Anharmonic Vibrational Wave Functions Research Articles

Characterization of the sulfur fluoride radical in the ground electronic state

The 2Π ground electronic state of the sulfur fluoride radical has been characterized by high-level ab initio methods, employing the CCSD(T) method with large, augmented correlation-consistent basis sets including aug-cc-pVTZ, aug-cc-pwCVTZ, and aug-cc-pVQZ. Anharmonic vibrational wave functions have been computed, and previously unavailable transition moments and infrared intensities have been obtained along with dipole moments for the lower vibrational states. A refined theoretical prediction is made for the dipole moment μ e, and the performance of the CCSD(T) method with the above basis sets is investigated for several other molecular constants for which experimental values are available.

Calculation of non-fundamental IR frequencies and intensities at the anharmonic level. I. The overtone, combination and difference bands of diazomethane, H 2CN 2

The experimental assignment of IR non-fundamental bands can be assisted by calculation of both frequencies and intensities, as shown in this work on diazomethane. The ab initio B3LYP method is used to obtain the anharmonic force fields up to the fourth order. The anharmonic vibrational wave functions have been calculated using a variation–perturbation algorithm. The dipole moment expansion needed in the evaluation of absolute intensities is limited to the first derivatives. The results, including those for overtone, combination and difference bands disagree with some experimental attributions and complement the available experimental data.

Simulation of à 1B1→X̃ 1A1 CF2 single vibronic level emissions: Including anharmonic and Duschinsky effects

CASSCF/MRCI/aug-cc-pVQZ(no g) and RCCSD(T)/aug-cc-pVQZ potential energy functions were reported for the à 1B1 and X̃ 1A1 states of CF2, respectively. Vibrational wave functions of the symmetric stretching and bending modes of the two states of CF2 were obtained in variational calculations, employing Watson’s Hamiltonian for a nonlinear molecule and anharmonic vibrational wave functions expressed as linear combinations of harmonic basis functions. Franck–Condon factors (FCFs) were computed for à 1B1→X̃ 1A1 CF2 single vibronic level (SVL) emissions and the SVL emission spectra were simulated with the computed FCFs. When compared with the observed spectra, the simulated spectra obtained in the present investigation, which include allowance for anharmonicity and the Duschinsky effect, were found to be significantly superior to those reported previously, based on the harmonic oscillator model. Using the iterative Franck–Condon analysis procedure, with the geometry of the X̃ 1A1 state fixed at the recently determined experimental equilibrium geometry, the geometry of the à 1B1 state of CF2, which gave the best match between simulated and observed spectra, was found to be re(CF)=1.317 Å and θe(FCF)=121.25 °.

A new method of calculation of Franck–Condon factors which includes allowance for anharmonicity and the Duschinsky effect: Simulation of the He I photoelectron spectrum of ClO2

A new method of Franck–Condon (FC) factor calculation for nonlinear polyatomics, which includes anharmonicity and Duschinsky rotation, is reported. Watson’s Hamiltonian is employed in this method with multidimensional ab initio potential energy functions. The anharmonic vibrational wave functions are expressed as linear combinations of the products of harmonic oscillator functions. The Duschinsky effect, which arises from the rotation of the normal modes of the two electronic states involved in the electronic transition, is formulated in Cartesian coordinates, as was done previously in an earlier harmonic FC model. This new anharmonic FC method was applied to the simulation of the bands in the He I photoelectron (PE) spectrum of ClO2. For the first band, the harmonic FC model was shown to be inadequate but the anharmonic FC simulation gave a much-improved agreement with the observed spectrum. The experimentally derived geometry of the X̃ 1A1 state of ClO2+ was obtained, for the first time, via the iterative FC analysis procedure {R(Cl–O)=1.414±0.002 Å, ∠O–Cl–O=121.8±0.1°}. The heavily overlapped second PE band of ClO2, corresponding to ionization to five cationic states, was simulated using the anharmonic FC code. The main vibrational features observed in the experimental spectrum were adequately accounted for in the simulated spectrum. The spectral simulation reported here supports one of the two sets of published assignments for this band, which was based on multireference configuration interaction (MRCI) calculations. In addition, with the aid of the simulated envelopes, a set of adiabatic (and vertical) ionization energies to all five cationic states involved in this PE band, more reliable than previously reported, has been derived. This led also to a reanalysis of the photoabsorption spectrum of ClO2.

A Vibrational Eigenfunction of a Protein: Anharmonic Coupled-Mode Ground and Fundamental Excited States of BPTI

We present detailed methodology and results for the approximate calculation of the anharmonic vibrational wave functions for a given structure of a protein. The ground state and the fundamental vibrationally excited states of hydrated BPTI are computed with an approach that is of very good accuracy for low-lying states. The eigenfunctions are used to predict quantum properties such as vibrational excitation frequencies and IR intensities as well as atomic mean square displacements at T = 0 K. The method treats diagonal anharmonic effects exactly up to fourth order in normal mode coordinates, while using a mean-field approximation, the vibrational self-consistent field (SCF) approximation for the mode−mode couplings. The inclusion of the diagonal effects (exact in fourth order) makes the potentials stiffer than quadratic. When the mode−mode coupling is included as a mean field, the system becomes even stiffer. These effects produce significant modifications of such observables as the vibrational spectrum and Debye−Waller factors. The results presented here should be considered essentially exact except for potentially important approximations: (a) that the potential energy used bears some resemblance to reality, (b) that the system is restricted to a single minimum in the potential energy surface, and (c) that the quartic expansion in normal modes, when truncated to fourth order, provides a good description of the potential. These issues will be addressed in the main text. The results show that the vibrational absorption spectrum is strongly affected by anharmonic and mode−mode coupling effects. Deviations from the corresponding harmonic absorption intensities are very large. Unlike our previous study that found only weak mode−mode coupling effects for the lowest 100 modes, the SCF corrections calculated here for the intermediate frequency modes are seen to be very significant. The results have important implications for the vibrational spectroscopy and other properties of proteins at low temperatures.

Calculation of rotational–vibrational energies of methane-shaped molecules directly from an anharmonic potential function

A method has been developed for calculating a selected number of the lowest rotational–vibrational energies of non-linear, semi-rigid molecules directly from the anharmonic potential function, without the use of the conventional contact transformation. Instead the matrix of the second-order Hamiltonian is set up in a large basis of anharmonic vibrational wavefunctions multiplied by symmetric-top rotational functions. This matrix is reduced by a van Vleck transformation, and the resulting submatrix diagonalized directly. A principle is given for selecting an optimum vibrational basis in the computer. The method has been realized as a computer program applicable to methaneshaped molecules. A discussion is given of the magnitude of the errors due to the various approximations involved, based on extensive test calculations. The largest errors are due to the use of a limited basis and to the truncation of the potential function rewritten in terms of normal coordinates.

Calculation of rotation-vibrational energies directly from an anharmonic potential function

A method has been developed for calculating a selected number of the lowest rotation-vibrational energies of nonlinear semirigid molecules directly from the anharmonic potential function. The use of the conventional contact transformation and the resulting intermediate spectroscopic constants is avoided. Instead the matrix of the untransformed, second-order Hamiltonian is set up in a systematically selected basis of anharmonic vibrational wavefunctions multiplied by symmetric top rotor functions. This matrix is block-diagonalized by a van Vleck transformation, and the relevant diagonal block is diagonalized directly. The method has been tested on the isotopic species CH 4, CH 2D 2, and CD 4, as well as on CF 4. Extensive test calculations indicate that a mean value of the errors at about 0.7 cm −1 for CH 4 and CH 2D 2, 0.1 cm −1 for CD 4, and 0.01 cm −1 for CF 4 may be obtained, computing the rotational states for J up to 20 for all the fundamental vibrational states and states of similar energies. These errors include the effects of the limited basis and of the van Vleck transformation, but not those of the Born-Oppenheimer approximation or of the truncation of the Hamiltonian operator. This method should be useful for checking potential functions computed by ab initio methods against spectroscopic data and eventually for determining the anharmonic potential functions of middle-size molecules directly from experimental data.

Infrared intensities of polyatomic molecules calculated from SCEP dipole-moment functions and anharmonic vibrational wavefunctions. I. Stretching vibrations of the linear molecules HCN, HCP and C 2N 2

Infrared intensities of transitions between a variety of stretching vibrational states of various isotopomers of HCN, HCP and C 2N 2 have been calculated from SCF and SCEP dipole-moment functions and anharmonic vibrational wavefunctions. Best agreement with experiment, usually with deviations of the order of 10% for the fundamentals, is obtained by CEPA when the dipole moment is calculated as an energy derivative. The performance of the familiar “double-harmonic approximation” is investigated and found to work well in the absence of vibrational resonances and strongly curved dipole-moment functions.

Anharmonic vibrational wave functions, infrared band intensities, and dipole moments of water

Abstract A new and simple method to expand the anharmonic vibrational wave functions with respect to the harmonic oscillator wave functions is proposed. The coefficients of the expansion are given as matrix elements of the S function of the contact transformation in the perturbation theory and the explicit expressions of these coefficients are given within the approximation to the second order in λ. As an example of the expansion, the wave functions of water molecules were calculated and applied to the calculation of infrared band intensities and average values of dipole moments in several states.

ChemInform Abstract: VIBRATIONAL FREQUENCIES FROM ANHARMONIC AB INITIO/EMPIRICAL POTENTIAL ENERGY FUNCTIONS. PART V. CYANOGEN. FREQUENCIES AND IR INTENSITIES OF STRETCHING VIBRATIONS

Abstract Low-lying stretching vibrational states of five isotopes of cyanogen have been calculated by diagonalisation of an approximate vibrational Hamiltonian which neglects the anharmonic interaction between stretching and bending coordinates. The underlying anharmonic potential energy function was obtained from ab initio SCF calculations and corrected for the most important errors (geometry shift and modification of the harmonic force field). The experimentally known band origins are reproduced with a standard deviation of 2.3 cm−1. The equilibrium geometry, equilibrium rotational constants and centrifugal distortion constants are predicted. Infrared intensities of the stretching fundamentals have been calculated from HF-SCF and SCEP dipole moment functions and the anharmonic vibrational wavefunctions. As previously observed for HCN, the HF-SCF method yields the wrong sign for the first dipole moment derivative with respect to the CN stretching coordinate.

Vibrational frequencies from anharmonicab initio/empirical potential energy functions IV. Diacetylene: frequencies and infrared intensities of stretching vibrations

An anharmonic potential energy function for the stretching vibrations of diacetylene has been constructed from ab initio SCF calculations and little experimental information. Low-lying vibrational states are calculated by diagonalization of an approximate vibrational hamiltonian which neglects the anharmonic coupling between stretching and bending motions. The potential energy terms of diacetylene show close resemblance with those obtained previously for cyanoacetylene. Equilibrium geometry, rotational and centrifugal distortion constants are predicted. Infrared intensities of several diacetylene isotopes are calculated from the anharmonic vibrational wavefunctions and SCEP dipole moment functions.

Vibrational frequencies from anharmonic ab initio/ empirical potential energy functions: Part V. Cyanogen. Frequencies and IR intensities of stretching vibrations

Low-lying stretching vibrational states of five isotopes of cyanogen have been calculated by diagonalisation of an approximate vibrational Hamiltonian which neglects the anharmonic interaction between stretching and bending coordinates. The underlying anharmonic potential energy function was obtained from ab initio SCF calculations and corrected for the most important errors (geometry shift and modification of the harmonic force field). The experimentally known band origins are reproduced with a standard deviation of 2.3 cm −1. The equilibrium geometry, equilibrium rotational constants and centrifugal distortion constants are predicted. Infrared intensities of the stretching fundamentals have been calculated from HF-SCF and SCEP dipole moment functions and the anharmonic vibrational wavefunctions. As previously observed for HCN, the HF-SCF method yields the wrong sign for the first dipole moment derivative with respect to the CN stretching coordinate.

Harmonic oscillator representation for anharmonic vibrational wavefunctions

It is shown that a harmonic oscillator basis set provides a compact analytic representation for the complete set of bound and continuum vibrational states of an anharmonic oscillator in the inner region. Calculations of vibrational excitation cross sections based on a harmonic oscillator model can thus be improved by choosing linear combinations of harmonic eigenfunctions such that the vibrational hamiltonian is diagonal.

The 4750-Å band system of chlorine dioxide. Rotational analysis, force field and intensity calculations

The structure of the ν 1 vibronic band in the 2A 2 ← 2B 1 electronic band system of chlorine dioxide has been rotationally analyzed as an A-type (parallel) band of a near-prolate asymmetric rotor. Excited state constants ν 0, A, B, C, τ aaaa , τ bbbb , τ aabb , and τ abab were obtained by a least-squares term value analysis based on about 730 assignments in the (100) ← (000) band, the results for 35ClO 2 being: ν 0 21 727.087±0.020 cm −1 τ aaaa(−0.1529±0.0010)×10 −3 cm −1, A 100 1.05633±0.00013 cm −1 τ bbbb(−0.0077±0.0003)×10 −3 cm −1, B 100 0.30950±0.00006 cm −1 τ aabb(0.0126±0.0015)×10 −3 cm −1, C 100 0.23815±0.00006 cm −1 τ abab(−0.0026±0.0007)×10 −3 cm −1 , The rotational analysis did not however distinguish between alternative sets of spin-rotation coupling constants, nor could a firm decision be reached from structural considerations. Quadratic, cubic, and quartic constants of the excited state force field have been evaluated using a model potential function to restrict the number of independent terms in the cubic and quartic portions of the field. The cubic constants are used to calculate “equilibrium” values of the excited state rotational constants from those observed for the (100) vibrational state, leading to a set of “equilibrium” moments of inertia, 35 ClO 2:I a=15.48, I b=54.16, I c=69.63 amu A ̊ 2 , and hence to an excited state structure, r( ClO)=1.619 A ̊ , ∠ OClO=107°0′ . Even-quantum changes (2-0 and 4-0 transitions) in the b 2 mode ν 3 are relatively prominent in the vibrational structure of the band system. In order to test the possibility that these bands “borrow” their intensity from transitions involving the totally-symmetrical modes (mainly, from bands of the ν 1 progression), relative intensities were calculated using second-order corrected anharmonic vibrational wavefunctions in which the harmonic frequencies and higher-order potential constants appear as coefficients of harmonic oscillator functions. These calculations give a good account of the observed intensities. Some further features, not readily explained by this approach, are noted.