Spectral Analysis of the HO2

Chapter 3

Spectral Analysis of the HO Molecule 2

1,3

2

Jörg Main , Christof Jung , and Howard S. Taylor

1

Downloaded by UNIV OF ARIZONA on January 11, 2013 | http://pubs.acs.org Publication Date: June 10, 1997 | doi: 10.1021/bk-1997-0678.ch003

1

2

Department of Chemistry, University of Southern California, Los Angeles, CA 90089 UNAMInstitutode Matematicas, Unidad Cuernavaca, Av. Universidad s/n, 62191 Cuernavaca, Mexico We present a scaling technique to analyse quantum spectra, i.e. to obtain from quantum calculations detailed information about the un derlying important classical motions and the statistical properties of level spacings. The method can be applied to systems without a clas sical scaling property as, e.g., the rovibrational motion of molecules. A demonstration on the conventionally unassignable vibrational spec trum of the HO radical reveals remnants of classical broken tori em bedded in the chaotic phase space and leads to a new assignment of spectral patterns in terms of classical Fermi resonances between the local mode motions. The nearest neighbor distribution of level spac ings undergoes a transition from mixed phase space behavior at low energies to the Wigner distribution characteristic for chaotic sytems at energies near the dissociation threshold. 2

Quantum spectra of classically regular systems are interpreted in terms of the un derlying classical dynamics that influence the quantum spectra, e.g., in molecules often the normal or local mode vibrational motions that take place on stable tori in phase space (1). Since many systems, especially upon excitation show classi cally chaotic motion, it is of fundamental importance to perform an analogous analysis for chaotic systems, i.e. to extract information about the important clas sical motions directly from quantum spectra and to assign spectral patterns in terms of broken tori or unstable periodic orbits (2). This is a nontrivial problem especially if the underlying classical dynamics of the quantum system changes significantly with energy. Exceptions are systems possessing a classical scaling property, i.e. the classical phase space structure is the same for all values of an appropriate scaling parameter which is usually some power of an external field strength or, for Hamiltonians with homogenous potentials, the energy and can be controlled in the laboratory. Examples are the three-body Coulomb system (3) Permanent address: Institut fur Theoretische Physik I, Ruhr-Universitàt Bochum, D-44780 Bochum, Germany © 1997 American Chemical Society In Highly Excited Molecules; Mullin, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1997.

39

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

40

or atoms in magnetic fields (4) where information about the underlying classical dynamics can be extracted from quantum spectra by application of scaled-energy spectroscopy (5, 6). In contrast, e.g., molecular vibrational Hamiltonians are prototypical examples of classically non-scaling systems. Here we present an extended version of the scaling technique which does not allow scaled-energy spectroscopy in the lab but can be applied in theoretical quantum calculations to provide information on the classical dynamics, periodic motions, and statistical properties of spectra at any given energy. Such informa tion can usually not be obtained from a single eigenstate φ at energy Ε = E but only from many states at energies around E . Analysing these states auto matically implies a smearing over the energy. The aim of the scaling technique is to obtain additional information by creating a family of quantum systems all of which have the same underlying classical dynamics at energy E. The eigen states of these systems can be analysed without any smearing in energy to create a diagram which highlights at any given energy, certain values of the periods. Classical trajectory calculations, by finding periodic orbits with matching peri ods, then reveal those motions and phase space structures that influence both the regular and irregular quantum spectra. The importance of the method is for chaotic dynamics as regular systems are well understood (1, 7).


η

n

The Concept of the Scaling Technique Consider a multidimensional system given by the Hamitonian Η = T + V'(q) with a general non-homogenous potential V(q). For the kinetic energy we assume for simplicity Τ = ρ /2m in the following, generalizations of the scaling technique to more complex representations of kinetic energy will be straightforward. The basic idea is to enlarge formally the parameter space of the potential V'(q) by introducing an additional parameter z. The dependence of the potential on ζ is defined by V'(q; z) = V(q/z; ζ = 1) = V(q/z) = V(q) (1) 2

where q = c\/ζ are scaled coordinates, i.e. ζ is a scaling parameter describing a blow up or shrinking of the potential, with ζ = 1 corresponding to the true physical situation. It follows that V'(q; z) is the value of the original potential at q/z. By considering the family of systems given by the Lagrangian 2

2

L = m q / 2 - V(q; z) = mz q /2 - V(q)

(2)

it is easy to show that ρ = ^p is the canonical conjugate momentum to q. We can now consider the parametric family of Hamiltonians ^=ip

2 +

V(q/,)

=

l

£

+

V( = l T q )

+

V(q)

.

(3)

The fact that the transformations of ρ and q are linear and canonical means that the Schrodinger quantisation rules can be applied directly to the new variables ρ and q. This also means that the classical motion and particularly the shape

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

3. MAIN E T AL.

Spectral Analysis of the H0 Molecule 2

41

of classical (periodic) orbits are not affected by a consistent scaling. Orbits just blow up and shrink in the same way as the potential. Considering now the quantum mechanics, obviously the systems have differ ent quantum spectra for various ζ values. Each quantum Hamiltonian H(z) = p /2mz -h V(q) has an effective H, h f[ = h/z (or alternatively an effective mass m ff = m z ) , i.e. the Hamiltonians H(z) describe a family of quantum systems with the same underlying classical dynamics. The semiclassical and classical lim its are reached as ζ —> oo (^ flf —•()). Note that so far the scaling parameter ζ is not a dynamical variable. 2

2

e

2

e

e


Scaled (E,z) Diagram of the H 0

2

Molecule

As an example to illustrate our scaling technique on a real system we study the vibrational spectrum of the hydroperoxyl radical, HO2 which shows a classically chaotic dynamics. Using the potential surface of Pastrana et al. (8) 361 bound states have been calculated recently at the physical value ζ = 1 (9, 10). Only about the 32 lowest states out of 361 bound states can be assigned (11) in the conventional regular spectra sense (1), i.e. methods of fitting regular spectra fail at higher energies and wavefunctions show no regular patterns (9). For the quantum calculation of eigenstates at various ζ values we use a D V R grid representation of basis functions and the computational method of filter diagonalization (10). A technical detail worth noting is that in a first step we diagonalize the Hamiltonian at a fixed value ζ = 2 with matrix dimension > 100000 and then in a second step we use about 2600 eigenfunctions at ζ = 2 as a small basis set to calculate eigenvalues at arbitrary ζ values. Details of the numerical procedure will be given elsewhere. A part of the resultant (E, ^)-diagram is shown in Fig. l a . For graphical purpose we plot in Fig. l b (E — Ε^)ζ vs. ζ where Eh -- —2.38 eV is the bottom energy of the H 0 potential surface. In the latter presentation the avoided crossings between levels become more pronounced and indicate the chaotic dynamics of the molecule. 2

Semiclassical Analysis of the Scaled Spectra The decisive step for the scaling technique is now to analyse the eigenstates in Fig. 1 not along a line of constant ζ but along lines of constant energy Ε (dashed lines in Fig. lb). By projecting all intersections of the lovi of levels with a line Ε = const on the ζ axis we obtain eigenvalues Zi(E), and from this the density of as a function of z. This is equivalent to considering states ρ (ζ) = Σίδ{ the scaling parameter ζ as a new dynamical variable and rewriting Schrodinger's equation Ε

ζ

V + V(q) \Φ) = Ε \ψ)


(4)



42

Figure 1: Scaled spectra for the H 0 molecule, (a) Energy Ε versus scale param eter z. (b) (E — Eb)z versus z. The dashed lines mark lines of constant energy at (from the left) Ε = -0.6 eV, Ε = -0.8 eV, and Ε = -1.0 eV. 2


3. MAIN E T AL.


43

for a particular fixed value of the energy Ε as a Hermitian eigenvalue equation for λ = z~ , 2

ι 2

1/2

Τ~ ' [Ε

- ν(ς)]Τ- \Φ)

= - ί | Φ ) ; |Φ> = Τ ^ Ι Φ )

(5)

2

where Τ = ρ / 2 m is the operator of scaled kinetic energy. The eigenvalue equa tion (5), which when diagonalized obviously also gives Fig. 1, yields the values of ζ for which Ε is an energy eigenvalue. The new Hamiltonian on the left can be used to formally set up the Greens function and density of states E

=

Ε

=


G (X) ρ (ζ)

1

1

2

1

(6)

E

Im Tr G (X)

ά

πζ

i

z

=

2

[λ-Γ- / ^-Vjr- ' ]"

£i( -,f) 2

.

(7) E

For the interpretation of the quant ally calculated Q (z) we can now formally com pare to semiclassical theories, i.e. the Berry-Tabor formula for semiclassical torus quantisation (12) or here Gutzwiller's periodic orbit theory for the semiclassical density of states of classically chaotic systems (2). Gutzwiller's trace formula gives the semiclassical approximation to Eq. 7 as Ε

ρ (ζ)

Ε

= ρ (ζ) + £ p.o.

A sin (zS k

k

k

V

- ^μΛ Z

(8) '

Ε

where ρ {ζ) is the mean level density, Ak is the amplitude related to the stability matrix of periodic orbit k, μι^ the Maslov index, and S = S/ζ the scaled classical action which can also be proven to be the canonically conjugate variable to z. Information about the classical dynamics can now be obtained from the quantum spectra by a Fourier transform of ρ (ζ). From Eq. 8 follows that each peak in the Fourier tansformed action spectra can be identifed either with an individual periodic orbit or with a braid of orbits with similar shapes and periods related to a broken resonance torus. For the Η 0 molecule the Fourier transform action spectra along lines of constant energy are presented in Fig. 2 in overlay form in the region —2.0 eV < Ε < —0.2 eV. We performed extensive classical calculations and the quantum recurrence spectra are superimposed on the classical bifurcation diagram (dashed lines in Fig. 2) obtained from a search for periodic orbits. The total number of periodic orbits in this threedimensional system is incredibly high and in the classical bifurcation diagram orbits are shown only which exist in longer energy ranges (i.e. do not undergo close in energy bifurcations). Details of the classical calculations will be given elsewhere. The quantum spectra show very sharp and detailed structures. The sharpness of peaks in the Fourier transform indicates Ε

2



Ο

10

20

30

40 S/2n

50

60

70

80 Ε

Figure 2: Magnitude square of the Fourier transformed scaled spectra ρ (ζ) of the HO2 molecule. The action spectra are superimposed on the classical bifurcation diagram (dashed lines).


3. MAIN E T AL.


45

10000 8000 CM_ CO

6000

-

4000


2000 0 0

10

20

30

40 50 60 70 80 8/2π Figure 3: Fourier transform density of states at constant energy Ε = —1,0 eV. Some peaks are assigned by frequency ratios of classical trajectories on broken resonance tori.

the importance of a particular orbit or braid of orbits along the whole range of ζ values (classical to quantum). The quantally obtained peak heights are related to the amplitudes A of periodic orbits in Gutzwiller's trace formula (Eq. 8). A direct semiclassical calculation of these peaks for a chaotic 3D system is a presently unachievable task requiring the finding of all periodic orbits up to a certain period. Indeed, a careful examination of Fig. 2 shows that not all quantum structures are explained by those orbits that we have computationally identified. On the other hand orbits not highlightened in Fig. 2, which are needed to converge the periodic orbit sum (Eq. 8), do not make major qualitative contributions to the quantum dynamics. The selected periodic orbit structures (which might be isolated orbits or broken resonance tori if families of similar periodic orbits exist under one peak) are the dynamics which determine the quantum spectra up to intermediate resolution. These structures are markers for regions in phase space where the dynamical potential causes trapping which is seen by quantum mechanics (13). The periods and amplitudes of the important structures highlightened in Fig. 2 depend on the energy. For example there is a weak resonance structure around Ε = —1.2 eV, 5 / 2 π = 45 whose intensity increases at higher energies to maxi mum strength around Ε = —1.0 eV, §/2π = 52 and then decreases again. Here highlighted periodic orbits show that such structures can be explained by classical trajectories moving on broken resonance tori. The individual peaks being asso ciated with recurrent motion have rational frequency ratios V\ : V2 : ^3 between k



46


2.2

2.4

2.6 2.8 R (0-0)

3 1.4 1.6 1.8 2 2.2 2.4 R (0-H)

60000 C\J_

S 40000

0.0005 0.001 0.0015 0.002 0.0025 0.003 ν Figure 4: (a) and (b): Projections of a periodic orbit at energy Ε = —1.0 eV in local mode coordinates #ΟΟΛ#ΟΗ, and bending angle 7. (c): Fourier transform of local mode motions showing a frequency ratio of 17 : 6 : 5.

the three local mode motions, viz. the O-H stretch, the O-O-H bending, and the 0 - 0 stretch. Interestingly most of the stronger peaks in Fig. 2 can be identified and assigned in this way, i.e. they represent classical Fermi resonances between the local modes. Fig. 3 shows as an example the assignments of the important structures in the scaled action spectrum at constant energy Ε = —1.0 eV. (If one or two local mode motions have nearly zero amplitude we assign these modes formally by frequencies ν = 0.) The strongest peak at 5 / 2 π = 52.5 belongs to a frequency ratio U\ : v '· ^3 = 17 : 6 : 5. This peak can be related to a family of similar periodic orbits localized in phase space, one member is shown in Fig. 4a,b in projections of the local mode coordinates. The Fourier transform of the local mode motions (Fig. 4c) clearly reveals the frequency ratio mentioned above. In fact in place of the labor intensive periodic orbit search used to identify the dynamics under the peaks, time segments of a long trajectory can be Fourier analysed to see if the frequencies of the three local mode motions are in rational 2


3. MAIN E T AL.


47


ratio. If they are the trajectory segment lies close to important periodic orbits and can be used to find them rather easily. V i b r o g r a m A n a l y s i s of Spectra. A method which supports similar analysis is a windowed Fourier transform (Gabor transform) of quantum spectra resulting in vibrograms being a function of energy and time (14)> This analysis has been applied to various classically non-scaling systems (15. 16) and has the advantage that it can be directly applied to experimental spectra. However, the resolution of vibrograms in energy and time is fundamentally restricted by Heisenberg's uncertainty principle (15). To improve the resolution and to relate structures to various classical orbits, k has to be reduced in theoretical calculations (16) which eliminates the experimental connection. In contrast the scaling technique introduced here works at single energies and resolution is not restricted by the uncertainty principle. Results obtained for the HO2 molecule go far beyond a vibrogram analysis which will be presented elsewhere but where diagrams are not defined enough to correlate its broad peaks to specific longer periodic orbits. Statistical A n a l y s i s of Nearest N e i g h b o r Spacings Another advantage of the scaling method is its application in a statistical analysis of eigenvalues. Statistical properties often depend strongly on energy but if the density of states is low short range energy intervals do not contain a sufficient number of states to carry out a statistical analysis. Here additional monoenergetic quantum information for such an analysis can be generated by application of the scaling technique. It is well established that the nearest neighbor spacing distribution of inte grable quantum systems after unfolding of the spectra to unit mean level spacing (s) = 1 is given by a Poisson distribution S

Ppoissonf*) = e~ ,

(9)

while quantum systems with a fully chaotic (ergodic) underlying classical dynam ics are characterized by the Wigner distribution 2

fWer(s) = y e — '

4

(10)

obtained from random matrix theory (17). In systems with a mixed regularchaotic classical dynamics the nearest neighbor spacing distribution can be phenomenologically described by the Brody distribution (18) PBrody(^9)-(9+l)^e-^

+ 1

(11)

where β=Γ

q+2

Spectral Analysis of the HO2

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