Variational method for the calculation of the vibrational structure of the electronic spectra of polyatomic molecules. II

A variational method has been developed to solve the vibrational problem in the excited electronic state and to calculate the vibrational structure of the electronic spectrum of polyatomic molecules. The properties and structural characteristics of the variational matrix have been analyzed and an effective algorithm has been proposed for its approximate diagonalization. The effectiveness of the method and the corresponding suite of programs for the personal computer have been analyzed via the results of model calculations for a number of molecular structures. The method has high precision (errors of about 5% for frequencies and 15% for relative intensities), is an order of magnitude faster than previously used methods, and provides the possibility for the effective solution of the electrono-vibrational problem for polyatomic molecules, including the reverse problem.K. A. Timiryazev Agricultural Academy. Translated from Zhurnal Strukturnoi Khimii, Vol. 34, No. 1, pp. 149–156, January–February, 1993.     

Variational method for the calculation of the vibrational structure of the electronic spectra of polyatomic molecules. I

A method is proposed to calculate the vibrational structures of the electronic spectra of polyatomic molecules based on the variational solution of the vibrational problem in the excited state with the vibrational wave functions of the ground state as basis set. The electrono-vibrational problem leads to an evaluated and diagonalized variational matrix. The elements of the variational matrix have a simple form which is easily evaluated, has a clear physical meaning and is directly interconnected with observed spectral effects. This allows preliminary estimation of spectral phenomena and correction of the molecular model to take account of experimental results. The use of contemporary methods of diagonalization of the variational matrix, which possesses a characteristic structure, facilitates a tenfold increase in the speed of the method in comparison with traditional methods.K. A. Timiryazev Agricultural Academy. Translated from Zhurnal Strukturnoi Khimii, Vol. 34, No. 1, pp. 141–148, January–February, 1993.     

On the non-crossing rule for potential energy surfaces of polyatomic molecules

It is shown that the non-crossing for the intersection of potential energy curves of diatomic molecule applies also to potential energy surfaces of polyatomic molecules.     

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.     

Semi-empirical calculation of potential surface for polyatomic systems

Potential energy formula of polyatomic systems include many multiple exchange integrals. These multiple exchange integrals can be decomposed into “diatomic” integrals by using the Mulliken approximation. The potential surface can then be calculated numerically. The result of this work applied to the H₃ system is presented.     

A new method to calculate overlap integrals of vibrational wave functions in the theory of vibronic spectra of polyatomic molecules

An approximate method to calculate overlap integrals of vibrational wave functions of combined electron states is proposed. It uses reducibility of the general transformation of normal coordinates and quasiorthogonality of the Dushinsky matrix. Simple analytical expressions and convenient recurrent relations for the desired integrals of Franck-Condon and Herzberg-Teller types are found. The errors of calculation are of the same order as those of the existing “accurate” methods (≤5%), and the speed of calculation is higher by more than two orders. The study was carried out under financial support of the Russian Fundamental Research Fund (93-02-3405). K. A. Timiryazev Moscow Agricultural Academy. Translated fromZhurnal Strukturnoi Khimii, Vol. 35, No. 2, pp. 16–23, March–April, 1994. Translated by L. Chernomorskaya     

Moment expansion method to calculate vibrational frequencies of molecules and solids

The vibrational properties of a quantum system are determined by the density response matrix. In linear response theory this quantity is connected to the polarizability matrix, which can be expressed in terms of a double summation over one-particle energies and wave functions. In has been shown that this expression is not useful in the calculation of vibrational frequencies because of the very slow convergence of the summation in terms of the unoccupied states. In this paper, a different but equivalent expression is presented using a continued fraction. The resulting expression contains only one summation over the occupied states, solving in this way all the problems connected with the sum-over-states expression of the polarizability matrix. The elimination of all the unoccupied states via the use of the moment formula turns out to be a crucial step in the solution of the problem of the first-principles calculation of the vibrational spectra of molecules and solids.     

The effect of collisions on vibrational energy distribution in polyatomic molecules

We approach the problem of the effect of collisions on infrared absorption by an experimental and modeling study of a well-understood system, absorption of CO₂ laser radiation by cold CF₄. The spectroscopy, collisional energy transfer rates, and laser characteristics are all known. We conclude that the dominant collisional processes that influence the vibrational energy distribution during infrared laser absorption are rotational energy transfer-rotational relaxation and pressure broadening.     

Semiclassical calculation of energy transfer in polyatomic molecules. XII. Organic molecules

A semiclassical method for energy transfer to the torsional motion of polyatomic molecules is presented. It is shown that a purely classical treatment of the torsional motion is problematic due to the zero-point vibrational energy which may migrate into other modes. State-to-state cross sections for the excitation of the CH₃ torsion in CH₃OH colliding with ⁴He are presented as a function of initial kinetic energy.     

On a semiclassical approach to energy transfer in polyatomic molecules

A semiclassical method for energy transfer in polyatomic molecules is suggested. The method is based on a partial quantization of the vibrational degrees of freedom. The remaining degrees of freedom are treated classically. As an example a system with three quantum coordinates is considered. The Coriolis coupling terms are taken into account in the quantum mechanical part of the system and discussed for the N₂ + CO₂ system.     

Model calculation on the pump-probe measurement of ultrafast electronic population decay in polyatomic molecules

We consider the common situation of strong vibronic coupling of an optically bright (in absorption from the ground state) excited electronic state to a lower-lying dark electronic state in a polyatomic molecule. It is shown that for sufficiently short pump and probe laser pulses a time-resolved experiment measures the total time-dependent population probability P(t) of the bright state. For a realistic model problem (representing the three lowest electronic states of the benzene cation) a conical intersection of the potential energy surfaces of the bright and the dark state causes an ultrafast initial decay of P(t) on a femtosecond time scale, followed by quasiperiodic recurrences. These recurrences show up as femtosecond quantum beats in the time-resolved pump-probe signal. The beating frequency is related to the vibrational frequency of the dominant accepting mode of the system.     

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.     

Fragment calculation of electronic structures of polyatomic molecules in the ground state. I. A method of intermediate fragment

A method is proposed for fragment calculation of electronic structures of polyatomic molecules in the ground state. The wave function of a molecule in the ground state in single-determinantal representation of a closed shell is employed. The concise formulation allows efficient calculation of the electronic structures of polyatomic molecules taking into account possible charge transfer between interacting molecular fragments. V.I. Vernadskii Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences. Translated fromZhurnal Struktumoi Khimii, Vol. 36, No. 3, pp. 387–394, May–June, 1995. Translated by I. Izvekova     

Ab initio potential energy surface and excited vibrational states for the electronic ground state of Li2H

A 285-point multi-reference configuration-interaction involving single and double excitations (MRS-DCI) potential energy surface for the electronic ground state of Li₂H is determined by using 6-311G (2df, 2pd) basis set. A Simons-Parr-Finlan polynomial expansion is used to fit the discrete surface with a X² of 4.64 × 10^-6. The equilibrium geometry occurs at R_e =0.172 nm and <LiHLi =94.10. The dissociation energy for reaction Li₂H(²A)⇑ Li₂(¹⌆_g)+H(²S) is 243.910 kJ/mol. and that for reaction Li₂H(²A)⇑HLi(¹be)+Li(²S) is 106.445 kJ/mol. The inversion barrier height is 50.388 kJ/mol. The vibrational energy levels are calculated using the discrete variable representation (DVR) method. Project supported by the National Natural Science Foundation of China (grant No. 29673029) and by the Special Doctoral Research Foundation of the State Education Commission of China.     

Ab initio potential energy surface and excited vibrational states for the electronic ground state of Li_2H

A 285-pomt multi-reference configuration-interaction involving single and double excitations ( MRS DCI) potential energy surface for the electronic ground state of L12H is determined by using 6-311G (2df,2pd)basis set.A Simons-Parr-Finlan polynomial expansion is used to fit the discrete surface with a x2 of 4.64×106 The equn librium geometry occurs at Rc=0.172 nm and,LiHL1=94.10°.The dissociation energy for reaction I2H(2A)→L12(1∑g)+H(2S) is 243.910 kJ/mol,and that for reaction L12H(2A')→HL1(1∑) + L1(2S) is 106.445 kl/mol The inversion barrier height is 50.388 kj/mol.The vibrational energy levels are calculated using the discrete variable representation (DVR) method.     

Efficient calculation of potential energy surfaces for the generation of vibrational wave functions

An automatic procedure for the generation of potential energy surfaces based on high level ab initio calculations is described. It allows us to determine the vibrational wave functions for molecules of up to ten atoms. Speedups in computer time of about four orders of magnitude in comparison to standard implementations were achieved. Effects due to introduced approximations--within the computation of the potential--on fundamental modes obtained from vibrational self-consistent field and vibrational configuration interaction calculations are discussed. Benchmark calculations are provided for formaldehyde and 1,2,5-oxadiazole (furazan).     

On the theory of collision-induced vibrational transitions in polyatomic molecules: The mode-matching model

A modification of the SSH theory for collision-induced vibrational transitions in polyatomic molecules is proposed. The breathing-sphere model assumptions are avoided by considering the angular relation between the direction of approach and the normal modes displacements of the atoms involved in the contract (mode-matching). The results, as compared to the breathing-sphere model, indicate a considerable, mode specific reduction of the transition probabilities. Intramolecular transitions of CH₃Cl are studied as an example.