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

Two new methods for computing vibrational energy levels

I review two new ideas for coping with the size of large product basis sets and large product grids when one computes vibrational energy levels. The first is based on a tensor reduction scheme. It exploits advantages of a sum-of-products potential. The key idea is to use a basis each of whose function is a sum of optimized products and to compress the number of terms in each basis function. When the potential does not have the sum-of-products form, it is usually necessary to use quadrature. The second idea uses a nondirect product grid that has structure and is therefore compatible with efficient matrix–vector products.

Controlled full adder-subtractor by vibrational computing

The implementation of a quantum-controlled full adder-subtractor of two binary digits and of a "carry in" or a "borrow in" is simulated by encoding four qubits in the vibrational eigenstates of a tetra-atomic molecule (trans-HONO). The laser field of the gate is computed using optimal control theory by treating dynamics in full dimensionality. A controlled qubit enforces the addition or the subtraction. The global unitary transformation that connects the inputs to the outputs is driven by a single laser pulse. This decreases the duration of the operation and allows for a better use of the optical resources and for an improvement of the fidelity (>97%). Initialization and reading out are discussed. The timescale of the sequence initialization, gate and read out is<100 ps.

A comparison of two methods for selecting vibrational configuration interaction spaces on a heptatomic system: ethylene oxide

Two recently developed methods for solving the molecular vibrational Schrodinger equation, namely, the parallel vibrational multiple window configuration interaction and the vibrational mean field configuration interaction, are presented and compared on the same potential energy surface of ethylene oxide, c-C(2)H(4)O. It is demonstrated on this heptatomic system with strong resonances that both approaches converge towards the same fundamental frequencies. This confirms their ability to tackle the vibrational problem of large molecules for which full configuration interaction calculations are not tractable.

Variational Calculation of Specific, Highly Excited Vibrational States in DFCO: Comparison with Experimental Data

A stimulated emission pumping spectra of jet-cooled DFCO performed by Crane et al. (J. Mol. Spectrosc. 1997, 183, 273) has provided a great number of ro-vibrational lines up to 9000 cm(-1) of excitation energy. By combining a Jacobi-Wilson (JW) approach with a Davidson scheme, we calculate the lines provided by the experiment up to 9000 cm(-1) using an ab initio global potential energy surface (PES) developed by Kato et al. (J. Chem. Phys. 1997, 107, 6114). Comparisons between experimental and calculated data provide a critical test of the quality of the PES used. We show that the variational calculated energies can be efficiently corrected by taking into account the error observed for the A' fundamental transitions nu(i) (i = 1, ..., 5) and the first overtone 2nu(6). A detailed analysis of the eigenstates obtained by the calculation allows one to quantify the coupling between the different modes. Such an information is essential to understand and predict the energy flow through a DFCO molecule that is initially excited.

Vibrational anharmonic calculations in solution: Performance of various DFT approaches

We report anharmonic spectra calculated for formaldehyde in acetonitrile solution using the quartic force field obtained for various DFT/solvent coupled models. A statistical study has been carried out for each mode by using several classes of DFT functionals and comparing them to the reference ab-initio CCSD(T)/cc-pVQZ calculations. Results lead to the recommended use of hybrid functionals associated with the 6-31+G** basis set and the Polarized Continuum model (PCM) to predict the expected shifts relative to the gas phase.

Calculation of highly excited vibrational states using a Richardson-Leja-Davidson scheme

An efficient computational scheme for calculating highly excited vibrational eigenstates is proposed, combining a Richardson-Leja spectral filter with a novel version of the Davidson method [J. Comput. Phys. 17, 87 (1975)]. Highly excited eigenstates of the Rb2 and H2O molecules are computed to test and verify the method. On the average less than 2.5 outer recursions per eigenstate are needed. For each outer Davidson recursion, less than 20 inner filter recursions per eigenstate are needed on the average.

Theoretical investigation of highly excited vibrational states in DFCO: Calculation of the out-of-plane bending states and simulation of the intramolecular vibrational energy redistribution

A previously developed modified Davidson scheme [C. Iung and F. Ribeiro, J. Chem. Phys. 121, 174105 (2005)] is applied to compute and analyze highly excited (nu2,nu6) eigenstates in DFCO. The present paper is also devoted to the simulations of the intramolecular vibrational energy redistribution (IVR) initiated by an excitation of the out-of-plane bending vibration (nnu6, n=2,4,6, . . . ,18, and 20). The multiconfiguration time-dependent Hartree method is exploited to propagate the corresponding six-dimensional wave packets. A comprehensive comparison with experimental data as well as with previous simulations of IVR in HFCO [G. Pasin et al. J. Chem. Phys. 124, 194304 (2006)] is presented.

New Theoretical and Experimental Infrared Results on Formaldehyde in Solution

An extension of our combined procedure to determine a complete quartic force field and to resolve a vibrational problem thanks to a variational treatment is proposed for quantitative calculations of vibrational spectra in solution. Energies and gradients are obtained through a polarizable continuum model (PCM), the so-called self-consistent isodensity (SCI)-PCM. We present in this paper new experimental results dealing with formaldehyde in solution in cyclohexane, chloroform, THF, acetonitrile, DMSO and water; the obtained vibrational spectra are then compared with CCSD(T)/cc-pVQZ calculations. In addition, density functional theory (DFT) calculations have been carried out with the aim of both anticipating and positioning these approaches for larger sized molecules.

Theoretical investigation of intramolecular vibrational energy redistribution in highly excited HFCO

The present paper is devoted to the simulations of the intramolecular vibrational energy redistribution (IVR) in HFCO initiated by an excitation of the out-of-plane bending vibration [nnu(6)=2,4,6,...,18,20]. Using a full six-dimensional ab initio potential energy, the multiconfiguration time-dependent Hartree (MCTDH) method was exploited to propagate the corresponding six-dimensional wave packets. This study emphasizes the stability of highly excited states of the out-of-plane bending mode which exist even above the dissociation threshold. More strikingly, the structure of the IVR during the first step of the dynamics is very stable for initial excitations ranging from 2nu(6) to 20nu(6). This latter result is consistent with the analysis of the eigenstates obtained, up to 10nu(6), with the aid of the Davidson algorithm in a foregoing paper [Iung and Ribeiro, J. Chem. Phys. 121, 174105 (2005)]. The present study can be considered as complementary to this previous investigation. This paper also shows how MCTDH can be used to predict the dynamical behavior of a strongly excited system and to determine the energies of the corresponding highly excited states.

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.