from the observation that the ground-state 1, and this energy as a function of 1/D becomes singular for D gives a major contribution at D = 3. As in the hydrogenic Z
-
+
-
-
- -
Symmetry Breaking and Bound States of the Hydride Ion
The Journal of Physical Chemistry, Vol. 92, No. 7, 1988 1819 TABLE I: Reduced Energy Parameters"
2.0
z
e,
'I
€1'
1 1.2 1.5 2
-0.500000 -0.5 I4 21 4 -0.595 486 -0.684 442
-0.647210 -0.687 51 1 -0.734553 -0.788 848
0.158 527 0.I32 55 0.105975 0.079 1 13
Limiting eigenvalues: e, obtained from classical electrostatic calculation (ref 10) and t l from &function potential (ref 21). Residue of first-order pole: e,' obtained from numerical calculations (ref 11).
--
monotonic, so that the D -,1 limit provides a lower bound and limit an upper bound. the D These limits can be combined in a systematic way by writing the total energy as the sum of the second-order pole term and the leading term of the 1/D expansion, with the coefficient of the latter chosen to agree with the known large-D behavior. This amounts to a resummation technique which corrects the large-D expansion at each order in 1/D by deducting the contribution from the pole.I2 Thus, as a first approximation we take
i
ED =
(=)tt' D-1
-
+ (%)'a,
(5)
where comparison with the D limit gives a, = e, - el. This yields a simple interpolation formula for the reduced energy D '
-0,526
1.220
1.225
20 1.23
1.23
1.240
Nuclear Charge, 2
Figure 4. Details of the symmetry-breaking transition in immediate neighborhood of Zo. Solid curves show 6, and corresponding I , and 8, at global minima, as in Figure 3. Dashed and dotted curves pertain to subsidiary local minima for symmetric and asymmetric configurations, respectively, as pictured in Figure 5. A
= €1
+ (1 - 6)'(€,
- €1)
(6) with 6 = 1/D. The formula can be improved by including further terms from expansions about either or both of the D 1 and D limits. The first correction to the D 1 limit has been evaluated (to about five significant figures) from results of variational calculations in the low-D regime." This correction is a first-order pole at D = 1, with residue el' = 0.158 527. Its contribution may be included by merely replacing el in eq 5 by e l + (1 - 6)t,'; this changes the interpolation formula to
-
-
-
=
+ (1 - 6)cI' + (1 - 6)'(€,
- €1 - €1')
(7) At the large-D limit, we have no additional information on the hydride ion itself, since for 2 1 that limit yields just a hydrogen atom plus a free electron. However, we can augment the interpolation by replacing a, in eq 5 by the 1/D expansion for the hydrogen atom minus the corresponding expansions from the hydride ion pole terms. This gives CD
€1
-
well lowest
Triple
equally low
Minim0
osymmelric wells lowest
.
Double Min;mo
LJ
Figure 5. Variation of Lewis structures in symmetry-breaking region (at left). Corresponding values of Z,r,, and 8, are shown. Solid lines indicate structures that pertain to global minima, and dashed lines indicate those that pertain to less stable local minima as pictured schematically (at right). limit of eq 3, the dominant singularity is a second-order pole. For finite 2, the residue of this pole can be determined as the eigenvalue of a one-dimensional problem in which all of the Coulombic interactions have been replaced by 6-functions.12,20 This greatly simplified two-electron problem has been solved2] (and predicts a strongly bound H- ion near the D 1 limit. The corresponding residue is tD e l = -0.6472102. Numerical calculations" indicate that the D dependence of the energy is
-
-
(20) Herrick, D. R.; Stillinger, F. H. Phys. Rev. A 1975, 11, 42. (21) Rosenthal, C. M. J . Chem. Phys. 1971, 55, 2474.
For generality, we have inserted the coefficients p k , which are all unity in the hydrogenic limit but for k > 1 differ from unity when 2 > 1. The values1° for helium are p o = 1, p1 = 1.lo6 30, p 2 = 1.07900, p 3 = 1.269 34, etc. When 2 > 1, the higher order p k become very large,13 so the 1 / D series diverges and must be truncated at low order.I2 Although the hydrogenic 1/D series converges for 161 < 1, its sum cannot represent the hydride ion except near the large-D limit, and we must likewise truncate the series after a few terms. Figure 6 shows the dimension dependence of the reduced energy tD for 1 d 2 d 2, as derived from the variational calculations.ll As a test of the interpolation procedure, we compare for 2 = 1 and 2 curves obtained from eq 6 and 8. Similar curves were found for the intermediate values of 2 but are omitted to avoid crowding. Table I lists the parameters el, el', and e,, and Table I1 compares values of tD from several variant approximations. The designation (n,m) indicates use of n terms from the D 1 limit, involving q or el and el', and M terms from the D limit, involving e, with up to three terms ( k S 2) of the series in eq 8. Variant ( l , l ) , just the zeroth-order result of eq 6, is seen to give only a rather rough approximation to the dimension dependence, especially for 2 = 1. The corresponding estimate for the energy a t D = 3 of the hydride ion is -0.582 au, while the true ground-state energy is -0.528 au. In part, this 10% error occurs
--
Doren and Herschbach
1820 The Journal of Physical Chemistry, Vol. 92, No. 7, 1988 n
Our interpolation formula can also be applied directly to predict the energy of the only other bound state of the hydride ion, the 2pz 3Peexcited state. The stability of this state pertains not to the ground-state asymptote but rather to that for H(2p) + e-, at -I/* au. For any two-electron atom, there is an exact interdimensional degeneracy'2v20between this excited state at D = 3 and the ground state at D = 5, and our dimension-scaled units make the corresponding asymptotes coincide at - l l 2 au. Thus, we obtain estimates of the excited-state energy by merely evaluating our formulas at D = 5, using the same parameters. We find that all variants (n 6 2,m 6 6) of the dimensional perturbation treatment do indeed predict this very weakly bound excited state to be stable, as indicated by reduced energy below -l/z au. The accurate result from extensive variational calculationsZ2is -0.5014au, below the asymptote by only 0.28%. Table I1 includes our results. For the simplest variant (1,l) the error is again 10% and for (2,l)it is 5.2%. The optimal truncation for H- is now (2,4),in error by only -0.09%, whereas for He variant (2,2)is slightly better. For the excited state, the increase in D (from 3 to 5) shrinks the higher order terms and hence shifts the optimal truncation upward. The correlation energy, given by the difference between exact (nonrelativistic) and Hartree-Fock results, has been found' to be nearly a linear function of 1/D;thus
U 00
10
5
3
2
1.5
1.2
I
I
I
I
0
0.5 6=1/D
1
1.0
At, =
Figure 6. Reduced energy q, as a function of 6 = 1 / D in low-Z regime.
Solid curves from highly accurate calculations employing HylleraasPekeris method (ref 11). Long-dashed portion for Z = 1 shows smooth extrapolation to saddle point energy at D m corresponding to isosceles symmetry. Short-dashed curves shown for Z = 1 and Z = 2 were obtained from approximation denoted ( l , l ) , using eq 6 with D m limit and the second-order pole at D 1. Dotted curves were computed from (2,2) approximation, using eq 8 truncated after the k = 1 term.
-
-
-
-C
D
N~
6AtI ~
+ (1
- ~)AE,
(9)
This was demonstrated for Z > 1.5 by variational calculations. We note that it is also fairly good for Z = 1, at least for D < 3, provided that At, is evaluated from the nominal limiting values without regard to symmetry breaking or ionization. The limiting results" for Z = 1 give AeI = -0.64721 - (-0.58333)= 0.06388 Aem = -0.44035 -(-0.41789) = -0.02246
TABLE 11: Approximations for Reduced Energies
291 2.2
CD
-0.581 78 -0.538 18 -0.530 91 -0.541 53 -0.546 56 -0.526 42 -0.5 12 44 -0.505 36 -0.527 751
-0.553 00 -0.51 5 32 -0.504 00 -0.520 40 -0.527 64 -0.51024 -0.503 00 -0.50097 -0.501 419
10.2, 10.3 2.0, 2.8 0.60, 0.52 2.6, 3.8 3.6, 5.2 -0.25, 1.8 -2.9, 0.32 -4.2. -0.09
-0.742 45 -0.733 07 -0.725 61 -0.736 11 -0.724 87 -0.727 21 -0.723 60 -0.71940 -0.725 93 1
-0.722 04 -0.713 92 -0.7 10 04 -0.725 56 -0.709 36 -0.71 1 40 -0.709 56 -0.708 33 -0.710500
2.3, 1.6 1.0, 0.5 -0.04, -0.05 1.4, 2.1 -0.15, -0.16 0.18, 0.12 -0.31, -0.14 -0.90. -0.31
'Approximation designated by (n,m), as defined in text. "Exact" values are from variational calculations (ref 11, 22). because the dimension dependence abruptly flattens out at the onset of ionization. Since tD actually extrapolates smoothly to the saddle point energy (long dashes in Figure 6,extending to -0.440 352),we also tried replacing t, = -1/2 with this value (in computing the short-dashed curve for 2 = 1). The corresponding energy at D = 3 is then -0.555 au, in error by 5%. For helium,Iz the error in eq 6 is 2.3%. Variant (2,l),the first-order approximation of eq 7,appreciably improves the dimension dependence. For the hydride ion a t D = 3, this yields -0.546 au, so the error is reduced to 3.6%. For helium the same approximation12 gives an error of only -0.15%. Variant (2,2),which includes just one further term from the series in eq 8,provides marked improvement for the hydride ion. It gives an energy of -0.526 au, in error by only -0.25%;the corresponding error for helium is 0.18%. Adding further terms worsens the approximation, so variant (2,2)proves to be optimal for the ground state of H- but not quite as good as (2,l)for the He atom.
From eq 9 we then find At3 N 0.0363 and tJHF -0.4915,in fair agreement with Hartree-Fock calculation^^^ which give Ae3 = -0.0398 and t3HF= -0.4880au. Such calculations seem to be lacking for the excited state corresponding to D = 5, but from eq 9 we obtain Ats N - 0.0307 and t3HFN -0.4707 au. This indicates that the fractional error in the Hartree-Fock result for the H- ground state (6.9%) is somewhat larger than for the excited state (6.1%). Again, these errors are substantially larger than for the He atom (1.5% and 1.3%, respectively).
V. Discussion Given its simplicity, the dimensional interpolation formula yields satisfactory results for the hydride ion. Both bound states are obtained with an accuracy of a few tenths of a percent even though in this case the D m limit corresponds to ionization (em = and thus the poles at D 1 become solely responsible for the stability of the atom. Of course, the accuracy is poor by current standards for electronic structure. At least for the ground state, better results can be obtained from relatively simple variational functions which also have the advantage of providing rigorous bounds.z4 Nevertheless, our dimensional perturbation results are encouraging because the same method used for higher members of the isoelectronic sequence proves to give reasonable accuracy for the hydride ion and yields both states in a single stroke. Unlike the Hartree-Fock method, it is not necessary to recast the approach in a major way just to get a crude approximation. For H- as well as larger two-electron atoms, even low-order dimensional perturbation treatments are superior to and simpler than the Hartree-Fock method. The basic virtue of dimensional continuation methods for electronic structure is that the complete Hamiltonian is solved for the limiting cases (here, D 1 and D m) used to construct the perturbation expansion or interpolation formula. The ability +
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+
-+
(22) Drake, G. W. F. Phys. Reo. &ti. 1970, 24, 126. Bhatia, A. Phys. Reo. A 1970, 2, 1667. (23) Banyard, K. E. J . Chem. Phys. 1968, 48, 2121. (24) Massey, H. S . W. Negative Ions, 3rd ed.; Cambridge Univ. Press Cambridge, U.K., 1976.
J. Phys. Chem. 1988, 92, 1821-1830
1821
of the method to represent the D = 3 system thus does not depend particle problems. Again, techniques based on Monte Carlo on the magnitude of the interaction but only on its dimension methods are applicable to tunneling in many degrees of freedom.32 dependence. Our results for the hydride ion add to the evidenThe multiple minima introduced by symmetry breaking are that major features of the electron correlation are included technically challenging but heuristically encouraging. As seen in both D 1 and D m limits, even in the lowest order apin Figure 5 , for H- these features suggest that something like shell proximations. In particular, much of the effect of short-range structure can appear even in the classical limit. The Lewis structure then indicates directly how the electrons are distributed electron repulsion apparently is contained in the residues of the among orbitals. A similar a n a l y s i ~ ~of~ the , ' ~ large-D limit for D = 1 poles. New methods are required if we aim for higher accuracy in the Li atomi4 finds a kindred symmetry breaking occurs for 2 < 2.29, with two equivalent inner electrons and a distinct outer applying dimensional continuation to strongly correlated and electron. Analogous effects appear at large D for molecules when weakly bound systems such as the hydride ion. For this purpose, either the nuclear charges or the internuclear distances are varied. extending the perturbative approach appears less promising than Thus, for the H2+and H2 molecules,34 the effective potential efficient semiclassical procedures for obtaining vibrational eigenstates using basis sets derived from classical trajectories.25 localizes the electron(s) in the plane bisecting the internuclear Direct semiclassical quantization may also prove f e a ~ i b l e . ~ ~ - ~ *axis when the separation between the nuclei is small but the potential acquires a double minimum when the separation between These methods should be readily applicable for large D, where m limit hence builds the nuclei becomes large enough. The D the effective potential becomes nearly harmonic. There are three into the zeroth-order model both chemical bonding and the proper chief issues: (1) Practical semiclassical techniques need to be developed for systems with many degrees of freedom. Monte Carlo dissociation behavior. Such features suggest the likely utility of methods now offer a promising approach.29 (2) When the efthe large-D regime in developing semiclassical interpretations of electronic structure. fective potential has no minimum in the large-D limit, scattering theory techniques may be needed to treat correlation effects. (3) Acknowledgment. We have enjoyed discussing many dimenWhen the effective potential has two or more equivalent or comsions of this study with John Loeser, David Goodson, and Don parable minima, tunneling among them must be accurately Frantz. We also thank Rudy Marcus for his incisive contributions evaluated. Pertinent approaches include supersymmetry techand exuberant advocacy of semiclassical methods which have niques recently applied to large-D expansions30and to semiclassical strongly influenced our work. t ~ n n e l i n g ,but ~ ' so far these methods deal just with simple one-
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Registry No. H-, 12184-88-2. (25) Frederick, J. H.; Heller, E. J. J . Chem. Phys. 1987, 87, 6592. (26) Strand, M. P.; Reinhardt, W. P. J . Chem. Phys. 1979, 70, 3812. (27) Maslov, V. P.; Fedoriuk, M. V. Semiclassical Approximations in Quantum Mechanics; Reidel: Boston, MA, 1981. (28) Delos, J. B. Adu. Chem. Phys. 1986,65, 161. Knudson, S. K.; Delos, J. B.; Nord, D. W. J . Chem. Phys. 1986,84, 6886. (29) See several papers in: J . Stat. Phys. 1986, 43, 729-1237. For an application to H-, see: McDowell, H. K.; Doll, J. D. Chem. Phys. Lett. 1981, 82, 127. (30) Imbo, T. D.; Sukhatme, U. P. Phys. Reu. Lett. 1985, 54, 2184.
(31) Bernstein, M.; Brown, L. S. Phys. Rev. Lett. 1984,52, 1933. Kumar, P.; Ruiz-Altaba, M.; Thomas, B. S . Ibid. 1986, 57, 2749. Keung, W. Y.; Kovacs, E.; Sukhatme, U. P. Ibid., in press. (32) Garett, B. C.; Truhlar, D. G. J . Chem. Phys. 1983, 79, 4931. Doll, J. D.; Coalson, R. D.; Freeman, D. L. Ibid. 1987, 87, 1641. Chang, J.; Miller, W. H. Ibid. 1987, 87, 1648. (33) Loeser, J. G., private communication. (34) Frantz, D. D.; Herschbach, D. R., to be submitted for publication.
Rotational-Vibrational Structure of a Quasi-Linear Molecule: CH,' Jae Shin Lee and Don Secrest* School of Chemical Sciences, University of Illinois, Urbana, Illinois 61 801 (Received: August 17, 1987)
A new potential energy function is obtained for the ground electronic state of CH2+through a Simons-Parr-Finlan (SPF) type expansion of an ab initio potential surface for this molecule. The SPF type potential is found to fit the a priori potential points extremely well and has reasonable physical properties along the vibrational coordinates of the molecule. The rotation-vibration states of this molecule are calculated for J = 0, 1,2, and 3 by using this potential function. The calculations were carried out using a linear molecule Hamiltonian. Assignments have been made to each vibrational state. It is possible to identify the (1 1 0) state through the rotational structure of the molecule. This level was assigned to (0 4 0) previously. As in the previous calculations on this molecule, the Renner-Teller effect was neglected.
Introduction Floppy molecules may undergo rather large excursions during normal vibrations. There have been a number of studies in the recent literature on molecules of this type.'J In particular, molecules with a low barrier to linearity are of interest because of the singularity in the Hamiltonian for the linear configuration. This has been handled by Carter and Handy3 by using a linear molecule Hamiltonian, and by Tennyson and S ~ t c l i f f eusing ~,~ (1) Bacic, Z.; Light, J. C. J . Chem. Phys. 1986, 85, 4594. (2) Bowman, J. M. Arc. Chem. Res. 1986, 19, 202. (3) Carter, S.; Handy, N. C. J. Mol. Spectrosc. 1982, 95, 9.
0022-3654/88/2092-1821$01,50/0
Jacobi coordinates and a technique related to close coupling techniques in scattering theory. Both of these calculations were performed on the CH2+ molecule which has a low barrier to linearity, using a potential computed by Carter and Handy.3 This potential is a sum of diatomic potentials with a three-particle term obtained by a fit to an ab initio calculation of Bartholomae, Martin, and Sutcliffe.6 (4) Tennyson, J.; Sutcliffe, B. T. J . Mol. Spectrosc. 1983, 101, 71. (5) Tennyson, J.; Sutcliffe, B. T. J . Chem. Phys. 1982, 77, 4061; 1983, 79, 43. (6) Bartholomae, R.; Martin, D.; Sutcliffe, B. T. J. Mol. Spectrosc. 1981, 87, 367.
0 1988 American Chemical Society
