Non-condon scheme of radiationless transitions for the morse oscillator model

The non-adiabatic coupling matrix elements responsible for radiationless deactivation of an electronically excited molecule are calculated without invoking the Condon approximation. Assuming the Morse potential surfaces for vibrational motion along the local and totally symmetric normal coordinates, the vibronic part of the radiationless rate constant is calculated. It is shown that the rate constant in the non-Condon scheme exceeds that obtained in the Condon approximation by about two orders of magnitude. The calculated dependence of the radiationless rate constant on the energy gap is in good agreement with the experimentally observed energy gap law for intersystem crossing T₁ → S₀ rate constants in aromatic hydrocarbons.     

Calculation of anharmonic force constants

Non-linear transformations from internal to cartesian displacements are considered for anharmonic calculation of vibrational frequencies. Force constants in normal coordinates, up to quartic terms, are related to Christoffel symbols. The latter are tabulated for valence angles or, for torsion and wagging out of plane, evaluated through expressions which avoid differentiation of explicit functional forms for each type of internal coordinate. The same symbols are used for cartesian tensors and redundancy coefficients up to the third order.     

Polyatomic,anharmonic, vibrational–rotational analysis. Application to accurate ab initio results for formaldehyde

A computer program SURVIB is described for calculating vibrational anharmonicity constants for polyatomic molecules. The program requires as input a grid of calculated energies in the vicinity of a stationary point. This grid is fit, in a least squares sense, to a polynomial function of the internal coordinates. This analytic representation of the energy surface is employed in a normal mode analysis, and the energy is reexpanded as a polynominal function of the normal mode coordinates (expressed as vectors in the mass-weighted atomic Cartesian coordinate space). The resulting coefficients are used in a second-order perturbation theory analysis to obtain the vibrational anharmonicity constants. Also reported is an application of this program to formaldehyde employing ab initio, RHF , MP 2, MP 3, and RHF -CI calculations. The spectroscopic constants obtained for H₂CO are in good agreement with experimentally derived values recently reported by Reisner.     

An analytical approximation for the number of states along the reaction coordinate

A variant of a new empirical method, enables one to express a collinear triatomic potential energy surface as a family of Morse curves along “natural” bond order coordinates orthogonal to the reaction coordinate. The procedure depends on a single adjustable parameter which is related to the barrier's height. Because an analytical expression for the number of vibrational states of a Morse oscillator is available, one has an analytical approximation for the number of states along the reaction coordinate. The extrema in the number of states are utilized in various versions of classical microcanonical variational transition state theory (among which is a new version, which is in better agreement with dynamical results), to estimate the probability of a collinear reactions, as a function of the total energy. The analytical expressions are also used to analyze the origins of the maximum and minima in the number of states.     

Normal coordinate ab initio force relaxation

A procedure for optimization of molecular geometries is presented, combining ab initio calculations with vibrational molecular data from spectroscopy or empirical force fields. Theoretical cartesian forces are transformed to vibrational normal coordinate forces from which geometry increments are calculated. Test results indicate that the method saves considerable effort compared to other optimization schemes.     

Tensor formalism in anharmonic calculations

A new method is presented to compute cartesian tensors in the expansion of curvilinear internal coordinates. Second- and higher-order coefficients are related to the metrics of the space of displacements. Components of the metric tensor are taken from existing tables of inverse kinetic energy matrix elements or, when rotations are involved, derived from general invariance conditions of scalars within a molecule. This leads to a tensor formalism particularly convenient in dealing with curvilinear coordinates in anharmonic calculations of vibrational frequencies. Formulae are given for elements of the potential energy matrix, related to quadratic and cubic force constants in terms of Christoffel symbols. The latter quantities are also used in the expansion of redundancy relations, with explicit coefficients given up to the third order.     

势阱中粒子能级与波函数微扰计算的代数递推公式

利用超位力定理(HVT)和Hellmann-Feynman 定理(HFT),导出了由有精确解的势阱的能级值用微扰法直接计算一维势阱的各级近似能级的普遍代数公式,并导出了由能级近似值计算定态波函数近似表达式的代数公式.给出了代数公式具体应用的几个典型一维势阱实例.此法可推广到二维势阱与三维势阱的情形.     

The PyPES library of high quality semi‐global potential energy surfaces

In this article, we present a Python‐based library of high quality semi‐global potential energy surfaces for 50 polyatomic molecules with up to six atoms. We anticipate that these surfaces will find widespread application in the testing of new potential energy surface construction algorithms and nuclear ro‐vibrational structure theories. To this end, we provide the ability to generate the energy derivatives required for Taylor series expansions to sixth order about any point on the potential energy surface in a range of common coordinate systems, including curvilinear internal, Cartesian, and normal mode coordinates. The PyPES package, along with FORTRAN, C, MATLAB and Mathematica wrappers, is available at http://sourceforge.net/projects/pypes-lib . © 2015 Wiley Periodicals, Inc.     

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.     

Grid‐based density functional calculations of many‐electron systems

Exploratory variational pseudopotential density functional calculations are performed for the electronic properties of many‐electron systems in the 3D cartesian coordinate grid (CCG). The atom‐centered localized gaussian basis set, electronic density, and the two‐body potentials are set up in the 3D cubic box. The classical Hartree potential is calculated accurately and efficiently through a Fourier convolution technique. As a first step, simple local density functionals of homogeneous electron gas are used for the exchange‐correlation potential, while Hay‐Wadt‐type effective core potentials are employed to eliminate the core electrons. No auxiliary basis set is invoked. Preliminary illustrative calculations on total energies, individual energy components, eigenvalues, potential energy curves, ionization energies, and atomization energies of a set of 12 molecules show excellent agreement with the corresponding reference values of atom‐centered grid as well as the grid‐free calculation. Results for three atoms are also given. Combination of CCG and the convolution procedure used for classical Coulomb potential can provide reasonably accurate and reliable results for many‐electron systems. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

The selection of coordinate systems for describing large-amplitude nuclear motion in triatomic molecules

A family of mass-scaled coordinate systems is presented for the triatom i75n which the kinetic energy operator for large-amplitude vibrational motion is diagonal (unlike that in current local-mode theories). For any particular potential, a procedure is described to choose the coordinate system in which the hamiltonian most closely approaches separability. This optimal system is applied to the calculation of vibrational excitation energies of water and related isotopic molecules. The relationship of the family of large- amplitude coordinates to normal coordinates is discussed.     

Ab Initio Equilibrium Geometry and Vibrational Spectroscopic Study of 25,26,27,28-tetrahydroxycalix[4]arene

The aim of this work was to present a comprehensive vibrational spectroscopic study of 25,26,27,28-tetrahydroxycalix[4]arene. For this purpose, quantum chemical calculations were carried out at the ab initio HF/4-31G* level, as a consequence of the great size of the molecule. In the frame of these calculations, the symmetry of the molecule was investigated. Trying C 4v , C 2v , and C s symmetries as input, the geometry optimization, however, pointed to the C 2 configuration. In the latter case, all the calculated vibrational frequencies were greater than zero, and therefore the equilibrium geometry could be identified. Medium and far infrared as well as Raman spectra of the compound were measured. On the basis of the calculated force constants and geometric parameters, normal coordinate analysis was applied for the interpretation of the experimental vibrational spectra. Problems arose with the choice of the internal coordinates of the molecule, which are important from the point of view of the internal macrocycle ("lower rim") of the molecule. On the basis of the theory of redundant coordinates, a program was written for choosing the coordinates of this 16-membered cycle. Full interpretation of the vibrational fundamentals of the compound is presented. Several force constant matrix elements have surprisingly high values. As a result of the normal coordinate analysis, the relative rigidity of the lower rim was concluded. Ab initio calculations and assignment of the vibrational spectra of 25,26,27,28-tetrahydroxycalix[4]arene based on the calculations are presented.     

Large configuration interaction calculations of nuclear spin-spin coupling constants: III. Vibrational effects in ammonia

Large configuration interaction calculations of the proton—proton coupling constant for several geometrical configurations of the ammonia molecules are reported. The analytical expressions for the energy surface and the coupling constant as functions of two cartesian displacement coordinates are fitted to the calculated values. The potential is used for the calculation of the vibrational wavefunctions for ¹⁵NH₃ and ¹⁵ND₃ species and the vibrational averaging of the coupling constant is carried out using these functions. Though the value of the coupling constants shows a very strong geometry dependence, the vibrational corrections are found to be small. A possible correlation of the proton—proton coupling constant with an angular parameter in the NH₂ group in RNH₂ compounds is indicated.     

Dipole moments of HDO in highly excited vibrational states measured by Stark induced photofragment quantum beat spectroscopy

We report here a measurement of electric dipole moments in highly vibrationally excited HDO molecules. We use photofragment yield detected quantum beat spectroscopy to determine electric field induced splittings of the J=1 rotational levels of HDO excited with 4, 5, and 8 quanta of vibration in the OH stretching mode. The splittings allow us to deduce mua and mub, the projections of dipole moment onto the molecular rotation inertial axes. We compare the measured HDO dipole moment components with the results of quantitative calculations based on Morse oscillator wave functions and an ab initio dipole moment surface. The vibrational dependence of the dipole moment components reflect both structural and electronic changes in HDO upon vibrational excitation; principally the vibrational dependence of the O-H bond length and bond angle, and the resulting change in orientation of the principal inertial coordinate system. The dipole moment data also provide a sensitive test of theoretical dipole moment and potential energy surfaces, particularly for molecular configurations far from equilibrium.     

Thermodynamic properties of diatomic molecule systems under SO(2,1)‐anharmonic Eckart potential

Due to one of the most representative contributions to the energy in diatomic molecules being the vibrational, we consider the generalized Morse potential (GMP) as a typical interaction for one‐dimensional microscopic systems, which describes local anharmonic effects. From the Eckart potential (EP) model, it is possible to find a connection with the GMP model, as well as obtaining the analytical expression for the energy spectrum because it is based on algebras. This gives the macroscopic properties such as vibrational mean energy U, specific heat C, Helmholtz free energy F, and entropy S for a heteronuclear diatomic system, as well as with the exact partition function and its approximation for the high temperature region. Finally, a comparison is made between the graphs of some thermodynamic functions obtained with the GMP and the Morse potential (MP) for molecules.     

Full-dimensional quantum dynamics study of vinylidene-acetylene isomerization: a scheme using the normal mode Hamiltonian

Full-dimensional quantum dynamics calculations of vinylidene-acetylene isomerization are performed and the state-specific resonance decay lifetimes of vinylidene(-d(2)) are computed. The theoretical scheme is a combination of several methods: normal coordinates are chosen to describe the nuclear motion of vinylidene, with both the parity and permutation symmetry exploited; phase space optimization in combination with physical considerations is used to generate an efficient discrete variable representation; the reaction coordinate is defined by us according to the three most relevant normal coordinates, along which a kind of optimal complex absorbing potential is imposed; the preconditioned inexact spectral transform method combined with an efficient preconditioner is employed to extract the energies and lifetimes of vinylidene. The overall computation is efficient. The computed energy levels generally agree with experiment well, and several state-specific lifetimes are reported for the first time.     

Study of vibrational spectra and their assignments for phenylphosphonic and phenylthiophosphonic acid and comparison to experiments

The structures and conformational stabilities of phenylphosphonic and phenylthiophosphonic acids are investigated using calculations mostly at the DFT/6-311G** level and ab initio ones at the MP2/6-311G** level (no frequency calculations in the latter case), because we know from our previous results that the addition of diffuse functions to a valence triple zeta basis with polarization functions might lead to an unbalanced basis set. Further, the experience tells that for large energy differences between conformers, DFT works very well. From the calculations the molecules are predicted to exist in a conformational equilibrium consisting of two non (near)-planar conformers that are identical by symmetry. Interestingly, in the internal rotation potential functions the planar conformer appears to be a stable minimum (also optimization converges to planar), however the vibrational frequencies were computed and the planar conformer exhibited an imaginary one, indicating that it is a maximum with respect to one of the internal coordinates. Only optimization without any restrictions and starting from a non (near)-planar structure converged to a real minimum with a non (near)-planar geometry. In the minimum structure, vibrational infrared and Raman spectra are calculated and those for phenylphosphonic acid are compared to experimental ones, showing satisfactory agreement. The rather low intensity of the OH bands in the experimental infrared spectrum (as compared to normal organic acids) indicates rather weak hydrogen bonding with at most dimers present. Normal coordinate calculations are carried out and potential energy distributions are calculated for the molecules in the non (near)-planar conformations providing a complete assignment of the vibrational modes to atomic motions in the molecules. From the rather low rotational barriers we conclude, in agreement with the results from the literature (for other P=O compounds) based on localized orbitals, that conjugation effects are absent — or at least negligible — as compared to electrostatic and steric ones.     

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.     

Normal mode calculation and IR band assignments of A-type zeolite

Normal mode calculation of A-type zeolite was carried out with the potential energy functions obtained from the constraint method. Mass-weighted cartesian coordinates and the pseudo-lattice method were used. The assignments of IR absorption bands were made with the calculated normal modes, by using the calculated absorption intensities of the modes and the degrees of contribution of the internal coordinates to the modes. The force constants of internal coordinate motions within the framework were also calculated and are compared with the empirical values.     

Molecular dynamic structure and dimerization of polyacetylene

The vibrational self-consistent-field approximation is used to calculate excited vibrational energy levels of the water molecule in hyperspherical coordinates. The calculations are made for a global realistic Sorbie–Murrell-type potential surface for which exactum quantum variational results are known for comparison. The coupled SCF equations are solved using the discrete variable representation (DVR ) method, which allows computation of the coupled multidimensional integrals in a very simple and efficient way. The results are in good agreement with exactum quantum calculations and are more accurate than SCF energy eigenvalues obtained using normal mode coordinates.