Van Vleck Perturbation Theory Research Articles

A theoretical study of the vibrational spectrum of the CS2 molecule

A theoretical study of the vibrational spectrum of the CS2 molecule is carried out. For that purpose, a new Morse-cosine potential energy function is determined by fitting to observed vibrational frequencies, using as a starting point an ab initio force field. Highly excited vibrational states for CS2 are then calculated up to 20 000 cm−1 using a DVR truncation-diagonalization method. Hyperspherical Radau coordinates, which are a set of normal curvilinear coordinates for linear symmetric triatomic molecules, are used in these calculations. The computed vibrational energy levels are shown to present an excellent agreement with the observed values up to 13 000 cm−1. Based on these calculations, some unassigned observed vibrational frequencies are identified and the assignments of others are reconsidered. Inspection is made of the vibrational wave functions computed for CS2 revealing a persistent regularity up to 13 000 cm−1. This regularity is found to be consistent with perturbation theory results for these energies. Van Vleck perturbation theory is used to derive effective Hamiltonians that contain polyad quantum numbers and that provide good agreement with the variational calculations. It is also shown that the asymmetric stretch is practically decoupled from the symmetric stretch and the bend in this range of energies. The nearest neighbor space distribution (NNSD) and the Δ3 spectral rigidity function show that the calculated vibrational spectrum of CS2 up to 20 000 cm−1 is essentially regular, in agreement with the most recent statistical analyses made of the spectroscopically observed frequencies.

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.

The vibrational overtones of SiH4 isotopomers: experimental wavenumbers, assignment, ab initio dipole moment surfaces and intensities

The overtone spectra of SiHD3, SiH2D2 and SiH3D in the range 2000–9000 cm−1 have been measured up to 720000 Pa m using a Bruker 120HR interferometer. The transition frequencies and intensities are reported. The observed band centers are found to be in good agreement with the predictions from high order canonical Van Vleck perturbation theory (CVPT) calculations. A four-dimensional Si–H stretching dipole moment surface (DMS) has been determined from coupled cluster [CCSD(T)] methods and expanded using a bond dipole model and a Cartesian model. The band intensities of the Si–H and Si–D stretching transitions have been computed employing these expansions and a local mode force field optimized to the observed band centers. The calculations agree reasonably well with observations for most bands. The vibrational intensity anomaly in the third local mode manifolds is partly reproduced and is attributed to the cancellation of linear and quadratic terms in the DMS.

A nine-dimensional high order perturbative study of the vibration of silane and its isotopomers

The vibrations of silane isotopomers with Td, C3v, and C2v symmetry are studied by means of high order canonical Van Vleck perturbation theory (CVPT). Transforming the quartic ab initio force field of Martin, Baldridge, and Lee [Mol. Phys. 95, 254 (1999)] into curvilinear normal coordinates, CVPT is used to calculate energies that agree well with experimental data. Both low energy stretch–bend combination bands and high energy stretch local mode bands of silane up to 12 000 cm−1 are well reproduced. The choice of polyad quantum number is discussed with respect to different molecules. Comparing sixth- to eighth-order level of theory, most of the levels agree to within 0.1 cm−1. Spectroscopic constants are given for all the major isotopomers. The construction of the full cubic and quartic resonance operators for symmetric top species are summarized. The Si–H and Si–D stretch modes of Td and C3v symmetry species are studied with a 4D stretch variational model using both the fitted three-parameter potential and the quartic ab initio stretch potential. Large energy level differences between these two calculations indicate that the fitted potential constants of the stretch variational model are different from those derived directly from the ab initio surface. By comparing with the nine-dimensional CVPT calculation, it is shown that the direct and indirect Fermi resonances are responsible for the inability of the four-dimensional model to fit some stretch levels.

Anharmonic force field and vibrational frequencies of tetrafluoromethane (CF4) and tetrafluorosilane (SiF4)

Accurate quartic anharmonic force fields for CF4 and SiF4 have been calculated using the CCSD(T) method and basis sets of spdf quality. Based on the ab initio force field with a minor empirical adjustment, the vibrational energy levels of these two molecules and their isotopomers are calculated by means of high order Canonical Van Vleck Perturbation Theory (CVPT) based on curvilinear coordinates. The calculated energies agree very well with the experimental data. The full quadratic force field of CF4 is further refined to the experimental data. The symmetrization of the Cartesian basis for arbitrary combination bands of Td group molecules is discussed using the circular promotion operator for the doubly degenerate modes, together with tabulated vector coupling coefficients. The extraction of the spectroscopic constants from our second order transformed Hamiltonian in curvilinear coordinates is discussed, and compared to a similar procedure in rectilinear coordinates.

High resolution spectroscopic study of CH3D in the region 5900–6100 cm−1

The vibrational overtone band of monodeuterated methane (CH3 in the region 5900–6100 cm−1 was recorded by a high resolution Fourier transform spectrometer. Rotational analyses were performed separately for the parallel 2v 4(A1) and perpendicular 2v 4(E) bands, whose origins were found to be 5980.41 cm−1 and 6022.21 cm−1, respectively. The A1 band displays J-resolved rotational structure while the K structure is resolved only at higher J values. A resonance Hamiltonian with the polyad number P = 2n 1 + 2n 4 + n 5 is derived using the fourth order Canonical Van Vleck Perturbation Theory (CVPT) to show that the 2v 4(A1 and 2v 4(E) are vibrationally perturbed by Fermi and Darling—Dennison resonances. The observation of rotational l-resonance in the 2v 4(E) band is a consequence and evidence of the mixing of the vibrational states with different l values.

A nine-dimensional perturbative treatment of the vibrations of methane and its isotopomers

The vibrations of methane isotopomers with Td, C3v, and C2v symmetry are studied by means of high order Van Vleck perturbation theory. The vibrational states up to 9000 cm−1 are investigated by combining the ab initio force field of Lee, Martin and Taylor [J. Chem. Phys. 95, 254 (1995)] with a fourth order perturbative treatment based on curvilinear normal coordinates. Implementation of the perturbation theory using both analytical and numerical expression of the kinetic energy operator is considered. The quadratic and select cubic and quartic force constants are refined via a nonlinear least squares fit to experimental data The fit force constants reproduce 130 experimental band centers with a root mean squares deviation of 0.70 cm−1. The choice of polyad quantum number is discussed with respect to different molecules. The convergence of the energy levels is discussed by carrying out the perturbation calculation up to eighth order.

Coupled cluster anharmonic force fields, spectroscopic constants, and vibrational energies of AIF3 and SiF3+

Anharmonic force fields and ro-vibrational spectroscopic properties of AlF3 and SiF3+ have been investigated in detail, using the coupled cluster method with single and double substitutions augmented by a perturbative treatment of triple excitations [CCSD(T)] with a triple zeta basis set of 124 contracted Gaussian-type orbitals (cGTOs). A complete set of ro-vibrational spectroscopic constants for each species has been calculated using second order perturbation theory. The geometry only was calculated with a quadruple zeta basis set (224 cGTOs). Our best estimates of the equilibrium bond distances re (Al–F) and re (Si–F) are 1.624 Å and 1.508 Å, respectively, based on the quadruple zeta CCSD(T) bond distances and corrections derived from spectroscopically related known molecules. The CCSD(T) fundamental frequencies are 689 cm−1(ν1),301 cm−1(ν2),951 cm−1(ν3), and 241 cm−1(ν4) for AlF3 and 853 cm−1(ν1),357 cm−1(ν2),1187 cm−1(ν3), and 307 cm−1(ν4) for SiF3+. Energies of a number of low lying vibrational states (⩽2400 cm−1) have been calculated by three methods: (1) standard second order perturbation theory formulas; (2) Van Vleck perturbation theory based on raising and lowering operators to second order and to fourth order; and (3) a variational method.

On the diagonalization of quantum Birkhoff–Gustavson normal form

An application to quantum mechanics of one of classical perturbation theory methods, the Birkhoff–Gustavson normal form (BGNF), is described. In the quantum case it results in the Van Vleck perturbation theory performed upon Wick normal ordered operators. Algebraic aspects of this procedure and formal construction of invariants (integrals of motion) for a perturbed system are considered. It turned out that a larger set of such operators existed in the quantum mechanics, rather than in the classical one. It is demonstrated that, according to general results of the quantum mechanical perturbation theory, the quantum BGNF may always be diagonalized, and two formal processes for such diagonalization are constructed. In the opposite case, the classical BGNF is, in general, nondiagonalizable. This reflects the fact that the classical perturbation theory cannot handle a system with two or more resonances. Possible reasons for such different behavior of two very close, in spirit, perturbation procedures are discussed. Results of the described procedure, entirely performed upon the Wick normal ordered operators, are equivalent to those of Rayleigh–Schrödinger perturbation expansion.

Five-dimensional local mode-Fermi resonance model for overtone spectra of ammonia

A five-dimensional local mode-Fermi resonance model for overtone spectra of the ammonia molecule has been constructed. The model Hamiltonian is expressed in terms of curvilinear internal valence coordinates and it includes the three stretching vibrations and the doubly degenerate bending vibration. The symmetric bending vibration associated with the inversion motion has been excluded. Thus the model is useful for energy levels with the totally symmetric bending vibration on its ground state. Vibrational energy levels have been calculated using van Vleck perturbation theory. Nonlinear least-squares method has been used to optimize potential energy parameters. Observed vibrational band origins for 14NH3, 14ND3, and 14NT3 have been employed as data. A fit with the standard deviation of 5.5 cm−1 has been obtained using one set of isotope invariant potential energy parameters. The optimised potential energy surface compare well with results of ab initio electronic structure calculations and with results of customary anharmonic force field calculations.

A combined use of perturbation theory and diagonalization: Application to bound energy levels and semiclassical rate theory

A new method, mixed diagonalization, is introduced in which an effective Hamiltonian operator acting on a reduced dimensional space is constructed using the similarity transformations of canonical Van Vleck perturbation theory (CVPT). This construction requires the characterization of modes into two categories, global and local, which in the bound vibrational problem are tantamount to the large and small amplitude vibrations, respectively. The local modes in the Hamiltonian are projected out by CVPT, and the resulting Hamiltonian operator acts only on the space of global modes. The method affords the treatment of energy levels of bound systems in which some vibrational assignments are possible. In addition, it systematically provides a reduced dimensional Hamiltonian which is more amenable to exact numerical solution than the original full-dimensional Hamiltonian. In recent work, a semiclassical transition state theory (SCTST) rate expression has been written in terms of a Hamiltonian operator parameterized by the imaginary action along the local reaction path in the transition state region [Chem. Phys. Lett. 214, 129 (1993)]. We show that the Hamiltonian constructed by mixed diagonalization has this form, and can be used to obtain more accurate semiclassical rate expressions.

Effects of X3 frame vibrations on the CH chromophore in CHX3 type molecules: Harmonic coupling in curvilinear internal coordinate approach

Effects of harmonic coupling between X3 frame vibrations and the CH stretching and bending vibration chromophore in CHX3 type molecules have been investigated using a curvilinear internal coordinate model. The CH chromophore Hamiltonian used consists of a Morse oscillator Hamiltonian for the CH stretching vibration, two-dimensional isotropic harmonic oscillator Hamiltonian for the CH bending vibration, anharmonic bending terms, and Fermi resonance interaction terms between the stretching and bending vibrations. Van Vleck perturbation theory is used to take the harmonic coupling terms between the CH chromophore and the X3 frame vibrations into account. Analytical matrix elements for the perturbation theory contributions have been derived. The model has been applied to vibrational energy level and infrared absorption intensity data for CHD3. Potential energy parameters for the CH chromophore have been optimized by a nonlinear least-squares method. The model reproduces the experimental data well. Only small changes in the potential parameters are observed when the coupling terms due to D atom motions are introduced into the model. The overall conclusion is that it is a good first approximation to ignore the effects of harmonic coupling between the CH chromophore and the X3 frame vibrations in the curvilinear internal coordinate model for the CH chromophore in CHX3 type molecules.

Third-order complete active space self-consistent field based generalized Van Vleck perturbation theory

A state-selective generalized Van Vleck perturbation theory (GVVPT), correct through third order, for the determination of the electronic structure of polyatomic molecules is presented. The method adapts Kirtman's GVVPT for use with complete active space self-consistent field (CASSCF), and orthogonal complement, zeroth-order many-electron functions. The tenability of the method is demonstrated by calculations on several well-known “benchmark” molecules: the Be atom, using a (12s5p1d)/[7s3pld] basis set, double zeta (DZ) H 2O; 6-31GN NH 2; the singlet—triplet and singlet—singlet energy separations of DZP CH 2; the excited states of DZP CH + 2; and the dissociation of DZP + F 2.

An algebraic approach to calculating rotation-vibration spectra of polyatomic molecules

An approach for calculating electric dipole intensities of transitions to highly vibrationally excited states in polyatomic molecules is presented. We apply canonical Van Vleck perturbation theory (CVPT) to the space fixed dipole moment operator, where both the rotational and vibrational coordinate dependencies are expressed in terms of boson creation and annihilation operators. This approach is general and can be applied to a large class of polyatomic molecules. Theoretical spectra are presented for H2O with T rot = 300 K and H2CO with T rot = 10 K.

Determining potential-energy surfaces from spectra: An iterative approach

A general method for iteratively fitting the coefficients of a Taylor-series expansion of the potential-energy surface for a polyatomic molecule to the observed transition frequencies and rotational constants is presented. This approach utilizes the efficiency of fourth-order Van Vleck perturbation theory for calculating these properties, scaled to the results of converged variational calculations. Three fits to the transition frequencies and rotational constants obtained from absorption and stimulated emission pumping spectra of HCN are presented. Comparison of these potentials sheds light on the uniqueness of fit potential surfaces, given this set of observables. We fit the vibrational energies for 71 states with a mean absolute deviation of 0.69 cm−1.

A simple curvilinear internal coordinate model for vibrational energy levels

A simple curvilinear internal coordinate Hamiltonian model for vibrational term values of well bent XY2 molecules is developed. The stretching vibrations are described in the zero-order picture by Morse oscillators and the bend by a harmonic oscillator. Coupling terms are approximated by harmonic oscillator formulas. Van Vleck perturbation theory is used to transform the Hamiltonian matrix to a block diagonal form. Analytical expressions for the matrix elements are derived. Fermi resonances between the stretches and the bend and local modes are taken into account by diagonalizing the block diagonal Hamiltonian matrix. Rotational parameters (α constants) are calculated with perturbation theory expressions. Potential energy parameters are optimized with the nonlinear least-squares method using vibrational term values and α constants as data. The model is applied to five isotopic species of hydrogen sulfide and to two isotopic species of sulfur dioxide. The potential energy parameters obtained agree well with parameters determined by other methods.

Rotation–vibration interactions in highly excited states of SO2 and H2CO

Canonical Van Vleck perturbation theory (CVPT) is used to investigate rotation–vibration mixing of highly excited vibrational states of SO2 and H2CO. For SO2 we find a nearly complete separation of the rotational and vibrational degrees of freedom, even for J=12 and Evib=11 000 cm−1. In contrast, for H2CO we observe extensive mixing between rotational and vibrational degrees of freedom at similar rotational excitation but with Evib=8000 cm−1. Although a-axis Coriolis coupling is pronounced, b- and c-axis Coriolis couplings play an important additional role in mixing states with different Ka quantum numbers. The implementation of CVPT, the choice of internal coordinates, and the convergence of the results are discussed in detail.

Calculation of infrared intensities of highly excited vibrational states of HCN using Van Vleck perturbation theory

Canonical Van Vleck perturbation theory (CVPT) is used to calculate electric dipole intensities for one-, two-, and three-dimensional models of HCN and a six-dimensional model of H2CO. Lehmann and Smith [J. Chem. Phys. 93, 6140 (1990)] have shown that the intensities of overtone transitions are sensitive to the details of the inner wall of the potential. Dipole intensities calculated for several, similar one-dimensional CH stretch potentials demonstrate that perturbation theory correctly predicts this sensitivity. The perturbation intensities of a two-dimensional ab initio dipole surface indicate the importance of selected stretch–stretch resonance interactions in interpreting the CH stretch overtone spectra of HCN. The inclusion of the bend confirms that this degree of freedom plays a significant role in weakening the intensity of the CN overtones. The CH stretch overtone spectra of H2CO is calculated to illustrate the utility of the perturbative approach for predicting the transition intensities for a system in which there are multiple Fermi interactions.

Perturbative calculations of vibrational (J=0) energy levels of linear molecules in normal coordinate representations

Canonical Van Vleck perturbation theory is used to transform curvilinear and rectilinear normal coordinate vibrational Hamiltonians of HCN, C2H2, and CO2 to block-diagonal effective Hamiltonians. Accurate energies as high as 11 000 cm−1 above the zero point are reported for all three molecules. In the absence of off-diagonal coupling terms in the effective Hamiltonians, these two coordinate systems yield identical perturbative expansions for the vibrational energies. Only when coupling terms are introduced do differences between the calculated energies in the two representations become apparent. In CO2, where there is pronounced configuration interaction between nearly degenerate states, we find that the perturbative energies obtained from the curvilinear normal coordinate Hamiltonian are converging significantly faster than those obtained in the rectilinear normal coordinate representation.

Perturbative approaches to highly excited molecular vibrations of H2O, D2O, and HDO

Molecular vibrations of water are studied using Van Vleck perturbation theory. In these calculations, the OH stretches are expressed in terms of the Morse coordinate, yi =[1−exp(−αri)]/α, and its conjugate momentum, while the bend is treated in a traditional manner. Nearly degenerate perturbative calculations are presented for bond and normal-mode representations. The eigenvalues agree well with those obtained variationally. The coefficients of the perturbatively derived, effective Hamiltonians are compared to those of spectroscopic Hamiltonians with similar structural forms. The former are derived from the potential of Halonen and Carrington [J. Chem. Phys. 88, 4171 (1988)], whereas the latter are treated as spectral fitting parameters. The fit parameters are accurately reproduced.