J. Phys. Chem. 1988, 92, 3122-3124
3122
A convincing interpretation of the physics which causes this resonance feature to appear, a t least weakly, in the angular dependence of photoejection but not in the photoabsorption cross section has not yet been given. This study has eliminated one possibility. Understanding the details of this particular failure of the static-exchange approximation will undoubtedly lead to a
better definition of its limitations in other applications. Acknowledgment. We thank Dr. s. Yabushita for helpful discussion concerning constructing the basis sets for K symmetries and providing the preliminary results of RPA calculations. This work was supported by NSF grants U-E-8607496 and CHE83 12286. Registry No. C 0 2 , 124-38-9; C, 7440-44-0; CO,’, 12181-61-2.
(34) Eland, J . D. H.; Berkowitz, J. J . Chem. Phys. 1977,67,2782.
Persistent Accidental Degeneracies for the Coffey-Evans Potential M. S. Child* and A. V. Chambers
Downloaded by NEW YORK UNIV on September 9, 2015 | http://pubs.acs.org Publication Date: June 1, 1988 | doi: 10.1021/j100322a023
Theoretical Chemistry Department, University of Oxford, Oxford OX1 3TG, England (Received: July 14, 1987; I n Final Form: November 28, 1987)
The Coffey-Evans potential, which is periodic, with two pairs of equivalent potential wells, gives rise to persistent accidental fourfold degeneracies which are traced by semiclassical arguments to a systematic exact coincidence between levels differing in quantum numbers by one unit in the two types of well, at all energies below the barrier maximum. The magnitude of tunneling splitting is also discussed.
1. Introduction Accidental degeneracies between symmetry-related levels that cannot interconnect by tunneling are commonplace, but the Coffey-Evans equation’ used to model dipolar coupling in polarizable liquids such as liquid crystal displaysZshows a remarkable new phenomenon. Reported levels3have alternating nondegenerate and triply near degenerate levels for large values of the coupling parameter b, although the underlying potential function V ( x ) = -2b cos 2x
+ b2 sin2 2x
(1)
has only twofold symmetry elements. We argue below that the alternating degeneracy pattern is a consequence of particular boundary conditions that pick out certain solutions from those that are fourfold near degenerate, due to a chance but persistent coincidence between the vth levels in the shallower pair of potential wells (see Figure 1) and the (v 1)th levels in the deeper pair. The purpose of this paper is to trace the origin of this coincidence. Symmetry considerations are discussed in section 2, and hints about the origin of the coincidences are obtained by the use of quadratic approximations in section 3. The main part of the paper is, however, the demonstration in section 4 that the two sets of levels are in exact coincidence for all subbarrier energies within the validity of semiclassical quantization and to the extent that tunneling is forbidden. Section 5 gives a simple rule for the predicted number of fourfold near degeneracies for a given b value, when tunneling is taken into account.
+
2. Symmetry Consideration The Coffey-Evans equation,’*2known also in the mathematical literature as Ince’s e q ~ a t i o nis, ~of SchrGdinger form
d2Y dx2
-+ [€
- V(x)]y = 0
( 1 ) Evans, M. W.; Coffey, W. T.; Pryce, J. D. Chem. Phys. Letr. 1979,63, 133. ( 2 ) Coffey, W. T.; Evans, M. W.; Gnigolini, P. Molecular Diffusion and Spectra; Wiley: New York, 1984. ( 3 ) Pryce, J. D. NAG Newsletter (Oxford) 1986,3, 4. (4) Arscott, F.M. Periodic Differential Equatiom; Pergamon: New York,
1964.
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TABLE I: Computed Eigenvalues with Boundary Condition (3) index b = 10 b = 20 index b = 10 b = 20 8 134.9394 283.4087 0 0.0000 -0.0008 77.9162 1 37.8059 9 153.7268 339.3707 2 69.7953 15 1.4628 10 174.1046 380.0949 3 70.5475 15 1.4632 11 196.5380 385.6448 4 71.4052 151.4637 12 221.1507 394.1303 13 247.8531 426.5246 5 96.2058 220.1542 6 110.6895 283.0948 14 276.6121 452.6312 15 307.4155 477.7145 7 120.2678 283.2507
with V ( x ) given by eq 1. The boundary conditions responsible for the pattern of alternating triply and nondegenerate levels, as illustrated in Table I, are y=O
at x = f n / 2
(3)
The full generality of the problem is, however, better displayed by imposing periodic boundary conditions Y(X) = Y(2* + x)
(4)
with the additional requirement that Y(X) =
-A*
(5)
- x)
when the solution satisfies eq 3. One then sees that V(x)in Figure 1 is symmetric under the operations 0,: x - - x
and
02: K - x
-
-(7r
-
x)
(6)
giving four types of solution overall (symmetric or antisymmetric with respect to 0 , or 0,) but no symmetry-determined degeneracies. Accidental double degeneracies associated with levels in either the two deeper wells (type I) solution or the two shallower wells (type 11) solution can, however, occur if tunneling is forbidden. The existence of higher degeneracies must then depend on a chance coincidence between the positions of the eigenvalues in the two separate wells. Such coincidences lead to fourfold degeneracies under periodic boundary conditions (eq 4), but eq 5 imposes the additional requirement that y(x) is antisymmetric under 02.Now of the two 0 1988 American Chemical Society
Degeneracies for the Coffey-Evans Potential
The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 3123
+ ~
t
It follows that Ae < O.O1(&/8v) for a*> 4b. Note by comparison between eq 10 and 13 that a2 measures the energy below the barrier maximum. Furthermore, the quadratic approximation overestimates the tunneling probability because the curvature of V ( x ) decreases as t decreases. It is evident from eq 7 and 13 that close degeneracies may be expected, at least within the quadratic approximation, provided b2 >> 1 and a2 > 4b.
4. Semiclassical Quantization
I
'
',
T
\ 0
X
TT
Figure 1. The Coffey-Evans potential for 6 = 20. The positions of the eigenvalues in Table I are indicated by solid and dashed lines, respectively, according to whether y(-x) = y ( x ) or y(-x) = -y(x). Splittings of the tripled levels, which are not shown to scale, are indicated to the right of the diagram. Note that according to eq 1 the deeper and shallower minima occur at V = -26 and V = 26, respectively. Downloaded by NEW YORK UNIV on September 9, 2015 | http://pubs.acs.org Publication Date: June 1, 1988 | doi: 10.1021/j100322a023
The harmonic approximation for el and e2 are clearly inadequate to account for all the close degeneracies in Table I because the eigenvalues deviate perceptibly from linearity even for v2 = 1. One can, however, apply the Bohr quantization conditions
type I solutions, one is always symmetric and the other antisymmetric under 02, but the two type I1 solutions are both symmetric under O2if the quantum number v2 is even or both antisymmetric under O2 if v2 is odd. Thus, the observed alternating 1, 3, 1, ... near degeneracy pattern can be explained if (a) tunneling is forbidden and (b) the eigenvalues for the two types of isolated solution accidentally coincide. The features of the potential V(x) responsible for this behavior are analyzed below.
$, =
1
8,
[e - V(X)]'/~dx = (v,
+ Y2)?r
(14)
at
to investigate the extent to which
- 42 = a as implied by u1
+I
- v2 = 1 in the harmonic limit.
The two integrals are conveniently transformed by the substitution z = b cos 2x
+1
(15)
to the forms
4l = ILt - xt
(16)
where
3. Quadratic Approximations It is readily shown by applying quadratic approximations around the two types of minimum that the low-lying eigenvalues for type I and type I1 solutions are given by tl N
-2b
+ 4(vl + '/,)[b(b + l)]'/'
f(z) = [(b + 1 - z)(b - 1 + z)(z - a)(z
and €2 N
2b
+ 4(V2 + j/J[b(b
- l)]'/'
Ae = a-1~-1(&/8U)
(7)
(8)
where K is the semiclassical tunneling factor, expressible in the quadratic approximation as
p ] = a , q1 = b + 1 p 2 = 1 - 6, q2 = -a
+1
(20)
the quantity a being defined by eq 12. The xi integrals, which are tabulated,6 reduce to a common result for i = 1 and 2
x1 = x2 = 2a2[(b + a + l)(b + a - l)]-1/2K(k)
(21)
where K ( k ) is the complete elliptical integral of the first kind, with argument
k = [(b- a where the factor 2 arises because in normal Schrodinger notation eq 1 implies that h 2 / m = 2. It follows on substituting
= b2
(19)
and the limits are given by
Hence, el N t2 for uI = v2 + 1 provided b2 >> 1. The occurrence of chance degeneracies in the low-energy regime is therefore readily explained. The magnitude of possible tunneling splittings must also be examined. It may be shown by standard semiclassical methodsS that they reduce in the present problem to
V,,
+
+ l)(b - a - l ) / ( b + a + l)(b + a - 1)]1/2 (22)
The integrals, which are more difficult, may be reduced to standard forms' by means of the identity
(10)
and on writing, for later convenience e = b2 - a2
that the ratio of level splitting for b2 >> 1, by
A6
+1
(12)
to level spacing at/& is given,
At/(&/av) = a-I exp(-aa2/4b) (5) Child, M. S. J . Mol. Spectrosc. 1974, 53,280.
(13)
where co and c1 are coefficients in the expansion f ( z ) = coz4
+ c1z3+ c2z2+ c3z + c4
(24)
(6) Grobner, W.; Hofreiter, N. Integraltafel II; Springer-Verlag: West Berlin, 1965. (7) Grobner, W.; Hofreiter, N. Integraltafel I; Springer-Verlag: West Berlin, 1965.
J . Phys. Chem. 1988, 92, 3124-3144
3124
The final results, obtained after considerable manipulation, are = [(b
+ a + l)(b + a - 1)]-’/2{-2a(a + b + 1)K(k) +
[ ( b + a)2 - 1 ] E ( k ) + 4aII(pl,k)) (25a)
42 = [ ( b+ u
+ l)(b + u - 1)]”2{2(b + 1 - a2 - ab)K(k) + [ ( b + a ) ’ - 1]E(k) - 4bn(p2,k)) (25b)
where K ( k ) , E ( k ) , and II(pi,k) are complete elliptic integrals of the first, second, and third kinds.6 The arguments pi are given by = -(b - u l ) / ( b u 1)
+
p2
= (b- a - l)/(b
+ + + a + 1)
(26)
Note that Abramowitz and Stegun* use a different sign convention for their equivalent to p i . It follows immediately that
d l - c,b2 = [ ( b + a
+ l)(b + a - l)]-’/2{-2(a + b + 1)K(k) + 4an(pi3k)
4bnb2,k)) (27)
Downloaded by NEW YORK UNIV on September 9, 2015 | http://pubs.acs.org Publication Date: June 1, 1988 | doi: 10.1021/j100322a023
which may be drastically further reduced by defining a quantity pI’
= -(k2
+ pl)/(l + p i ) = ( b - u + l ) / ( b + u - I )
(28)
and noting that P I ’ P ~=
k2
(29)
Equations 28 and 29 then allow one to exploit the identities6
n(P23k) + n ( k 2 / p * , k )=
K ( k ) + ( ~ / 2 ) ~ 2 ” ~ / [+( 1p2)(k2 + P Equation 30 may be used to show that 4Un(pl,k) = 4bn(pl’,k) - 2(b
J I ” ~ (31)
- U - 1)K(k)
(32)
while eq 31 yields
m-J2,k) + W J l ’ , k ) = K ( k ) + (*/2b)[(b
+ a + l)(b + a - 1)]1’2
(33)
Thus overall, on combining eq 27, 32, and 33
41 - 42 =
(34)
which is an exact result.
5. Conclusion The remarkable feature of the Coffey-Evans potential, implied by eq 34, is therefore that the 0th energy level in each supposedly isolated shallow well coincides exactly with the (v + 1)th level in the deeper one, within the accuracy of the Bohr quantization condition. Hence, the splittings of the implied fourfold degeneracies (with periodic boundary conditions) depend only on tunneling, and eq 13 shows that such splittings will be at least 2 orders of magnitude less than the level spacings if a2 > 4b. A measure of the number of near-fourfold degeneracies for a given b value may therefore be obtained by using eq 12 to replace a2 by E and approximating e by eq 7. Thus, the level v2 will be unsplit by tunneling provided b > 4(u* 2) (35)
+
and
The computed levels in Table I in fact suggest high near degeneracies for v2 = 1 with b = 10 and v 2 = 3 with b = 20, the first of which suggests that eq 13 slightly overestimates the tunneling probability, as discussed in section 3.
(8) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover: New York, 1965.
Acknowledgment. We are grateful to Dr. P. Pajunen for drawing this problem to our attention. A.V.C. acknowledges financial support from SERC.
(1 +
+ Pl)n(/Jl,k) = Pl(1 - k*)nbl’,k)
+ k2(l + Pl)K(k) (30)
Phase Space Structure in Classically Chaotic Regions and the Nature of Quantum Eigenstates Michael J. Davis Theoretical Chemistry Group, Chemistry Division, Argonne National Laboratory, Argonne, Illinois 60439 (Received: July 29, 1987; In Final Form: November 9, 1987)
We review recent work concerning phase space structure relevant to energy relaxation and chemical reactions. Another example is shown, and that is for the local to normal mode transition in classically chaotic regions. The main purpose of the paper is to investigate the nature of the quantum eigenstates under such a condition, and we investigate what happens to the local and normal mode character of eigenstates when there is widespread chaos and there is extensive exchange of energy between two local modes and local and normal modes classically. We show that the quantum local-normal transition does not change significantly as classical chaos spreads. We relate this to the size of the flux across various phase space structures and present evidence that the quantum mechanics is essentially regular even when chaos is so widespread that, near the dissociation energy, chaotic regions can support approximately 10 quantum states. To further demonstrate this regularity, we quantize some of the “noninvariant” phase space structure.
I. Introduction One of the reasons chemists became interested in nonlinear dynamics’ about 15 years ago2 was the way simple systems showed a distinct transition between mostly regular motion at low energy (1) Lichtenberg, A. J.; Lieberman, M. A. Regular and Stochastic Motion; Springer: New York, 1983. (2) For reviews, see: (a) Rice, S. A. Adu. Chem. Phys. 1981,47, 117. (b) Brumer, P. Ibid. 1981, 47, 201. (c) Noid, D. W.; Koszykowski, M. L.; Marcus, R. A. Annu. Rev. Phys. Chem. 1981, 32, 267.
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and mostly chaotic motion at high energy. This suggested a means for understanding the transition between the spectroscopic molecular regime at low energy and the kinetic or statistical regime at high energy. However, the connection between chaotic motion and statistical behavior in a chemical sense is still not clear at this point and is still being pursued t ~ d a y . ~In. ~addition, the con(3) For a recent review, see: Truhlar, D. L.; Hase, W. L.; Hynes, J. T. J . Phys. Chem. 1983, 87, 2664. (4) Dumont, R.; Brumer, P. J . Phys. Chem. 1986, 90, 3509.
0 1988 American Chemical Society
