Molecular vibrations: Iterative solution with energy selected bases

EXACT AND CONSTRAINED KINETIC ENERGY OPERATORS IN POLYSPHERICAL COORDINATES

In this paper the derivation of exact and constrained kinetic energy operators in curvilinear coordinates is presented in a very general context. The polyspherical approach allows us to provide simple and explicit expressions of these operators which are well adapted to the numerical approaches used to solve the Schrödinger equation. These expressions can be exploited to treat a lot of problems such as the calculation of infrared or photoabsorption spectra as well as the study of intramolecular vibrational relaxation of molecules or the study of reactive scattering systems. Special emphasis is placed on concrete applications.

ACCURATE AND HIGHLY EFFICIENT CALCULATION OF THE O(1D)HCl VIBRATIONAL BOUND STATES, USING A COMBINATION OF METHODS

Hypochlorous acid, HOCl, is an important intermediate in the O (1D) HCl reactive system. Due in part to a large number of vibrational bound states (over 800), extremely large direct product basis sets (around 300,000) are required to compute the energy levels just below the dissociation threshold. This situation, combined with a very high density of states, results in difficult convergence for iterative methods — e.g. Lanczos requires 50,000 iterations, and filter diagonalization uses 60,000 iterations. In contrast, using new methodologies, we are able to compute the highest-lying bound states with only 271 iterations, although the CPU cost per iteration is substantially greater. Lower lying states are also computed, for a fraction of the CPU cost of the highest energy calculation.

Constant pressure ab initio molecular dynamics with discrete variable representation basis sets

The use of discrete variable representation (DVR) basis sets within ab initio molecular dynamics calculations allows the latter to be performed with converged energies and, more importantly, converged forces. In this paper, we show how to carry out ab initio molecular dynamics calculations in the isothermal-isobaric ensemble with fully flexible simulation boxes within the DVR basis set framework. In particular, we derive the appropriate DVR based expression for the pressure tensor when the electronic structure is represented using Kohn-Sham density functional theory, and we examine the convergence of this expression as a function of the basis set size. An illustrative example using 64 silicon atoms in a fully flexible box using a combination of the Martyna-Tobias-Klein [Martyna et al., J. Chem. Phys. 101, 4177 (1994)] and Car-Parrinello [Car and Parinello, Phys. Rev. Lett. 55, 2471 (1985)] algorithms is presented to demonstrate the efficacy of the approach.

Ultrafast excited-state dynamics of tetraphenylethylene studied by semiclassical simulation

Detailed simulation study is reported for the excited-state dynamics of photoisomerization of cis-tetraphenylethylene (TPE) following excitation by a femtosecond laser pulse. The technique for this investigation is semiclassical dynamics simulation, which is described briefly in the paper. Upon photoexcitation by a femtosecond laser pulse, the stretching motion of the ethylenic bond of TPE is initially excited, leading to a significant lengthening of ethylenic bond in 300 fs. Twisting motion about the ethylenic bond is activated by the energy released from the relaxation of the stretching mode. The 90 degrees twisting about the ethylenic bond from an approximately planar geometry to nearly a perpendicular conformation in the electronically excited state is completed in 600 fs. The torsional dynamics of phenyl rings which is temporally lagging behind occurs at about 5 ps. Finally, the twisted TPE reverts to the initial conformation along the twisting coordinate through nonadiabatic transitions. The simulation results provide a basis for understanding several spectroscopic observations at molecular levels, including ultrafast dynamic Stokes shift, multicomponent fluorescence, viscosity dependence of the fluorescence lifetime, and radiationless decay from electronically excited state to the ground state along the isomerization coordinate.

Optimal internal coordinates, vibrational spectrum, and effective Hamiltonian for ozone

In this paper the authors use the optimal internal vibrational coordinates previously determined for the electronic ground state of the ozone molecule to study the vibrational spectrum of the molecule employing the second empirical potential energy surface calculated by Tyuterev et al. [Chem. Phys. Lett. 316, 271 (2000)]. First, the authors compute variationally all the bound vibrational energy levels of the molecule up to the dissociation limit and state the usefulness of the optimal coordinates in this respect, which allows us to converge all the bound levels using relatively small anharmonic basis sets. By analyzing the expansion coefficients of the wave functions, they show then that a large portion of the vibrational spectrum of O3 can be structured in nearly separable polyadic groups characterized by the polyad quantum number N=n1+n2+n(theta) corresponding to the optimal internal coordinates. Accordingly, they determine an internal effective vibrational Hamiltonian for O3 by fitting the effective Hamiltonian parameters to the experimental vibrational frequencies, using as input parameters in the fit those extracted from an analytical second-order Van Vleck perturbation theory calculation. It is finally shown that the internal effective Hamiltonian thus obtained accurately describes the vibrational spectrum of ozone in the low and medium energy regimes.

Spectral convergence of the quadrature discretization method in the solution of the Schrödinger and Fokker-Planck equations: Comparison with sinc methods

Spectral methods based on nonclassical polynomials and Fourier basis functions or sinc interpolation techniques are compared for several eigenvalue problems for the Fokker-Planck and Schrodinger equations. A very rapid spectral convergence of the eigenvalues versus the number of quadrature points is obtained with the quadrature discretization method (QDM) and the appropriate choice of the weight function. The QDM is a pseudospectral method and the rate of convergence is compared with the sinc method reported by Wei [J. Chem. Phys., 110, 8930 (1999)]. In general, sinc methods based on Fourier basis functions with a uniform grid provide a much slower convergence. The paper considers Fokker-Planck equations (and analogous Schrodinger equations) for the thermalization of electrons in atomic moderators and for a quartic potential employed to model chemical reactions. The solution of the Schrodinger equation for the vibrational states of I2 with a Morse potential is also considered.

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.

Ro-Vibrational States of Triplet H2D+

We present rotational term values for J < or = 3 of the vibrational states with up to twofold excitation of H2D+ in the lowest electronic triplet state (a3sigma(u)+). The calculations were performed using the method of hyperspherical harmonics and our recent accurate double many-body expansion potential energy surface.

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.

Improving the calculation of rovibrational spectra of five-atom molecules with three identical atoms by using a C3υ(G6) symmetry-adapted grid: Applied to CH3D and CHD3

In this paper we report two improvements on the approach we have used to compute rovibrational levels of methane and apply the new ideas to calculate rovibrational levels of two methane isotopomers CH3D and CHD3. Both improvements make the bend calculation better. The first improvement is a G6-invariant (or C3upsilon-invariant) grid which is designed such that each point on the grid is mapped to another point on the grid by any of the G6 operations. The second improvement is the use of fast Fourier transform (FFT) to compute the bend potential matrix-vector products. The FFT matrix-vector product is about three and ten times faster than the previous sequential summation method for the J=0 and J>0 cases, respectively. The calculated J=1 rovibrational levels of CH3D and CHD3 on the Schwenke and Partridge [Spectrochim. Acta, Part A 57, 887 (2001)] ab initio potential are in good agreement (within 6 cm(-1) for the levels up to 3000 cm(-1)) with the experimental data. The agreement is even better (within 0.1 cm(-1) for the levels up to 6000 cm(-1)) if the associated J=0 energies are subtracted.

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.

Vibrational dynamics of the floppy LiNC∕LiCN molecular system

Modern spectroscopical techniques allow the efficient experimental investigation of highly excited vibrational states in molecular systems. On the theoretical side, powerful computational methods have also been developed for the calculation of the corresponding energy levels and wave functions, and their interpretation. In this paper we use a combination of two such methods, namely, the distribution of zeroes in the Husimi function and energy-level correlation diagrams, to discuss a classification scheme, for the lowest hundred vibrational levels of the LiNC/LiCN floppy molecular system, based on their dynamical characteristics.

Theoretical and experimental studies of the infrared rovibrational spectrum of He2–N2O

Rovibrational spectra of the He(2)-N(2)O complex in the nu(1) fundamental band of N(2)O (2224 cm(-1)) have been observed using a tunable infrared laser to probe a pulsed supersonic jet expansion, and calculated using five coordinates that specify the positions of the He atoms with respect to the NNO molecule, a product basis, and a Lanczos eigensolver. Vibrational dynamics of the complex are dominated by the torsional motion of the two He atoms on a ring encircling the N(2)O molecule. The resulting torsional states could be readily identified, and they are relatively uncoupled to other He motions up to at least upsilon(t) = 7. Good agreement between experiment and theory was obtained with only one adjustable parameter, the band origin. The calculated results were crucial in assigning many weaker observed transitions because the effective rotational constants depend strongly on the torsional state. The observed spectra had effective temperatures around 0.7 K and involved transitions with J < or =3, with upsilon(t) = 0 and 1, and (with one possible exception) with Deltaupsilon(t)=0. Mixing of the torsion-rotation states is small but significant: some transitions with Deltaupsilon(t) not equal 0 were predicted to have appreciable intensity even assuming that the dipole transition moment coincides perfectly with the NNO axis. One such transition was tentatively assigned in the observed spectra, but confirmation will require further work.

Computing resonance energies, widths, and wave functions using a Lanczos method in real arithmetic

We introduce new ideas for calculating resonance energies and widths. It is shown that a non-Hermitian-Lanczos approach can be used to compute eigenvalues of H+W, where H is the Hamiltonian and W is a complex absorbing potential (CAP), without evaluating complex matrix-vector products. This is done by exploiting the link between a CAP-modified Hamiltonian matrix and a real but nonsymmetric matrix U suggested by Mandelshtam and Neumaier [J. Theor. Comput. Chem. 1, 1 (2002)] and using a coupled-two-term Lanczos procedure. We use approximate resonance eigenvectors obtained from the non-Hermitian-Lanczos algorithm and a very good CAP to obtain very accurate energies and widths without solving eigenvalue problems for many values of the CAP strength parameter and searching for cusps. The method is applied to the resonances of HCO. We compare properties of the method with those of established approaches.

On the use of optimal internal vibrational coordinates for symmetrical bent triatomic molecules

The use of generalized internal coordinates for the variational calculation of excited vibrational states of symmetrical bent triatomic molecules is considered with applications to the SO2, O3, NO2, and H2O molecules. These coordinates depend on two external parameters which can be properly optimized. We propose a simple analytical method to determine the optimal internal coordinates for this kind of molecules based on the minimization with respect to the external parameters of the zero-point energy, assuming only quadratic terms in the Hamiltonian and no quadratic coupling between the optimal coordinates. The optimal values of the parameters thus obtained are shown to agree quite well with those that minimize the sum of a number of unconverged energies of the lowest vibrational states, computed variationally using a small basis function set. The unconverged variational calculation uses a basis set consisting of the eigenfunctions of the uncoupled anharmonic internal coordinate Hamiltonian. Variational calculations of the excited vibrational states for the four molecules considered carried out with an increasing number of basis functions, also evidence the excellent convergence properties of the optimal internal coordinates versus those provided by other normal and local coordinate systems.

Isomerization and dissociation dynamics of HCN in a picosecond infrared laser field: A full-dimensional classical study

We report a full-dimensional study of the classical dynamics of HCN-->HNC isomerization and of HCN rovibrational dissociation driven by a strong but nonionizing picosecond infrared laser field. The dynamics of the isolated molecule and of the molecule in liquid Ar have both been studied. Our theoretical and numerical results show that when all degrees of freedom are accounted for the field induced molecular dynamics can be totally different from what was found in previous studies, where the HCN molecule is restricted to a plane containing the external field. It is shown that as HCN is driven by an infrared laser field, the rotation of the H atom around the C-N bond provides an important and highly efficient energy absorption mechanism. In the presence of a monochromatic picosecond infrared laser field with an intensity of 10(13) W/cm(2), this energy absorption mechanism generates considerable HCN-->HNC isomerization yield or high rovibrational dissociation yield without molecular preorientation or prealignment. Our study of the field induced isomerization and dissociation dynamics of the same system in liquid Ar shows that the picosecond isomerization dynamics is insignificantly affected by the surrounding atomic liquid whereas the dissociation yield may be greatly suppressed in a high density liquid. The implications of this study for full-dimensional quantum dynamics of multiphoton rovibrational excitation and dissociation of triatomics are briefly discussed.

How to choose one-dimensional basis functions so that a very efficient multidimensional basis may be extracted from a direct product of the one-dimensional functions: Energy levels of coupled systems with as many as 16 coordinates

In this paper we propose a scheme for choosing basis functions for quantum dynamics calculations. Direct product bases are frequently used. The number of direct product functions required to converge a spectrum, compute a rate constant, etc., is so large that direct product calculations are impossible for molecules or reacting systems with more than four atoms. It is common to extract a smaller working basis from a huge direct product basis by removing some of the product functions. We advocate a build and prune strategy of this type. The one-dimensional (1D) functions from which we build the direct product basis are chosen to satisfy two conditions: (1) they nearly diagonalize the full Hamiltonian matrix; (2) they minimize off-diagonal matrix elements that couple basis functions with diagonal elements close to those of the energy levels we wish to compute. By imposing these conditions we increase the number of product functions that can be removed from the multidimensional basis without degrading the accuracy of computed energy levels. Two basic types of 1D basis functions are in common use: eigenfunctions of 1D Hamiltonians and discrete variable representation (DVR) functions. Both have advantages and disadvantages. The 1D functions we propose are intermediate between the 1D eigenfunction functions and the DVR functions. If the coupling is very weak, they are very nearly 1D eigenfunction functions. As the strength of the coupling is increased they resemble more closely DVR functions. We assess the usefulness of our basis by applying it to model 6D, 8D, and 16D Hamiltonians with various coupling strengths. We find approximately linear scaling.

Accurate quantum calculation of the bound and resonant rovibrational states of Li−(H2)

In a recent paper [B. Poirier, Chem. Phys. 308, 305 (2005)] a full-dimensional quantum method for computing the rovibrational dynamics of triatomic systems was presented, incorporating three key features: (1) exact analytical treatment of Coriolis coupling, (2) three-body "effective potential," and (3) a single bend angle basis for all rotational states. In this paper, these ideas are applied to the Li-(H2) electrostatic complex, to compute all of the rovibrational bound state energies, and a number of resonance energies and widths, to very high accuracy (thousandths of a wave number). This application is very challenging, owing to the long-range nature of the interaction and to narrow level spacings near dissociation. Nevertheless, by combining the present method with a G4 symmetry-adapted phase-space-optimized representation, only modest basis sizes are required for which the matrices are amenable to direct diagonalization. Several new bound levels are reported, as compared with a previous calculation [D. T. Chang, G. Surratt, G. Ristroff, and G. I. Gellene, J. Chem. Phys. 116, 9188 (2002)]. The resonances exhibit a clear-cut separation into shape and Feshbach varieties, with the latter characterized by extremely long lifetimes (microseconds or longer).

Iterative solutions with energy selected bases for highly excited vibrations of tetra-atomic molecules

The use of energy selected bases (ESB) with iterative diagonalization of the Hamiltonian matrix is described for vibrations of tetra-atomic systems. The performance of the method is tested by computing vibrational states of HOOH below 10,000 cm(-1) (1296 A+ symmetry states) and H(2)CO below 13,500 cm(-1) (729 A(1) symmetry states). For iterative solutions, we tested both the implicitly restarted Lanczos method (IRLM) and the standard (nonreorthogonalizing) Lanczos approach. Comparison with other contracted basis approach as well as direct product grid representation shows superior performance of the ESB/IRLM approach. Of the two systems, H(2)CO is found to be more challenging than HOOH since it has much stronger couplings among vibrational modes, which leads to a drastically larger primitive basis set. For H(2)CO we also discuss some interesting behavior of the molecule in the high internal energy regime.

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.