J . Phys. Chem. 1987, 91, 2258-2267
2258
and D 2 0 complex bands are then in part determined by this “inversion” motion. For the H D O complex, the out-of-plane shear vibration of the hydrogen (deuterium) bond is considerably higher for the hydrogen-bonded than for the deuterium-bonded form,12 and the complex is trapped in the D-bonded form. When the interaction energy increases with increasing methyl substitution, the “inversion barrier” will also increase. Since the intermolecular zero-point vibration energy of a D 2 0 complex is lower than that of an H 2 0 complex, the effective barrier will be higher for D 2 0 than for H20. When the inversion ceases to be rapid for D 2 0 , the shape of vl(DOD.C,H,) will become similar to that of vl(HOD.C,H,). This appears to happen with the xylenes. Finally, when the inversion ceases to be important for H 2 0 , the shapes of all three bands become similar. The observed temperature effect on the water-benzene bands (Figure 6) may perhaps be attributed to a motional averaging process, similar to what has been observed for v2(NH3)in solid nitrogen. At low temperatures the ammonia inversion makes v2(NH3)appear as a pair of sharp lines; when the temperature is increased, the two bands coalesce and at 15-20 K a single sharp band is observed midway between the two low-temperature bands.l3 In the present case, one would assume that a t 25-30 K the effects of the inversion on the HOH.C6H6 spectrum are averaged away, and therefore the band shape of vl(HOH-C6H6) is similar to that of Y1(HOD*C6H6) (Figure 6). The observations from the water-benzene experiments in nitrogen and krypton matrices (Figure 5) do not exclude the possibility that the structure of the bands is due to relative motions of the complex-forming molecules. Krypton.may be expected to allow relative motions similar to argon, and therefore the band shapes in the two matrices should be similar. Nitrogen, on the other hand, is expected to modify the relative motions drastically, giving entirely different band shapes. (12) Engdahl, A.; Nelander, B. J . Chem. Phys., in press. (13) Fredin, L.; Nelander, B. Chem. Phys. 1981, 60, 181.
The observation that the structure of v3(DOD.C,&) may differ from that of vl(DODC,H,) can be rationalized if one notes that the transition dipole moments of the two bands have different orientations. Therefore, the relative intensities of the relative motion components of the two bands may differ, giving each band its own shape It seems reasonable to assume that a given complex may be trapped in more than one trapping site. Since each trapping site will have its own band position and band shape, the observed band will be the superposition of bands from all trapping sites occurring in the matrix. One could in principle obtain any of the band shapes observed here as a superposition of simple symmetric peaks, one for each site. Such a model would easily explain the differences between different matrix materials (Figure 5). The temperature effects could perhaps be attributed to reversible changes in the site populations with temperature. The observation that the HDO band shape differs from that of H 2 0 or D 2 0 for the benzene complex may possibly be due to the different orientation of the transition dipole of HDO relative to the matrix, compared to those of H20or D20. The matrix shifts may therefore be different for H D O complexes than for those of H 2 0 and D 2 0 . It seems difficult, however, to explain how the shapes of the H 2 0 and D 2 0 complex bands can differ assuming that the band shapes are solely determined by site effects. It seems unlikelythat the differences in vibrational amplitudes of H20and D 2 0 make the trapping sites of H 2 0 and D 2 0 complexes different (see especially Figure 2). We may therefore conclude that relative motions of the complex-forming molecules have a decisive influence on the shape of the intramolecular absorption bands of the water-methylbenzene complexes, but we cannot exclude important contributions from complexes trapped in different sites. The results of this paper support the conclusion of ref 2, that water bound to a benzene ring has an unusual freedom to move relative to the ring.
Acknowledgment. This work was supported by the Swedish Natural Science Research Council and by Knut and Alice Wallenberg’s Foundation.
Use of Classical Fourier Amplitudes as Quantum Matrlx Elements: A Comparison of Morse Oscillator Fourler Coefflcients with Quantum Matrlx Elements Randall B. Shirts Department of Chemistry, University of Utah, Salt Luke City, Utah 84112 (Received: October 13, 1986)
It is well-known that classical Fourier amplitudes can approximate quantum matrix elements. A limited number of comparisons have previously been done numerically and, in some cases, analytically. In this paper, the analytic comparison for Morse oscillatorsis done for several different operators of interest. Several classical expectationvalues are found to be exact. Exceptions are the coordinate variable and a dipole moment parametrization. Classical off-diagonal matrix elements are not exact but are remarkably accurate except for the dipole moment parametrization. A large discrepancy for the expectation value of the coordinate for marginally bound states is understood in terms of quantum tunneling into the classically forbidden region. Other discrepancies between classical Fourier amplitudes and quantum matrix elements are also understood in terms of tunneling. In this case, perhaps surprisingly, the semiclassical picture is most inaccurate at the highesr quantum numbers.
Introduction
The classical mechanics of anharmonic oscillators has been studied intensively in recent The analytic dynamics (1) Noid, D. W.; Kmzykowski, M. L.; Marcus, R. A. Annu. Rev. Phys. Chem. 1981, 32, 267. Rice, S. Adu. Chem. Phys. 1981, 47, part 1, 117.
(2) Berry, M. J. Topics in Nonlinear Dynamics, AIP Conference Proceedings No. 46, Jorna, s., Ed.; American Institute of Physics: New York, 1978; p 16. (3) Hedges, Jr., R. M.; Reinhardt, W. P. J. Chem. Phys. 1983, 78, 3964. (4) Sumpter, B. G.; Thompson, D. L. J . Chem. Phys. 1985, 82, 4557. (5) Hutchinson, J. S.; Sibert, E. L. III; Hynes, J. T. J . Chem. Phys. 1984, 81, 1314. (6) McDonald, J. D.; Marcus, R. A. J . Chem. Phys. 1976, 65, 2180.
0022-3654/87/2091-2258$01.50/0
of such systems are important in studies of mode-mode energy transfer3-I6 (Le. intramolecular energy redistribution) as well as (7) Sibert, E. L. 111; Reinhardt, W. P. Chem. Phys. Lett. 1982, 92, 455. (8) Hutchinson, J. S.;Hynes, J. T.; Reinhardt, W. P. Chem. Phys. Lett. 1984, 108, 353. (9) Voth, G. A.; Marcus, R. A. J . Chem. Phys. 1985, 82, 4064. (10) Kellman, M. E. Chem. Phys. Lett. 1984, 208, 174. (1 1) Sibert, E. L. III; Reinhardt, W. P.; Hynes, J. T. J . Chem. Phys. 1982, 77, 3595. (12) Jaff€, C.; Brumer, P. J . Chem. Phys. 1980, 73, 5646. (13) Sibert, E. L. III; Hynea, J. T.; Reinhardt, W. P. J . Chem. Phys. 1982, 77, 3583. (14) Clodius, W. B.; Shirts, R. B. J . Chem. Phys. 1984, 81, 6224.
0 1987 American Chemical Society
Classical Fourier Amplitudes as Quantum Matrix Elements
The Journal of Physical Chemistry, Vol. 91, No. 9, 1987 2259
and the kinetic energy;38however, some of these calculations have in the interaction of molecules with infrared radiation (e.g. infrared been limited to particular Fourier amplitudes, and none of the absorption The Morse oscillator is an important reports have exhibited the method by which they were obtained. model system in such studies because it is one of the few anThis suggests that a general method for obtaining Morse oscillator harmonic systems that has an analytic solution. Fourier components is desirable. The quantum dynamics of Morse oscillators is of similar interest The matrix elements for a number of operators between Morse because, as is the case for other classical systems with analytic oscillator quantum states have also been obtained analyticalsolutions, there are also analytic solutions for the quantum wave 1y.27,31-33,36.38,39,5~56 Although N a c ~ a c h and e ~ ~Koszykowski et functions of the Morse oscillator.2633 Quantum applications have al.39 showed how excellent agreement is obtained between quantum included both intramolecular energy redistributi~n~~-~~*~+~~ and matrix elements and classical Fourier amplitudes, a comprehensive infrared a b s o r p t i ~ n . ' ~ - ~ ~ ~ ~ ~ and systematic comparison has not been made. The idea that quantum matrix elements are given by or at least Aside from the ability to use classical quantities without having approximated by classical Fourier amplitudes is an old one going to evaluate numerical matrix elements, Fourier coefficients are back at least to Heisenberg in 1930.37 In fact, it was the mulalso useful in classical, semiclassical, and quantum resonance tiplication properties of classical Fourier amplitudes which, when analysis of coupled oscillator systems.9,12,'3,22-25,57,58 combined with the Bohr frequency condition, led Heisenberg to This paper has three purposes: (1) to collect together a number matrix mechanics in the first place.37 This correspondence is of useful classical Fourier amplitudes for the Morse oscillator (a sometimes called Heisenberg's form of the correspondence pringeneral method for evaluating these and similar amplitudes is ciple. Recently, this idea has been emphasized by N a ~ c a c h e , ' ~ discussed in an Appendix), ( 2 ) to derive some Fourier amplitudes not previously obtained, and (3) to compare the classical Fourier Marcus and c o - w ~ r k e r s and ,~~~ in hopes of using amplitudes with quantum matrix elements to clearly identify the classical quantities instead of hard-to-calculate quantum meaccuracy of approximation and the reasons for discrepancy. chanical quantities. Another problem found by the uninitiated reader is that previous The periodic classical motion of a bound Morse oscillator may be expressed analytically in action-angle v a r i a b l e ~ . . ' ~ J ~ J ~ J ~workers , ~ ~ - ~ have ~ used varying notations. We here attempt to pull together different notations as a convenience. A number or researchers have reported Fourier amplitudes for the coordinate variable,'z~38~39*48~49 the momentum ~ a r i a b l e , " J ~ + ~ ~Review of Morse Oscillator Formulas The Hamiltonian for a Morse oscillator is given by the following formula:26 (15) Lawton, P. T.; Child, M. S. Mol. Phys. 1981, 44, 709. (16) Uzer, T. Chem. Phys. Lett. 1984, 110, 356. (17) Christoffel, K. M.; Bowman, J. M. J . Phys. Chem. 1981, 85, 2159. (18) Davis, M. J.; Wyatt, R. E. Chem. Phys. Lett. 1982, 86, 235. (19) Stine, J. R.; Noid, D. W. J . Phys. Chem. 1982, 86, 3733. (20) Dardi, P. S.; Gray, S.K. J. Chem. Phys. 1982, 77, 1345. (21) Dardi, P. S.; Gray, S.K. J . Chem. Phys. 1984, 80, 4738. (22) Gray, S. K. Chem. Phys. 1983, 75, 67. (23) Gray, S.K. Chem. Phys. 1984,83, 125. (24) Shirts, R. B.; Davis, T. F. J . Phys. Chem. 1984, 88, 4665. (25) Voth, G. A,; Marcus, R. A. J . Phys. Chem. 1985, 89, 2208. (26) Morse, P. M. Phys. Reu. 1929, 34, 57. (27) Watson, I. A.; Henry, B. R.; Ross, I. G. Spectrochimia Acta, Part A 1981, 37, 857. (28) Lawton, R. T.; Child, M. S. Mol. Phys. 1980, 40, 773. (29) Greenawalt, E. M.; Dickinson, A. S.J. Mol. Spectrosc. 1%9,30,427. (30) Parkinson, J. R.; Birtwistle, D. T. Comput. Phys. Commun. 1972, 4, 257. (31) Sage, M. L. Chem. Phys. 1976, 35, 375. (32) Efremov, Yu. S. Opt. Spectrosc. 1978, 43, 695. (33) Wallace, R. Chem. Phys. Lett. 1976, 37, 115. (34) Sage, M. L.; Williams, J. A. I11 J . Chem. Phys. 1983, 78, 1348. (35) Buch, V.; Gerber, R. B.; Ratner, M. A. J. Chem. Phys. 1982, 76, 5397. (36) Halonen, L.; Child, M. S.Mol. Phys. 1982, 46, 239. (37) Heisenberg, W. The Physical Principles of Quantum Theory; Dover: New York, 1930; pp 105-123. For an excellent account, see also Tomonaga, S.I. Quantum Mechanics, Vol. 1; North-Holland;Amsterdam, 1962; Chapter 5. (38) Naccache, P. F. J . Phys. B 1972, 5, 1308. (39) Koszykowski, M. L.; Noid, D. W.; Marcus, R. A. J. Phys. Chem. 1982, 86, 2113. (40) Wardlaw, D. M.; Noid, D. W.; Marcus, R. A. J. Phys. Chem. 1984, 88, 536. (41) Stine, J. R.; Noid, D. W. J . Phys. Chem. 1983, 78, 1876. (42) Heller, E. J. Physica 1983, 07, 356. (43) Klein, A. J. Math. Phys. 1978, 29, 292. (44) Clark, A. P.; Dickinson, A. S.;Richards, D. Adu. Chem. Phys. 1977, 36, 63.
(45) Percival, I. C.; Richards, D. Adu. A t . Mol. Phys. 1975, 11, 2. (46) Sibert, E. L., 111 J . Chem. Phys. 1985, 83, 5092. (47) Rankin, C. C.; Miller, W. H. J. Chem. Phys. 1971, 55, 3. (48) Sazonov, I. E.; Zhirnov, N . I. Opt. Spectrosc. 1973, 34, 254. (49) Tipping, P. H. J . Mol. Spectrosc. 1974, 53, 402.
-2
P
H=-
2m
+ D [ l - exp(-aq)12
where q and p are the coordinate and conjugate momentum, D is the dissociation energy of the oscillator, m is the mass, and a is the length scale parameter. Any one-dimensional Hamiltonian is integrable,2 but the Morse oscillator Hamiltonian has a particularly simple form12-25*47 in action angle variables:59
H = h0(1 -
2)
where I is the oscillator action, and the harmonic frequency is ~ . anharmonicity of the Morse given by wo = c ~ ( 2 D / m ) ' / The oscillator is demonstrated by noting that its frequency changes with increasing action:zJz
where I,, = 2D/w0 and X is defined by eq 3 or 5 . The frequency decreases linearly from wo (when I = 0 and E = 0) to zero (when I = I,.,,,, = 2D/w0 and E = D ) . It is also useful to note that a semiclassical quantization of the Morse oscillator made by rein eq 2 yields exact quantum eigenvalues!' placing I by h(n 'Iz) The momentum and coordinate variables may be expressed in terms of the action and angle variables as follows:12*47-60
+
q =
a-l
In [(l
+ b cos 6)/X2];
p =
mwoXb sin 6 + b cos 6 ) (4)
a(1
(50) Gallas, J. A. C. Phys. Rev. A 1980, 21, 1829. (51) Gallas, J. A. C. Lett. Nuouo Cimenro 1980, 28, 29. (52) Tipping, R. H.; Ogilvie, J. F. J . Chem. Phys. 1983, 79, 2537. (53) Jedrzejek, C.; Freed, K. F. Surf.Sci. 1981, 109, 191. (54) Kellman, M. F. J . Chem. Phys. 1985,82, 3300. (55) Requena, A.; Pena, R. An. Quim. 1983, 9, 508. (56) Carney, G. D.; Porter, R. N. J . Chem. Phys. 1976, 65, 3547. (57) Sibert, E. L.III; Hynes, J. T.; Reinhardt, W. P. J. Chem. Phys. 1982, 77, 3595. (58) Shirts, R. B. J . Chem. Phys. 1986,85,4949. Int. J. Quantum Chem., in press. (59) Goldstein, H. Classical Mechanics, 2nd ed; Addison Wesley: Reading, MA, 1950.
2260 The Journal of Physical Chemistry, Vol. 91, No. 9, 1987
MORSE OSClLlATOR ACTION-ANGLE COORDINATES 1.o
Shirts We can express any function of the coordinate and/or momentum as a Fourier series in the angle variable e, which has period 27r (Fourier's theorem). The Fourier amplitudes and Fourier series are expressed in real form as follows:63 A0 f ( q , p ) = - C [ A , cos (ne) + B, sin (ne)] (9) 2 n-l
+
0.5
where A, =
E
0 Pi U
a
-
L12kq,p)(ne) de, COS
7 r o
n = 0, 1 , 2 , 3,
B, = L 1 2 k q , p )sin (ne) de, n = 1 , 2, 3,
0.0
T
2
O
...
...
(10)
and where q and p are expressed in terms of 0 and I by using eq 4. Note that A , and B,, are functions of I (or equivalently of w , E , b, A, or a ) for Morse oscillators, as well as of n.
-0.5
Evaluation of Useful Fourier Amplitudes Fourier Amplitudes of the Momentum. The evaluation of the Fourier amplitudes of the momentum is performed by evaluating the following integral:64 -1.0
-1
I
0
3
2
ALPHA*q
4
5
Pn=--
1. Level curves of Morse oscillator action-angle coordinates. Solid curves are curves of constant action at values corresponding to E = 0.2, 0.4,0.6,0.8,and 0.98 D. Dashed curves are curves of constant angle with values of *30°, 160°,*90°, &120°, f150°,and f175O measured from the positive q axis.
where X and b are convenient parameters which are related to the energy of the oscillator, the action variable, or the frequency:
X = 1 - - =I 1 - 0 = Iw Imax 20
(1
- E/D)'/' = w/wO = ( 1 - b2)'/'
mwoXb a7r
x
2r
sin @ s i n(ne) de 2mooX = -(-a),, n= 1 b cos e a 1 , 2 , 3, ... (11)
+
These coefficients represent the B, coefficients in eq 9; the A , coefficients vanish by symmetry. Fourier Amplitudes of the Kinetic Energy. The kinetic energy of a Morse oscillator is T = p2/2m. The Fourier coefficients of this quantity can be evaluated in the following integral? , /.2* Y-2 - COS (ne) de = T , = : J0 2m
(5)
L ( ? ) Z X
2n
sin2 0 cos (ne) dB
2?rm
(1
= 2D(-1)"+'Xan(nX - l), n = 1 , 2, 3, = 2DX(1 - A), Parameter b is called P in ref 48 and 49. Note that X decreases from 1 to 0, and b increases from 0 to 1 as energy increases from 0 to D. Another useful parameter, which will be used later, we denote a, following JaffE and Brumer:I2
+ b cos ...
n =0
(12)
Fourier Amplitudes of the Coordinate. The Fourier coefficients of the coordinate variable may be determined by evaluation of the following integral: 1 21 In [(l b cos e)/X2] COS (ne) de = q, = ? r ~ y
+
7ra
J2'ln
(1
2 In X2 + b cos 0) cos (ne) dB - 06,;, CY
0, 1, 2, 3,
This parameter increases from 0 to 1 as energy increases. Note also that X = (1 - a2)/(1 a2),b = 2a/( 1 c?), and E = D[2a/(1 + a2)I2.The inverse of transformation (4) may be displayed as follows:
+
I = Zmax[l- ( 1
+
n =
... (13)
where 6, is the Kronecker delta, and we have used the properties of the logarithm to obtain the last equality. These represent the A, coefficients in eq 9. The B,, coefficients vanish by symmetry. The value of the integral (1 3) is given by? 2 qn=-lnA, n = O ff
- E / D ) 1 / 2 ]'6; = p(i
tan-'
-E/D)I/~
(2Dm)'lZ(1- Z - E / D )
(8)
where E = H coqJuted from (l), 2 = exp(-aq), and ?r must be added to or subtracted from 8 in (8) if 2 > X2 [aq < -In (1 E / D ) ].61 Figure 1 illustrates some of the contours of I and 8. (60) Notation differs from author to author. We have also taken a convention that has the angle 0 increasing in time in the clockwise direction and measured from the positive coordinate axis. Reference 12, for example, chooses the angle measured from the negative coordinate axis.
(61) The quadrant for 0 depends on the signs o f p and 2 - X2. In addition, any arbitrary function of the action can be added to 0 (see ref 62). We have here chosen 0 to be measured from the positive z axis for all values of I . (62) Skodje, R. T.; Borondo, F. Chem. Phys. Lett. 1985, 118, 409. (63) Several conventions are possible for the definition of Fourier series, e.g. Williamson, D. E.; Crowell, R. H.; Trotter, H. F. Calculus of Vector Funcrionc, 2nd d, Rentice-Hall: Englewood C l i s , NJ, 1968. Note also that the multiplication properties of amplitudes (eq 17) is much simpler in complex form. (64) Grahhteyn, T. S.; Ryzhik, I. W. Tables of Integrals, Series and Products; Academic: New York, 1965; formula 3.613.3, p 366. (65) Reference 64, formula 3.616.7, p 369.
Classical Fourier Amplitudes as Quantum Matrix Elements
+
where A = ( 1 X)/2X2. Fourier Amplitudes of Powers of the Coordinate. The Fourier amplitudes of qm, m = 2, 3 , 4, ... may be evaluated as (qm),, = A,, = =
1 -
The Journal of Physical Chemistry, Vol. 91, No. 9, 1987 2261 a4(q4),= 2 in4 A
- XZr{ln [ ( l + b COS e ) / X 2 ] ) m cos (ne) dB *am m
(r) (-ln
am j = o
X2yJ,(m - j ) , n = 0, 1 , 2, ... ( 1 5 )
-In3 A
n
-
a2/
+ 24C([ln A - 2Fl(r>I2- 2F2(I)) 1-1 12
+ 3F(n) ln2 A - 3 In A[(Fl(n))'- F2(n)]+
where
J,,o') =
1 JZr[ln T
O
( I +b cos e)]' cos (ne) de
(16) F2(n) + 3(Fl(r))2- 3F2(r>+ 6FI(I)F;(n+ I )
+
A Laurent series expression for In ( 1 b cos 0) is developed in Appendix A which allows one to immediately write down a Laurent series for powers of In ( 1 b cos e). One can then use contour integral methods to evaluate the needed integrals. This method has been implemented and used for j (and thus m) equal to 2 and 3. However, another method has been found to be easier to implement. Once we have the Fourier coefficients for m = 1 from eq 14, one can just multiply two Fourier series together to get Fourier series for higher powers of q. If two even periodic functions are given by ?(e) and s(e) where
+
+
then using some trigonometric identities and elementary sum manipulations, one obtains the following for the Fourier coefficients of the product r(e) s(0) = t(0):
n = 1 , 2, 3,
...
(22) Higher powers of the coordinate have more complicated Fourier coefficients and have not been attempted. Fourier Amplitudes of the Potential Energy. It is useful to have Fourier amplitudes of the quantity 2 = exp(-aq), from which Fourier amplitudes of the potential can be obtained. These are obtained from the following integral:67 2, = R
2*X2 cos (ne) dB = 2X(-a)", n = 0 , 1 , 2, (1 + b COS e)
... (23)
The Fourier components for Z? are also needed for the potential energy. These can be evaluated directly6*or by using eq 17 with r = s = Z . The result is
( Z 2 ) , = 2X(-a)"(l
+ nX),
n = 0, 1, 2, 3 ,
...
(24) The final result of this section is to combine eq 23 and 24 to give the Fourier coefficients for the potential energy:
...
n = 1 , 2, 3 , Using r = q and s = q, we obtain:
(18)
+ 4E-
V,, = D[26,,0 - 2 2 ,
= 2DX(-a)"(nX - l ) , n = 1 , 2, 3 ,
,2/
a2(q2)o = 2 ln2 A
4(-a)" "' 7-In A + Fl(n) + nE/=11(1 + n )
= -(an[ln 4(-1)"
X
n
]
(19b)
where
F,(n) = 0, n = 0 n-1
CP, n = 1 , 2, 3, ...
/=I
(20)
Note that the functions F,(n) can be related to poly-y functions, e.g. Fl(n) = +(n) + y where y = 0.57721566 ... . Using r = q and s = q2 in eq 18, we obtain: ~ r ' ( q ~=)2~ln3 A
a3(q3)n =
-
n =0 (25) It is now an easy exercise to show that the Fourier components of T V = E vanish for n # 0 and give 2D(1 - X2) for n = 0, which is the correct expression for the energy. Fourier Components of a Convenient Dipole Moment Operator. The function p = C(q Ro)e-aqmay profitably be used as the dipole moment function of a diatomic molecule where C and Ro are parameters. Although this form has not been used previously, it is convenient because it is somewhat linear in the region of the equilibrium position but then asymptotically approaches zero as it should for dissociation into neutral fragments. If Ro = Re, 1 also vanishes for R = 0 (q = -Re), or, alternately, Rocan be used as an adjustable parameter to fit experimental data. In studies of how a molecule interacts with a radiation field using dipole interaction terms, it is useful to have the Fourier coefficients of this function. We can obtain these quantities using eq 14, 18, and 23:
+
+
+ Fl(n)] + a-"
=
...
= 2D(1 - A),
/-I 12
a2(q2)n=
+ (Z2),]
.2/
+ 12E-(ln /=I 12
A - 2Fl(I))
-6(-a)"
In2 A - 2 In A Fl(n) + (Fl(n)I27
-
1, 2, 3,
.2/
F2(n) + 2nC -[Fl(l) + Fl(n + 1) - In A ] / = I l(1 + n ) n = 1 , 2, 3 ,
...
Using r = q2 = s, we obtain: (66) Reference 64, formula 4.398.2, p 594.
(21)
... ( 2 6 )
where the sum in the last formula may be replaced by aW2"[-ln ( 1 - a2)- z E : ( a 2 ' / I ) ]for values of a which are not too small. Discussion of the Fourier Amplitude Formulas. The formulas of this section have been checked numerically and converge to machine precision. The rate of convergence is controlled by the (67) Reference 64, formula 3.613.1, p 366. (68) Reference 64, formula 3.616.2, p 368.
2262 The Journal of Physical Chemistry, Vol. 91, No. 9, 1987
size of the parameter a. For small values of a (low energy), only a few Fourier coefficients are needed to get a good representation of the function. For the same reason, relatively few terms are needed in the infinite sum. Conversely, the infinite sum and the Fourier series itself converge more slowly as the parameter a approaches unity. The two forms of eq 19 (b and c) or eq 26 are useful in different regions. The first form is more efficient for small values of a. The second form is more efficient for large values of a. The switchover point can be chosen when the second form stops losing precision due to cancellation between the sum and the logarithm. Comparison of Classical Fourier Amplitudes with Quantum Matrix Elements We now compare the classical Fourier amplitudes evaluated in the last section with quantum matrix elements. If they agree, it may be easier in some applications to use the classical expressions. If they are different, it is instructive to discuss these differences. We will first compare expectation values of the operators we have examined and then proceed to examine the off-diagonal matrix elements and their classical analogues. Expectation Values. When one makes the semiclassical and notes that 0 0 / 2 represents the identification Z = h(n time average of any quantity 0, one can make direct comparisons of classical time averages with analytic formulas for quantum expectation values derived by Sage3' and Carney and Parker.56 The following operators have exact agreement between the quantum expectation values and the classical time averages for quantizing actions:
-
logarithmically as X 0 (classical prediction), the quantum expectation value (ulqlu) goes up as l/Xk. This nonclassical tunneling effect cannot be recovered by any classical treatment. These points are explored in Appendix B. The expectation values of p are given by classical
Po
h.(n + Y 2 h o (ulZlu) = - = h(n) = 1 (ref 31, eq 8) 2 20
ZO
+
(ula~lu= ) C(k - 2u - l)[Roa In
k - $(k - u ) ] / k
(28) These two formulas are again not identical but the classical expression is a very good approximation to the quantum version. The differences are of order k2.An analytic comparison may be made for both q and p expressions by expanding the $ function for large k.69 Off-Diagonal Matrix Elements. Koszykowski, Noid, and Marcus39 (KNM) argue that semiclassical approximation to off-diagonal matrix elements of quantum operators can be obtained from the classical Fourier components using the following rule: (u+jlOlu) = O,(Z); Z = h(u
+ j / 2 + Y2)
(29)
where 0, is the j t h complex Fourier component of the classical variable corresponding to the quantum operator 0. In terms of the notation for real Fourier components, 0, = (A, - iB,)/2. Using the methods of Sage,31we now compare the matrix elements of Z, Z2,p , q, and p, where X and a are functions of Z evaluated according to eq 29: classical
y2z,= (-1)JXd
quantum
( u + j l Z ( u ) = (-1)JK
classical
yg, = -i(-ly(2Dm)1/2Xd
quantum (ref 3 1 , eq 9 )
I)$(
2 = a[Roa + ln 2
quantum
+
(ulplu) = - = 0 (trivially) 2
Shirts
classical
(u
+ j l p l u ) = -i(-l)J(2Dn1)'/~K Y2(Zz), = (-1)JXd(l + jX)
(30)
(31)
quantum (u
(ref 3 1, eq 3)
+ jlZ*lu) = (-l)J[(j + I)(k - u - j ) - u ( j - 1)]K/k (32)
classical
y2aqj
= (-l)J+'d/J
quantum ( u + jlaqlu) = (-1)J+IkK/u(k - 2u - j - I)]
(33)
classical
quantum The expectation values for q have the following forms:
a(u + j l p l u ) = (-1)"'CK
$(k - j - u )
+ F I G + 1) - In k -
classical quantum (ulaqlu) = In k
+ $(k - u ) - $(k - 20) - $(k - 1 - 2u) =
In k - $ ( k - 1 - 20)
+ +C"-l k - Iu - I
(ref 56, eq 32) (27)
where k = 4 D / ( h u o ) = 2ZmaX/h,and $ ( x ) is the logarithmic derivative of the I' function, F(x). Note that eq 9 of ref 31 and eq 32 of ref 56 can be shown equivalent, but the latter is simpler. In contrast to the previously discussed expectation values, the classical and quantum expectation values for q are not exactly equivalent. They agree very well, however, for all except the highest value of u (see Figure 2c). The difference is of order k-2 for small u. When the classical turning point becomes very large as is especially true for k slightly larger than odd integer values and u the maximum allowed quantum number, the particle is only just barely bound. In this case, it can have a large probability of being in the classically forbidden region. Instead of going up
+
where K = ( [ I ' ( k - u - j ) ( u j ) ! ( k - 2u - 2 j - l ) ( k - 2u- l)]/[I'(k - u ) ~ ! ] ) ' / ~KNM39 /k. compared these expressions for the operator q and noted that the quantum expression can be made to agree exactly with the classical expression when the product of j terms given by the ratio of r functions is replaced by j factors of the arithmetic mean of the terms and likewise with the product of j terms from the ratio of factorials. This same substitution of arithmetic means for geometric means also produces exact agreement between classical and quantum expressions for the operators 2,Z2,and p . However for the operator p, no such replacement has been found to give exact agreement even in simple special cases. (69) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover: New York, 1965; p 259.
Classical Fourier Amplitudes as Quantum Matrix Elements
The Journal of Physical Chemistry, Vol. 91, No. 9, 1987 2263
n-
I
0.0
0.2
0.8
0.6
0.4
NORMALIZED ACTION
1 .o
1.8 1.6 n
al >r
$ 1.4 0
W
3
I
1.2
LL
O W
3 6
1.0
0.8
z
E4
0.6
I-
o
xw
0.4 0.2 0.0 0.0
1
1 0.2
I
I
I
I
I
I 0.8
1.o NORMALIZED ACTION Figure 2. Expectation values of Morse oscillator operators plotted vs. I/I,-. Solid lines are classical time averages. Quantum diagonal matrix elements are positioned along the abscissa according to eq 29. Solid dots ( 0 ) are for k = 46 and crosses (+) are for k = 10 showing that quantum-classical agreement is not strongly dependent on the number of states. Plot a shows expectation values of Z = exp(-aq). Plot b shows values for Z2. Plot c shows values for aq. Plot d shows values for /.t with parameters chosen appropriate for HF molecule.
In order to allow visual appreciation of the accuracy of classical expressions as approximations to quantum matrix elements, we note that the classical expressions are all functions of the variable Illmax.This “normalized action” is used as an abscissa in Figures 2 and 3 in which the quantum matrix elements are plotted as points over the classical curves. A value of I,, = 23h (k = 46) was chosen as an example. Figure 2 exhibits expectation values, and Figure 3 exhibits coupling elements. In each figure, Z matrix elements are plotted in window (a) (matrix elements of p are proportional to those of Z);$ matrix elements are plotted in window (b); q matrix elements are plotted in window (c), and p matrix elements are plotted in window (d). Parameters for p were chosen to match experimental data for HF molecule (C = 2.915 D/bohr, R,, = 0.610 bohr, a = 1.174 bohr-’ so that ~ ( 0= ) 1.778 D and p’(0) = 0.827 D/bohr). Table I summarizes the accuracy
0.4
0.6
of the classical Fourier amplitudes as approximations to quantum matrix elements. In general, classical and quantum expectation values 0’ = 0) are in much better agreement than the agreement between classical Fourier amplitudes and quantum off-diagonal matrix elements 0’ > 0). However, the agreement i s surprisingly good for offdiagonal elements for lower states. The accuracy diminishes for larger j . The agreement can be improved, however, by choosing a slightly different classical action than given by eq 29. N a c c a ~ h e ~ ~ has suggested the following recipe: D
Z=h(&)
+j)!
l/J
(35)
This recipe agrees with eq 29 for j = 1, but for j I2 it gives small
2264 The Journal of Physical Chemistry, Vol. 91, No. 9,1987
Shirts
TABLE I: Average and Maximum Relative Error of Classical Fourier Amplitudes as Approximations to Quantum Matrix Elements for the Morse Oscillator (k = 46) j = 0‘
operator
j = 1
maxC
avb
j = 3
j = 2
j = 5
j = 4
av
max
av
max
av
max
av
max
av
max
Z
O.Od
0.0
0.011
0.155
0.033 0.029e
0.342 0.341
0.064 0.049
0.512 0.521
0.103 0.075
0.668 0.680
0.152 0.094
0.814 0.824
2 2
0.0
0.0
0.011
0.155
0.033 0.029
0.342 0.342
0.064 0.051
0.512 0.524
0.103 0.076
0.668 0.686
0.152 0.102
0.814 0.836
T
0.0
0.0
0.011
0.155
0.033 0.032
0.342 0.346
0.064 0.056
0.512 0.517
0.103 0.080
0.668 0.664
0.152 0.109
0.814 0.767
4
0.002
0.02
0.011
0.155
0.033 0.027
0.342 0.341
0.064 0.046
0.512 0.509
0.103 0.065
0.668 0.659
0.152 0.086
0.814 0.794
IJ
4
5 x 0-5
0.890
x 10-5
11.1
1.27 1.27
11.94 11.92
1.53 1.53
13.6 13.6
1.87 1.85
17.9 17.9
9.83 9.80
15.9
15.9
‘Denotes expectation value. *av denotes average relative error for 23 - j matrix elements. emax denotes maximum relative error among 23 - j matrix elements. dFirst line for each operator uses actions described by eq 29. eSecond line for each operator uses actions described in eq 35. corrections to eq 29. It should be noted that one should still use I = h(v + I/*) for diagonal elements (j = 0) although the limiting value of eq 35 is not far off ( h [ v+ exp($(l))] = h(u + 0.5615...)). Wardlaw et a1.@have noted that evaluation of Fourier amplitudes at actions given by eq 35 gives better agreement with quantum matrix elements; they also noted, however, that eq 29 gives exact frequencies for the Morse oscillator whereas eq 35 does not. Note also that eq 35 preserves the Hermiticity of matrix elements that originally led K N M to eq 29.39 It is curious that eq 35 gives good agreement; indeed, Naccache gives no derivation to explain why this is so. We will now show why eq 35 is correct for small values of u. The problem is to find classical actions such that the classical and quantum versions of eq 29-34 agree in the limit of u and j small. We will substitute m which means that the classical and the equivalent limit k quantum curves should agree on the left-hand side of Figure 3a-d. We will demonstrate that eq 35 gives this agreement for the coordinate operator, eq 33, but similar calculations give the identical solution for all six operators. We desire to find the classical action, I , such that d = k K / ( k - 2v - j - l), where u is given by eq 7, and K is given following eq 34. Squaring the desired equality gives
-
I’(k - v - j)(v
-
+ j ) ! ( k- 2u - 2j - l ) ( k - 2v - 1) -
I’(k - v)v!(k- 2~ - j - 1)2 ( v j ) ! ( k - 2v - 2j - l)(k - 2v - 1)
+
( k - v - l ) ( k - v - 2 ) ...(k - v - j ) u ! ( k - 2~ - j - 1)2 now letting k >> (v + j , u j ) , and consequently,,Z that k = 21max/hwe obtain in the limit: j
(U
(36)
>> I , and noting
+j)!
kJv!
bidden region than corresponding lower states. In other words, a lower state has an exponentially smaller tunneling probability than a higher state, and, to this extent, the lower state cannot overlap with the higher state to give a nonzero matrix element. This fact is most clear in the limit. For states at the extreme right of the figures which are marginally bound, the tunneling probability approaches unity (see Appendix A). Such a state obviously cannot overlap with a strongly bound state even when multiplied by an operator like e. For this reason, the quantum off-diagonal matrix elements limit to zero on the right-hand side of Figure 3a-d even when the classical limit is nonzero. It is pertinent to discuss the comparatively poor agreement between classical Fourier amplitudes and quantum matrix elements for the dipole moment parameterization. This form was partially picked as a worst-case comparison. This failure is entirely due to the q exp(-aq) part of h. This operator exhibits a cancellation between the increasing function q and the decreasing function exp(-aq). Because of this difference, the signs of the off-diagonals of the matrix for q alternate in an opposite order than those of the matrix for exp(-aq) causing loss of accuracy from only small errors in the component matrices. This should be the case with any nonmonotonic product of monotonic functional operators. It should also be noted that off-diagonal matrix elements of q exp(-aq) bear a surprising relationship with those of q. It might seem reasonable that these matrix elements should be asympm (harmonic limit), but this is only the totically equal as k case w h e n j = 1. The ratio between the matrix elements can be shown to be j F I G + 1) which increases with increasing j and is already three for j = 2. This fact is surprising when one regards m as holding the potential constant while letting the limit k the mass increase. In this scenario, the wave functions become more and more localized near the bottom of the well in the region where the operators q and q exp(-aq) agree more and more closely. However, the matrix elements for j > 1 result from cancellations of positive and negative regions of the integrand, and the two operators above agree best in the region where the integrand is small. The largest disagreement between the operators is in the region near the classical turning points where the integrand is large with signs appropriate such that a slight change in the integrand gives a large change in the total value. This effect is not, however, a quantum effect since the classical Fourier amplitudes show the correct behavior. The reason for this discussion is to point out the necessity of using correct functional forms for operators, especially for multiple quantum transitions. The parameters for fi in Figures 2d and 3d were chosen to give the experimental values of and db/dq at the equilibrium position for HF. However, the values of the matrix elements are considerably in error, especially for Aj > 1.41 Indeed, the expectation values of p (see Figure 2d) decrease monotonically instead of reaching a maximum. This warning can also be used to great advantage by noting the ability that multiple quantum jump matrix elements have to distinguish sensitively between similar operators. The agreement between classical and quantum quantities also
(37)
from which eq 35 can be obtained. When this substitution is made, the agreement between classical Fourier amplitudes and quantum matrix elements is much improved on the left of Figure 3a-d but not much changed on the right of the figures. Since the major errors are on the right half of each figure, the entries in Table I are not substantially improved when eq 35 is used. The second line for each operator in Table I was obtained by using eq 35. As was true for expectation values, the discrepancies between classical Fourier amplitudes and quantum matrix elements are believed to be due to behavior in the classically forbidden region. This fact is difficult to pin down since the classically forbidden region has no sharp boundary for an off-diagonal matrix element (the two states have different classical turning points). It is easy to note, however, that quantum matrix elements always become smaller in magnitude than the classical prediction as the maximum quantum number is approached. This seems to be because each higher state has more probability of being in the classically for-
-
Classical Fourier Amplitudes as Quantum Matrix Elements
0.35
The Journal of Physical Chemistry, Vol. 91, No. 9, I987 2265
0.0 '
I
'
I
I
I
'
I
I
a
0.30
-0.5
?
g
0.25
-1.0
U
L
LL
0 -1.5
0
; 0.20 I
I-
z
=Y
W
5
-2.0
W
xLY 0.15 t
-2.5
3
z0.10
-
-3.0
3
-3.5
(3
0.05 0.00
0.0
-4.0
0.2
0.4 0.6 NORMALIZED ACTION
0.8
0.0
0.2
0.4
0.6
NORMALIZED ACTION
0.8
1 .o
0.1 n 0,
nh
g
0.0
W
3
I
0.3
5u
-0.1
0
5 y -0.2
I
W
Gi 0.2
5w
X
E4
-0.3
I -
+ +
F 4
0.1
I
-0.4
n.n -.-
I
I
I
I
1 0.8
I
-0.5
0.0 0.2 0.4 0.6 0.8 1.o NORMALIZED ACTION NORMALIZED ACTION Figure 3. Matrix elements for Morse oscillator operators plotted vs. l / l - Solid curves are classical Fourier amplitudes with j = 1,2, 3,4, 5 . Absolute values are shown in (a) and (b), log of the absolute value is shown in (c). Quantum matrix elements for k = 46 are positioned along the abscissa according 0.0
0.2
0.4
0.6
1.o
to eq 29. Better agreement could be obtained for small values of the quantum number by using eq 35. Solid dots (*) are for j = 1, (+) for j = 2, (*) for j = 3, ( 0 ) for j = 4, and (x) for j 3: 5 . Plot a is for operator Z = exp(-aq); plot b for 3,plat c for aq; and plot d for p. Discrepancies between classical and quantum values are understood in terms of tunneling into the classically forbidden region at large q.
suggests the utility of semiclassical inversion procedures such as that of Stine and Noid for dipole moment functions? Classical approximationsto quantum matrix elements may have an important advantage from a numerical point of view. Vasan and Cross70have pointed out that many of the analytic forms for quantum matrix elements of Morse oscillators are numerically inconvenient due to round-off error. This is less of a problem with the formulas based on Sage's3' work (i.e. eq 30-34), but the simple classical expressions reported here are remarkably accurate. If one makes the replacement of geometric mean for j factors of one parameter, then these formulas are even better. Thus,the analytic formulas presented here, combined with other similar ones which (70) Vasan, V.
S.;Cross,R. J. J. Chem. Phys. 1983, 78, 3869.
can be derived by using the methods suggested here, may be of use to those interested purely in numerical values for matrix elements. The exception to this argument is that class of functions like q exp(-aq) which are nonmonotonic. More work should be done before trusting the classical Fourier amplitudes of these functions.
Summary and Comment We have collected together a number of formulas for classical Fourier amplitudes for Morse oscillators. In Appendix A, we have demonstrated that these amplitudes can be obtained by general contour integral techniques. We have graphically exhibited the excellent agreement between classical Fourier amplitudes and quantum matrix elements for a number of operators. It is argued
2266
The Journal of Physical Chemistry, Vol. 91, No. 9, 1987
that disagreement can be blamed on quantum mechanical tunneling into the classically forbidden region at large coordinate values. It was also shown that even though two functional operators are very similar, their matrix elements for multiquantum transitions and overtone Fourier components can be very different, thus suggesting that semiclassical inversion procedures should take advantage of multiquantum transitions in fitting experimental data. We also confirmed and derived the ability of choosing the geometric mean for the "quantum number" of the classical action at which classical quantities are calculated for multiquantum transitions instead of the usual arithmetic mean. It is not yet clear how general our observations are. What is true for Morse oscillators may not necessarily be true for general anharmonic oscillator systems like large polyatomic molecules. Numerical results, however, have been encouraging.3g41 There is no evidence as yet to suggest that the present results, where generalizable to other systems, are not equally as accurate. Further work is encouraged. One usually assumes that classical mechanics and semiclassical methods are more accurate at high quantum n~mbers.4~ We have demonstrated here that this comment is not necessarily true. One well-known exception is the class of problems with a potential barrier in which tunneling through the barrier may be important.71)72 However, there is no such barrier in a single Morse oscillator. For Morse oscillators, classical methods can quantitatively recover nearly all observable quantities. This is especially true for low quantum numbers. The only quantum effects that were not recoverable from classical mechanics were the tunneling effects near the dissociation limit, that is for high quantum numbers. It may be argued that the disagreement between quantum and classical variables near the dissociation limit is associated with the separatrix which divides bound and unbound motion. In systems of several degrees of freedom, chaotic classical motion is first observed near the separatrices. This breakdown of the quasiperiodicity on which semiclassical methods are based is thought to be due to the overlapping of resonances between two or more classical f r e q ~ e n c i e s . ~These ~ classical resonances may cause strong perturbations and mixing between zero-order quantum states that are near degenerateess There are no such resonances in the example examined here because it is only one-dimensional. It may also be seen that near a separatrix between two types of motion, there is a shift in the type of zero-order basis set necessary to describe the motion.74 In such a case, neither basis set may be accurate in the vicinity of the separatrix. Again, basis set problems are not responsible for the problems noted above because we are examining exact analytic wave functions. The precise role of the separatrix in this system might be a subject for further clarification, but we believe the physical basis behind the disagreement between classical and quantum results near dissociation to be tunneling of a high-quantum-number wave function into the classically forbidden region. This arises because the energy of the particle is nearly enough for it to be unbound, and thus the exponential decay of the wave function is slow at large coordinate values. The observation of classical-quantum disagreement at high quantum numbers poses an interesting quandary. Chemists have studied classical dynamics for a number of years because the quantum mechanics was intractable for polyatomic systems at hi& levels of excitation. However, as we become more adept at classical mechanics, it appears that in some cases it is precisely at the limit of high quantum numbers near dissociation that quantum effects may become most important.
Shirts of Utah, in evaluating contour integrals for Appendix A and for useful discussions. The author also thanks the following for useful comments: Prof. R. A. Marcus, Prof. W. H. Miller, Prof. W. P. Reinhardt, Prof. R. T. Skodje, Prof. C. JaffE, Prof. E. L. Sibert, 111, and conscientious referees. This work was supported by N S F Grant CHE85-11164.
Appendix A Direct Evaluation of Fourier Amplitudes of the Coordinate. The Fourier coefficients of the coordinate variable may be determined by finding the integrals in tables.% However, it is useful to derive the result directly to demonstrate how this may be done for functions which are not found in tables. Also, the tabulated form may not be the most convenient form,7s making extraction of the desired answer inconvenient. To evaluate the integral in eq 13, we make the substitution, z = exp(iO), which converts the definite integral into a contour integral: J2'ln
(1
+ b cos j ) cos (ne) dO=
where the contour is the unit circle. Integral A1 can be simplified by noting that 22 22 - 1 = (2 - zl)(z - z2) (-42) b
+ +
where z1 and z2 are given by 1 z1 = --[I - (1 - b2)1/2] = -a;
b
1
z2 = --[l b
+ (1 - b2)'/2] = -a-I (A3)
where a is given in eq 7. Note that lzll < 1 and lzzl > 1 for E < D. The contour integral can be evaluated by the method of resid u e ~however, ; ~ ~ the evaluation is simpler if we first consider the logarithm in the integrand:
1
- zl)(z - z2) = In
(-?)+
In ( 1 -
:)
+ In ( 1
-
:)
(A4)
Since lzll < lzl and Izl < lzzl on the unit circle contour, we can express the last two logarithms in eq A4 as power series.77 This allows us to express eq A4 conveniently as a Laurent series as follows: In (1
+ b cos 0) =
m
C cjzj j=-m
(-45)
where
('46) This form allows a simple evaluation of the contour integral which has one pole at z = 0. 1 2 In h2 qn = -(cn + c-") - a
a
2 =-lnA, a
n=O
Acknowledgment. The author acknowledges the assistance of Prof. James P. Keener, Department of Mathematics, University (71) Truhlar, D. G.; Hase, W. L.; Hynes, J. T. J. Phys. Chem. 1983,87, 2664. (72) Hase, W. L. J. Phys. Chem. 1986, 90, 365. (73) Chirikiv, B. Phys. Rep. 1979, 52, 265. (74) E.g. JaffO, C.; Reinhardt, W. P. J. Chem. Phys. 1982, 77, 5191.
(75) E.g. ref 64 is not precisely like eq 1 1 . (76) Carrier, G. F.; Krook, M.; Pearson, C. E. Functions of a Complex Variable; McGraw-Hill: New York, 1966; Chapter 2 and 3. (77) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover: New York, 1965; p 68.
Classical Fourier Amplitudes as Quantum Matrix Elements
+
where A = (1 X)/2X2. This completes the direct evaluation of eq 13. Direct Evaluation of Fourier Amplitudes of Powers of the Coordinate. Since we already have a Laurent series expression for In (1 b cos e) in eq A5, one can immediately write down a Laurent series for powers of In ( 1 + b cos 0 ) as
+
The Journal of Physical Chemistry, Vol. 91, No. 9, 1987 2261 Equation 28 can be rewritten in the form (4)" = In k
+ $(v + 1 + c) - $ ( l +
-
=
-$(e)
+ iE- $(e) -I1 + e 0
In k
1
(Bl)
-
In the limit o f t 0, the second term takes the value Fl(u + 1 ) and the last term has the form $(e) - ( l / t ) - y + O(t). Substituting these into eq B1 gives 1 ( q ) u - In k Fl(v 1) y (B2) t
-
where di is
t)
+
+
+ +
+
where the sum is over all those values of P I such that C p I = i. This method has been implemented and used for j (and thus m) equal to 2 and 3 . The sums in eq A9, however, become tedious and difficult for j > 3 , and the alternative multiplication method discussed in the text was found to be easier. Direct Evaluation of Fourier Amplitudes for Other Functions. In order to evaluate Fourier amplitudes of the other functions tabulated in this paper, it is only necessary to evaluate integrals of two forms:
We also note that since X = 1 - (2u l ) / k , e = Xk. We now demonstrate that the discrepancy between classical and quantum expectation values for q arises because of quantum tunneling into the classically forbidden region. The outer classical turning point marks the beginning of the classically forbidden region. If z = k e w , then the z value corresponding to the turning point is t, = k [ 1 - ( 1 - X 2 ) 1 / 2 ] = kX2/2 for X 0. This means that the outer classical turning point blows up logarithmically as -2 In A. Let us first calculate the probability that a quantum particle is in the classically forbidden region in the limit that X 0. The quantum wave function is of the form:31
-
-
$(z) = N,e'/2z(k-2~1)/ZL,(z) or
1 7
Sns, (1
sin (ne) d0
+ b cos 6)"'
('410)
(B3)
where &(z) is an nth order associated Laguerre polynomial with constant term given by S = I'(k - v ) / [ v ! r ( k- 2u)],z = k exp(-aq), and N,2 = a ( k - 2v - l ) v ! / r ( k- v ) . Now let k = 2v 1 c and integrate $*$ over the classically forbidden region:
+
+
When one makes the substitution, z = exp(i6), one obtains:
where the contour is again the unit circle. For the cosine integral in eq A10, one uses the upper sign and s = 1 ; for the sine integral in eq A10, one uses the lower sign and s = 2. This type of integral may again be evaluated by residues. If the function, f,has no poles inside the contour, there is a pole of order m at z 1 and a pole of order n 1 - m (for n > m - 1 ) at the origin. Thus, if g ( z ) is the integrand, the integral in eq A1 1 has the value:
+
(
i)m(-i)pl[Res(zl+ ) Res(O)]
+ +
(A121
where the residues are of g ( z ) a t the indicated 10cation.'~ This method is appropriate for all of the functions of this paper and many other functions of interest.
Appendix B We will demonstrate here why the expectation value of q differs from the classical value for those states that are only marginally bound. The same result will also help to explain the discrepancies between classical Fourier amplitudes and quantum off-diagonal matrix elements. First let us examine the analytic matrix element of q to find the asymptotic form. The limit we desire to take is to let k = 2v + 1 t where v is an integer labeling the highest bound state and c is a small real number.
+
Here we approximated the product of the exponential and polynomial in the integration range by its asymptotic value of unity. Note that even though the turning point is becoming very large, the probability that the particle is in the large classical region is vanishingly small. Now let us calculate the integral of aq in the forbidden region with k = 2v 1 t:
-k't[l
+ $(v + l ) e ] [ l + $ ( l ) ~ ] -"[In ~t $(v
x,
-
:]
+ 0+ 27 + Ink + ... ( B 5 )
Thus comparing eq B2 and B 5 , we see that the dominant contribution to ( q ) is from the classically forbidden region where most of the probability is. The classically allowed region contributes 0. This fact explains the a negligible amount in the limit E 1 in Figure 2c. failure of classical expectation values for X
- -
