. I Phys. . Chem. 1988, 92, 602-607
602
Similarly, Me2Si0 and Me(H)SiO were observed in Me2SiH2/03 experiments, and Me(H)SiO and H2Si0were produced in MeSiH3/03 experiments. Both H2 and CH4 eliminate from excited methylsilanols cleaving Si-H or Si-C bonds; however, the relative yields of the silanone products show that Si-H cleavage is favored. No silanones were observed in similar Me4Si/03 experiments, but new bands appeared that likely belong to the carbinol resulting from 0 atom insertion into a C-H bond of Me4% The argon matrix is a convenient vehicle for first-step oxidation of reactive compounds like the silanes with infrared spectroscopic identification of the products.
Acknowledgment. We gratefully acknowledge financial support from N S F Grant 85-16611 and helpful discussion with C. A. Arrington. Registry No. 03,10028-15-6; MeSiH,, 992-94-9; Me(H)Si=O, 43435-60-5; '*03,21424-26-0; H 2 S i 0 , 22755-01-7; M e S i H 2 0 H , 11 1935-69-4; MeSiD,, 1066-43-9; MeSiD20D, 11 1935-70-7; SiH3SiH,OH, 87963-93-7; Si2H6, 1590-87-0; Me2SiH2, 11 11-74-6; Me2Si=0, 47956-45-6; Me2SiD2, 1066-41-7; Me2SiHOH, 5906-76-3; Me2SiDOD, 11 1935-71-8; Me,SiH, 993-07-7; Me3SiD, 18026-91-0; Me(HO)Si= CH2, 11 1935-72-9; Me,Si, 75-76-3; M e 3 S i C H 2 0 H ,3219-63-4; SizDs, 13537-08-1; 0, 17778-80-2.
A Quasi-Classical Approximation for Autocorrelation Functlons Seon-Woog Cho Department of Chemistry, Kansas State University, Manhattan, Kansas 66506
and Kenneth G. Kay* Department of Chemistry, Kansas State University, Manhattan, Kansas 66506, and Department of Chemistry, Bar-Ilan University, Ramat-Gan 52100, Israel (Received: May 26, 1987; In Final Form: August 24, 1987)
A quasi-classical approximationis presented for the calculation of time autocorrelation functions of properties of bound molecular systems. In this treatment, classical motion is restricted to quantized invariant tori on a discretized energy shell. The dynamics on these tori is obtained by the adiabatic switching technique. When applied to the Henon-Heiles system, the quasi-classical approximation is sometimes found to yield significantly better agreement with the quantum results than that obtained from conventional classical calculations.
I. Introduction Time correlation functions are of central importance for the description of motion in isolated, molecular systems. Somewhat different versions of such functions are used to determine optical spectra,'-) to calculate probabilities for transitions among internal states,4d to formulate stochastic theories of intramolecular dynamics,' and to detect the presence of chaotics or ergodic beha~ior.~-'~ Since a full quantum mechanical calculation of such functions is usually not feasible, it is desirable to develop various classical approximations. The goal of this paper is to present one such approximation that is sometimes capable of yielding good agreement with the exact quantum results. The original motivation for this work came from an earlier comparisonlo between quantum and analogous classical autocorrelation functions for the Henon-Heiles14 system. Although the quantum and classical quantities were generally found to agree well when the energy of the system was high, they were sometimes found to disagree significantly when the energy of the system was lower. The explanation offered for these discrepancies implied that the agreement could be improved if the classical calculations were confined to a set of discrete invariant tori corresponding to states contributing to the quantum dynamics. This idea forms the basis for the present work. We organize the remainder of this paper as follows. In section 11, we describe our basic method for simulating quantum correlation functions. In section 111, we present more details about the application of this technique to the Henon-Heiles system. In section IV we compare autocorrelation functions computed by the present method to exact quantum and conventional classical autocorrelation functions. Finally, in section V, we make some concluding remarks. ~
* Address ~~
reprint requests to Bar-Ilan University.
0022-3654/88/2092-0602$01.50/0
11. Approach
The time-autocorrelation functions we explore here are of the form
CA(t)= Tr[p'(E - %)A(t)ta]/Tr[pu(E - &)at2] (1)
act)
a
where is a quantum operator, 3 this operator at time t in the Heisenberg representation, and 7f is the Hamiltonian for the system. The quantity p* is a "broadened delta function" which projects onto states in the energy shell of half-width 6 about energy E . Specifically, we use the expression p " ( ~
&) = exp[-(E
- &)2/2d]
(2)
although others may be applied. The form of CA(t)defined above is general enough to be useful in many physical cases. For example, if u is chosen to be very (1) Breene, R. G. Theories of Spectral Line Shape; Wiley-Interscience: New York, 1981. (2) Koszykowski, M. L.; Noid, D. W.; Marcus, R. A. J . Phys. Chem. 1982, 86, 2113. (3) Wardlaw, D. M.; Noid, D. W.; Marcus, R. A. J. Phys. Chem. 1984, 88, 536. (4) Kay, K. G. J . Chem. Phys. 1980, 72, 5955. (5) Hutchinson, J. S.;Wyatt, R. E. Phys. Reu. A 1981, 23, 1567. (6) Davis, M. J.; Heller, E. J. J . Chem. Phys. 1984, 80,5036. (7) Mukamel, S. J. Chem. Phys. 1979, 71 2012. (8) Noid, D. W.; Koszykowski, M. L.; Marcus, R. A. J. Chem. Phys. 1977, 67, 404. (9) Kay, K. G. J. Chem. Phys. 1983, 79, 3026. (10) Ramachandran, G.; Kay, K. G. J . Chem. Phys. 1985, 83, 6316. (11) Ramachandran, B.; Kay, K. G. J. Chem. Phys. 1987,86, 4628. (12) Koszykowski, M. L.; Noid, D. W.; Tabor, M.; Marcus, R. A. J . Chem. Phys. 1981, 74, 2530. (13) Hamilton, I.; Carter, D.; Brumer, P. J. Phys. Chem. 1982, 86, 2124. Hamilton, I.; Brumer, P. J. Chem. Phys. 1983, 78, 2682. (14) Henon, M.; Heiles, C. Astron. J . 1964, 69, 73.
0 1988 American Chemical Society
The Journal of Physical Chemistry, Vol. 92, No. 3, 1988 603
Autocorrelation Functions small and 2 is identified as the dipole moment, CA(t)may represent the Fourier transform of the spectroscopic line shape for a pure state. If u is taken to have an intermediate value, CA(t) is the quantum analog of a classical microcanonical autocorrelation function"' and can be used to investigate the ergodic-like behavior of the system. If u is taken to be infinite and A is chosen to be a density operator, c A ( t ) becomes the survival probability for the state represented by A . We are concerned with a system that is bound at energies near E . We can, therefore, evaluate C A ( t ) by inserting an essentially complete set of energy eigenstates Im) with discrete eigenvalues E , into eq 1. The result is
initially restricted to lie on these tori. This is reminiscent of the quasi-classical procedure2' for treating collisional problems where the initial conditions place phase points for the internal motion of the colliding species on quantized tori. It is for this reason that we call the present technique a quasi-classical approximation. Our quasi-classical treatment is but one member of a family of dynamical approximations that can be derived from the Heisenberg correspondence principle. For example, another, presumably more accurate, approximation can be obtained by applying eq 9 to the off-diagonal matrix elements appearing in eq 5. Then the autocorrelation function is once again given by eq 3 but with GAm(t)expressed as G,m(t) = CIAn-m(J,)12 n
e x ~ [ i ( m- n).a(Jmn)tI
(1 1)
where
where (41 (5)
and where
is a Fourier component of A and w(J,) is the vector of classical To explore the relationship frequencies on the torus J = J,. between the two approximations introduced above, we Fourier expand the functions A appearing in eq 10 in terms of the angle variables to obtain G,m(t)
The classical analogue of the autocorrelation function can be obtained by applying the Wigner-Weyl correspondence ruleI5 to the trace appearing in eq 1. The result is found to be9
= CIAn(Jm)12 exp[-i~o(J,)t] n
(13)
The choice of a new summation index then leads to the equivalent expression G,m(t) = CIAwm(Jm)12 e x ~ [ i ( m- n).a(Jm)tI n
(14)
which shows that the quasi-classical treatment and the approximation of eq 11 differ in the values of the actions that are used to calculate the frequencies and Fourier components of A . This difference becomes unimportant near the classical limit where classical quantities are insensitive to variations in the action where %(p,q) is the classical Hamiltonian function, 4= A(p,q) variables on the order of h . is the classical function corresponding to operator A , and A ( t ) We note that both approximations introduced above are closely = A(p(t), q(t)) is the function A propagated to time t . related to the semiclassical Liouville eigenfunction treatment The quasi-classical approximation we introduce in this paper by Jaffe, Kanfer, and B r ~ m e r . ' ~ .All ~ ~ of these is based on the Heisenberg correspondence p r i n ~ i p l e ~ ~ ~ ~ ' ~introduced -~~ methods discretize action space by replacing certain quantum density operators by sums of delta functions on tori with coefficients that are given by the exact quantum density matrix elements. The major difference between the present approximations and that of ref 22 lies in the choice of density operators to be represented in this manner. Although we explore only the quafor quantum matrix elements. In eq 9, J and 0 are the action and si-classical treatment in the present calculations, we point out that Jn)/2, J, are the angle variables for the system, J, = (J, the approximations presented in eq 11 and ref 22 can be implequantized values for the actions associated with sta_teIm), B(p,q) mented numerically in a way that is very similar to that applied is the classical function corresponding to operator B, and N is the here. number of degrees of freedom of the system. To obtain our To perform our calculations, we use the adiabatic switching quasi-classical approximation, we use eq 3 for CA(r)but apply m e t h ~ d ~to ~ -determine ~' the classical time evolution of functions eq 9 to express the expectation value appearing in eq 4 as a phase A on the quantized tori J and then apply the Monte-Carlo method space average over the torus characterized by J = J,. The reto evaluate the phase space integrals GAm(t)of eq 10. This sulting expression for CA(t) is, thus, eq 3 with GAn(t)now given procedure is reminiscent of that adopted in very recent work by by Smith, Shirts, and Patterson28who used adiabatic switching to provide initial conditions for classical calculations of isomerization in H C N . However, unlike that study, we are not primarily interested here in the details of the purely classical evolution thus Our approximation, thus, replaces the phase space integrals over obtained. Instead, we are mainly concerned with the relationship the entire energy shell appearing in the classical expression, eq of classical dynamics on quantized tori to the analogous quantum 8, with phase space integrals over a discrete set of tori in the energy behavior. shell. Since the tori are invariant, the autocorrelation function can be calculated by examining the dynamics of phase points
+
(1 5 ) Wigner, E. Phys. Rev. 1932, 40, 749. Balescu, R. Equilibrium and Nonequilibrium Statistical Mechanics; Wiley: New York, 1975. (16) Clark, A. P.; Dickenson, A. S.; Richards, D. Adu. Chem. Phys. 1977, 36, 63. (17) Percival, I. C.; Richards, D. Adv. At. Mol. Phys. 1975, 11, 2. (18) Heller, E. J. Physica D (Amsterdam) 1983, 7 0 , 356. (19) Jaffe, C.; Brumer, P. J . Chem. Phys. 1985, 82, 2330. (20) Shirts, R. B. J . Phys. Chem. 1987, 91, 2258.
(21) Porter, R. W.; Raff, L. M. In Dynamics of Molecular Collisions; Miller, W. H., Ed.; Plenum: New York, 1976. (22) Jaffe, C.; Kanfer, B.; Brumer, P. Phys. Rev. Lett. 1985, 54, 8 . (23) Solov'ev, E. A. Sou. Phys.-JETP (Engl. Transl.) 1978, 48, 635. (24) Skodje, R. T.; Borondo, F.; Reinhardt, W. P. J . Chem. Phys. 1985, 82, 4611. (25) Johnson, B. R. J . Chem. Phys. 1985,83, 1024. (26) Grozdonov, T. P.; Saini, S.; Taylor, H. S. Phys. Rev. A 1986, 33, 55. (27) Skodje, R. T.; Borondo, F. Chem. Phys. Lett. 1985, 118, 409. (28) Smith, R. S.; Shirts, R. B.; Patterson, C. W. J . Chem. Phys. 1987, 86, 4452.
604 The Journal of Physical Chemistry, Vol. 92, No. 3, I988 111. Numerical Implementation We test the quasi-classical treatment for the Henon-Heiles14 system described by the Hamiltonian
with h set equal to 1. This system has a total of 99 states with energies below the dissociation limit at 13.33. It is conventional to assign the pair of (generally approximate) quantum numbers (n,l ) to label these states, where n (=O, 1, 2, ...) is a principal quantum number and 1 (=-a, -n 2, ..., n) is an angular momentum quantum number. The states can also be classified rigorously according to symmetry species of the C3, point group. States with 1 = 0, f 3 , f 6 , ..., are of A, or A2 symmetry while the remaining states are of E symmetry. We now discuss details of the technique for generating quantized trajectories which are needed for the quasi-classical calculations. We recall that the adiabatic switching method proceeds by dividing describing a the full Hamiltonian into a zero-order term Po, separable system, and a perturbation, 7f‘ = 7f - 7fo. Then, a new, time-dependent Hamiltonian % ( t ) is constructed as % ( t ) = 7 f O + s(t)%’ (16)
+
where the switching function s(t) varies smoothly from 0 at time t = 0 to 1 at time t = T. Since the zero-order system is separable, its action variables are easily determined. If the perturbation is switched on slowly enough (e.g., the switching time Tis sufficiently large), the adiabatic principle leads us to expect that these actions will be conserved. Thus, it should be possible to prepare trajectories on an invariant torus J, of the fully coupled system by selecting phase space points with actions J, for the zero-order system and propagating these points from time t = 0 to time t = T using Hamiltonian % ( t ) . Actually, some care must be exercised in choosing the zero-order Hamiltonian. To avoid large nonadiabatic effects associated with passage through low-order resonances, it is important to select 7foso that the zero-order tori have the same topology as the exact ones,24*26Thus, we must consider the topologies of tori for the Henon-Heiles system. Noid and Marcus29have established that there are two main kinds of tori for this system. Librating tori bear trajectories of low angular momentum which are confined to regions near the C, symmetry axes of the potential. Such ton occur in sets of three since the Henon-Heiles potential has three such axes. Precessing tori, on the other hand, bear trajectories of higher angular momentum which circulate around the potential. These tori occur in pairs, corresponding to circulation in the clockwise and counterclockwise senses. The adiabatic switching method requires different 7f,’s to generate the librating and precessing tori. To form the precessing tori, we follow Skodie et al.” and choose the zero-OrderHamiltinian to be 7 f O = f/z(p;
+ xz + pyz + y2)
and identify the zero-order actions as
J, = r - ’ $ p r
dr
+ (2r)-’$p8
dB
and JI = (2n)-’$pe
dB
where r a n d 0 are conventional polar coordinates. these action variables by applying the conditions J , = n + 1 (n = 0, 1, 2, ...) and J / = I (I = f n , f ( n
- 2),
...)
where positive and negative values of quantum number I are associated with precessional motion in the clockwise and counterclockwise directions. We use the generating function presented (29) Noid, D.W.; Marcus, R. A. J. Chem. Phys. 1977, 67, 559
Cho and Kay by Martens and Ezra30 to determine the polar or Cartesian coordinates and momenta that correspond to desired values of the zero-order actions and angles. These serve as initial conditions for the adiabatic switching calculations. To obtain librating tori, we follow another procedure discussed, but not extensively applied, by Skodje et alaz4 We choose the zero-order Hamiltonian to be and quantize the zero-order actions
J, = (2r)-’$px
dx
and
by applying the conditions and J y = ny +
(ny = 0, 1, 2, ...)
These conditions determine only one of the three symmetry-related librating tori. We obtain the remaining two by rotating the coordinates and momenta for the adiabatically switched torus generated above by 120° and 240O. Of course, the Cartesian coordinates and momenta associated with these action-angle variables are easy to obtain from the usual transformation relat i o n ~for~ harmonic ~ oscillators. Grozdanov et a1.26 have proposed an alternative, zero-order Hamiltonian for treating librating motion. Their P odescribes the system as consisting of uncoupled harmonic and cubic oscillators. We have performed some calculations by using that choice for the Hamiltonian and found the results to be very similar to those based on the procedure described above. We have, therefore, used the simpler, quadratic, zero-order Hamiltonian of eq 22 for the calculations presented here. None of the above quantization procedures successfully generates librating tori associated with states (n,1 = 0) when n = 0, 2, and 4. To obtain these tori, we use the “straight-line” approximation, developed by Noid and Marcusz9 and adapted to the adiabatic switching technique by Skodje et aLZ4 In this treatment, the zero-order Hamiltonian is again taken to have the form given in eq 22, but different quantization conditions are applied: J , and Jy are now set to n 1 and 0, respectively. As before, this procedure generates only one of three symmetry-related librating tori. The remaining two tori are obtained by rotating the coordinates and momenta of the above torus by 120° and 240°. It is still necessary to determine which of the two kinds of tori should be associated with each energy eigenstate contributing to the summations in eq 3 and, thus, which quantization procedure should be used to construct these tori. By examination of rootmean-square deviations of adiabatic switching energies, earlier s t ~ d i e shave ~ ~ ,established ~~ that states (n, Ill), 111 > 3 are more accurately described by precessing tori while stat‘es (n,O)for n even and states (n,Ill), for n odd and 25 are more accurately represented by librating tori. Uncertainties, however, arise concerning states (n,121) and states (n,131) with n 2 5 . On the one hand, comparison of adiabatic switching energies with exact ene r g i e ~ ~and ~ , ’ examination ~ of root-mean-square energy deviat i o n ~ ~suggest ~ , ’ ~ that some of these states are more accurately described by precessing tori. On the other hand, some of these states should correspond to symmetry-related partners of the To make librating tori used to describe states (n,0) or (n the issue clearer, consider a case where n is even. Then, the state (n,0) of A, symmetry is knownz6to be associated with the librating torus generated from initial condition n, = n, ny = 0 and the two other symmetry-related tori. However, we expect the number of
+
(30) Martens, C. C.; Ezra, G. S . J. Chem. Phys. 1987, 86, 279. (31) Goldstein, H. Classical Mechanics, 2nd ed.; Addison-Wesley: Reading, MA, 1980; p 462.
Autocorrelation Functions states to be equal to the number of corresponding quantized tori. Thus, a total of three states should be determined by this quantization condition: the aforementioned state (n,0) and the pair of states (n,+2) and (n,-2) of E symmetry. A similar situation occurs when n is odd. There, the torus generated from n, = n, nu = 0 and its two symmetry-related partners should determine not only the two states (n,1) and (n, -1) of E symmetry but the additional state ( n , 131) of A, symmetry as well. Indeed, we may recall that superpositions of at least three states (one of A symmetry and two of E symmetry) are required to produce wavepackets that are localized in classical librating regions.I0J1 In these calculations, we consistently identify the (n, 121) and the (n, 131) states as precessing or librating according to which quantization procedure more accurately reproduces the quantum energy. Thus, for example, we treat A, and A2 states ( 5 , 131), E states (6, 121), and the A2 state (1 1, 131) as precessing, while we treat E states (10, 121) and the A, state (1 1, 131) as librating.32 We adopt this procedure because we are trying to simulate quantum behavior by classical mechanics and not the other way around. Thus, we represent the quantum dynamics by the classical trajectories that most closely describe the behavior that is being modeled. Indeed, we find that this procedure yields results that are in much better agreement with the exact quantum behavior than are obtained by alternative assignments. It is still a bit disturbing that the number of states associated with precessing and librating tori is sometimes smaller than the number of such tori. However, this result seems impossible to avoid completely, regardless of how we assign the states. As discussed by Grozdanov et the fundamental problem we are encountering can be traced to the shallowness of the effective potential wells which bound the classical librating motion. The states with 111 = 1 and 2 and low n have energies near the maxima of these wells and the difficulties described above may reflect the need for a semiclassical uniformization procedure33to treat these states accurately. Nevertheless, the success of the calculations reported below suggests that the simple approach we adopt here is adequate for our purposes. We are now in a position to describe the overall quasi-classical procedure for evaluating the autocorrelation functions. The calculation requires that we perform the following steps for each term in eq 3, associated with a particular energy eigenstate ln,l): 1. Assign this state as precessing or librating, based on the considerations of the previous paragraphs. 2. Identify the appropriate zero-order Hamiltonian and quantization conditions for the state. 3. Generate points randomly, with a uniform weighting of angles on the interval (0,27r], on each of the symmetry-related zero-order tori associated with the quantization conditions. 4. Propagate these points from time 0 to time T by using the adiabatic switching Hamiltonian 7 f ( t ) . 5. Propagate these points for an additional time interval, using the Hamiltonian H, to determine the dynamics of the fully coupled system on the time scale of interest. 6. Use these trajectories on the final tori and the Monte-Carlo method to evaluate the autocorrelation function for each of the symmetry-related tori. 7. Average the autocorrelation functions thus obtained over the symmetry-related tori. This quantity is the contribution from state InJ) to the summations in eq 3. Obviously, since the same set of symmetry-related tori are usually used to describe more than one state, trajectories on these tori need to be calculated only once and the contribution per state, obtained in step 7, can be multiplied by the number of states to obtain the overall contribution from this set. Step 3 requires a few words of explanation. The reason for forming uniform distributions on the zero-order tori is that these
The Journal of Physical Chemistry, Vol. 92, No. 3, 1988 605 1
D
.oo
60 80 100 120 TIME Figure 1. Autocorrelation function CA(t)for property D of the HenonHeiles system with average energy E = 5.70: solid lines, quantum; long dashes, classical; short dashes, quasi-classical. 0
20
40
H O 1
.oar
0 . 9 6 I ' ' 0 20
I
'
I
'
'
60 80 TIME Figure 2. Same as Figure 1 for property go. 40
'
I
100
' '
120
can be shown34to evolve to uniform distributions on the final tori. This, in turn, simplifies the Monte-Carlo evaluation of the autocorrelation functions. Note that to prepare such distributions it is necessary to know the transformation of zero-order angle variables as well as action variables to conventional coordinates and momenta. Although the procedure described above is based on the implicit assumption that the classical motion of the system is regular so that true invariant tori exist, the adiabatic switching method is known to predict quantum energy levels successfully even when the classical behavior is ~ h a o t i c In . ~these ~ ~ ~cases, ~ the adiabatic switching procedure is believed to produce "vague tori"35 with properties analogous to those of the quantum states. It is, therefore, possible to apply our treatment to all states without regard for the regular or irregular nature of the underlying classical dynamics. We have carried out quasi-classical calculations of autocorrelation functions CA(t)for the same two values of the average energy E that were previously considered in ref 10, 5.70 and 12.67. At the lower energy the classical dynamics is predominately regular, while at the higher energy it is mostly chaotic. As in the previous work, we have considered the following six properties:
and (32) The energies and root-mean-square dispersions obtained in our adiabatic switching calculations are very similar to those of ref 24 (for precessing and straight-line initial conditions) and 26 (for librating initial conditions). The data presented in these references may be consulted to verify our identification of states as precessing or librating. (33) Jaffe, C.; Reinhardt, W. P. J . Chem. Phys. 1982, 77, 5191.
(34) This follows from the results derived in ref 27. (35) Shirts, R. B.; Reinhardt, W. P. J . Chem. Phys. 1982, 77, 5204.
606 The Journal of Physical Chemistry, Vol. 92, No. 3, 1988
D2 1 'O0[
-+-
a
0
t 0.401 0
'
I
20
'
40
" 60
" 80
TIME
Figure 3. Same as Figure 1 for property D2.
'
I 100
'
'
120
Cho and Kay As Figures 3 and 4 show, the improvement obtained by replacing the classical with the quasi-classical approximation is more modest for properties Dzand 7YX. Still, except for the interval (30 < t < 60), the quasi-classical curves generally lie closer to the quantum results than do the classical curves. In addition, the quasi-classical curve for 02 imitates the oscillations in the quantum function more faithfully than does the classical curve. In our comparison of the various curves, we consider the most significant indicator of the relative accuracy of the classical and quasi-classical results to be the agreement with the early-time quantum behavior. We can anticipate that large discrepancies will almost always appear between quantum results and classically based approximations if the dynamics are followed for sufficiently long times. These arise due to tunneling, nonclassical recurrences, and certain aspects of wavepacket spreading which cannot be described without a calculation on at least the semiclassical level. Indeed, even the good agreement between the quasi-classical and quantum curves exhibited in Figure 1 breaks down for times longer than those considered here. However, it is reasonable to expect a classically based approximation to at least describe the short-time quantum behavior accurately. We therefore regard the improved short-time agreement between quasi-classical and quantum results as the feature of Figures 1-4 that most convincingly indicates that our procedure is more accurate than the classical treatment. We have not presented autocorrelation functions for properties L and Lz at E = 5.70 nor have we presented autocorrelation functions for any properties at E = 12.67. In none of those cases is there a substantial difference between the classical result, presented in ref 10, and the present quasi-classical result. Adopting the quasi-classical approximation for these properties and energies results neither in major improvements nor in detectable deterioration in the agreement with the quantum results. It must be pointed out, however, that the original classical autocorrelation functions are already in good agreement with the quantum results in these cases so that there is not as much room for improvement as for the examples illustrated in Figures 1-4.
V. Concluding Remarks The success of the quasi-classical treatment for property D at E = 5.70 supports the explanation presented in ref 10 for the poor quantum-classical agreement in this case. To restate the argument from the present perspective, we first note that the autocorrelation function of D is especially sensitive to the precessing versus librating nature of the motion. In the case of purely librating behavior, the sign of the autocorrelation function remains positive for all times, while in the case of purely precessing motion, the sign of this function changes from positive to negative as the axis of precession rotates. Of course, both the classical and quasiclassical autocorrelation functions described here are weighted averages of contributions from both kinds of motion in the energy shell. However, by discretizing the actions, the quasi-classical method achieves a different weighting than does the classical method. In particular, the quasi-classical approximation treats four of the six states with n = 5 (which dominate the energy shell at E = 5.70) as precessing and thus assigns a weight'of only about one-third to regions in the energy shell with librating character. Since the resulting quasi-classical autocorrelation function lies below the classical one and sometimes becomes negative, we may deduce that the quasi-classical method assigns a greater weight to precessing regions than does the classical approximation. The agreement between the quasi-classical and the quantum autocorrelation functions then suggests that the quasi-classical and quantum weightings are similar and that the classical approximation is inaccurate because it cannot duplicate them. The method we have used to perform the quasi-classical calculations can be simplified when the dynamics is known to be predominately regular (as is the case for E = 5.70 but not for E = 12.67). One can then use of the invariance of the tori and the quasi-periodicity of the motion to greatly reduce the number of trajectories required for the computation. For example, neglecting problems associated with nonadiabaticity, only a single, long, adiabatically switched trajectory need be propagated for each final
J. Pkys. Chem. 1988, 92, 607-614 torus. The correlation function on the torus GAm(t)can then be evaluated by averaging the quantity A(t - to)*A(to)over different time origins to along this trajectory. Alternatively, G,,(t) can be calculated by Fourier analyzing A ( t ) along this trajectory and applying eq 13. Clearly, the quasi-classical technique cannot bring classical mechanics into complete agreement with quantum mechanics. It is not able to describe such long-time phenomena as tunneling and purely quantal wavepacket spreading. It is, however, capable of improving the short-time agreement between quantum and classical mechanics, and sometimes this improvement is very significant. The related approximation presented in eq 11 may
607
yield even better agreement with quantum behavior and deserves further study. However, the present technique is computationally simpler. It may be readily applied to a variety of different kinds of correlation functions including survival and transition probabilities that describe the evolution of nonstationary states. The quasi-classical approximation thus deserves consideration in many cases where a completely classical treatment is suspected of inaccuracy.
Acknowledgment. We thank Mr. B. Ramachandran for a critical reading of the manuscript. This work was supported by National Science Foundation Grant CHE-8418 170.
Correlation between the Electron Paramagnetlc Resonance Spectra, Structure, and Bonding of the Bis(biuretato)cuprate( I I ) Dianion Saba M. Mattad Department of Chemistry, McCill University, 801 Sherbrooke St., West, Montreal, Quebec, Canada H3A 2K6 (Received: June 2, 1987)
The X a scattered wave self-consistent field method is used to investigate the electronic structure and excited states of the bis(biuretato)cuprate(II) dianion. The relationship between the spin Hamiltonian tensor components and the MO expansion coefficients for the dianion is discussed. Detailed expressions for the g tensor components are derived and are evaluated. They are also determined from the experimental solid-state electron paramagnetic resonance spectrum by simulation. The computed and experimentalvalues for the hyperfine and g tensors are found to be in good agreement. The,g and gw anisotropies are quite similar and are induced by the b2, and b3gorbitals representing the out-of-plane %-type” interactions. In contrast, the g,, anisotropies are directly linked to the coupling of the ’B,,,ground state and the excited A, states. The electronic absorption spectra and the computed electronic transitions for this complex are compared and discussed.
Introduction Alkaline media containing complexes formed by the reaction of Cu(I1) ions and biuret are characterized by their intense colors ranging from red to blue, depending on their method of preparation.’ When the biuret ligand is substituted by proteins, these solutions still retain their intense colors. This property was widely used to determine the total protein content in clinical assays.’ Another significant property of these solutions is their ability to dissolve cellulose.’ Their dissolution power is comparable to, if not greater than, standard inorganic solvent^.^ In order to understand the nature of these complexes, their mechanism of formation, and their interaction with carbohydrates, a systematic isolation and study of each individual complex under different conditions is necessary. As the first part of such a comprehensive study, the electron paramagnetic resonance (EPR) spectra and their relation to the electronic structure and bonding of bis(biuretato)cuprate(II) in solid and liquid solutions are investigated. The EPR spectra of this complex display copper nuclear hyperfine structure together with nitrogen ligand hyperfine splitting patterns very similar to those of copper(I1) porphyrins and phthalocyanins.s Due to the large number of overlapping resonance components of these spectra, a computer simulation of the spectral line shapes is necessary. With this technique the principal components of the magnetogyric (g) and copper hyperfine and nitrogen ligand hyperfine tensor components for this complex are determined. It will be demonstrated that the analysis of the EPR line shapes by simulation reveals that the fourth copper hyperfine component overlaps with the other three via its spatial perpendicular component. This phenomenon is found not to be restricted to the EPR spectra of the bis(biuretato)cuprate(II) complex alone Present address: Department of Chemistry, University of New Brunswick, Bag Service No. 45222, Fredericton, NB, Canada E3B 6E2.
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but also generally occurs in the X-band EPR spectra of copper(I1) porphyrins and copper(I1) phthalocyanins. Expressions for the components of the spin-Hamiltonian tensors are derived in terms of the molecular orbital coefficients for the ground and excited states of this dianion. These coefficients are numerically determined by using the X a scattered wave method. Most of the computed tensor components are found to be in good agreement with the experimental values determined by simulation. Experimental and Computational Details Biuret (Eastman Organic Chemicals) was recrystallized twice from water, dried at 393 K, and used in the anhydrous form (melting point 465 K). The solid dipotassium bis(biuretat0) complexes of copper and nickel were prepared by the method of McLellan and Melson.6 Cupric hydroxide was prepared by dissolving cupric sulfate pentahydrate (125 g) in distilled water and ammonium hydroxide (specific gravity 0.9) was added with constant stirring until the contents were neutral. The greenish blue precipitate formed was washed by decantation until free of sulfate ions. It was then suspended in water and a solution of sodium hydroxide (425 mL, 244 g/L) was added at 293 K. The resulting blue precipitate of cupric hydroxide was then washed and dried under vacuum at 310 K for several days. (1) Kurzer, F. Chem. Rev. 1956, 56, 95. (2) Kingsley, G. R. J . Biol. Chem. 1940, 133, 731. Cassatt, J. C. J. Biol. Chem. 1973, 284, 6129. Herbertz, G.Med. Lab. 1976, 29, 281. Later, R.; Quincy, C. J. Chromatogr. 1979, 174, 131. Kharlamov, I. P.; Popov, B. A. Zauod. Lab. 1985, 51, 13. Molnar, I. Elelmiszervizsgalati Kozl. 1982, 28, 195.
(3) Jayme, G.; Lang, F. Kolloid-2. 1957, 150, 5 . (4) Jayme, G. High Polym. 1971, 5 (Part IV), 381. ( 5 ) Lin, W. C. The Porphyrins; Dolphin, D., Ed.; Academic: New York, 1979; Vol. 4, p 235. (6) McLellan, A. W.; Melson, G. A. J. Chem. SOC.A 1967, 137.
0 1988 American Chemical Society
