Sergey V. Krasnoshchekov

Numerical-Analytic Implementation of the Higher-Order Canonical Van Vleck Perturbation Theory for the Interpretation of Medium- Sized Molecule Vibrational Spectra

Anharmonic vibrational states of semirigid polyatomic molecules are often studied using the second-order vibrational perturbation theory (VPT2). For efficient higher-order analysis, an approach based on the canonical Van Vleck perturbation theory (CVPT), the Watson Hamiltonian and operators of creation and annihilation of vibrational quanta is employed. This method allows analysis of the convergence of perturbation theory and solves a number of theoretical problems of VPT2, e.g., yields anharmonic constants y(ijk), z(ijkl), and allows the reliable evaluation of vibrational IR and Raman anharmonic intensities in the presence of resonances. Darling-Dennison and higher-order resonance coupling coefficients can be reliably evaluated as well. The method is illustrated on classic molecules: water and formaldehyde. A number of theoretical conclusions results, including the necessity of using sextic force field in the fourth order (CVPT4) and the nearly vanishing CVPT4 contributions for bending and wagging modes. The coefficients of perturbative Dunham-type Hamiltonians in high-orders of CVPT are found to conform to the rules of equality at different orders as earlier proven analytically for diatomic molecules. The method can serve as a good substitution of the more traditional VPT2.

Criteria for first- and second-order vibrational resonances and correct evaluation of the Darling-Dennison resonance coefficients using the canonical Van Vleck perturbation theory

The second-order vibrational Hamiltonian of a semi-rigid polyatomic molecule when resonances are present can be reduced to a quasi-diagonal form using second-order vibrational perturbation theory. Obtaining exact vibrational energy levels requires subsequent numerical diagonalization of the Hamiltonian matrix including the first- and second-order resonance coupling coefficients. While the first-order Fermi resonance constants can be easily calculated, the evaluation of the second-order Darling-Dennison constants requires more complicated algebra for seven individual cases with different numbers of creation-annihilation vibrational quanta. The difficulty in precise evaluation of the Darling-Dennison coefficients is associated with the previously unrecognized interference with simultaneously present Fermi resonances that affect the form of the canonically transformed Hamiltonian. For the first time, we have presented the correct form of the general expression for the evaluation of the Darling-Dennison constants that accounts for the underlying effect of Fermi resonances. The physically meaningful criteria for selecting both Fermi and Darling-Dennison resonances are discussed and illustrated using numerical examples.

Polyad quantum numbers and multiple resonances in anharmonic vibrational studies of polyatomic molecules

In the theory of anharmonic vibrations of a polyatomic molecule, mixing the zero-order vibrational states due to cubic, quartic and higher-order terms in the potential energy expansion leads to the appearance of more-or-less isolated blocks of states (also called polyads), connected through multiple resonances. Such polyads of states can be characterized by a common secondary integer quantum number. This polyad quantum number is defined as a linear combination of the zero-order vibrational quantum numbers, attributed to normal modes, multiplied by non-negative integer polyad coefficients, which are subject to definition for any particular molecule. According to Kellmans method [J. Chem. Phys. 93, 6630 (1990)], the corresponding formalism can be conveniently described using vector algebra. In the present work, a systematic consideration of polyad quantum numbers is given in the framework of the canonical Van Vleck perturbation theory (CVPT) and its numerical-analytic operator implementation for reducing the Hamiltonian to the quasi-diagonal form, earlier developed by the authors. It is shown that CVPT provides a convenient method for the systematic identification of essential resonances and the definition of a polyad quantum number. The method presented is generally suitable for molecules of significant size and complexity, as illustrated by several examples of molecules up to six atoms. The polyad quantum number technique is very useful for assembling comprehensive basis sets for the matrix representation of the Hamiltonian after removal of all non-resonance terms by CVPT. In addition, the classification of anharmonic energy levels according to their polyad quantum numbers provides an additional means for the interpretation of observed vibrational spectra.

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.

Determination of the Eckart molecule-fixed frame by use of the apparatus of quaternion algebra

The problem of determining the Eckart molecule-fixed frame for an arbitrary molecule with nuclei displaced from the equilibrium positions is considered. The solution of the problem is formulated by minimizing the sum of mass-weighted squared deviations (MWSD) of the nuclei of a displaced configuration from the nuclei of the equilibrium configuration. A mathematical proof of the equivalence of Eckart conditions and the minimum of MWSD is given. It is shown that the extrema of the sum of MWSD coincide with eigenvalues of a special 4 × 4 symmetric matrix. Its minimal eigenvalue corresponds to the desired solution, and the respective eigenvector can be treated as the quaternion containing the necessary information for rotating the original coordinate system and aligning its axes with the molecule-fixed coordinate system. A detailed scheme for an efficient numerical implementation of the method is provided, and a numerical example is given.

Calculation of anharmonic intensities in vibrational spectra of raman scattering and full interpretation of the vibrational spectrum of trans-1,3-butadiene

The anharmonic model of vibrations of polyatomic molecule makes it possible, using the secondorder perturbation theory, to interpret vibrational spectra in detail with resonances taken into account and to calculate the band intensities in spectra for fundamental frequencies, overtones, and combination frequencies. For molecules possessing a center of symmetry (to which trans-1,3-butadiene belongs), many vibration modes have zero intensities of absorption in IR spectra due to the mutual exclusion rule. For meaningful analysis of vibrations of such molecules, measurement of Raman scattering (RS) spectra is necessary, as is developing a theoretical model for calculating the anharmonic intensity. Perturbation theory (PT) in the form of contact transformations (CTs) has proven applicable for calculating the anharmonic intensities in RS spectra. The FORTRAN software program ANCO has been developed, which allows the calculation of vibration frequencies and IR-RS intensities of fundamental vibrations, overtones, and combination frequencies on the basis of second-order PT in form of CTs using the polynomial representation of potential energy, dipole moment and polarizability surfaces. The obtained frequencies and modes of anharmonic vibrations are provided, as well as interpretation of the experimental spectrum of the trans-1,3-butadiene molecule. A procedure for calculating the scale factors of the anharmonic force field is proposed. We also show that within the anharmonic vibration model, these factors are close to unity.

Determination of accurate semiexperimental equilibrium structure of proline using efficient transformations of anharmonic force fields among the series of isotopologues

ABSTRACT The complete semiexperimental (SE) equilibrium structure of proline (45 degrees of freedom) is determined using the mixed estimation method. The cubic force fields for the parent and eight isotopologues of proline molecule are evaluated at the MP2-FC/cc-pVTZ level in Cartesian coordinates. The accuracy of the SE structure is verified by optimising the structure with the CCSD(T) model and a basis set of quadruple-ζ quality. A significantly more accurate equilibrium structure of proline is obtained when compared to the previous one. It is shown that the employed technique is efficient for the determination of SE equilibrium structures of rather large molecules. A simple transformation of anharmonic force fields between normal coordinate and Cartesian coordinate representations is proposed. The suggested technique allows efficient evaluation of the rotation–vibration interaction constants for a number of isotopologues, once the cubic force field of any species is found either in normal or Cartesian coordinates.

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.

Vibrational spectroscopy of tolane; Coriolis coupling between Raman-active modes of g symmetry

ABSTRACT Vibrational spectroscopy of tolane (diphenylacetylene), which has 66 normal modes, has been advanced. Anharmonic wavenumber predictions were made with the quartic potential energy surface obtained with B3LYP/cc-pVTZ model and the second-order perturbation theory (VPT2). Infrared (IR) intensity and Raman activities were computed at the harmonic level. The IR spectrum of the crystal and Raman spectra of the liquid and the crystal tolane were newly recorded. The lingering problem of an excess of polarised Raman bands at wavenumbers appropriate for fundamentals, other than ag modes, has now been attributed to Coriolis coupling within modes of g symmetry species. Consequently, D2h point symmetry group has been confirmed for a planar tolane molecule. Assignments for almost all fundamentals of tolane are now secure. The assignment for ν32 remains questionable. Remaining unassigned fundamentals are: ν34 and ν35, which, as au symmetry species, are IR- and Raman-inactive transitions, and ν59(b2u), which is predicted to have a very low wavenumber. GRAPHICAL ABSTRACT