Contracted basis Lanczos methods for computing numerically exact rovibrational levels of methane

Comparison of curvilinear coordinates within vibrational structure calculations based on automatically generated potential energy surfaces

For floppy molecules showing internal rotations and/or large amplitude motions, curvilinear internal coordinates are known to be superior to rectilinear normal coordinates within vibrational structure calculations. Due to the myriad definitions of internal coordinates, automated and efficient potential energy surface generators necessitate a high degree of flexibility, supporting the properties arising from these coordinates. Within this work, an approach to deal with these challenges is presented, including key elements, such as the selection of appropriate fit functions, the exploitation of symmetry, the positioning of grid points, or elongation limits for different coordinates. These elements are tested for five definitions of curvilinear coordinates, with three of them being generated in an automated manner. Calculations for semi-rigid molecules, namely H2O, H2CO, CH2F2, and H2CNH, demonstrate the general functionality of the implemented algorithms. Additional calculations for the HOPO molecule highlight the benefits of these curvilinear coordinates for systems with large amplitude motions. This new implementation allowed us to compare the performance of these different coordinate systems with respect to the convergence of the underlying expansion of the potential energy surface and subsequent vibrational configuration interaction calculations.

A two-step quadrature-based variational calculation of ro-vibrational levels and wavefunctions of CO2 using a bisector-x molecule-fixed frame

In this paper, we propose a new two-step strategy for computing ro-vibrational energy levels and wavefunctions of a triatomic molecule and apply it to CO2. A two-step method [J. Tennyson and B. T. Sutcliffe, Mol. Phys., 1986, 58, 1067] uses a basis whose functions are products of K-dependent "vibrational" functions and symmetric top functions. K is the quantum number for the molecule-fixed z component of the angular momentum. For a linear molecule, a two-step method is efficient because the Hamiltonian used to compute the basis functions includes the largest coupling term. The most important distinguishing feature of the two-step method we propose is that it uses an associated Legendre basis and quadrature rather than a K-dependent discrete variable representation. This reduces the cost of the calculation and simplifies the method. We have computed ro-vibrational energy levels with J up to 100 for CO2, on an accurate available potential energy surface which is known as the AMES-2 PES and present a subset of those levels. We have converged most levels up to 20 000 cm-1 to 0.0001 cm-1.

Using Collocation to Solve the Schrödinger Equation

We review the collocation approach to the solution of the Schrödinger equation and its uses in applications. Interrelations between collocation and other methods are highlighted. We also stress advantages and disadvantages of the rectangular collocation formulation. Using collocation makes it possible to use any, e.g. optimized, coordinates and basis functions, including nonintegrable basis functions, and provides a straightforward way of dealing with singularities in the potential. In addition, we stress that using collocation facilitates tuning the shape of basis functions and the placement of points, both of which can be done with machine-learning methods. Applications to electronic and vibrational problems are reviewed focusing on calculations for molecules on surfaces for which there are few variational calculations. Collocation has advantages when potential energy surfaces are unavailable, in particular, for molecule-surface systems, and for systems for which standard direct product quadrature grids, often used with variational methods, are costly.

Computing excited OH stretch states of water dimer in 12D using contracted intermolecular and intramolecular basis functions

Due to the ubiquity and importance of water, water dimer has been intensively studied. Computing the (ro-)vibrational spectrum of water dimer is challenging. The potential has eight wells separated by low barriers, which makes harmonic approximations of limited utility. A variational approach is imperative, but difficult because there are 12 coupled vibrational coordinates. In this paper, we use a product contracted basis whose functions are products of intramolecular and intermolecular functions computed using an iterative eigensolver. An intermediate matrix F facilitates calculating matrix elements. Using F, it is possible to do calculations on a general potential without storing the potential on the full quadrature grid. We find that surprisingly many intermolecular functions are required. This is due to the importance of coupling between inter- and intra-molecular coordinates. The full G₁₆ symmetry of water dimer is exploited. We calculate, for the first time, monomer excited stretch states and compare P(1) transition frequencies with their experimental counterparts. We also compare with experimental vibrational shifts and tunneling splittings. Surprisingly, we find that the largest tunneling splitting, which does not involve the interchange of the two monomers, is smaller in the asymmetric stretch excited state than in the ground state. Differences between levels we compute and those obtained with a [6+6]D adiabatic approximation [Leforestier et al. J. Chem. Phys. 137 014305 (2012)] are ∼0.6 cm^-1 for states without monomer excitation, ∼4 cm^-1 for monomer excited bend states, and as large as ∼10 cm^-1 for monomer excited stretch states.

Exact quantum dynamics developments for floppy molecular systems and complexes

Molecular rotation, vibration, internal rotation, isomerization, tunneling, intermolecular dynamics of weakly and strongly interacting systems, intra-to-inter-molecular energy transfer, hindered rotation and hindered translation over surfaces are important types of molecular motions. Their fundamentally correct and detailed description can be obtained by solving the nuclear Schrödinger equation on a potential energy surface. Many of the chemically interesting processes involve quantum nuclear motions which are 'delocalized' over multiple potential energy wells. These 'large-amplitude' motions in addition to the high dimensionality of the vibrational problem represent challenges to the current (ro)vibrational methodology. A review of the quantum nuclear motion methodology is provided, current bottlenecks of solving the nuclear Schrödinger equation are identified, and solution strategies are reviewed. Technical details, computational results, and analysis of these results in terms of limiting models and spectroscopically relevant concepts are highlighted for selected numerical examples.

Performance of a black-box-type rovibrational method in comparison with a tailor-made approach: Case study for the methane-water dimer

The present work intends to join and respond to the excellent and thoroughly documented rovibrational study of X. G. Wang and T. Carrington, Jr. [J. Chem. Phys. 154, 124112 (2021)] that used an approach tailored for floppy dimers with an analytic dimer Hamiltonian and a non-product basis set including Wigner D functions. It is shown in the present work that the GENIUSH black-box-type rovibrational method can approach the performance of the tailor-made computation for the example of the floppy methane-water dimer. Rovibrational transition energies and intensities are obtained in the black-box-type computation with a twice as large basis set and in excellent numerical agreement in comparison with the more efficient tailor-made approach.

A variational calculation of vibrational levels of vinyl radical

We report the vibrational energy levels of vinyl radical (VR) that are computed with a Lanczos eigensolver and a contracted basis. Many of the levels of the two previous VR variational calculations differ significantly and differ also from those reported in this paper. We identify the source of and correct symmetry errors on the potential energy surfaces used in the previous calculations. VR has two equivalent equilibrium structures. By plotting wavefunction cuts, we show that two tunneling paths play an important role. Using the computed wavefunctions, it is possible to assign many states and thereby to determine tunneling splittings that are compared with their experimental counterparts. Our computed red shift of the hot band at 2897.23 cm^-1, observed by Dong et al. [J Chem. Phys. 128, 044305 (2008)], is 4.47 cm^-1, which is close to the experimental value of 4.63 cm^-1.

Full-dimensional quantum mechanical calculations of the reaction probability of the H + CH4 reaction based on a mixed Jacobi and Radau description

A full-dimensional time-dependent wave packet study using mixed polyspherical Jacobi and Radau coordinates for the title reaction has been reported. The non-reactive moiety CH₃ has been described using three Radau vectors, whereas two Jacobi vectors have been used for the bond breaking/formation process. A potential-optimized discrete variable representation basis has been employed to describe the vibrational coordinates of the reagent CH₄. About one hundred billion basis functions have been necessary to achieve converged results. The reaction probabilities for some initial vibrational states are given. A comparison between the present approach and other methods, including reduced and full-dimensional ones, is also presented.

Computational Study of the Rovibrational Spectra of CH2D+ and CHD2. J Phys Chem A 2019;123:10281-10289. [PMID: 31657568 DOI: 10.1021/acs.jpca.9b09045] [Citation(s) in RCA: 0] [Impact Index Per Article: 0]

In this paper, we present rovibrational energy levels of CH₂D⁺ and CHD₂⁺. They are computed with a large basis and the Lanczos algorithm. CH₂D⁺ and CHD₂⁺ are believed to play an important role in interstellar space, but so far, there are no definitive observations. The predictions of this paper should facilitate detection. For CH₂D⁺, two CH stretch bands have been studied at high resolution. Compared to our calculated energies, the root-mean-square error is 0.08 cm^-1. For CHD₂⁺, one CH stretch band has been studied at high resolution. Compared to our calculated energies, the root-mean-square error is 0.5 cm^-1. Errors are larger, for both isotopologues, for bend states. We attribute these errors to the potential energy surface. Wave function and probability distribution plots are used to make assignments. The ν₁ band of CHD₂⁺ is significantly perturbed, and according to our calculations, the 3ν₃ state is closest and might be the most important perturber.

Computational study of the ro-vibrational spectrum of CO-CO2

An accurate ab initio ground-state intermolecular potential energy surface (PES) was determined for the CO-CO₂ van der Waals dimer. The Lanczos algorithm was used to compute rovibrational energies on this PES. For both the C-in and O-in T-shaped isomers, the fundamental transition frequencies agree well with previous experimental results. We confirm that the in-plane states previously observed are geared states. In addition, we have computed and assigned many other vibrational states. The rotational constants we determine from J = 1 energy levels agree well with their experimental counterparts. Planar and out-of-plane cuts of some of the wavefunctions we compute are quite different, indicating strong coupling between the bend and torsional modes. Because the stable isomers are T-shaped, vibration along the out-of-plane coordinates is very floppy. In CO-CO₂, when the molecule is out-of-plane, interconversion of the isomers is possible, but the barrier height is higher than the in-plane geared barrier height.

Highly excited vibrational levels of methane up to 10 300 cm-1: Comparative study of variational methods

In this work, we report calculated vibrational energy levels of the methane molecule up to 10 300 cm^-1. Two potential energy surfaces constructed in quite different coordinate systems with different analytical representations are employed in order to evaluate the uncertainty of vibrational predictions. To calculate methane energy levels, we used two independent techniques of the variational method. One method uses an exact kinetic energy operator in internal curvilinear coordinates. Another one uses an expansion of Eckart-Watson nuclear motion Hamiltonian in rectilinear normal coordinates. In the Icosad range (up to five vibrational quanta bands-below 7800 cm^-1), the RMS standard deviations between calculated and observed energy levels were 0.22 cm^-1 and 0.41 cm^-1 for these two quite different approaches. For experimentally well-known 3v₃ sub-levels, the calculation accuracy is estimated to be ∼1 cm^-1. In the Triacontad range (7660-9188 cm^-1), the average error of the calculation is about 0.5 cm^-1. The accuracy and convergence issues for higher energy ranges are discussed.

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.

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.

Vibrational energy levels of difluorodioxirane computed with variational and perturbative methods from a hybrid force field

We have computed vibrational energy levels of difluorodioxirane (CF2O2). For the potential, a Taylor expansion in normal coordinates is used. The CCSD(T) and MP2 methods and correlation consistent basis sets of quadruple-zeta quality are used to determine the force constants. The vibrational Schrödinger equation was solved using both a variational method and second order perturbation theory. The Watson kinetic energy operator and a discrete variable representation were used with the DEWE (E. Mátyus, G. Czakó, B.T. Sutcliffe and A.G. Császár, J. Chem. Phys. 127 (2007) 084102) computer program to do the variational calculations. For the variational calculations, the average absolute deviation of fundamentals, with respect to experimental values, is less than 3 cm(-1). Perturbative results are almost as good. About 300 vibrational levels were computed. (16)O→(18)O isotopic shifts have also been calculated variationally for the lowest 75 vibrational energy levels and are compared to experimental results.

Calculating and assigning rovibrational energy levels of (15N2O)2, (15N14NO)2, 14N2O-15N2O and 15N14NO-15N2O

In this paper we report transition frequencies and rotational constants computed for several isotopologues of the nitrous oxide dimer. A previously reported intermolecular potential, the symmetry adapted Lanczos algorithm and an uncoupled product basis set are used to do the calculations. Rotational transition frequencies and rotational constants are in good agreement with experiment. We calculate states localized in both polar and nonpolar wells on the potential surface. Two of the four isotopologues we study have inequivalent monomers. They have wavefunctions localized over a single polar well.