A simple method for derivation of the rovibrational Hamiltonian: application to orthogonal radau coordinate systems as special cases

The molecular Hamiltonian of polyatomic molecules has been obtained. A general choice of internal coordinates depending on external parameters was considered. The rovibrational Hamiltonian for this set of coordinate system was derived in general terms as a function of the external parameters a and b. This procedure is also applicable to various kinds of internal coordinates in a straightforward way. The rovibrational Hamiltonian of triatomic molecules is considered as an application of this general formulation. In addition, orthogonal Radau coordinates are considered as cases of this new approach     

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.     

Calculating energy levels of isomerizing tetra-atomic molecules. II. The vibrational states of acetylene and vinylidene

A general, full-dimensional computational method for the accurate calculation of rotationally and vibrationally excited states of tetra-atomic molecules is further developed. The resulting computer program may be run in serial and parallel modes and is particularly appropriate for molecules executing wide-amplitude motions and isomerizations. An application to the isomerizing acetylene/vinylidene system is presented. Large-scale calculations using a coordinate system based on orthogonal satellite vectors have been performed in six dimensions and vibrational term values and wave functions for acetylene and vinylidene states up to approximately 23 000 cm(-1) above the potential minimum have been determined. This has permitted the characterization of acetylene and vinylidene states at and above the isomerization barrier. These calculations employ more extensive vibrational basis sets and hence consider a much higher density of states than in any variational calculations reported hitherto for this system. Comparison of the calculated density of states with that determined empirically suggests that our calculations are the most realistic achieved for this system to date. Indeed more states have been converged than in any previous study of this system. Calculations on lower lying excited states of acetylene based on HC-CH diatom-diatom coordinates give nearly identical results to those based on orthogonal satellite vectors. Comparisons are also made with calculations based on HH-CC diatom-diatom coordinates.     

改进的三维分离变量表象方法计算振动光谱

The modified discrete variable representation for three-dimension (DVR3D) method was applied to the determination of the vibrational energy levels of the fundamental electronic state of H2S and H2O. The Hamiltonian was expressed in Jacobi coordinates and developed on a DVR basis for each internal coordinate. The angular coordinate used a DVR based on Legendre polynomials and the radial coordinates utilized a DVR based on sine basis functions. Successive diagonalization and truncation technique was used to reduce the size of the final Hamiltonian matrix to be diagonalized. Calculations were presented for H2S and H2O to demonstrate the accuracy of these algorithms.     

Variational calculation of vibrational energies of triatomic molecules using SCF optimized modes

A self-consistent field optimization of the vibrational coordinates for nonlinear triatomic molecules is presented. The optimal coordinates are obtained by making a three-dimensional rotational transformation of the normal modes and determining the rotation angles as those for which the SCF energy is stationary. The utility of the optimized coordinates in full variational calculations of vibrational energies is studied for the molecules of H₂O, O₃, H₂D⁺, H₂T⁺, and D₂T⁺. For H₂O and O₃, the optimization procedure leads to the local mode representation. It is shown that the use of the optimal coordinates in variational calculations allows a large reduction of the dimension of the Hamiltonian matrix to be diagonalized in order to reach convergence.     

First-principle calculation of reduced masses in vibrational analysis using generalized internal coordinates: some crucial aspects and examples

In this paper we present and analyze the most essential aspects of reduced masses along generalized internal coordinates. The definition of reduced masses in the internal coordinate formalism is established through the Wilson G-matrix concept and includes sophisticated relations between internal and Cartesian coordinates. Moreover, reduced masses in internal coordinates are, in general, no longer constant but coordinate-dependent. Based on the approach presented earlier [Stare, J.; Balint-Kurti, G. G. J. Phys. Chem. A 2003, 107, 7204-7214] and on our experience with reduced masses discussed in this paper, we have developed a robust program for the calculation of Wilson G-matrix elements and their functional coordinate dependence. The approach is based on the first principles and can be used in virtually any (internal) coordinate set. Since the program allows for projection of any kind of nuclear motion on the selected internal coordinates, the method is particularly suitable for ab initio or DFT potential energy functions calculated by partial geometry optimization. Moreover, reduced masses obtained by this program can be used as a decision tool for selecting the most appropriate internal coordinates for the considered vibrational problem and for the inclusion or omission of the kinetic coupling terms in the vibrational Hamiltonian.     

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.     

A reaction surface Hamiltonian study of malonaldehyde

We report calculations using a reaction surface Hamiltonian for which the vibrations of a molecule are represented by 3N-8 normal coordinates, Q, and two large amplitude motions, s(1) and s(2). The exact form of the kinetic energy operator is derived in these coordinates. The potential surface is first represented as a quadratic in Q, the coefficients of which depend upon the values of s(1),s(2) and then extended to include up to Q(6) diagonal anharmonic terms. The vibrational energy levels are evaluated by solving the variational secular equations, using a basis of products of Hermite polynomials and appropriate functions of s(1),s(2). Our selected example is malonaldehyde (N=9) and we choose as surface parameters two OH distances of the migrating H in the internal hydrogen transfer. The reaction surface Hamiltonian is ideally suited to the study of the kind of tunneling dynamics present in malonaldehyde. Our results are in good agreement with previous calculations of the zero point tunneling splitting and in general agreement with observed data. Interpretation of our two-dimensional reaction surface states suggests that the OH stretching fundamental is incorrectly assigned in the infrared spectrum. This mode appears at a much lower frequency in our calculations due to substantial transition state character.     

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.     

The calculation of vibrational energy levels of polyatomic molecules including anharmonic effect using contact transformation perturbation method

Using contact transformation perturbation method based on the Taylor expansion of the potential energy function in terms of dimensionless normal coordinates up to sixth‐order, the vibrational energy levels in terms of force constants are derived. The contact transformation theory has been applied to simplify the calculation of perturbation effects. To calculate the second‐order vibrational energy correction, the third and fourth‐order terms of potential function have been placed in the first‐order perturbation Hamiltonian and the second‐order Hamiltonian contains hexatic ones. We present expressions which give relations between the fourth‐ and sixth‐order terms in dimensionless normal coordinates of the potential and the anharmonicity coefficients. For illustration, a set of vibrational energies levels of SO₂, and H₂O molecules including anharmonic effects has been calculated. © 2013 Wiley Periodicals, Inc.     

Molecular structure calculations: a unified quantum mechanical description of electrons and nuclei using explicitly correlated Gaussian functions and the global vector representation

We elaborate on the theory for the variational solution of the Schro?dinger equation of small atomic and molecular systems without relying on the Born-Oppenheimer paradigm. The all-particle Schro?dinger equation is solved in a numerical procedure using the variational principle, Cartesian coordinates, parameterized explicitly correlated Gaussian functions with polynomial prefactors, and the global vector representation. As a result, non-relativistic energy levels and wave functions of few-particle systems can be obtained for various angular momentum, parity, and spin quantum numbers. A stochastic variational optimization of the basis function parameters facilitates the calculation of accurate energies and wave functions for the ground and some excited rotational-(vibrational-)electronic states of H(2) (+) and H(2), three bound states of the positronium molecule, Ps(2), and the ground and two excited states of the (7)Li atom.     

Ab initio calculations of the ultraviolet resonance Raman spectra of uracil

An equation been derived to calculate, ab initio, the frequencies and intensities of a resonant Raman spectrum from the transform theory of resonance Raman scattering. This equation has been used to calculate the intensities of the ultraviolet resonance Raman spectra from the first π-π* excited state of uracil and 1,3-dideuterouracil. The protocol for this calculation is as follows: (1) The force constant matrix elements in Cartesian coordinate space, the vibrational frequencies, and the minimum energy ground and excited state geometries of the molecule are calculated ab initio using the molecular orbital program Gaussian 92, (2) the force constants in Cartesian coordinates are transformed into force constants in the space of a set of 3N – 6 nonredundant symmetrized internal coordinates, (3) the G matrix is constructed from the energy minimized ground state Cartesian coordinates and the GFL = LΛ eigenvalue equation is solved in internal coordinate space, (4) the elements of the L and L^?1 matrices are calculated, (5) the changes in all of the internal coordinates in going from the ground to the excited state are calculated, and (6) these results are used in combination with the transform theory of resonance Raman scattering to calculate the relative intensities of each of the 3N – 6 vibrations as a function of the exciting laser frequency. There are no adjustable parameters in this calculation, which reproduces the experimental frequencies and intensities with remarkable fidelity. This indicates that the Dushinsky rotation of the modes in the excited state of these molecules is not important and that the simplest form of the transform theory is adequate. © 1995 John Wiley & Sons, Inc.     

Discrete variable representation method applied to the determination of rotation-vibration bound states of NO2

The discrete variable representation method is applied to the determination of the rotation-vibration energy levels of the fundamental electronic state of NO₂. The Hamiltonian is expressed in Johnson hyperspherical coordinates and developed on a DVR basis for each internal coordinate, while parity-adapted linear combinations of Wigner functions are used to describe the rotational motion. The diagonalization of the Hamiltonian matrix is performed using the Lanczos algorithm for large symmetric and Hermitian matrices. Results for rovibrational states up to J = 11 for the first five vibrational energy levels are presented. © 1997 John Wiley & Sons, Inc.     

Spectroscopy of highly excited vibrational states of HCN in its ground electronic state

An experimental technique based on a scheme of vibrationally mediated photodissociation has been developed and applied to the spectroscopic study of highly excited vibrational states in HCN, with energies between 29,000 and 30,000 cm(-1). The technique consists of four sequential steps: in the first one, a high power laser is used to vibrationally excite the sample to an intermediate state, typically (0,0,4), the nu3 mode being approximately equivalent to the C-H stretching vibration. Then a second laser is used to search for transitions between this intermediate state and highly vibrationally excited states. When one of these transitions is found, HCN molecules are transferred to a highly excited vibrational state. Third, a ultraviolet laser photodissociates the highly excited molecules to produce H and CN radicals in its A 2Pi electronic state. Finally, a fourth laser (probe) detects the presence of the CN(A) photofragments by means of an A-->B-->X laser induced fluorescence scheme. The spectra obtained with this technique, consisting of several rotationally resolved vibrational bands, have been analyzed. The positions and rotational parameters of the states observed are presented and compared with the results of a state-of-the-art variational calculation.     

Six-dimensional tunneling dynamics of internal rotation in hydrogen peroxide molecule and its isotopomers

The six-dimensional torsion-vibration Hamiltonian of the H₂O₂ molecule and its H/D- and ¹⁸O/¹⁶O-isotopomers is derived. The Hamiltonian includes the kinetic energy operator, which depends on the tunneling coordinate, and the potential energy surface represented as a quartic polynomial with respect to the small-amplitude transverse coordinates. Parameters of the Hamiltonian were obtained from DFT calculations of the equilibrium geometries, eigenvectors, and eigenfrequencies of normal vibrations at the stationary points corresponding to the ground state and both the cis- and trans-transition states, carried out with the B3LYP density functional and 6-311+G(2d,p) basis set. The quantum dynamics problem is solved using the perturbative instanton approach generalized for the excited states situated above the barrier top. Vibration-tunneling spectra are calculated for the ground state and low-lying excited states with energies below 2000 cm^–1. Strong kinematic and squeezed potential couplings between the large-amplitude torsional motion and bending modes are shown to be responsible for the vibration-assisted tunneling and for the dependence of tunneling splittings on the quantum numbers of small-amplitude transverse vibrations. Mode-specific isotope effects are predicted.     

The potential energy surfaces and the radiationless dynamics of excited states of benzene and pyrazine

The potential energy surfaces of the lowest excited states of benzene and pyrazine are investigated as a function of some of the symmetry-adapted internal coordinates by means of the INDO/S method. A large stabilization of the T₂ (ππ*) state of pyrazine (≈ 0.5 eV) along the S_8b vibrational coordinate is found. The calculated potential energy in some excited states (T₁ in benzene, T₂ and S₂ in pyrazine) is a very flat function of the S_16b vibrational coordinate, leading to a crossing with the potential energy of the ground state at relatively small excess of vibrational energy (≈ 1 eV). Thus the ν_16b vibrational mode is postulated to play an important role in the radiationless relaxation to the ground states of these systems. No such crossing has been found near the “channel three” threshold of benzene.     

Generalized approximation to the reaction path: the formic acid dimer case

A set of mass-weighted internal coordinates was derived and applied to the double proton transfer reaction in the formic acid dimer (FAD). The coordinate set was obtained starting from the Hirschfelder "mobile" by an optimization procedure consisting of a sequence of kinematic rotations. In FAD, the optimization procedure leads to three coordinates that do change significantly along the reaction path. These coordinates span the reaction space, whereas the remaining modes are treated in a harmonic approximation. The effect that the dimer dissociative motion has on the ground and excited vibrational states dynamics was explored. In the frequency region corresponding to the symmetric OH-stretch vibration four doublets have been identified with splittings of 2.76, 0.07, 0.60, and 4.03 cm(-1).     

Use of the normal coordinates in variational and perturbative ab initio handling of the vibronic and spin–orbit couplings in Π electronic states of linear tetra‐atomic molecules

Avariational and a perturbative approach are developed to handle the combined effect of the vibronic and spin–orbit couplings in Π electronic states of tetra‐atomic molecules with linear equilibrium geometry. Both of them are based on the use of the normal vibrational bending coordinates. The perturbative treatment is carried out via two schemes for partition of the model Hamiltonian: In the first, the spin–orbit coupling term is treated as a perturbation; in the second, it is included in the zeroth‐order Hamiltonian. It is demonstrated that both perturbative approaches lead to the same second‐order formulae when the spin–orbit coupling constant is small compared to the bending frequency, but much larger than the splitting of potential surfaces upon bending. These approaches are used to calculate the vibronic and spin–orbit structure in the X²Π electronic state of HCCS by employing the ab initio‐computed potential energy surfaces. Complete numerical equivalence of the results obtained with the present variational approach and those generated by the algorithms employing internal vibrational coordinates is demonstrated. The restrictions concerning the applicability of the perturbative approaches are discussed in terms of the agreement between the results obtained by means of them with those generated in the corresponding variational computations. The general reliability of the model employed is checked by comparing the theoretical results with the available experimental data. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003