Extraction of eigenstates from an optically prepared state by a time-dependent quantum-mechanical method. Toward simulation of “intermediate case” radiationless transitions

Neural network iterative diagonalization method to solve eigenvalue problems in quantum mechanics

The neural network iterative diagonalization structure for computing the eigenstates of complex symmetric or Hermitian matrices.

A complex guided spectral transform Lanczos method for studying quantum resonance states

A complex guided spectral transform Lanczos (cGSTL) algorithm is proposed to compute both bound and resonance states including energies, widths, and wavefunctions. The algorithm comprises of two layers of complex-symmetric Lanczos iterations. A short inner layer iteration produces a set of complex formally orthogonal Lanczos polynomials. They are used to span the guided spectral transform function determined by a retarded Green operator. An outer layer iteration is then carried out with the transform function to compute the eigen-pairs of the system. The guided spectral transform function is designed to have the same wavefunctions as the eigenstates of the original Hamiltonian in the spectral range of interest. Therefore, the energies and/or widths of bound or resonance states can be easily computed with their wavefunctions or by using a root-searching method from the guided spectral transform surface. The new cGSTL algorithm is applied to bound and resonance states of HO2, and compared to previous calculations.

QUANTUM DYNAMICAL CALCULATION OF ALL ROVIBRATIONAL STATES OF HO2 FOR TOTAL ANGULAR MOMENTUM J = 0–10

The energy levels and wavefunctions for all rovibrational bound states of HO2 are systematically computed, for all total angular momentum values J = 0–10. The calculations are performed using ScalIT, a suite of software modules designed to enable quantum dynamics and related calculations to be performed on massively parallel computing architectures. This is the first-ever application of ScalIT to a real (and very challenging) molecular application. The codes, and in particular, the algorithms (optimal separable basis, preconditioned inexact spectral transform, phase space optimized discrete variable representation basis) are so efficient that in fact, the entire calculation can be performed on a single CPU — although parallel scalability over a small number of CPUs is also evaluated, and found to be essentially perfect in this regime. For the lowest 11 vibrational states, the rotational levels for J = 0–10 fit fairly well to a rigid rotor model, with all vibrational-state-dependent rotational constants, B eff (v), close to values obtained from a previous calculation for J = 0 and 1 [J Chem Phys107:2705, 1997]. However, comparatively larger discrepancies with the rigid-rotor model are found at the higher J values, manifesting in the observed K-splitting (along the O–O bond) of rovibrational levels. This supports earlier work [J Chem Phys113:11055, 2000] suggesting that Coriolis coupling is quite important for this system.

CALCULATION OF SELECTED HIGHLY EXCITED VIBRATIONAL STATES OF POLYATOMIC MOLECULES BY THE DAVIDSON ALGORITHM

We described an improved version of a modified Davidson scheme previously introduced (F. Ribeiro, C. Iung and C. Leforestier, Chem. Phys. Lett.362, 199 (2002)), aimed at computing highly excited energy levels of polyatomic molecules. The key ingredient is a prediagonalization-perturbation step performed on a subspace of a curvilinear normal modes basis set (including diagonal anharmonicities). The efficiency of the method is demonstrated by computing the lowest 350 vibrational states of A′ symmetry of the HFCO molecule. Also shown is the possibility to restrict the calculation to selected energy levels, based on their zero-order description. This State Filtered Diagonalization method is illustrated on a high overtone (7ν5) of the OCF bend, and on the few energy levels (20) which have been experimentally assigned up to 5000 cm -1 of excitation energy.

Non-normal Lanczos methods for quantum scattering

This article presents a new complex absorbing potential (CAP) block Lanczos method for computing scattering eigenfunctions and reaction probabilities. The method reduces the problem of computing energy eigenfunctions to solving two energy dependent systems of equations. An energy independent block Lanczos factorization casts the system into a block tridiagonal form, which can be solved very efficiently for all energies. We show that CAP-Lanczos methods exhibit instability due to the non-normality of CAP Hamiltonians and may break down for some systems. The instability is not due to loss of orthogonality but to non-normality of the Hamiltonian matrix. While use of a Woods-Saxon exponential CAP-as opposed to a polynomial CAP-reduced non-normality, it did not always ensure convergence. Our results indicate that the Arnoldi algorithm is more robust for non-normal systems and less prone to break down. An Arnoldi version of our method is applied to a nonadiabatic tunneling Hamiltonian with excellent results, while the Lanczos algorithm breaks down for this system.

Long time wave packet dynamics from energy eigenfunctions: nonuniform energy resolution via adaptive bisection fast Fourier transformation

This article presents a new approach to long time wave packet propagation. The methodology relies on energy domain calculations and an on-the-surface straightforward energy to time transformation to provide wave packet time evolution. The adaptive bisection fast Fourier transform method employs selective bisection to create a multiresolution energy grid, dense near resonances. To implement fast Fourier transforms on the nonuniform grid, the uniform grid corresponding to the finest resolution is reconstructed using an iterative interpolation process. By proper choice of the energy grid points, we are able to produce results equivalent to grids of the finest resolution, with far fewer grid points. We have seen savings 20-fold in the number of eigenfunction calculations. Since the method requires computation of energy eigenfunctions, it is best suited for situations where many wave packet propagations are of interest at a fixed small set of points--as in time dependent flux computations. The fast Fourier transform (FFT) algorithm used is an adaptation of the Danielson-Lanczos FFT algorithm to sparse input data. A specific advantage of the adaptive bisection FFT is the possibility of long time wave packet propagations showing slow resonant decay. A method is discussed for obtaining resonance parameters by least squares fitting of energy domain data. The key innovation presented is the means of separating out the smooth background from the sharp resonance structure.

Comparison of Perturbative and Variational Treatments of Molecular Vibrations: Application to the Vibrational Spectrum of HFCO up to 8000 cm-1†

We calculated highly excited states of the HFCO molecule, comparing results from two methods. In the first method, Van Vleck perturbation theory is used to transform away all off-diagonal couplings except those between nearly degenerate states. This perturbative transformation leads to a matrix representation where eigenvalues are obtained with relatively small matrices. In the second method, variational eigenvalues are obtained by combining the Jacobi-Wilson approach with the block-Davidson scheme. The key ingredient here is a prediagonalized-perturbative scheme applied to a subspace of a curvilinear normal-mode basis set. Comparisons of the two methods provide a critical test of the less time-consuming perturbation theory. Two different coordinate sets are used to test the sensitivity of the results to coordinate choice. Perturbation theory also requires a polynomial fit to the potential. The implications of this restriction are investigated.

Combination of perturbative and variational methods for calculating molecular spectra: Calculation of the υ=3–5 CH stretch overtone spectrum of CHF3

Highly excited states of the CHF3 molecule belonging to the third, fourth, and fifth Fermi polyad are calculated using a combination of the Van Vleck perturbation theory and a variational treatment. The perturbation theory preconditions the Hamiltonian matrix by transforming away all couplings except those between nearly degenerate states. This transformation is implemented so that eigenvalues can be found with significantly smaller matrices than that which would be needed in the original normal mode representation. Even with preconditioning, at the energies as high as 3-5 quanta in the CH stretch, it is not possible to directly diagonalize the Hamiltonian matrix due to the large basis sets required. Iterative methods, particularly the block-Davidson method, are explored for finding the eigenvalues. The methods are compared and the advantages discussed.

Calculation of specific, highly excited vibrational states based on a Davidson scheme: Application to HFCO

We present the efficiency of a new modified Davidson scheme which yields selectively one high-energy vibrationally excited eigenstate or a series of eigenstates. The calculation of a highly vibrationally excited state psi located in a dense part of the spectrum requires a specific prediagonalization step before the Davidson scheme. It consists in building a small active space P containing the zero-order states which are coupled with the zero-order description of the eigenstate of interest. We propose a general way to define this active space P which plays a crucial role in the method. The efficiency of the method is illustrated by computing and analyzing the high-energy excited overtones of the out-of-plane mode [formula: see text] in HFCO. These overtone energies correspond to the 234th, 713th, and 1774th energy levels in our reference basis set which contains roughly 140,000 states. One of the main advantages of this Davidson scheme comes from the fact that the eigenstate and eigenvalue convergence can be assessed during the iterations by looking at the residual [formula: see text]. The maximum value epsilon allowed for this residual constitutes a very sensitive and efficient parameter which sets the accuracy of the eigenvalues and eigenstates, even when the studied states are highly excited and are localized in a dense part of the spectrum. The physical analysis of the eigenstates associated with the 5th, 7th, and 9th out-of-plane overtones in HFCO provides some interesting information on the energy localization in this mode and on the role played by the in-plane modes. Also, it provides some ideas on the numerical methods which should be developed in the future to tackle higher-energy excited states in polyatomics.

A Jacobi-Wilson description coupled to a block-Davidson algorithm: An efficient scheme to calculate highly excited vibrational levels

We present a new approach based on the block-Davidson scheme which provides eigenvalues and eigenvectors of highly excited (ro) vibrational states of polyatomic molecules. The key ingredient is a prediagonalized-perturbative scheme applied to a subspace of a curvilinear normal-mode basis set. This approach is coupled to the Jacobi vector description recently developed by our group [C. Leforestier, A. Viel, F. Gatti, C. Munoz, and C. Iung, J. Chem. Phys. 114, 2099 (2001)], and applied to the HFCO and H2CO molecules, which represent the main difficulties of such calculations for any available method. The first one presents a significant state density because of its low symmetry and the presence of a fluorine atom, while strong resonances and intermode couplings occur in H2CO. This study establishes the robustness, the numerical efficiency, and the versatility of the method which is compared to the regular Lanczos and Davidson schemes. It is also shown that the eigenvectors can be obtained within a given accuracy easily set by the user. This point constitutes one of the main advantages of the method as very few potential-energy surfaces achieve an accuracy of the order of a wave number for highly excited states. Furthermore, this method allows one to restrict the calculations to selected energy levels based on their zero-order descriptions.

Methods for calculating vibrational energy levels

This article reviews new methods for computing vibrational energy levels of small polyatomic molecules. The principal impediment to the calculation of energy levels is the size of the required basis set. If one uses a product basis the Hamiltonian matrix for a four-atom molecule is too large to store in core memory. We discuss iterative methods that enable one to use a product basis to compute energy levels (and spectra) without storing a Hamiltonian matrix. Despite the advantages of iterative methods it is not possible, using product basis functions, to calculate vibrational spectra of molecules with more than four atoms. A very recent method combining contracted basis functions and the Lanczos algorithm with which vibrational energy levels of methane have been computed is described. New ideas, based on exploiting preconditioning, for reducing the number of matrix-vector products required to converge energy levels of interest are also summarized.Key words: vibrational energy levels, kinetic energy operators, Lanczos algorithm, contracted basis functions, preconditioning.

Full-dimensional quantum calculations of vibrational spectra of six-atom molecules. I. Theory and numerical results

Two quantum mechanical Hamiltonians have been derived in orthogonal polyspherical coordinates, which can be formed by Jacobi and/or Radau vectors etc., for the study of the vibrational spectra of six-atom molecules. The Hamiltonians are expressed in an explicit Hermitian form in the spatial representation. Their matrix representations are described in both full discrete variable representation (DVR) and mixed DVR/nondirect product finite basis representation (FBR) bases. The two-layer Lanczos iteration algorithm [H.-G. Yu, J. Chem. Phys. 117, 8190 (2002)] is employed to solve the eigenvalue problem of the system. A strategy regarding how to carry out the Hamiltonian-vector products for a high-dimensional problem is discussed. By exploiting the inversion symmetry of molecules, a unitary sequential 1D matrix-vector multiplication algorithm is proposed to perform the action of the Hamiltonian on the wavefunction in a symmetrically adapted DVR or FBR basis in the azimuthal angular variables. An application to the vibrational energy levels of the molecular hydrogen trimer (H2)3 in full dimension (12D) is presented. Results show that the rigid-H2 approximation can underestimate the binding energy of the trimer by 27%. Finally, it is demonstrated that the two-layer Lanczos algorithm is also capable of computing the eigenvectors of the system with minor effort.

Spectral filters in quantum mechanics: A measurement theory perspective

We present the time-domain theory of spectral filters, starting with the basic propositions of the theory of measurement in quantum mechanics, and develop its parameter-free implementation in the traditional correlation function as well as the filter diagonalization (FD) form. The present study unifies all the time-domain spectral filter algorithms in the literature, under a single theme which is based on the notion of selective measurements. For specific numerical purposes, we have selected Chebyshev polynomials for developing the time propagator and this permits us to carry out the relevent time integrals fully analytically and obtain FD equations in a numerically convenient form. We also argue that the FD method is a particular realization of the general spectral filter goal and it is constrained, in general, by the time-energy uncertainty regime at least as much as the correlation-function-based method. To contrast the performance of the correlation function and the FD methods, we have carried out the detailed numerical experiments on a model system, which suggest that the FD method needs almost as much time propagation as the correlation function method, in order to identify the correct spectrum. The difference lies in the procedure for the exact location of eigenvalue positions, for which the FD method employs a diagonalization step while the correlation function method involves the location of zeros.