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Back | Show only these items: | Iterative quantum computations of HO2 bound states and resonances for J = 4 and 5

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Iterative quantum computations of HO₂ bound states and resonances for J = 4 and 5

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj1

Bound and resonance states of HO₂ have been calculated by both the complex Lanczos homogeneous filter diagonalisation (LHFD) method^1,2 and the real Chebyshev filter diagonalisation method^3,4 for non-zero total angular momentum J = 4 and 5.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj1

For bound states, the agreement between the two methods is quite satisfactory; for resonances while the energies are in good agreement, the widths are only in general agreement.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con1

The relative performances of the two iterative FD methods have also been discussed in terms of efficiency as well as convergence behaviour for such a computationally challenging problem.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj2

A helicity quantum number Ω assignment (within the helicity conserving approximation) is performed and the results indicate that Coriolis coupling becomes more important as J increases and the helicity conserving approximation is not a good one for the HO₂ resonance states.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con2

Introduction

Exact calculations for total angular momentum J > 0 are essential in fully understanding quantum reaction dynamics.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac1

However, these J > 0 calculations are still very challenging even for triatomic reactions, especially when dealing with complex-forming systems.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac1

The major reason for this situation is the so called ‘angular momentum catastrophe’:⁵ many J > 0 calculations have to be performed, and the size of the Hamiltonian matrix increases linearly with J.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac1

For these non-zero J calculations, it is apparently impractical to employ conventional direct diagonalisation methods due to the requirement of a significant computer core memory.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

On the other hand, iterative methods are well suited to solve this large-scale eigenvalue problem.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

Among different iterative methods, two of the most powerful methods which have emerged are the real Chebyshev filter diagonalisation method (RCFD)^3,4,6 and complex Lanczos filter diagonalisation method.^1,2,7–9

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met1

The basic idea of filter diagonalisation¹⁰ is to act with operator functions f(E_i − Ĥ) onto a generic initial wavepacket to generate filtered states at a sequence of chosen filter energies E_i.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met1

Eigenstates in the energy window can then be obtained by diagonalising a small-sized Hamiltonian matrix in a subspace spanned by the filtered states.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met1

However, a significant difficulty presents itself in that a large storage requirement is needed for storing the filtered states in the primary representation in earlier FD versions.

Type: Method | Advantage: No | Novelty: Old | ConceptID: Met1

Later, Wall and Neuhauser proposed an auto-correlation function based filter diagonalisation scheme,¹¹ in which eigen-values in the energy windows can be obtained from a single sequence of auto-correlation functions without the need to explicitly construct the filtered states.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met2

This essentially eliminates the memory bottleneck.

Type: Method | Advantage: Yes | Novelty: Old | ConceptID: Met2

Mandelshtam and Taylor³ combined this scheme with their modified real Chebyshev propagation approach, and implemented a box-like low storage filter diagonalisation formalism.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met3

The most important advantage associated with this approach is that one can employ a real algorithm with a single, extended Chebyshev vector recursion.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met3

This leads to substantially greater efficiency in comparison with step-by-step propagation of a complex wavepacket.

Type: Method | Advantage: Yes | Novelty: Old | ConceptID: Met3

Very recently, Li and Guo proposed a very efficient scheme of doubling Chebyshev correlation functions for calculating narrow resonances,¹² with numerical validation of their proposal for two molecular systems.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met4

Meanwhile, Neumaier and Mandelshtam derived a pseudo-time Schrödinger equation and provided a rigorous proof that an exact doubling formula exists for damped Chebyshev propagation.¹³

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met5

Another attractive approach is to use iterative methods based primarily upon the Lanczos algorithm of tri-diagonalisation.^14,15

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met6

Lanczos-based FD methods have been introduced by Yu and Smith^7–9 and by Chen and Guo.¹⁶

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met6

These methods are especially useful when one only considers a small section of the entire spectrum as in most applications.

Type: Method | Advantage: Yes | Novelty: Old | ConceptID: Met6

In addition, they have the desirable property of avoiding most of the ghost eigenvalues that will appear in a normal Lanczos diagonalisation.

Type: Method | Advantage: Yes | Novelty: Old | ConceptID: Met6

A key feature of Smith et al.’s Lanczos representation FD methods^8,17 is that the entire calculation is carried out within the Lanczos subspace.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met7

This has the major advantage that one does not incur the memory cost of having to explicitly store the filtered states in the primary representation.

Type: Method | Advantage: Yes | Novelty: Old | ConceptID: Met7

Consequently a single Lanczos subspace serves to analyse the entire spectrum, since the tridiagonal representation of the Hamiltonian makes it a straightforward exercise to generate a new set of filtered states in the next energy window by solving QMR or MINRES equations implicitly within the Lanczos subspace.

Type: Method | Advantage: Yes | Novelty: Old | ConceptID: Met7

Very recently, we have developed a simpler and more efficient Lanczos homogeneous filter diagonalization (LHFD) algorithm based on a very simple homogeneous filtering recursion within the Lanczos representation.¹

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met8

Using the LHFD method, J = 1–3 bound and resonance calculations on HO₂ have been performed in a previous paper.¹⁸

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met8

In that paper we compared our calculations with Wu and Hayes’s results¹⁹ for low lying bound states for J = 1–3, but for resonances no other calculations have been available except our own Lanczos results.

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot1

For J = 4 and 5, to our knowledge, no calculations have been published for either resonance or bound states, so we will use the two most powerful FD methods presently available (namely, the Chebyshev and Lanczos approaches) to perform the J = 4 and 5 calculations.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj1

The computational tasks are demanding for a system with a deep well such as this, in particular for the purpose of resolving individual resonances.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac3

It will also be of some interests to compare the relative efficiency of Lanczos-based and Chebyshev-based approaches to the computation of molecular bound states and resonances for more challenging non-zero J calculations.

Type: Object | Advantage: None | Novelty: None | ConceptID: Obj2

Some such comparative studies have appeared for bound state problems,^20–24 and for resonance calculations.^25,26

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met8

In the above comparative studies,^23,24,26 it was found that in RCFD the convergence behaviour can be accurately predicted by the spectral range and the physical density of states.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met8

For resonance calculations, the Chebyshev algorithm required roughly the same number of iterations as the Lanczos method if doubling schemes^12,26 for computing the autocorrelation functions are employed, but due to its use of real algebra the Chebyshev FD methods may gain some advantages.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met8

In the present work we present a comparison of Lanczos-based and Chebyshev-based FD methods for the much more demanding problem of the resonance calculations of the HO₂ radical for J = 4 and 5.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj2

Our objective is to examine how the relative efficiency of the Lanczos and Chebyshev iterative approaches scales with the order of difficulty of the calculation.

Type: Goal | Advantage: None | Novelty: None | ConceptID: Goa1

It is now well known^1,9 that convergence of the very narrow resonances requires ca. 40 000 complex-symmetric Lanczos iterations for J = 0 HO₂ case, and it will be interesting to see the convergence behaviour for higher J values of HO₂ for the two iterative methods.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

Due to the computational constraints, approximate quantum methods such as adiabatic rotation (AR),²⁷J-shifting²⁸ and helicity conserving (HC)²⁹ approximations are commonly used for non zero J calculations.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac5

The key issue in these approximations is whether a reasonably good quantum number Ω associated with the projection of total angular momentum on a body fixed axis exists.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac5

If the substates Ω of the wave function for J > 0 are heavily coupled, the Coriolis coupling between the states cannot be ignored and any attempts to assign the helicity quantum number Ω will fail.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac5

Instead of explicitly computation of bound and resonance states within these approximations, we will examine this issue by a helicity quantum number Ω assignment for both bound and resonance states.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj3

The rest of this article is organised as follows.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met9

In section 2 we briefly describe the theoretical methods needed to characterize both bound and resonance states for non-zero total angular momentum.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met9

In section 3 we shall present the results of J = 4 and 5 bound and resonance calculations and give discussions concerning the relative performances of the two iterative FD methods and the helicity Ω assignment.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res1

Section 4 concludes.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con3

Methodology

In general, we treat the three internal Jacobi coordinates (R,r,γ) in discrete variable representation (DVR), while the three Eulerian angles (θ,ϕ,ψ) are described in a basis set.^30–32

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met10

This procedure is very efficient because the potential part of the Hamiltonian matrix is diagonal, which can reduce the memory requirement substantially.

Type: Method | Advantage: Yes | Novelty: Old | ConceptID: Met10

Using symmetry-adapted symmetric top eigenfunctions to expand the total wave function, one can get the Hamiltonian matrix explicitly represented in DVR (for more details the reader is referred to ^{ref. 18}):withIn eqn. (1), we have used Ω-dependent DVR for γ coordinate, which is obtained by either diagonalizing the coordinate, operator (x = cos γ) matrix or by a Gauss–Jacobi quadrature scheme.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met11

We have compared the two DVR schemes, and the DVR points as well as the transformation matrix T from the two methods are nearly the same.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res2

We calculate at the outset and then store the neighbouring Coriolis coupling matrices [see the last two terms in eqn. (1) for the details of the matrix elements] for each Ω component.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met12

The memory requirement for the coupling matrices is not large, and whenever they are needed in the iterations, we use them to perform the Hamiltonian matrix vector multiplications directly within the DVR.

Type: Method | Advantage: Yes | Novelty: Old | ConceptID: Met12

For R and r coordinates, we have used potential optimised DVR.³³

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met13

In the two iterative methods, the basic propagation is quite similar in principle, which is a three-term recursion.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met13

In Lanczos iteration, we use the basic Lanczos algorithm for complex-symmetric matrices³⁴ to project the non-Hermitian absorbing potential augmented Hamiltonian into a Krylov subspace.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met14

We need to store the two complex vectors in Lanczos iterations for later FD analysis to extract physical information such as energies and widths.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met14

Similarly, in Chebyshev propagation, a real modified Chebyshev polynomial propagation proposed by Mandelshtam and Taylor^35–37 has been employed.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met15

Again, we will store the correlation functions in the Chebyshev propagation for later FD analysis.

Type: Method | Advantage: None | Novelty: None | ConceptID: Met15

After setting up tridiagonal Lanczos subspace, we perform LHFD inside the subspace to extract the bound and resonance information.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met16

The key issue in LHFD is to solve the homogeneous linear system(E_j − T_M)|ϕ(E_j)〉 = 0to generate filtered state ϕ(E_j) at different filter energies E_j.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met16

Here a backward three-term substitution recursion is employed.¹⁸

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met16

Then we construct the overlap matrix with elements S_jj′ = (ϕ(E_j)|ϕ(E_j′)) and subspace Hamiltonian matrix with elements W_jj′ = (ϕ(E_j)|T_M|ϕ(E_j′)).

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met16

The complex energies, {ε} can be obtained through solving the small-size generalized complex-symmetric eigenvalue problem WB = SBε.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met16

The above procedures can be repeated for any chosen energy windows.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met16

Several FD versions based on the Chebyshev recursion exist, such as Wall and Neuhauser’s version,¹¹ Mandelshtam and Taylor’s version³ and Chen and Guo’s version.⁶

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met17

We note that Guo et al.’s version is essentially identical to Mandelshtam and Taylor’s version using a box filter to compute resonances.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met17

In this paper, we will employ Mandelshtam’s real Chebyshev FD version.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met18

For more details about RCFD, the reader is referred to ^{refs. 3,4}.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met19

Results

The triatomic HO₂ Hamiltonian matrix was set up in terms of reactant Jacobi coordinates, and the HO₂ DMBE IV PES³⁸ was employed as we did for previous J = 0–3 bound and resonance calculations.^1,2,18,25

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

For the two radial coordinates, a potential-optimized discrete variable representation³³ (PODVR) was utilized to reduce the size of the Hamiltonian matrix and the details can be found in ^{ref. 18} For the γ variable, Ω-dependent symmetry-adapted DVR functions were employed to take account of the odd O−O exchange parity.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

The resulting direct product basis set was further contracted by discarding those points whose potential energies were higher than the cut-off energy V_cutoff = 2.0 eV, resulting in the final basis size of approximately 110700(J + 1) for even spectroscopic symmetry and approximately 110700J for odd spectroscopic symmetry.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

The details of the absorbing or damping potential is similar to those used in .^{ref. 18}

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

No stabilisation procedure³⁹ has been attempted due to the huge computational resources required, and this will have some effects on resonance widths (see below).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

The bound state energies as well as the resonance energies and widths for two chosen energy windows for J = 4 and 5 have been computed.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj1

The first energy window is for the lowest bound state energies from −0.08 eV to 0.92 eV.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res3

Here the zero energy point is referred to as the ground state energy of HO₂ for J = 0, which is −2.015861 eV relative to the H + O₂ dissociation limit.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res3

This energy window is relatively easy to calculate and a Lanczos subspace size M = 5000–10 000 is enough to converge all the energies in this window.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res3

In RCFD, Chebyshev iteration number needed is roughly n = 20 000–30 000.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res3

In Tables 1–4 we have listed the selected lowest bound state energies for each symmetry of J = 4 and 5 from both LHFD and RCFD methods for comparison.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res4

Inspection of the energies shows that the agreement between them is quite satisfactory and 4 digits of relative accuracy have been achieved for most of the energies.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res4

The second energy window we have chosen is close to and above dissociation threshold, namely, the highest lying bound state energies and lowest resonance energies and widths from 2.10 eV to 2.18 eV.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met20

Since these are the first calculated results, the convergence has been carefully tested and in Fig. 1 we have plotted the convergence behaviour for one resonance at E = 2.1227 eV for J = 5 (even symmetry) case from the two iterative FD methods.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res5

From Fig. 1(a) one can see that M > 1 60 000 Lanczos iterations can well converge the resonances and in all our calculations we have used a larger M = 2 20 000 Lanczos subspace size for this energy range.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res6

Similarly, Fig. 1(b) indicates that M > 5 00 000 Chebyshev iterations are needed to converge the resonance and in all our calculations we have used a larger M = 6 00 000 Chebyshev iteration number for the second energy window.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res6

These results show that the convergence is very slow for resonances and almost monotonic.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con4

The difficulty of convergence for HO₂ system is related to both the changes of spectral range and the physical densities of states as J increases.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res7

By comparison with previous J = 0–3 results, we also found that the iteration number needed to converge the resonances increases as J increases.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res7

For example, in RCFD method, 1 00 000 iterations are enough to converge most of the resonances for J = 0 case; similarly, in LHFD method, 40 000 iterations are enough to converge most of the resonances for J = 0 case.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res7

It is understandable because there are more energy levels to converge for high J values and we believe that this behaviour is general for higher J values.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con5

In Fig. 2 we have plotted the resonance widths versus energies for J = 4–5 for both even and odd symmetries from the two methods.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res8

Inspection of the resonances in these plots, one can see that while for resonance energies the agreement is good, for resonance widths the differences are relatively large.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res8

In principle, the complex resonance energies should be the same for the two methods if a perfect absorbing potential and a perfect damping operator are respectively used.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res8

However, it is very challenging to obtain the perfect absorbing or damping parameters using stabilisation method³⁹ or optimizing method²⁶ for the HO₂ system, especially for J > 0 calculations and we do not attempt this in the present work.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con6

For LHFD, we did only 6 test calculations by varying absorbing strength parameter V₀ in J = 5 even symmetry case.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met21

The results indicate that the best V₀ should fall in between 0.02 eV and 0.06 eV and we take V₀ = 0.02 eV in this paper.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res9

For RCFD, the damping potential parameters are based upon previous J = 0 calculations.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met22

Since approximate absorbing boundary conditions have been imposed here via two quite different methods it will be apparent that the resonance widths reported in this paper for either method are simply estimations of the true resonance widths.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con7

100

Having said this, observation of the plots from Fig. 2(a) to Fig. 2(d) indicates that the agreement for most resonances between the two methods is not too bad, albeit not perfect.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con7

101

For the HO₂ system, other investigators have also pointed out^40,41 that the resonances widths in particular are sensitive to many details of the calculations (e.g., basis set size, grid size, cut off energy), which adds to the complexity of accurately calculating the resonance widths.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met23

102

In principle, a stabilisation procedure³⁹ or optimizing approach²⁶ can be employed to determine more accurate resonance widths in the future.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con8

103

Here it is interesting to note that although the resonance widths do not agree very well, the general variation trends with energy are still similar for the two methods.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res10

104

In Tables 1–4 an Ω assignment has been given for the low lying bound states, supposing that the helicity conserving or adiabatic rotation approximation holds for most of them (because there exist near degeneracy for the same Ω components from both symmetries, it is possible to assign them by comparing the calculated energies from even and odd symmetries).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res11

105

The purpose of Ω assignment is to investigate the importance of the Coriolis coupling for this system.

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot2

106

If this assignment is successful, then helicity conserving calculations or even much simpler adiabatic rotation approximations should be accurate, which will save quite a lot of computational time.

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot2

107

We note that two bound states at about 0.318641 eV from the J = 5 (even symmetry) case in Table 3 are too degenerate to be distinguished and we have put them together.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res12

108

In J = 5 (odd symmetry) case, the two bound states are still distinguishable.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res12

109

For the high-lying bound states as well as for the resonances, we have failed to assign them unambiguously.

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110

For example, we have analysed the high lying bound state energies near the dissociation threshold from J = 4–5 calculations for both even and odd spectroscopic symmetries, respectively (the results for the high lying bound state energies are not shown here, and they can be acquired from us upon request).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res12

111

While only several of them can be assigned tentatively, most of them cannot be assigned with confidence.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res12

112

The indication is that the mixing of different Ω components is so strong that Ω is no longer a good quantum number.