Rectilinear Normal Coordinates Research Articles

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.

Fourth-order vibrational perturbation theory with the Watson Hamiltonian: Report of working equations and preliminary results.

A derivation of fourth-order vibrational perturbation theory (VPT4) based on the Watson Hamiltonian in dimensionless rectilinear normal coordinates is presented. Terms that are linear and cubic in the (nk + ), with nk being the zeroth-order harmonic oscillator quantum numbers, appear in fourth order and extend the much simpler second-order vibrational perturbation theory model. The rather involved expressions for the fourth-order terms are derived with Rayleigh-Schrödinger perturbation theory, the process of verifying their correctness is described, and a computer code to generate the VPT4 constants from the potential energy surface derivatives is provided. The paper concludes with numerical examples featuring the H2O, Si2C, and cyclic-C3H2 molecules.

Comparing the accuracy of perturbative and variational calculations for predicting fundamental vibrational frequencies of dihalomethanes.

Three dihalogenated methane derivatives (CH2F2, CH2FCl, and CH2Cl2) were used as model systems to compare and assess the accuracy of two different approaches for predicting observed fundamental frequencies: canonical operator Van Vleck vibrational perturbation theory (CVPT) and vibrational configuration interaction (VCI). For convenience and consistency, both methods employ the Watson Hamiltonian in rectilinear normal coordinates, expanding the potential energy surface (PES) as a Taylor series about equilibrium and constructing the wavefunction from a harmonic oscillator product basis. At the highest levels of theory considered here, fourth-order CVPT and VCI in a harmonic oscillator basis with up to 10 quanta of vibrational excitation in conjunction with a 4-mode representation sextic force field (SFF-4MR) computed at MP2/cc-pVTZ with replacement CCSD(T)/aug-cc-pVQZ harmonic force constants, the agreement between computed fundamentals is closer to 0.3 cm-1 on average, with a maximum difference of 1.7 cm-1. The major remaining accuracy-limiting factors are the accuracy of the underlying electronic structure model, followed by the incompleteness of the PES expansion. Nonetheless, computed and experimental fundamentals agree to within 5 cm-1, with an average difference of 2 cm-1, confirming the utility and accuracy of both theoretical models. One exception to this rule is the formally IR-inactive but weakly allowed through Coriolis-coupling H-C-H out-of-plane twisting mode of dichloromethane, whose spectrum we therefore revisit and reassign. We also investigate convergence with respect to order of CVPT, VCI excitation level, and order of PES expansion, concluding that premature truncation substantially decreases accuracy, although VCI(6)/SFF-4MR results are still of acceptable accuracy, and some error cancellation is observed with CVPT2 using a quartic force field.

Anharmonic Vibrational Analysis of the Infrared and Raman Gas-Phase Spectra of s-trans- and s-gauche-1,3-Butadiene.

A quantum-mechanical (hybrid MP2/cc-pVTZ and CCSD(T)/cc-pVTZ) full quartic potential energy surface (PES) in rectilinear normal coordinates and the second-order operator canonical Van Vleck perturbation theory (CVPT2) are employed to predict the anharmonic vibrational spectra of s-trans- and s-gauche-butadiene (BDE). These predictions are used to interpret their infrared and Raman scattering spectra. New high-temperature Raman spectra in the gas phase are presented in support of assignments for the gauche conformer. The CVPT2 solution is based on a PES and electro-optical properties (EOP; dipole moment and polarizability) expanded in Taylor series. Higher terms than those routinely available from Gaussian09 software were calculated by numerical differentiation of quadratic force fields and EOP using the MP2/cc-pVTZ model. The integer coefficients of the polyad quantum numbers were derived for both conformers of BDE. Replacement of harmonic frequencies by their counterparts from the CCSD(T)/cc-pVTZ model significantly improved the agreement with experimental data for s-trans-BDE (root-mean-square deviation ≈ 5.5 cm(-1)). The accuracy in predicting the rather well-studied spectrum of fundamentals of s-trans-BDE assures good predictions of the spectrum of s-gauche-BDE. A nearly complete assignment of fundamentals was obtained for the gauche conformer. Many nonfundamental transitions of the BDE conformers were interpreted as well. The predictions of multiple Fermi resonances in the complex CH-stretching region correlate well with experiment. It is shown that solving a vibrational anharmonic problem through a numerical-analytic implementation of CVPT2 is a straightforward and computationally advantageous approach for medium-size molecules in comparison with the standard second-order vibrational perturbation theory (VPT2) based on analytic expressions.

Vibrational Energy Levels via Finite-Basis Calculations Using a Quasi-Analytic Form of the Kinetic Energy.

A variational method for the calculation of low-lying vibrational energy levels of molecules with small amplitude vibrations is presented. The approach is based on the Watson Hamiltonian in rectilinear normal coordinates and characterized by a quasi-analytic integration over the kinetic energy operator (KEO). The KEO beyond the harmonic approximation is represented by a Taylor series in terms of the rectilinear normal coordinates around the equilibrium configuration. This formulation of the KEO enables its extension to arbitrary order until numerical convergence is reached for those states describing small amplitude motions and suitably represented with a rectilinear system of coordinates. A Gauss-Hermite quadrature grid representation of the anharmonic potential is used for all the benchmark examples presented. Results for a set of molecules with linear and nonlinear configurations, i.e., CO2, H2O, and formyl fluoride (HFCO), illustrate the performance of the method and the versatility of our implementation.

An adaptive density-guided approach for the generation of potential energy surfaces of polyatomic molecules

We present an adaptive density-guided approach for the construction of Born–Oppenheimer potential energy surfaces (PES) in rectilinear normal coordinates for use in vibrational structure calculations. The procedure uses one-mode densities from vibrational structure calculations for a dynamic sampling of PESs. The implementation of the procedure is described and the accuracy and versatility of the method is tested for a selection of model potentials, water, difluoromethane and pyrimidine. The test calculations illustrate the advantage of local basis sets over harmonic oscillator basis sets in some important aspects of our procedure.

Equilibrium structure of chloroform

The quadratic, cubic and semi-diagonal quartic force field of chloroform has been calculated at the MP2 level of theory employing a basis set of triple-ζ quality and in the frozen core approximation. A semi-experimental equilibrium structure has been derived from experimental ground state rotational constants and vibration–rotation interaction constants calculated from the ab initio force field determined in rectilinear normal coordinates. This structure is in excellent agreement with the ab initio structure calculated at the CCSD(T) level of theory using a basis set of quintuple-ζ quality and a core correlation correction. Experimental mass-dependent r m structures are also determined and are found to be much less accurate.

Automatic generation of potential energy and property surfaces of polyatomic molecules in normal coordinates

A procedure for the automatic construction of Born-Oppenheimer (BO) potential energy and molecular property surfaces in rectilinear normal coordinates is presented and its suitability and accuracy when combined with vibrational structure calculations are assessed. The procedure relies on a hierarchical n-mode representation of the BO potential energy or molecular property surface, where the n-mode term of the sequence of potentials/molecular properties includes only the couplings between n or less vibrational degrees of freedom. Each n-mode cut of the energy/molecular property surface is first evaluated in a grid of points with ab initio electronic structure methods. The ab initio data are then spline interpolated and a subsequent polynomial fitting provides an analytical semiglobal representation for use in vibrational structure programs. The implementation of the procedure is outlined and the accuracy of the method is tested on water and difluoromethane. Strategies for improving the proposed algorithm are also discussed.

Vibrational and thermal effects on the dipole polarizability of methane and carbon tetrachloride from vibrational structure calculations

We present a theoretical study of vibrational and thermal effects on the dipole polarizability of methane and carbon tetrachloride. Using a fourth order Taylor expansion in rectilinear normal coordinates of the potential and property surfaces we solve the vibrational problem using vibrational structure theory, e.g., through vibrational self-consistent-field or vibrational configuration-interaction theory. For each vibrational state we calculate in addition the vibrational state average polarizability. Constructing the vibrational partition function by "brute force" allows for prediction of thermal effects on the dipole polarizability. The method is not restricted in any way to polarizabilities nor to the specific representation of the potential and property surfaces employed in this work. Any molecular property with a suitable normal coordinate representation may be considered. We discuss the performance of vibrational self-consistent field as compared to vibrational configuration interaction and study in detail the convergence of the former method with respect to the number of vibrational states included in the thermal averaging. Based on calculations including up to 170,000 vibrational self-consistent-field states we present thermal effects on the dipole polarizability of methane and carbon tetrachloride in the temperature ranges 0-1100 and 0-500 K, respectively. The predicted thermal effect on the dipole polarizability of methane is found to be approximately 0.8% which compare well with previous experimental measurements.

A perturbative calculation of the rovibrational energy levels of methane

The rovibrational energy levels of methane are determined from a quartic ab initio potential energy force field where the expansion coordinates are the Morse coordinates for the stretches and extension coordinates for the bends. Energies are calculated using canonical Van Vleck perturbation theory. Results are obtained for both rotation–vibration Hamiltonians expressed as functions of curvilinear and rectilinear normal coordinates. Second, fourth, and sixth order curvilinear results are compared with experimental results, and fourth order results for the rectilinear and curvilinear Hamiltonian are compared to each other. The calculated rovibrational levels are in good agreement with the experimental values for low J levels. The calculated rotational level splittings are in even better agreement with the experiment. In particular, the ground state tetrahedral splittings, which are as small as 10 −4 cm −1, are well reproduced by our calculations at sixth order.

Derivation of rotation-vibration Hamiltonians that satisfy the Casimir condition

A general prescription for deriving rotation-vibration Hamiltonians that satisfy the Casimir condition is presented. This condition, achieved using the Eckart constraints, is that there is no vibrational angular momentum in the molecular equilibrium configuration. The Eckart condition, while useful for studying rotation-vibration interactions, is difficult to apply for coordinates other than rectilinear normal coordinates. The present derivation allows one to derive Hamiltonians in curvilinear coordinates, yet still take advantage of the most relevant property of the Eckart frame, this being that Coriolis coupling is minimized in the limit of small vibrations.

Fermi resonances and local modes in stibine, SbH3: A Fourier interferometric and laser photoacoustic study of the overtone spectrum

The third stretching overtone region of a natural sample of stibine, SbH3, has been studied with high resolution infrared spectroscopy and the fifth and the sixth overtone region with Ti:Sapphire ring laser intracavity photoacoustic spectroscopy. The third overtone consists of a local mode pair of bands (400A1/E) which have been rotationally assigned both for 121SbH3 and 123SbH3 with a vibration-rotation model based on rectilinear normal coordinates. The vibrational dependencies of the model parameters are explained well with a simple block diagonal vibrational model. An extension of the standard vibration-rotation model is used to show that the upper state rotational energy level structures of both isotopic species are close to the rotational structure of an asymmetric rotor. High resolution laser spectrum of the fifth overtone consisting of a local mode pair of bands (600A1/E) shows severe perturbations in the upper state rotational structure. The (510A1/E) and (700A1/E) bands have been recorded with low resolution. All experimentally known vibration-rotation band origins of 121SbH3 have been reproduced well with a curvilinear internal valence coordinate system based Fermi resonance local mode model. The potential energy surface obtained agrees well with recent ab initio results.

Investigating optimal coordinates for describing vibrational motion

Results of independent mode models of molecular vibrations of SO2 and H2O are compared to converged eigenvalues and eigenfunctions. Energies of self-consistent-field and adiabatic wave functions are calculated and compared to the eigenvalues; Brueckner functions and natural modals are compared to the eigenfunctions. These comparisons are made for a class of normal coordinates of varying curvature. Of the coordinates we considered, an independent mode model based on rectilinear normal coordinates provides the best description of the energetics yet provides the poorest description of the eigenfunctions. The ramifications of this finding are discussed for both independent mode models and perturbative descriptions of molecular vibrations.