12178
J. Phys. Chem. 1996, 100, 12178-12182
Matrix Fluctuation-Dissipation Theorem: Application to Quantum Relaxation Phenomena M. Gruebele School of Chemical Sciences and Beckman Institute for AdVanced Science and Technology, UniVersity of Illinois, Urbana, Illinois 61801 ReceiVed: February 13, 1996; In Final Form: May 8, 1996X
We recently derived a matrix fluctuation-dissipation (MFD) theorem, which directly relates the spectral intensities to the eigenvalue fluctuations of a quantum system. Here additional properties of the MFD theorem are presented. MFD is a microcanonical version of the fluctuation-dissipation theorem for a single highenergy state embedded in a dissipative quantum-mechanical bath. This is useful in applications to vibrational relaxation, which can be exactly described by the MFD formula if a single initial state carries all the oscillator strength. MFD versions of formulas for the dilution factor (σ) and fraction of occupied phase space (F) are derived, which can be used to compute these quantities exactly without knowledge of the eigenfunctions. Computational applications are given in a companion paper.
I. Introduction Generally in quantum mechanics, eigenvalues and eigenfunctions of an operator are treated as distinct aspects of the operator’s properties. The purpose of this paper is to discuss in detail the MFD theorem,1 which relates these properties in a simple equation. In a Green function representation, it will be seen that the MFD formula is related to the standard fluctuation-dissipation theorem for a single high-energy quantum state embedded in a dissipative “dark” bath of states. In this formulation, it is also related to the RRGM formalism developed by Wyatt and co-workers.2,3 This naturally leads to an application of the MFD theorem to problems in intramolecular vibrational energy redistribution (IVR), where a bright state interacts with a complex anharmonic bath. Of particular interest is the question whether certain quantities usually derived from the intensity distribution of the spectrum can be written down directly in terms of the eigenvalues. Formulas for the dilution factor (σ)4 and the fraction of occupied phase space (F)5 are derived here as examples. Since the MFD theorem is exact for a given Hamiltonian, the results apply for systems ranging from isolated resonances to quantum ergodic behavior. We briefly discuss some computational applications such as spectral sensitivity analysis here. Applications to large realistic models of molecules undergoing IVR are presented along with the details of a new IVR model in a companion paper.6 II. The MFD Formula: Basic Properties Consider a Hermitian operator H with eigenbasis {|n〉} and eigenvalues En. Let {|j〉} be an arbitrary basis in which H has matrix elements 〈j|H|k〉 ) Hjk and expectation values 〈j|H|j〉 ) E(0) j . Then it was shown in ref 1 that the square modulus of all eigenvector components |〈n|i〉|2 for a given basis state |i〉 ∈{|j〉} can be evaluated without calculating the eigenvectors. One proceeds by splitting H into three parts:
H ) H(0) + λiHi(1) + λi′Hi(1)′
where Pi is the projection operator onto state |i〉. λi′H(1) i ′ includes all the remaining off-diagonal matrix elements of H in the basis {|j〉}. The structure of the operator’s matrix representation is summarized in Figure 1. The parameters λi and λi′ allow one to arbitrarily scale the coupling of the selected basis function |i〉 to the remaining basis states, and the coupling of the remaining states among themselves. When λi ) 1 and λi′ ) 1, H is taken to represent the actual operator of interest, at “full coupling strength”. Using eq 2 it was shown that1
|cni|2 ) |〈n|i〉|2 )
λi 〈n|Hi(1)|n〉 λi ∂En ) ) (0) (0) 2 E -E 2(En - E0 ) ∂λi n i λi ∂ ln |En(λi) - E(0) 0 | (3) 2 ∂λi
(Evaluation bars at λi ) 1, λi′ ) 1 are implicit in eq 3 and elsewhere as appropriate.) Variation of λi induces fluctuations in the eigenvalues En, which in turn lead to nonvanishing eigenvector components. For λi′ ) 0, we have a single state coupled to a prediagonalized manifold. The usefulness of eq 3 in spectroscopic applications arises when H represents a molecular Hamiltonian. In many problems, only certain components of the eigenvectors of the Hamiltonian matrix are required: for instance, when one of the basis states |i〉 ∈ {|j〉} is a spectrally “bright” state that carries all the oscillator strength, only the components |〈n|i〉|2 for all n are required to obtain the spectrum, or the decay of amplitude from the initially prepared state |i〉, which is given by the well-known relation
Pi(t) ) |〈i(0)|i(t)〉|2 ) ∑ |cni|2|cn′i|2 e-i(En-En′)t/p
(4)
n,n′
(1)
H(0) is the zero-order operator with basis {|j〉} and eigenvalues (1) E(0) j . λiHi couples the chosen basis state |i〉 to the other basis states: X
λiHi(1) ) PiH(I - Pi) + (I - Pi)HPi ) PiH + HPi - 2PiHPi (2)
Abstract published in AdVance ACS Abstracts, July 1, 1996.
S0022-3654(96)00442-X CCC: $12.00
This is often the case in IVR. As λi is increased from 0 to 1, eq 3 links the fluctuations of the energy levels due to stronger coupling of the bright state to the other basis states, to the diffusion of intensity into the broadening spectrum. For sparse matrices, eq 3 eliminates the need for computing whole eigenvectors, resulting in substantial computational savings. © 1996 American Chemical Society
MFD Theorem: Application to Quantum Relaxation Phenomena
J. Phys. Chem., Vol. 100, No. 30, 1996 12179 tion” of an initially prepared state |i〉 into the dark manifold {|j〉, j * i}. This is caused by fluctuations in the energy level structure as the coupling of |i〉 to the dark manifold is increased by increasing λ, resulting in multiple avoided crossings and amplitude transfer at those crossings.10 However, the relation to the fluctuation-dissipation theorem of statistical mechanics is more explicit than that.
Figure 1. Matrix partitioned into three contributions for the zero-order operator, special state |i〉 off-diagonals, and all other off-diagonals.
Effectively, a Hermitian matrix can be diagonalized “sideways”, instead of “vertically”, eigenvector by eigenvector. A further interesting result is obtained when we take the derivative of En with respect to the complementary parameter λi′ instead of λi. From the Hellman-Feynman theorem a derivation analogous to ref 1 immediately yields
∂En ∂λi′
+
∂En ∂λi
) 〈n|H - H(0)|n〉 ) En - ∑ Ej(0)|〈n|j〉|2 (5) j
A special case of the quantity on the right hand side was derived previously.1 The sum of the eigenvalue fluctuations due to bright and dark states measures the deviation of the total Hamiltonian from the zero-order Hamiltonian, as quantified by the eigenbasis expectation values of H - H(0). It was also previously discussed how the second derivative ∂2En/∂λi2 allows a sensitivity analysis of the spectrum due to internal or external perturbations:7 2
Consider the picture for the states of a quantum system shown in Figure 2. We will assume that all the oscillator strength is carried by one basis state |i〉, which can be populated from state |g〉 (Figure 3a). The corresponding transition dipole operator is
µˆ ) |i〉µ〈g| + |g〉µ〈i|
In the eigenstate picture (Figure 3b), a number of eigenstates |n〉, |n′〉 ... can be excited from |g〉. This is due to the mixing of |i〉 with the remaining states in {|j〉}, and spreads out spectral intensity, as indicated by the different lengths of the levels in Figure 2b. (We assume that the “ground state” |g〉 is shared by both bases.) This is a common case, for example for vibrational overtone transitions in a spectral region where multiple bands do not overlap. To show how eq 3 is related to the fluctuation-dissipation theorem, consider the frequency-domain formulation of the fluctuation-dissipation theorem for the dipole operator in eq 7,11
χ(i)(E) ) (1/2)(1 - e-βE)C+(E)
∂ En
1 ∂ |〈n|i〉|2 ) - 2|〈n|i〉|4 + |〈n|i〉|2 (6) 2 (0) ∂λi ∂λ 2(En - Ei ) i evaluated at λi ) 1. The dependence of spectral intensity on coupling strength requires only one more evaluation of the eigenvalues if the second derivative is computed via the usual three-point formula (y+- 2y0 + y-)/∆x2. An example application to a system based on the model in ref 6 is shown in Figure 2. In this model, the basis states are coupled by a network of third to ninth order anharmonic matrix elements. The elements are constrained by spectroscopically motivated distribution statistics.6 The structure of the resulting matrix corresponds to clusters of coupled levels in quantum number space,8 but can be correlated with the usual tier picture.9 The bright state is located at E ) 0 cm-1, and the density of states is F ) 32 per cm-1. For λ J 0, the bright state has the expected negative sensitivity, as its amplitude leaks to IVR gateway states. States that are located in “gateway” regions of the spectrum rapidly gain their maximum intensity, then lose it (zero crossing of eq 6) to states that correlate with high-tier wave functions. Other states, which correlate with high-tier basis states, go through the zero crossing later, or not at all for values of λi < 1. Some eigenstates which adiabatically correlate to gateway states crucial for the IVR process do not show up in the spectrum, their amplitude having leaked to more remote tiers. III. Relation to the Single-Level Fluctuation-Dissipation Theorem We now consider why the name MFD or “matrix fluctuationdissipation” theorem is appropriate for eq 3: The spectral intensity |〈n|i〉|2 is clearly related to the dephasing or “dissipa-
(7)
(8a)
) (π/Z) ∑ (e-βEg - e-βEn)〈g|µˆ |n〉 × g,n
〈n|µˆ |g〉δ(En - Eg - E) (8b) χ(i)(E) is the imaginary part of the susceptibility (yielding the absorption/stimulated emission spectrum), the dipole correlation function C+(E) is defined by eq 8b in the usual manner,13 |g〉 and |n〉 are molecular eigenstates, Z is the partition function, and broadening of spectral transitions is neglected. We wish to recast eq 3, which involves basis states, in the form of eq 8. Using the dipole operator µˆ in eq 7, we can let β f ∞ and assume excitation from a single state |g〉 at Eg to the bright state |i〉. To rederive eq 3 in the desired form, we start with the Green operator
G(E,γ) )
1 H - E - iγ
(9)
whose diagonal matrix elements in the eigenbasis |n〉 satify the usual relationship
lim Im{Gnn(E,γ)} ) πδ(En - E) γf0
(10)
(i) (E) based on eq 3, which can Consider a susceptibility χMFD be manipulated as follows by using eqs 3, 9, and 10,
(i) (E) ) πµ2 ∑ χMFD n
1
∂En
2(En - Ei(0)) ∂λi
δ(En -Eg - E)
(11)
12180 J. Phys. Chem., Vol. 100, No. 30, 1996
Gruebele
Figure 2. Sensitivity analysis: (A) excerpt of the spetral sensitivity from a fragmented IVR spectrum,6 with three negative and many positive components; (B) behavior of the eigenstates adiabatically correlating to the bright state (squares), gateway states (circles), and “higher tier” states (diamonds and triangles). States near gateways show early zero crossings in the spectral sensitivity, indicating that the amplitude they have collected from the bright state is rapidly being funneled to terminal states in higher tiers. The hatched curve shows how the positive and negative parts of the sensitivity balance at low couplings (λi < 1) for states near gateways, as all their intensity is transferred to other states.
Indeed, eq 13 is just the desired recasting of the MFD formula in the form of eq 8 in the special case β f ∞. With the proper β f ∞ limit of C+(E) in eq 8, and eq 7, Fourier transformation yields
C+(t) ) µ2 ∑ n
Figure 3. (A) Basis manifold before diagonalization, showing special state |i〉. (B) Diagonalized manifold showing correlated state |ni〉; both systems share the same ground state.
1 ) µ2 ∑ Gnn(Ei(0),0) 〈n|Hi(1)|n〉 lim Im{Gnn(E + γf0 2 n Eg,γ)} 1 ) lim Im{Tr[µG(Ei(0),0)Hi(1)G(E + Eg,γ)µ]} (12) 2 γf0 since G is diagonal in the eigenbasis of H. Evaluating the trace in the basis {|j〉} and rotating the operators in the trace as needed, it follows that (i) χMFD (E) ) lim Im{〈i|µG(E + Eg)µ|i〉} ) γf0
πµ2 ∑ |〈i|n〉|2δ(En - Eg - E) (13a) n
) π ∑ |〈g|µˆ |n〉〈n|µˆ |g〉δ(En - Eg - E)
(13b)
n
The middle of eq 13a is analogous to the expression for the absorption spectrum first derived by Harris in 1963,12 except that the bright state |i〉 appears instead of the ground state |g〉 (i) due to the definition of µˆ in eq 7. χMFD is clearly the imaginary part of the susceptibility obtained by scanning through the spectrum generated by |i〉 immersed in the dark manifold {|j〉, j * i}. It was stated in a footnote by Harris12 that his version of eq 13 “is related to Kubo’s formula in the zero temperature limit.”
e-i(En-Eg)t ∂En En -
Ei(0)
∂λi
) µ2〈i(0)|i(t)〉eiEgt ) 〈µˆ (t)‚µˆ (0)〉 (14)
the zero-temperature dipole correlation function. This could of course have been obtained directly by multiplying the left of eq 3 by a δ-function line shape and summing, but the above derivation more clearly shows how the MFD theorem is related to the Green function formalism for absorption intensities and the fluctuation-dissipation theorem. The fluctuations of the system autocorrelation function in a dark bath for a state at sufficiently high energy are linearly proportional to the fluctuations of eigenenergies induced by the system state as a functions of its coupling λi to the bath. Comparing eq 12 to ref 2, it is also clear that the MFD formula relies on the same extraction of intensities |〈n|i〉|2 (residues in ref 2) via Green functions, although the MFD formula can be proved without recourse to the Green function formalism.1 The MFD and RRGM formalisms are also closely related computationally. The latter requires only one Lanczos iteration, allowing faster calculation of the spectrally strongest states,3 whereas the former can be started with any arbitrary Lanczos vector given a specific state |i〉, potentially allowing faster convergence of spectrally weak states, if required. Computationally, the two approaches are comparable. IV. Relating Eigenvalue and Eigenfunction Properties in Quantum Ergodic Systems13 While eqs 3 and 14 are exact descriptions of the spectral and temporal behavior of any quantum system initiated in a state |i〉, it is interesting to ask what they imply about spectral and temporal properties at different energies. Of particular concern for IVR is the question: How extensively and on what time scale does the initial state |i〉 explore the available phase space? A number of important results, related to earlier work in the statistics of random matrices and nuclear physics,14 have been derived in the literature to answer this question.5 It was first discussed by Pechukas that the average number of eigenstates Ni participating in IVR from a prepared state |i〉 is given by15
MFD Theorem: Application to Quantum Relaxation Phenomena
1
t dt′ P(t′) ) σi ) ∑In2 ∫ 0 tf∞ t
Ni-1 ) lim
J. Phys. Chem., Vol. 100, No. 30, 1996 12181
(15)
n
The second half of eq 15 is the dilution factor introduced by McDonald,4 and the equality results because both of these quantities are given by ∑|〈n|i〉|4. Consider a spectrum In ) |〈n|i〉|2 (as a function of En), as shown in Figure 4. Ni states are active under this envelope, in accordance with eq 15. This corresponds to a rate ki, which in practice will be evaluated by inserting the intensities In into eq 4, and finding the 1/e point in P(t). This is practical because for an ideal Lorentzian line shape, the only case where an exponential decay rate is strictly defined, it yields a ki equal to the exponential decay rate. We can superpose on this spectrum In a smoothed, normalized envelope, also yielding rate ki, as shown in Figure 4. This is the reference line shape discussed by Stechel and Heller, such that the mean deviation of In from Iav is zero.5 The maximum number of participating eigenstates Nmax > Ni compatible with the actual rate ki is then calculated by setting the eigenstate intensities equal to Iav, as shown in Figure 4b. The ratio F ) Ni/Nmax represents the fraction of phase space explored.5 If we write In ) Iav(En) + ∆I(En), then using eq 15, F is clearly given by
F)
Ni Nmax
∑Iav(En)2
)
∑Iav(En)2
)
∑[Iav(En) + ∆I(En)]
∑Iav(En)2 + ∆I(En)2
2
(16)
where the second half results from the fact that the average intensity fluctuation about the envelope is zero. A particularly common case, which we will use in ref 4, occurs when the average line shape is Lorentzian:
[∑(Γi2 + (En - Ei(0))2)-1]-1 Iav )
n
(17) Γi + (En 2
Ei(0))2
In that case, we can easily evaluate Nmax for a mean level spacing F-1. (Γi ∼ ki is the line width in cm-1 if F is in 1/cm-1.) Simply approximate states |n〉 to have constant density F and envelope Iav ∼ 1/[Γi2 + (n - ni)2/F2]:
Nmax )
2g2 cosh2(g2) cosh(g2) sinh(g2) + g2
≈ 2g2
(18)
g2 ) πFΓ, and in the Golden-Rule limit g ∼ FVrms, where Vrms is the root-mean-square coupling strength. Clearly, the spectral fluctuations of the broadened spectrum of state |i〉 allow one to read off the extent of IVR undergone by |i〉. Generally, these intensity properties are considered separately in the quantum ergodicity literature from eigenvalue properties. In fact, this is not so, as the two are intimately connected by eqs 3 and 14 in the spectral and time domains. Intensities and P(t)’s are directly related to eigenvalue fluctuations as a function of the parameter λi:
Ni
-1
)
1 4
∑n
(
)
∂ ln |En - E(0) 0 | ∂λi
2
(15b)
Figure 4. (A) Actual IVR line shape In with superposed average line shape Iav such that average fluctuation about the envelope is zero. (B) “Dressed up” spectrum where all intensities have been set to the average value; if the spacings were also evened out, this line shape would yield Nmax in eq 18 for the Ni in Figure 3A. (0) 〈n|H(1)|n〉 1 ∂ ln |En - E0 | ) ∆I(En) + Iav(En) ) ) 2 ∂λI 2(En - E(0) 0 )
〈|
|〉
1 H - H(0) n n (3b) 2 H - E(0) 0 The right side of 3b holds for a prediagonalized dark manifold, and shows how the intensity depends on the scaled deviation of H from H(0) as measured by the eigenfunction expectation values. We can therefore calculate the accessible phase space fraction h (1) + ∆H(1), purely from the eigenlevel fluctuations. Let H(1) ) H (1) where H h has all identical matrix elements, equal to the rms average of the elements of H h (1). (This ensures the correct behavior in the Golden-Rule limit; for a prediagonalized h (1) manifold and weak coupling H h (1), the Hamiltonian H(0) + H clearly leads to the Lorentzian reference line shape eq 17.) ∆H(1) describes the matrix element-by-matrix element deviations from the true perturbation Hamiltonian coupling |i〉 to {|j〉,j * i}. h (1) Then H h (1) generates the average intensity in 3b, and ∆H generates the intensity fluctuations. Combining eqs 3b and 16,
1/F ) ∑ n
〈n|H h (1) + ∆H(1)|n〉2 (E h n - Ei(0))2 〈n|H h (1)|n〉2
)
(En - Ei(0))2
∑n
(
)
∂(ln |En - Ei(0)|)/∂λi
∂(ln |E h n - Ei(0)|)/∂λi
2
(19)
allows one to compute F with four eigenvalue evaluations. E hn is the energy obtained when H h (1) is used instead of the full h (1) + ∆H(1). The use of eq 19 coupling Hamiltonian H(1) ) H will be further discussed in ref 6. We conclude this section by noting that eq 3 provides an alternate point of view for the intensity distribution of a quantum ergodic system. The intensity statistics for a GOE ensemble are well-known to follow a Porter-Thomas14 distribution (χ2 with 1 degree of freedom). Alhassid and Levine have provided a derivation guided purely by information-theoretic arguments, which makes minimal assumptions about the Hamiltonian other than “sufficient randomization”.16 It would thus be reasonable to assume that if a system has a GOE level spacing distribution (close to a Wigner distribution17), it will have a Porter-Thomas intensity distribution; i.e., the distributions are “universal” at sufficiently strong couplings, irrespectiVe of the details of the Hamiltonian matrix. (See numerical computations in ref 6.) In order to discuss intensity statistics, the line shape must be scaled by the reference Lorentzian Iav to provide a distribution of intensities invariant under energy shifting. Using eq 3,
12182 J. Phys. Chem., Vol. 100, No. 30, 1996
ISC )
In λi〈n|H(1)|n〉(Γi2 + (En - Ei(0))2) ) Iav 2c(E - E(0)) n
Gruebele
(20)
i
where c is the normalization factor in the numerator of eq 17. The numerical application of eq 20 is discussed in ref 6. For the wings of the line and Lorentzian scaling, the quantity to consider is then
ISC ∼ In(En - Ei(0))2 )
(En - Ei(0)) 〈n|λiH(1)|n〉 2
(21)
By using a second-order perturbation wave function for n, it is easily shown that this quantity is approximated by
ISC ) λi|〈i|H(1)|n0〉|2
(22)
just the off-diagonal matrix element of H that connects the bright state |i〉 to the basis state |n0〉 which correlates with |n〉. This is exactly what would be expected for a bright state coupled to a dark manifold, such as obtained from a general matrix by LKL inversion.18,19 (Note that by doing the perturbation expansion in the intensity directly, we obtain an expression linear in λi.) If the matrix elements of H(1) are real and Gaussian distributed, the In must be distributed χ2 with 1 degree of freedom. If the matrix elements of H(1) are complex with statistically independent, Gaussian distributed real and imaginary parts, the In must be distributed χ2 with 2 degrees of freedom since λi|〈i|H(1)|n0〉|2 ) λi(Re[〈i|H(1)|n0〉]2 + Im[〈i|H(1)|n0〉]2), as derived by other approaches.5,16 It will be interesting to see whether dark state matrix elements for realistic molecular models are Gaussian distributed, or not.6 An important question from the point of view of IVR is how does the “freeness” of vibrational energy flow correlate with ergodicity, as measured by level spacing distributions, eq 19, or eq 20. The companion paper considers these and other question via large scale numerical simulations using a new model for IVR.6 V. Summary The MFD theorem as applied to a single “bright” state |i〉 is seen to be an exact microcanonical quantum version of the fluctuation-dissipation for a high-lying state embedded in a dark bath of states. It can be used as a computational tool in the calculation of spectra, bypassing the need for eigenvector
computation in sparse systems. By evaluating its derivative, a sensitivity analysis of spectral features is also possible efficiently. Equation 3 is particularly useful when applied to IVR problems, which are often modeled as a bright state interacting with a dark manifold. Since it directly relates eigenvector properties (e.g., intensities) to eigenvalue properties (i.e., their fluctuations), it can be used to provide formulas for the dilution factor and fraction of occupied phase space directly in terms of the molecular energy levels. It also allows direct calculation of scaled intensity statistics, for comparison with both experiment and very general statistical matrix models. Acknowledgment. This work was funded by the David and Lucile Packard Foundation and a National Young Investigator grant of the National Science Foundation (CHE-9457970). The author thanks the UIUC Center for Advanced Studies for a fellowship during the period of this work, and D. Leitner, N. Makri, and the referee for a careful reading of the manuscript. References and Notes (1) Gruebele, M. J. Chem. Phys. 1996, 104, 2453. (2) Nauts, A.; Wyatt, R. E. Phys. ReV. A 1983, 30, 872. (3) Wyatt, R. E. AdV. Chem. Phys. 1989, 73, 231. (4) Stewart, G. M.; McDonald, J. D. J. Chem. Phys. 1983, 78, 3910. (5) Stechel, E. B.; Heller, E. J. Annu. ReV. Phys. Chem. 1984, 35, 563, and references cited therein. (6) Gruebele, M. J. Phys. Chem. 1996, 100, 12183. (7) Note that in ref 1, an additive term |〈n|i〉|2 is missing in the corresponding equation. (8) Bigwood, R.; Gruebele, M. Chem. Phys. Lett. 1995, 235, 604. Note that the exponent n is used instead of n - 3 in that reference; in table four, the anharmonic coefficients should thus have read read c ) 0.4S3 and c′ ) 0.1S3; the correct values were used in the calculations. (9) Sibert, E. L., III; Reinhardt, W. P.; Hynes, J. T. J. Chem. Phys. 1984, 81, 1115; J. Chem. Phys. 1984, 81, 1135. (10) Wilkinson, M. Phys. ReV. A 1990, 41, 4645. (11) Weiss, U. Quantum DissipatiVe Systems; World Scientific: Singapore, 1993. (12) Harris, R. A. J. Chem. Phys. 1963, 39, 978. (13) We take the meaning of quantum ergodic in the “weak” sense as discussed in ref 5; the size of the F parameter indicates the extent of quantum ergodicity, and approaches 1/3 for real time reversal-invariant, strongly coupled quantum systems. (14) Porter, C. E. Statistical Theories of Spectra: Fluctuations; Academic: New York, 1965. (15) Pechukas, P. Chem. Phys. Lett. 1982, 86, 553. (16) Alhassid, Y.; Levine, R. D. Phys. ReV. Lett. 1986, 57, 2879. (17) Mehta, M. L. Random Matrices, 2nd ed.; Academic Press: Boston, 1991. (18) Lawrance, W. D.; Knight, A. E. W. J. Phys. Chem. 1985, 89, 917. (19) Lehmann, K. K. J. Phys. Chem. 1991, 95, 7556.
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