A new method for the derivation of exact vibration-rotational kinetic energy operator in internal coordinates

An efficient angular momentum method is presented and used to derive analytic expressions for the vibration-rotational kinetic energy operator of polyatomic molecules.The vibration-rotational kinetic energy operator is expressed in terms of the total angular momentum operator J,the angular momentum operator J and the momentum operator p conjugate to Z in the molecule-fixed frame Not only the method of derivation is simpler than that in the previous work,but also the expressions ot the kinetic energy operators arc more compact.Particularly,the operator is easily applied to different vibrational or rovibrational problems of the polyatomic molecules by variations of matrix elements Gn of a mass-dependent constant symmetric matrix     

Vibrational energy levels with arbitrary potentials using the Eckart-Watson Hamiltonians and the discrete variable representation

An effective and general algorithm is suggested for variational vibrational calculations of N-atomic molecules using orthogonal, rectilinear internal coordinates. The protocol has three essential parts. First, it advocates the use of the Eckart-Watson Hamiltonians of nonlinear or linear reference configuration. Second, with the help of an exact expression of curvilinear internal coordinates (e.g., valence coordinates) in terms of orthogonal, rectilinear internal coordinates (e.g., normal coordinates), any high-accuracy potential or force field expressed in curvilinear internal coordinates can be used in the calculations. Third, the matrix representation of the appropriate Eckart-Watson Hamiltonian is constructed in a discrete variable representation, in which the matrix of the potential energy operator is always diagonal, whatever complicated form the potential function assumes, and the matrix of the kinetic energy operator is a sparse matrix of special structure. Details of the suggested algorithm as well as results obtained for linear and nonlinear test cases including H(2)O, H(3) (+), CO(2), HCNHNC, and CH(4) are presented.     

Rovibrational interactions in linear triatomic molecules: a theoretical study in curvilinear vibrational coordinates

Variational solution of the rovibrational problem in curvilinear vibrational coordinates has been implemented and used to investigate the nuclear motions in several linear triatomic molecules, like HCN, OCS, and HCP. The dependence of the rovibrational energy levels on the rotational quantum numbers and the l-doubling has been studied. Two approximations to the rovibrational Hamiltonian have been examined, depending on the level of truncation of the potential energy operator. It turns out that the truncation after the fifth order in the potential is sufficient to produce vibrational energies of high accuracy. An interesting feature of the present formulation of the problem in terms of the curvilinear vibrational coordinates is the explanation of the l-doubling of the rovibrational levels, which in this picture is interpreted as the result of the inequivalency of the average rotational constants in mutually perpendicular planes, rather than as the effect of the Coriolis-type interactions between the vibrational and rotational motions. The present theoretical results are compared with the available experimental data from high-resolution spectroscopy, as well as with other ab initio calculations.     

Rovibrational molecular hamiltonian in mixed bond-angle and umbrella-like coordinates

A new exact quantum mechanical rovibrational Hamiltonian operator for molecules exhibiting large amplitude inversion and torsion motions is derived. The derivation is based on a division of a molecule into two parts: a frame and a top. The nuclei of the frame only are used to construct a molecular system of axes. The inversion motion of the frame is described in the umbrella-like coordinates, whereas the torsion motion of the top is described by the nonstandard torsion angle defined in terms of the nuclear vectors and one of the molecular axes. The internal coordinates chosen take into account the properties of the inversion and torsion motions. Vibrational s and rotational Omega vectors obtained for the introduced internal coordinates determine the rovibrational tensor G defined by simple scalar products of these vectors. The Jacobian of the transformation from the Cartesian to the internal coordinates considered and the G tensor specify the rovibrational Hamiltonian. As a result, the Hamiltonian for penta-atomic molecules like NH2OH with one inverter is presented and a complete set of the formulas necessary to write down the Hamiltonian of more complex molecules, like NH2NH2 with two inverters, is reported. The approach considered is essentially general and sufficiently simple, as demonstrated by derivation of a polyatomic molecule Hamiltonian in polyspherical coordinates, obtained by other methods with much greater efforts.     

Hamiltonian for rovibrational spectra of “hot” molecules

A variational method based on the determination of rotation as a state with a constant angular momentum has been proposed for calculating rovibrational energy levels of a polyatomic molecule. By using this method, energy level values can be determined for any vibrational state with arbitrary values of vibrational quantum numbers. This makes it possible to calculate the rovibrational energy levels of the “hot” molecules.     

The role of axis embedding on rigid rotor decomposition analysis of variational rovibrational wave functions

Approximate rotational characterization of variational rovibrational wave functions via the rigid rotor decomposition (RRD) protocol is developed for Hamiltonians based on arbitrary sets of internal coordinates and axis embeddings. An efficient and general procedure is given that allows employing the Eckart embedding with arbitrary polyatomic Hamiltonians through a fully numerical approach. RRD tables formed by projecting rotational-vibrational wave functions into products of rigid-rotor basis functions and previously determined vibrational eigenstates yield rigid-rotor labels for rovibrational eigenstates by selecting the largest overlap. Embedding-dependent RRD analyses are performed, up to high energies and rotational excitations, for the H(2) (16)O isotopologue of the water molecule. Irrespective of the embedding chosen, the RRD procedure proves effective in providing unambiguous rotational assignments at low energies and J values. Rotational labeling of rovibrational states of H(2) (16)O proves to be increasingly difficult beyond about 10,000 cm(-1), close to the barrier to linearity of the water molecule. For medium energies and excitations the Eckart embedding yields the largest RRD coefficients, thus providing the largest number of unambiguous rotational labels.     

CVRQD ab initio ground-state adiabatic potential energy surfaces for the water molecule

The high accuracy ab initio adiabatic potential energy surfaces (PESs) of the ground electronic state of the water molecule, determined originally by Polyansky et al. [Science 299, 539 (2003)] and called CVRQD, are extended and carefully characterized and analyzed. The CVRQD potential energy surfaces are obtained from extrapolation to the complete basis set of nearly full configuration interaction valence-only electronic structure computations, augmented by core, relativistic, quantum electrodynamics, and diagonal Born-Oppenheimer corrections. We also report ab initio calculations of several quantities characterizing the CVRQD PESs, including equilibrium and vibrationally averaged (0 K) structures, harmonic and anharmonic force fields, harmonic vibrational frequencies, vibrational fundamentals, and zero-point energies. They can be considered as the best ab initio estimates of these quantities available today. Results of first-principles computations on the rovibrational energy levels of several isotopologues of the water molecule are also presented, based on the CVRQD PESs and the use of variational nuclear motion calculations employing an exact kinetic energy operator given in orthogonal internal coordinates. The variational nuclear motion calculations also include a simplified treatment of nonadiabatic effects. This sophisticated procedure to compute rovibrational energy levels reproduces all the known rovibrational levels of the water isotopologues considered, H(2) (16)O, H(2) (17)O, H(2) (18)O, and D(2) (16)O, to better than 1 cm(-1) on average. Finally, prospects for further improvement of the ground-state adiabatic ab initio PESs of water are discussed.     

The vibrational levels of ammonia

A new six-dimensional potential energy function (PEF) of ammonia expressed in internal coordinates is determined by fitting to points evaluated by Density Functional Theory with the B97-1 functional. The C3v and D3h structures are treated on an equal footing. The inversion barrier is 1820 cm(-1), which is in very good agreement with the experimental value of 1834 cm(-1). The minimum 'reaction path' is well defined by the analytic function up to 40 degrees for the umbrella angle. Using this PEF, the vibrational levels are calculated variationally using three different methods. The first employs the internal kinetic energy operator developed for ammonia by Handy, Carter and Colwell (Mol. Phys. 96 (1999) 477). The second uses the code MULTIMODE (J. Chem. Phys. 107 (1997) 10458), which involves the kinetic energy operator as expressed in normal coordinates by Watson. The third uses an implementation of the reaction path hamiltonian (J. Chem. Phys. 72 (1980) 99) within the MULTIMODE code. All three approaches give similar energies for the vibrational energies of ammonia, and these agree with experiment to within 15 cm(-1) for the fundamental vibrations.     

The rotation–vibration coupling equations for polyatomic molecules in internal coordinates

An exact vibration–rotation kinetic energy operator for polyatomic molecules has been obtained on the basis of Sutcliffe's method, in terms of curvilinear internal coordinates and rotational angular moment operators. This operator is derived from the kinetic energy operator in Cartesian coordinates by the successive transformations using the chain rule. This kinetic energy operator can be used not only for the system of any triatomic and tetraatomic molecules and common polyatomic molecules in chemistry, but also for the investigation of the collision problems between two molecules after some modifications. Finally, using this Hamiltonian, the rotation–vibration coupling equations of polyatomic molecules have been derived and discussed. © 1992 John Wiley & Sons, Inc.     

Quantum dynamics of rovibrational transitions in H2-H2 collisions: internal energy and rotational angular momentum conservation effects

We present a full dimensional quantum mechanical treatment of collisions between two H(2) molecules over a wide range of energies. Elastic and state-to-state inelastic cross sections for ortho-H(2)?+ para-H(2) and ortho-H(2)?+ ortho-H(2) collisions have been computed for different initial rovibrational levels of the molecules. For rovibrationally excited molecules, it has been found that state-to-state transitions are highly specific. Inelastic collisions that conserve the total rotational angular momentum of the diatoms and that involve small changes in the internal energy are found to be highly efficient. The effectiveness of these quasiresonant processes increases with decreasing collision energy and they become highly state-selective at ultracold temperatures. They are found to be more dominant for rotational energy exchange than for vibrational transitions. For non-reactive collisions between ortho- and para-H(2) molecules for which rotational energy exchange is forbidden, the quasiresonant mechanism involves a purely vibrational energy transfer albeit with less efficiency. When inelastic collisions are dominated by a quasiresonant transition calculations using a reduced basis set involving only the quasiresonant channels yield nearly identical results as the full basis set calculation leading to dramatic savings in computational cost.     

Natural coordinates for polyatomic reactions

The curvilinear natural coordinates (NRC) are considered to describe the polyatomic reaction dynamics. An invariant equation is found for the reaction path curve (RP), allowing its calculation in an arbitrary initial coordinate system. Apart from common properties, well known for triatomic reactions, the polyatomic NRC take explicit account of a smooth change of normal vibrational modes of a chemical system during its evolution along RP. This generates special terms in the kinetic energy operator that induce nonadiabatic vibrational transitions and do not vanish in a classical limit, no matter how small the curvature of the RP is. For the case when the amplitudes of vibrations perpendicular to the RP are small it is shown that, as a first approximation, the vibrational transitions can be treated independently of the rotational degrees of freedom. This conclusion is obtained as a generalization of Eckart theory stating the possibility of vibration—rotational separation for stable molecules.     

Imaging the state-specific vibrational predissociation of the C2H2-NH3 hydrogen-bonded dimer

The state-to-state vibrational predissociation (VP) dynamics of the hydrogen-bonded ammonia-acetylene dimer were studied following excitation in the asymmetric CH stretch. Velocity map imaging (VMI) and resonance-enhanced multiphoton ionization (REMPI) were used to determine pair-correlated product energy distributions. Following vibrational excitation of the asymmetric CH stretch fundamental, ammonia fragments were detected by 2 + 1 REMPI via the B1E' <-- X1A1' and C'1A1' <-- X1A1' transitions. The fragments' center-of-mass (c.m.) translational energy distributions were determined from images of selected rotational levels of ammonia with one or two quanta in the symmetric bend (nu2 umbrella mode) and were converted to rotational-state distributions of the acetylene co-fragment. The latter is always generated with one or two quanta of bending excitation. All the distributions could be fit well when using a dimer dissociation energy of D0 = 900 +/- 10 cm(-1). Only channels with maximum translational energy <150 cm(-1) are observed. The rotational excitation in the ammonia fragments is modest and can be fit by temperatures of 150 +/- 50 and 50 +/- 20 K for 1nu2 and 2nu2, respectively. The rotational distributions in the acetylene co-fragment pair-correlated with specific rovibrational states of ammonia appear statistical as well. The vibrational-state distributions, however, show distinct state specificity among channels with low translational energy release. The predominant channel is NH3(1nu2) + C2H2(2nu4 or 1nu4 + 1nu5), where nu4 and nu5 are the trans- and cis-bend vibrations of acetylene, respectively. A second observed channel, with much lower population, is NH3(2nu2) + C2H2(1nu4). No products are generated in which the ammonia is in the vibrational ground state or the asymmetric bend (1nu4) state, nor is acetylene ever generated in the ground vibrational state or with CC stretch excitation. The angular momentum (AM) model of McCaffery and Marsh is used to estimate impact parameters in the internal collisions that give rise to the observed rotational distributions. These calculations show that dissociation takes place from bent geometries, which can also explain the propensity to excite fragment bending levels. The low recoil velocities associated with the observed channels facilitate energy exchange in the exit channel, which results in statistical-like fragment rotational distributions.     

Expressing kinetic‐energy for vibration–rotation motions in general rotating systems of axes and introducing quasirectilinear vibrational coordinates to simplify Hamiltonian forms

To understand how the internal and rotational motions of a polyatomic system depend on which rotating system of axes is selected, we derived the explicit form of the atomic velocities determined by an observer stationed on the general rotating system of axes. Using the derived velocities, we formulated the kinetic energy expression for vibration–rotation motions with respect to the rotating system of axes. From this expression, we clarified covariant metric tensors under zero angular momentum, which have been confused with an erroneous expression even in the professional literature, and the relationship between the kinetic energy expression and the rotating system of axes. Furthermore, to simplify the Hamiltonian form, we introduced quasirectilinear vibrational coordinates to describe the Hamiltonian. The resulting Hamiltonian form is superior to those of the previous studies in that the kinetic and potential energy expressions are simple and the vibrational frequencies are independent of the original internal coordinates used. In fact, we show that its application for three examples is useful. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 83: 22–29, 2001     

A nine-dimensional calculation of the vibrational OH stretching and HOH bending spectrum of the water trimer

We have studied the vibrational high-frequency spectrum of the water trimer computationally. We expand an earlier study [J. Chem. Phys. A 2009, 113, 9124-9132] where we approximated the water trimer as three individually vibrating water monomer units. Some intramolecular potential energy coupling terms are now included in the previous model. The six OH bond lengths and the three HOH bending angles are used as the internal coordinates. The kinetic energy operator is a sum of the kinetic energy operators of the monomer units. We use the coupled cluster method with single, double, and perturbative triple excitations method [CCSD(T)] with augmented correlation consistent polarized valence triple-ζ (aug-cc-pVTZ) basis set to calculate the potential energy surface (PES). The counterpoise correction is included in the one-dimensional part of the PES. We calculate the vibrational energy eigenvalues using the variational method. The corresponding eigenfunctions are used to obtain the absorption intensities.     

A new exact quantum mechanical rovibrational kinetic energy operator for penta-atomic systems in internal coordinates

The concrete molecule-fixed (MF) kinetic energy operator for penta-atomic molecules is expressed in terms of the parameterδ, the matrix element G_(?), and angular momentum operator (?). The applications of the operator are also discussed. Finally, a general compact form of kinetic energy operator suitable for calculating the rovibrational spectra of polyatomie molecules is presented.     

Rovibrational energy levels of hydrogen peroxide, studied by MULTIMODE with a reaction path Hamiltonian

Recently, Carter and Handy [J. Chem. Phys. 113 (2000) 987] have introduced the theory of the reaction path Hamiltonian (RPH) [J. Chem. Phys. 72 (1980) 99] into the variational scheme MULTIMODE, for the calculation of the J=0 vibrational levels of polyatomic molecules, which have a single large-amplitude motion. In this theory the reaction path coordinate s becomes the fourth dimension of the moment-of-inertia tensor, and must be treated separately from the remaining 3N-7 normal coordinates in the MULTIMODE program. In the modified program, complete integration is performed over s, and the M-mode MULTIMODE coupling approximation for the evaluation of the matrix elements applies only to the 3N-7 normal coordinates. In this paper the new algorithm is extended to the calculation of rotational-vibration energy levels (i.e. J>0) with the RPH, following from our analogous implementation for rigid molecules [Theoret. Chem. Acc. 100 (1998) 191]. The full theory is given, and all extra terms have been included to give the exact kinetic energy operator. In order to validate the new code, we report studies on hydrogen peroxide (H2O2), where the reaction path is equivalent to torsional motion. H2O2 has previously been studied variationally using a valence coordinate Hamiltonian; complete agreement for calculated rovibrational levels is obtained between the previous results and those from the new code, using the identical potential surface. MULTIMODE is now able to calculate rovibrational levels for polyatomic molecules which have one large-amplitude motion.     

Discrete variable approaches to tetratomic molecules: part I: DVR(6) and DVR(3) + DGB methods

We present two discrete variable representation (DVR) based methods for the determination of the vibrational energy levels of tetratomic molecules. Both methods are designed for orthogonal internal coordinates in a body-fixed reference frame and make use of the DVR of three angular variables. The angular DVRs allow the construction of a fixed-angle three-mode Hamiltonian for the stretching vibrations. For each of the angular triples, the stretching eigenvalue problems are solved by employing 3D radial DVRs in the DVR(6) approach and real three-dimensional distributed Gaussian functions in the DVR(3) + DGB method. The angular degrees of freedom are taken sequentially into account in conjunction with a contraction scheme resulting from several diagonalization/truncation steps. Vibrationally averaged geometries, expectation values of rotational constants, and several adiabatic projection schemes developed in this work for tetratomic molecules are used to characterize the vibrational levels calculated by the DVR(6) and DVR(3) + DGB.