Filter Diagonalization - ACS Publications - American Chemical Society

Chapter 2

Downloaded by NORTH CAROLINA STATE UNIV on January 11, 2013 | http://pubs.acs.org Publication Date: June 10, 1997 | doi: 10.1021/bk-1997-0678.ch002

Filter Diagonalization A General Approach for Calculating High-Energy Eigenstates and Eigenfunctions and for Extracting Frequencies from a General Signal Daniel Neuhauser Department of Chemistry, University of California, Los Angeles, CA 90095-1569

Filter-Diagonalization was introduced in 1990 (D. Neuhauser, J. Chem. Phys. 93, 2611) as a general approach for extracting high-energy eigenstates. The method combines simultaneous calculation of a rough Filter at many energies with a subsequent small-matrix diagonalization stage. The hybrid method combines the advantages of the pure-matrix and the diagonalization approaches, without their deficiencies. Recently, we have been able to reformulate the method so that it can extract frequencies and eigenfunctions at any energy range from a fixed set of residues, without large storage requirements. Given the residues, repetitive calculations of spectra and eigenfunctions at arbitrary multiple energy ranges are trivial. We review the achievements of the approach, as well as a a new feature of it, the ability to extract frequencies from a short-time segment of any signal, even if the signal is not due to a quantum correlation function. Introduction A general problem in chemistry and other fields is the extraction of highly ex cited eigenstates and eigenvalues of large sparse matrices. Recently, the author developed the Filter-Diagonalization approach [l]-[6], which furnishes a new way to extract eigenstates for the sparse molecular Hamiltonians encountered in quan tum molecular spectra and dynamics simulations. The method has been adapted by several authors [8]-[17]. It combines the good features of the filter-technique for extracting highly excited eigenstates, while avoiding long propagation times by adding a diagonalization step. Further, Filter-Diagonalization solves another problem: efficient extraction of frequencies and damping factors out of a general time-dependent signal, without long time-propagations or large-matrix diagonalizations. In this short review we outline the Filter-Diagonalization method and discuss recent developments and applications. For concreteness, we start with the problem of eigenvalue and eigenfunction extraction for an Hamiltonian H (or simply a 26

© 1997 American Chemical Society In Highly Excited Molecules; Mullin, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1997.

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NEUHAUSER

27

Filter Diagonalization

general matrix) acting on functions with Ν points. For concreteness, we assume that H is either real-symmetric, or, more generally, complex symmetric: T

H

= H.

(1)

(The final results are also valid for Hermitian Hamiltonians.) The one feature we need to recall on complex symmetric Hamiltonians is that they have orthogonal eigenstates, φ , fulfilling {Φη\Φτη) = ).

(5)

0

There are different filters one can choose and different approaches for calculating the action of each on φ , as elucidated later. More important however is the general property of a filter: a high quality filter, F (E), chooses out of φ the one eigenfunction φ which has an associated e that is the closest in energy to E: Φη « F (E^ , (6) 0

accurate

η

0

n

accurate

Q

and the eigenvalue e can then be determined from the expectation value of H , n

e = (φ \Η\φ )/(φ \φ ). n

η

η

η

η

(7)

It is clear however that for F (E) to choose accurately an eigenfunction of Η which happens to be close in energy to other eigenfunctions, it needs to be a high quality filter; this generally would translate to a requirement many terms are necessary in the evaluation of F . For example, for typical iterative filters (as discussed below) the numerical effort is inversely proportional to the the spacing between the levels that are the closest to the sampling energy. This is simply a accurate

accurate

In Highly Excited Molecules; Mullin, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1997.

HIGHLY E X C I T E D M O L E C U L E S

28

manifestation of the uncertainty principle: high accuracy requires long "times", i.e., much numerical effort. Filter-Diagonalization avoids this caveat by first noting that typically, the effort in producing a filter at several energies is only negligibly larger than the effort of producing the filter at one energy, and therefore it pays to use a filter at multiple energies. We pick a filter, F(E), which is now "rough" (as explained below) and apply it on the same initial wavefunction \φ ) but at several ("£") energies; this produces a set of vectors, each of size TV (see Figure 1): 0


10) — F{Ei)\th).

(8)

Here, Ej would be an arbitrary set of nearby energies; typically, an equi-spaced set within a desired energy range [i^mim ^max] is used. The roughness of the filter means that each vector, £j, encompasses many near-by eigenfunctions φ rather than one isolated eigenfunction, i.e., η

(i = E η

^

-

(9)

where the Dj are numerical coefficients. In most of the discussion the filter F(Ej) is a function of the difference between the energy and the Hamiltonian, n

F{Ej)

= f(Ej - H).

(10)

For such a filter, the numerical coefficients in Eq.(9) are simply equal to the strength of the filter at the difference between the eigenfunctions and the sampled energies: D = f{E - en). (11) in

s

The crucial point is then that even a rough filter restricts the sum in Eq. (9) to extend only over a few eigenfunctions, i.e., the Dj coefficients are appreciable only when e are relatively close in energy to Ej even though they may not be peaked at one frequency. Filtered functions at different nearby energies (Ej) would be covered by approximately the same subset of eigenfunctions. When the number of vectors (j is sufficiently large, they therefore span the desired energy range and we can then use them as a basis for those eigenfunctions which have eigenvalues in the energy range [Ε^, E \: n

n

max

Φη = Σ

Β

^ >

(12)

3

and the expansion coefficients can be determined from the requirement that φ is an eigenvalue of H. The basic algorithm is then simple: pick an energy range and an initial wavefunction and construct with the filter a set of vectors (j. When the set of vectors is sufficiently large, orthogonalize it explicitly, and construct an L x L Hamiltonian matrix, H * . = ( 0 1 * 10')· The eigenvalues of the small matrix Η are exactly those eigenvalues of Η in the desired energy range, and the eigenfunctions of it are the Β coefficients in Eq.( 12). η



2.

NEUHAUSER


Fi 1I cr- D i agon al i zat i on

Range o f Extraction

Figure 1. A schematic outline of Filter-Diagonalization. Each Filtered func tion encompasses a large range of eigenstates.


29


30

The advantage of the combined Filter-Diagonalization approach lies in the "di vide and conquer" strategy: it is relatively easy to act with a rough filter which has a broad energy resolution, and the next stage, the entangling of eigenstates from the broad filter, is handled very efficiently by a small-matrix diagonalization. The basic algorithm suffers however from one seeming deficiency, i.e., the cal culation has to be repeated for each new desired energy range. In order to avoid this problem, we have recently recasted the algorithm [4] as follows. We leave the filtered vectors unorthogonalized. The statement that φ is an eigenstate of i f , coupled with the expansion (12), leads to the generalized eigenvalue problem: η


H B = SBe,

(13)

where e is the diagonal L x L eigenvalue matrixes, and Sij = (^|F(J5,-)TO)l^o)

(14)

H = (0 |TO#TO)llM(15) There are several filters for which the product F(Ei)F(Ej) can be simplified, leading to eventual expressions involving terms of the form (φ \Ε(Ε{)\φο). These forms are easy to evaluate even when multiple energy windows are used, and they generally require very little storage (explicit vectors are not required; only correlation functions). The more technical details of the formalism are discussed in the next sections. The ability to extract explicit eigenstates and eigenvalues at any energy range makes Filter-Diagonalization favorable also for other situations. One case is reac tive scattering in the presence of resonances. Straightforward iterative calculations can run in such cases into difficulties due to the presence of the narrow resonances. The long calculation are avoided [2, 6] by noting that at late times, a scattering wavefunction transforms to a sum of a few resonance wavefunctions which are efficiently extracted by Filter Diagonalization. Finally, the mathematical derivations of Filter-Diagonalization lead,as shown below, to an unexpected feature. The formalism allows the determinations of frequencies and damping factors for a time-dependent signal, regardless of whether the signal is inherently due to a time-dependent Hamiltonian or whether it is due to, e.g., a classical simulation. The normal mode frequencies can be extracted even when a very long signal containing possibly millions of frequencies is used, since only a selective set of sampling energies (at the desired energy range) is used. This is done without the long times required for extracting signals by pure Fourier transforms, and the method is equally efficient for signals containing overlapping resonances. t j

o

0

F i l t e r s and Propagators Different filters were employed by us and other group. Most filters are a function of the Hamiltonian (Eq. 10). We note however that other choices are possible, e.g., the pre-Lancosz filter by Wyatt and his colleagues [17]: F{E ) 3

= - J — , Hfj

—

£1Q

where HQ is a solvable zero-order approximation to the Hamiltonian Η.


(16)

2. NEUHAUSER

31


Of the filters of form f(E — H), we have employed a Gaussian filter: 2

f(Ej -H)

= e-l"-*^' ,

(17)

where the parameter Τ is chosen relatively short (much shorter than the minimum of the inverse level spacing, A n additional filter is the inverse Fourier transform: [8, 14, 19] u


f(Ej - H)

=" *6{Ej -H)

i ~ Ej-H~~

= ImG+iEj)

i Ej-H*'

where is the Green's function. The delta-function equality sign is symbolic here; in practice one does not evaluate the full delta function, but rather a limited series approximation to it. A third filter would be simply m - H Y

="Ë^H,

where again one stops at a finite series approximation to / . The f(Ej — H) filters are easily evaluated by iterative approaches: Lancosz 20], Chebyshev [19], Newton [21], Modified Chebyshev as pursued by Kouri and loffmann [14] or the Damped Chebyshev approach of Mandelshtam and Taylor 8]. In these approaches, the action of the filter on the Hamiltonian is described }y a polynomial expansion: f{Ej - Η)φ

= £ α (Ε)Ρ (Η)φ ,

0

η

η

(18)

0

η

where α (Ε) is a set of numerical coefficients. P (H) is a polynomial in Η and its action on φο is evaluated by recursion, e.g., η

n

Ρ {Η)φ η

= α^ΗΡη.^φο

0

+ βη-χΡη^Φο,

(19)

where α are numerical coefficients (or damping functions [8]). This form is feasible for describing Chebyshev-type and Lancosz series; other, more general expressions are possible. The terms a (E) can be easily obtained in different formalisms. For the explicit orthogonalization schemes, the working formulae are simple: the vectors f(Ej — Η)φ are constructed for several energies Ej in a desired range, the vectors are then orthogonalized, and the Hamiltonian matrix is then obtained from orthogonalizing. A more efficient use of the formalism is, as mentioned, the construction of the Η and S matrices, involving products of the form f(E{ — H)f(Ej — H). There are several filters for which the calculation of this product can be simplified. First, a Gaussian filter (Eq. 17) for which the product of two filters yields another Gaussian filter, leading to: η

n

0

Sy = expi-iEi

-

Ε^Τ'β^φοΙβ-^-^^ΙΦο) 2

2

= exp(-(£?i - ΕΛ Τ β)Σάη((Ε

{

+ £;)/2)Λ , η

η


(20)

32


where R are the residues defined as n

Rn =

(φο\Ρη(Η)\φ ), 0

and a are the coefficients associated with a Gaussian filter with a parameter y/2T. Other filters can also lead to simple expressions; for example, for a filter of the form f(Ej -H) = \/{Ε - Η), n

Ά


[φο\/(Η - Ei)f(H

ΙΦΟ)

- ΕΛ\φο) = (ΨοΙ-^

~ Ε^Η {ΕΓ~Η

~

Ε~~Η)'

and similar expressions in terms of the residues can be obtained if an iterative approach is used. (Note that this particular equation is not limited to iterative evaluations of the Hamiltonian, and can be used, e.g., also with l/(H — E) filters.) Even for filters in which the product f(E — Hj)f(E{ — H) cannot be simplified, the method can still be used when the Chebyshev polynomials [19] T (H) are employed in Eq. (18) [4]. The property of the Chebyshev polynomials: 0

n

2T (H)T (H) n

m

= T + (H) + T | _ ( # ) , n

m

n

H

implies readily that for a general filter: ),

(21)

nm a sum which can be evaluated efficiently by convolutions (or in selected cases can be evaluated analytically, as recently shown by Mandelshtam and Taylor [8]). The analytical equations for S, and similar equations for H , lead thus to a very simple prescription for Filter Diagonalization when used with iterative procedures. First, a set of residues, jR , is prepared. Next, for any desired energy window one selects a small (typically L=100) set of energies, evaluates the a (Ej) (or a ((Ej + Ει)I'2)) coefficients, and uses these to calculate the small S and H matrices, which are solved by canonical diagonalization, and used to extract the eigenvalues and eigenfunctions. Applications of this formalism included studies by our group of a two-dimensional L i C N model [3] and an H C N triatomic molecule [7]. Mandelshtam and Taylor have used their modified Chebyshev propagator with this approach, and produced large scale simulations of HO2 [9] as well as H3 [10]. The approach emerges as a suc cessful alternative to Lancosz approaches, and is especially useful for resonance studies (where the Hamiltonian contains non-hermitian absorbing-potential parts simulating the outgoing nature of the wavefunction) since it allows the extrac tion of overlapping resonances (where the difference betweem the real part of the eigenvalue is smaller than the imaginary part). n

n

n

Time-Dependent Series An unexpected feature of Filter-Diagonalization is that it can be used to extract frequencies from any signal. Thus, assume we are given a signal C(t) which can be written as a sum of fluctuating terms, possibly with a decaying magnitude: i

C(t) =

nt

rnt/

J2d e- " - \ n

η


(22)

2. NEUHAUSER

33


where d are (possibly complex) constants. Our goal would be to extract the ω and the Γ for a given frequency range. We first note that the following Hamiltonian n

η

η

Η = Σ\ΦΜΦ^Ι

(23)

η

(where φ is an arbitray orthogonal basis) with a wavefunction of the form η

ά

Φη = Σ "\Φη)

(24)


η

formally gives rise to this signal: ίΗ

C(t) = (ψ \ε- '\φο).

(25)

0

Next, we apply the Gaussian filter, and resolve it to a time-dependent integral to obtain (see Ref. [4] for derivation): S = {j

J -^Ε )φ -^ΐΑΤ- ^

e - ^ - E ^ / ^ d o

β

3

6

0

( 2 6 )

A similar equation applies for H : Hy = i - ( £ . - ^ / 4 . * o J £ . e

(

+

E

j

+

i^jW+BiWe-t/WcWdt.

(27)

The important feature about this equation is that it does not involve Η explicitly, only C(i). Thus, it can be applied to extract frequencies and damping coefficients from a general signal, in spite of the fact that Η is unknown. The signal not be quantum in origin - a correlation function of a classical origin [5] applies equally well. We emphasize the difference between this approach and other methods for ex tracting signals. Thus, the simplest approach for extracting frequencies from a signal is from the peaks of the power spectrum: iEt

2

P(E) = I j (t)e C(t)dt\ 9

(28)

where g(t) is a damping function. The difficulty with this approach is first that long propagation times (slowly decaying g(t)) are required. Further, overlapping resonances cannot be resolved. An alterate is furnished by a host of methods (Prony's approach, MUSIC, maxi mum entropy [22, 23, 24]) which extract the full set of frequencies by solving linear algebra equations. Here however we note that for systems with a very large set of frequencies, the required algebra can be formidable. As in the quantum dynam ics problem, filter-diagonalization bridges the gap between the single-propagation and the large-matrix diagonalization methods, retaining the advantage of each approach. We have applied the frequency extraction approach in various contexts. In quan tum dynamics studies, it was used [25] for extracting Floquet frequencies in timedependent studies within the t,t' formalism [26]. We have also tested in on ran dom signals containing 10000 frequencies. Additionally, it was used for a classical



34

molecular dynamics simulation of Ar clusters [5]. Wells has applied the code for extracting overlapping resonance frequencies in laser-molecule reactions [13]. Resonance Functions and Scattering Another application of Filter Diagonalization is as an aid for reactions involving resonances. Thus, the most fundamental quantity in scattering is the energy resolved scattering function, E

H

*(JS) = i J Jl - H dt,

(29)


0

where H is assumed to contain the proper absorbing boundary condition. Straight forward application of approaches to extract $(E) (using either the time-dependent or time-independent version in Eq. 29) would therefore require much propagation time. A key for solving this difficulty was outlined in Ref. [2] and improved and exemplified in recent work with G. D. Kroes [6]. The key observation is that even in the presence of narrow resonances, most of the wavefunction is damped after a short "direct-reaction" time. Thus, we divide the effort into two parts. We propagate and find the resonances which are long-lived, and then use them to describe the behavior of the wavefunction at long times. (The relevant formulae are discussed in greater detail in [6].) The first part is the evaluation of the resonance wavefunction. These are ex tracted, e.g., by applying Filter-Diagonalization with a weight function g(t) = - ^ e

2

'

2

T

\

(30)

where t is a time by which most resonances have decayed, and Τ is typically taken as t /A. This time-dependent weight function, sampling late times, is completely equivalent to a Gaussian energy filter 0

0

f(E

-H)

=

i{E-H)t -(H-E)^/2.do

e

0e

(

3

1

)

The Filter-Diagonalization matrices take then the form S« = exp(-(Ei

2

E

H

M

- E ) Tyi.dQ){to\^ -W*e-l -^r'^ \& ), j

(32)

i

with a similar expression for H . The matrix elements in Eq. 32 can be evaluated using any desired propagation scheme; the final result again reads: S

y

= exp(-(Ei - EtfP/i.dO)

Σ àn((Ei + £;)/2)fl„,

(33)

η

where â are numerical coefficients; when a Chebyshev propagator is used â (E) n

= (2 - ί

η 0

) Η Γ / e^-^MtA^e-^^'/^dt.

(34)

where Η and Δ # are the Chebyshev parameters [19] and we introduced the Bessel polynomials. We have outlined one approach for calculating the a coefficients in Ref. [4]; an alternate, very efficient algorithm has been recently found [7] so that n


2.

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35


the these coefficients take negligible computation time. Again, similar equations follow for H . One difficulty in this calculation is that in scattering, due to the need to absorb the wavefunction, an absorbing potential is inserted into H, making the Chebyshev propagation unstable. There are various ways to overcome this difficulty; the simplest is to use damped Chebyshev propagations, as developed by Mandelshtam and Taylor [8]. We have used this approach successfully [6] to extract resonance states in a multidimensional H + C u scattering. Other approaches are possible, including breaking the time-integration into several distinct parts [1] or using generalized Chebyshev Hamiltonians [14], as well as Newton Propagations [21]. Once we have the S and H matrices, we diagonalize the resulting eigenvalue equation (Eq. 13) and use them to find the resonance wavefunctions, φ . These are then used to shorten the propagation times, as follows. We rewrite the expression for Φ(Ε) in terms of a time-dependent expression as a sum of contributions up to ίο and from t , which readily yields (with r = t — to)


2

η

0

ftQ E

H

iEt

Φ ( £ ) = / e^ - H dt Jo

+e °

0

ΛCO / e^ - ^ {t ), Jo E

H

0

0

(35)

or in the equivalent time-independent form: 1

1 _

F»° = Ε - Η

J{E-H)t

ι

0

* „ Η-Ε

Φο + τ Λ τ β ^ - ^ Φ ο Ε-Η υ

(36)

The physical assumption that the non-resonant scattering vanishes after to implies that the second term in the right hand side of the preceding equations can be evaluated analytically as a sum over the few resonance states, Β

_1_β··ί -^Φ

0

=^ ^ -

¥

— \ φ

η

) .

(37)

We further note that typically one does not require the full scattering wavefunction at all grid points but only at a few grid points (where the flux orflux-amplitudeis evaluated). This then implies that the resonance states need to be extracted only at a few grid points, avoiding storage difficulties. As an example of the achievements of this formalism, we show in Figure 2 how Filter-Diagonalization enabled the reduction of the total propagation time by a factor of 8, for an H + C u model [6]. 2

Discussion and Conclusion Filter Diagonalization emerges as a general approach for extracting high-energy eigenvalues and eigenfunctions from matrices and operators, or, equivalently, ex tracting frequencies from a time-dependent signal. We outlined the use of the method and have shown that it only requires a small number of residues (or a short segment of the correlation function) for extracting the energy spectrum, part by part. The method is stable and allows for the extraction of overlapping resonances. The success of the method stems from the fact that in constructing the small matrices to be diagonalized, the residues are effectively "filtered", i.e., summed at the desired energy range. Thus, one restricts the sampling to a small portion of the spectrum.



ο-

0.30

0.20

0.50

0.50

0.40

0.40 Ekin (eV)

0.60

0.60

Q-

0.20

0.45

1.00

0.20

0.5

0.30

0.30

0.40 Ekin (eV)

0.40

0.50

0.50

0

F i g u r e 2. A demonstration of the application of Filter-Diagonalization within an Hi -f Cu scattering model [6]. Parts (a-c) show the result of direct prop agation to 0.6, 2.4, and 19.6 psec; Part (d) shows and with extraction of the resonance states the total propagation time can be reduced to t = 0.6psec (with reduction of the total effort by a factor of 8).

0.30

0.20

0.5


0.60

0.60


2.

NEUHAUSER


37

Technically, new developments in Filter-Diagonalization are still emerging. For example, most recently Mandelshtam and Taylor have derived an alternate equa tions to (14,15), also appropriate for signal processing but using a 1/(H — E) filter rather than a Gaussian Filter. The use of Lancosz techniques after a FilterDiagonalization stage can also be anticipated. We also note that the method may prove a useful tool for extracting frequencies from real-time path integral simulations, as it does not require long-time portions of the full wavefunction. Finally, we expect that the method would have impact in areas where the dy namics is only approximately of an eigenvalue form, i.e., in regions where the strict frequency sum of Eq. 25 is only approximately valid. These include, for example, approximate quantum dynamics calculations (e.g., T D S C F [27] or more complicated approaches, as pursued recently by Manthe and Matzkies [11]) where the method would enable extraction of approximate eigenvalues (analogous to the classical normal mode calculation in Ref. [5]). The method also allows the study of the transition, in classical mechanics, from instantenous normal-mode (T —• 0) to long-time averaged modes [28]. Acknowledgements I gratefuly acknoweldge discsussions with R. Baer, R. Kosloff, V . A . Mandelsh tam and H . S. Taylor. Many of the ideas reporeted here were developed in discus sions with my students J . Pang and M . Wall, and through a frutiful collaboration with Dr. G . D. Kroes of Leiden University. Financial support of the NSF, the NSF Early Career Award program and the Sloan Foundation is acknowledged.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

D. Neuhauser, J. Chem. Phys.93, 2611 (1990). D. Neuhauser, J. Chem. Phys.95, 4927 (1991). D. Neuhauser, J. Chem. Phys.100, 5076 (1994). M. R. Wall and D. Neuhauser, J. Chem. Phys.102, 8011 (1995). J. W. Pang and D. Neuhauser, Chem. Phys. Lett.252, 173 (1996). G. D. Kroes, J. W. Pang and M. R. Wall, submitted for publication (1996). D. Neuhauser, J. W. Pang and M. R. Wall, to be submitted. V. A. Mandelshtam, T. P. Grozdanov and H. S. Taylor, J. Chem. Phys.103, 10074 (1995). V. A. Mandelshtam, H. S. Taylor and W. H. Miller, J. Chem. Phys.105, 496 (1996). V. A. Mandelshtam and H. S. Taylor, J. Chem. Phys., submitted. U. Manthe and F. Matzkies, Chem. Phys. Lett.252, 71 (1996). M.J. Bramley, J.W. Tromp, T. Carrington Jr. and B.T. Sutcliffe, J. Chem. Phys. 98, 10104 (1993). J. C. Wells, submitted for publication, 1996. D.J. Kouri, W. Zhu, G. Parker, and D.K. Hoffman, Chem. Phys. Lett. 238, 395 (1995).


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[15] [16] [17] [18] [19] [20]

S.K. Gray, J.Chem.Phys.96, 6543 (1992). R. Q. Chen and H. Guo, J. Chem. Phys.105, 1311 (1996). R. E. Wyatt, Phys. Rev. Ε 51, 3643 (1995). Ν. Moiseyev, P. R. Certain and F. Weinhold, Mol. Phys. 36, 1613 (1978). R. Kosloff, J. Phys. Chem. 92, 2087 (1988). J.K. Cullum and R.A. Willoughby, Lanczos Algorithms for Large Symmetric Eigenvalue Computations, birkhäuser, Boston, 1985. [21] M. Berman, R. Kosloff and H. Tal-Ezer, J. Phys. A 25, 1283 (1992). S. M. Auerbach and C. Leforestier, Comp. Phys. Comm. 78, 55-66 (1993). [22] R. Roy, B. G. Sumpter, G. A. Pfeffer, S. K. Gray and D. W. Noid, Phys. Rep. 205, 109 (1991). [23] S. K. Gray, J. Chem. Phys.96, 6543 (1992). [24] F. Remade, R. D. Levine and J. L. Kinsey, Chem. Phys. Lett.205, 267 (1993). R.D. Levine, J. Phys. A 13, 91 (1980). [25] J. W. Pang, D. Neuhauser and N. Moiseyev, J. Chem. Phys., submitted (1996). [26] U. Peskin and N. Moiseyev, J. Chem. Phys.99, 4590 (1993). [27] B. Vekhter, M. A. Ratner and R. B. Gerber, J. Chem. Phys.99, 7916 (1993). [28] R. Stratt, private communication.


Filter Diagonalization - ACS Publications - American Chemical Society

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