Six-dimensional variational calculation of the bending energy levels of HF trimer and DF trimer

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.

Using nondirect product Wigner D basis functions and the symmetry-adapted Lanczos algorithm to compute the ro-vibrational spectrum of CH4-H2O

By doing calculations on the methane-water van der Waals complex, we demonstrate that highly converged energy levels and wavefunctions can be obtained using Wigner D basis functions and the Symmetry-Adapted Lanczos (SAL) method. The Wigner D basis is a nondirect product basis and, therefore, efficient when the kinetic energy operator has accessible singularities. The SAL method makes it possible to exploit symmetry to label energy levels and reduce the cost of the calculation, without explicitly using symmetry-adapted basis functions. Line strengths are computed, and new bands are identified. In particular, we find unusually strong transitions between states associated with the isomers of the global minimum and the secondary minimum.

Computational study of the rovibrational spectrum of CO2-N2

The CO2-N2 complex is formed from two key components of Earth's atmosphere, and as such, has received some attention from both experimental and theoretical studies. On the theory side, a potential energy surface (PES) based on high level ab initio data was reported [Nasri et al., J. Chem. Phys., 2015, 142, 174301] and then used in more recently reported rovibrational calculations [Lara-Moreno et al., Phys. Chem. Chem. Phys., 2019, 21, 3550]. Accuracy of about 1 percent was achieved for calculated rotational transitions of the ground vibrational state of the complex, compared with previously reported microwave spectra. However, a very recent measurement of the geared bending mode frequency [Barclay et al., J. Chem. Phys., 2020, 153, 014303] recorded a value of 21.4 cm-1, which is wildly different from the corresponding calculated value of 45.9 cm-1. To provide some insight into this discrepancy, we have constructed a new more accurate PES, and used it to perform highly converged variational rovibrational calculations. Our new results yield a value of 21.1 cm-1 for that bending frequency, in close agreement with the experiment. We also obtain significantly improved predicted rotational transitions. Finally, we note that a very shallow well, previously reported as a distinct second isomer, is not found on our new PES, but rather a transition structure is seen in that location.

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.

The vibration-rotation-tunneling levels of N2-H2O and N2-D2O

In this paper, we report vibration-rotation-tunneling levels of the van der Waals clusters N2-H2O and N2-D2O computed from an ab initio potential energy surface. The only dynamical approximation is that the monomers are rigid. We use a symmetry adapted Lanczos algorithm and an uncoupled product basis set. The pattern of the cluster's levels is complicated by splittings caused by H-H exchange tunneling (larger splitting) and N-N exchange tunneling (smaller splitting). An interesting result that emerges from our calculation is that whereas in N2-H2O, the symmetric H-H tunnelling state is below the anti-symmetric H-H tunnelling state for both K = 0 and K = 1, the order is reversed in N2-D2O for K = 1. The only experimental splitting measurements are the D-D exchange tunneling splittings reported by Zhu et al. [J. Chem. Phys. 139, 214309 (2013)] for N2-D2O in the v2 = 1 region of D2O. Due to the inverted order of the split levels, they measure the sum of the K = 0 and K = 1 tunneling splittings, which is in excellent agreement with our calculated result. Other splittings we predict, in particular those of N2-H2O, may guide future experiments.

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.

Approximate first-principles anharmonic calculations of polyatomic spectra using MP2 and B3LYP potentials: comparisons with experiment

Anharmonic vibrational spectroscopy calculations using MP2 and B3LYP computed potential surfaces are carried out for a series of molecules, and frequencies and intensities are compared with those from experiment. The vibrational self-consistent field with second-order perturbation correction (VSCF-PT2) is used in computing the spectra. The test calculations have been performed for the molecules HNO3, C2H4, C2H4O, H2SO4, CH3COOH, glycine, and alanine. Both MP2 and B3LYP give results in good accord with experimental frequencies, though, on the whole, MP2 gives very slightly better agreement. A statistical analysis of deviations in frequencies from experiment is carried out that gives interesting insights. The most probable percentage deviation from experimental frequencies is about -2% (to the red of the experiment) for B3LYP and +2% (to the blue of the experiment) for MP2. There is a higher probability for relatively large percentage deviations when B3LYP is used. The calculated intensities are also found to be in good accord with experiment, but the percentage deviations are much larger than those for frequencies. The results show that both MP2 and B3LYP potentials, used in VSCF-PT2 calculations, account well for anharmonic effects in the spectroscopy of molecules of the types considered.

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.

Constrained molecular vibration-rotation Hamiltonians: contravariant metric tensor

Here, I present a practical recipe for obtaining contravariant vibration-rotation metric tensors, and thus the kinetic energy operators, when some degrees of freedom are constrained rigidly. An element of the contravariant metric tensor is obtained as a sum of dot products of contravariant measuring vectors, which are obtained from their unconstrained counterparts by adding a frozen mode correction. The present method applies in principle for any choice of shape coordinates and a body-frame for which the contravariant measuring vectors can be evaluated. In contrast to the existing methods, the present method does not involve evaluation of covariant metric tensors, matrix inversions, chain rules of derivation, or numerical differentiation. It is applied in the sequel paper [L. Partanen, J. Pesonen, E. Sjöholm, and L. Halonen, J. Chem. Phys. 139, 144311 (2013)] to study the effects of several different approximations to the kinetic energy operator, when the two large-amplitude OH-torsional motions in H2SO4 are of interest.

CALCULATION OF TRANSITION AMPLITUDES WITH A SINGLE LANCZOS PROPAGATION

We review in this article a recently proposed energy-global method that is capable of calculating the entire transition amplitude matrix with a single Lanczos propagation. This method requires neither explicit computation nor storage of the eigenfunctions, rendering it extremely memory efficient. Procedures are proposed to handle situations where "spurious" eigenvalues aggregate around true eigenvalues due to round-off errors. This method is amenable to both real-symmetric and complex-symmetric Hamiltonians. Applications to molecular spectra and reactive scattering are presented. Its relationships with other methods are also discussed.

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.

Theoretical study of the rovibrational spectrum of He2–OCS

We report calculated microwave and infrared rovibrational transitions of the van der Waals complex He2–OCS. The calculations were done using a product basis, a Lanczos eigensolver, and potentials built from He–OCS, and He–He potential functions taken from the literature. All five of the large amplitude coordinates are treated exactly and calculations are done for J values up to five. All rovibrational levels are converged to 0.001 cm–1 by using basis sets with as many as 87 million funcions. Good agreement is found with previously reported experimental results. Although we assume that the dipole moment is along the OCS axis, we find transitions with appreciable intensity between different torsion states.

Theoretical and experimental study of the rovibrational spectrum of He(2)-CO

We report calculated microwave and infrared ro-vibrational transitions of the van der Waals complex He(2)-CO. The calculations were done using a product basis and a Lanczos eigensolver, together with He-CO and He-He potential functions taken from the literature. The results are found to be in good agreement with previously reported experimental results, and they enable the experimental assignments to be clarified, augmented, and (in one case) corrected. Unlike some other van der Waals complexes with two He atoms such as He(2)-N(2)O, it is not possible to associate a set of energy levels with the "torsional" motion of the two He atoms on a ring encircling the dopant (in this case CO). Although we assume that the dipole moment is along the CO axis we find nonetheless that many transitions have appreciable intensity due to ro-vibrational coupling.

Using neural networks to represent potential surfaces as sums of products

By using exponential activation functions with a neural network (NN) method we show that it is possible to fit potentials to a sum-of-products form. The sum-of-products form is desirable because it reduces the cost of doing the quadratures required for quantum dynamics calculations. It also greatly facilitates the use of the multiconfiguration time dependent Hartree method. Unlike potfit product representation algorithm, the new NN approach does not require using a grid of points. It also produces sum-of-products potentials with fewer terms. As the number of dimensions is increased, we expect the advantages of the exponential NN idea to become more significant.

Cooperative organic hydrogen bonds: The librational modes of cyclic methanol clusters

Intermolecular hydrogen bond libration modes of isolated cyclic methanol trimers (approximately 613 cm(-1)) and tetramers (695 and 760 cm(-1)) are observed in pulsed jet Fourier transform infrared spectra and found to exhibit sizeable anharmonicity and mode coupling effects, opening the way for a microscopic interpretation of the broad librational bands of alcohols. The correlation of experimental OH stretching and OH libration band intensities provides important constraints for theoretical band strengths, cluster densities, and size assignments.

Calculating vibrational energies and wave functions of vinylidene using a contracted basis with a locally reorthogonalized coupled two-term Lanczos eigensolver

We use a contracted basis+Lanczos eigensolver approach to compute vinylidene-like vibrational states of the acetylene-vinylidene system. To overcome problems caused by loss of orthogonality of the Lanczos vectors we reorthogonalize Lanczos vector and use a coupled two-term approach. The calculations are done in CC-HH diatom-diatom Jacobi coordinates which make it easy to compute states one irreducible representation at a time. The most costly parts of the calculation are parallelized and scale well. We estimate that the vinylidene energies we compute are converged to approximately 1 cm(-1).

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.