Iterative Franck Condon Analysis Procedure Research Articles

A general analytical expression for evaluation of an arbitrary n-dimensional Franck-Condon overlap integral including the Duschinsky effect

Evaluations of the multidimensional Frank-Condon (FC) overlapping integrals are needed theoretically to investigate the mechanism of radiative and non-radiative processes among the different electronic states of polyatomic molecules. We extended our recently reported approach to calculate an arbitrary n-dimensional Franck–Condon overlapping integrals based on harmonic oscillator approximation and taking into account the Duschinsky effects. An analytical formula was derived straightforwardly based on the Hermite polynomials expanding properties and directly evaluating Gaussian integrals. A method to determine the coefficient parameter matrices of four-dimensional Franck-Condon integrals by matrix operations was presented. In addition, as a practical application, we simulated and elucidated the vibrationally resolved peaks of the photodetachment spectra of SeO2‾. Moreover, in the spectral simulations, employing our iterative Franck-Condon analysis procedure, fairly reliable geometries of the ground state X~2B1 of the negative ion SeO2‾ are obtained.

A general analytical expression for the three-dimensional Franck–Condon integral and simulation of the photodetachment spectrum of the [formula omitted] anion

Calculations of Franck–Condon factors are crucial for interpreting vibronic spectra of molecules and studying nonradiative processes. We have derived straightforwardly a more general analytical expression for the calculation of the three-dimensional Franck–Condon overlap integrals on the basis of harmonic oscillator approximation under the influence of mode mixing effects. This new analytical expression was applied to study the photoelectron spectra of PO2-. The theoretical spectrum obtained by employing CCSD(T) values is in excellent agreement with the observed one. An ‘irregular spacing’ observed in the experimental photoelectron spectrum of PO2- is interpreted as contributing from a hot-band sequence of the bending vibration ω2 and combination bands of the stretching vibration ω1 and the bending vibration ω2. In addition, the equilibrium geometry parameters, r(O–P)=1.495±0.005Å and ∠(O–P–O)=119.5±0.5°, of theX∼1A1 state of PO2-, are derived by employing an iterative Franck–Condon analysis procedure in the spectral simulation.

A combined ab initio and Franck–Condon simulation study of the photodetachment spectrum of the [formula omitted] anion

A general analytical expression for the calculation of the two-dimensional four-mode Franck–Condon overlap integrals was presented and applied to scrutinize the photoelectron spectroscopy of ClO 2 - . Geometry optimization and harmonic vibrational frequency calculations have been performed on the X ∼ 2B 1 state of ClO 2 and X ∼ 1A 1 state of ClO 2 - by using the density functional theory (B2PLYP functional) and the coupled cluster singles and doubles with perturbative triples [CCSD(T)] methods with various basis sets up to aug-cc-pVQZ. The adiabatic electron affinity of ClO 2 was computed by using the CCSD(T) method extrapolated to the complete basis set limit with aug-cc-pVXZ (X = D, T, Q). Franck–Condon analyses and spectral simulation were carried out on the ClO 2( X ∼ 2B 1)–( ClO 2 - X ∼ 1A 1) photodetachment process. The simulated photoelectron spectrum of ClO 2 - are in accord with the experiment. While the Duschinsky effect plays a minor role and can be neglected. In addition, the equilibrium geometry parameters, R(Cl–O) = 1.572 ± 0.005 Å and ∠ (O–Cl–O) = 112.5 ± 0.5°, of the X ∼ 1A 1 state of ClO 2 - , are obtained by employing an iterative Franck–Condon analysis procedure in the spectral simulations.

A combined ab initio and Franck–Condon factor simulation study on the photodetachment spectrum of HfO2−

Restricted-spin coupled-cluster single-double plus perturbative triple excitation {RCCSD(T)} potential energy functions (PEFs) of the X[combining tilde]1A1 state of HfO2 and the X[combining tilde]2A1 state of HfO2- were computed, employing the quasi-relativistic effective core potential, ECP60MWB, and an associated contracted [13s6p6d4f3g2h] basis set designed for Hf, and the augmented correlation-consistent polarized valence quadruple-zeta (aug-cc-pVQZ) basis set for O. Based on the differences between the computed r0 and re geometrical parameters obtained from the PEF, and available experimentally derived r0 geometrical parameters of the X[combining tilde]1A1 state of HfO2, the experimental re geometrical parameters of the X[combining tilde]1A1 state of HfO2 were estimated as:-re(HfO)=1.7751 A and thetae(OHfO)=107.37 degrees. In addition, Franck-Condon factors for the HfO2 (X[combining tilde]1A1)+e<--HfO2- (X[combining tilde]2A1) electron detachment process, which include allowance for Duschinsky rotation and anharmonicity, were calculated using the computed RCCSD(T) PEFs, and were used to simulate the HfO2 (X[combining tilde]1A1)+e<--HfO2- (X[combining tilde]2A1) photodetachment band of HfO2-. Employing the estimated experimental re geometrical parameters of the X[combining tilde]1A1 state of HfO2 obtained in this work, the iterative Franck-Condon analysis (IFCA) procedure was carried out to optimize the geometrical parameters of the X[combining tilde]2A1 state of HfO2- until the simulated spectrum matched best with the experimental 355 nm photodetachment spectrum of W. Zheng, et. al., J. Phys. Chem. A, 1998, 102, 9129. The equilibrium geometrical parameters of X[combining tilde]2A1 state of HfO2- derived via the IFCA procedure are re(HfO)=1.823 A and thetae(OHfO)=111.5 degrees. Further calculations on low-lying triplet states of HfO2 gave adiabatic electronic energies (Te's) of, and vertical excitation energies (Tvert's) to, the ã 3B2, b[combining tilde]3B1, c[combining tilde]3A1 and d[combining tilde]3A2 states of HfO2 (from the X[combining tilde]1A1 state of HfO2), as well as electron affinities (EAs) and vertical detachment energies (VDEs) to these neutral states from the X[combining tilde]2A1 state of HfO2-.

Ab initio calculations and Franck–Condon analysis of photoelectron spectra of [formula omitted

Geometry optimization and harmonic vibrational frequency calculations have been performed on the X ˜ 2 A 1 state of PO 2 radicals and X ˜ 1 A 1 state of PO 2 - anions. Franck–Condon analysis and spectral simulations were carried out on the PO 2 ( X 2 A 1 ) – PO 2 - ( X 1 A 1 ) photodetachment process. The theoretical spectra obtained by using CCSD/6-311+G(2p, d) values are in agreement with the observed one. In addition, the equilibrium geometry parameters, r(PO) = 1.504 ± 0.005 Å and ∠(OPO) = 119.0 ± 0.2°, of the X ˜ 1 A 1 state of PO 2 - are derived by employing an iterative Franck–Condon analysis procedure in the spectral simulation. Furthermore, through the different methods, our best calculated electron affinity is 78.80 cal mol −1, which is in excellent agreement with the experimental value and other theoretical values.

Franck–Condon simulation of photoelectron spectroscopy of [formula omitted]: Including Duschinsky effects

Geometry optimization and harmonic vibrational frequency calculations have been performed on the X ˜ 1 A 1 state of O 3 and X ˜ 2 B 1 state of O 3 - . Franck–Condon analyses and spectral simulation were carried out on the first photoelectron band of O 3 - . The theoretical spectrum obtained by employing CCSD(T)/6-311+G(2d, p) values are in excellent agreement with the observed one. In addition, the equilibrium geometry parameters, r e(OO) = 0.1355 ± 0.0005 nm and θ e(O–O–O) = 113.5 ± 0.5°, of the X ˜ 2 B 1 state of O 3 - , are derived by employing an iterative Franck–Condon analysis procedure in the spectral simulation.

Franck–Condon analysis of the photoelectron spectra of S 2O −: Including Duschinsky effects

A program developed to calculate multidimensional Franck–Condon factors including Duschinsky effects is described and used to simulate the vibrationally resolved photoelectron spectra of S 2O − combining with ab initio molecular orbital calculations. Geometry optimizations and harmonic vibrational frequency calculations have been performed on the X ˜ 1 A ′ state of S 2O and X ˜ 2 A ″ state of S 2O −. Franck–Condon analysis and spectral simulations were carried out on the first photoelectron band of S 2O −. The simulated spectra obtained by using B3LYP/6-311+G(3df) are in excellent agreement with the experiment. In addition, the accurate equilibrium geometric parameters, r(ss) = 0.1996 ± 0.0005 nm and r(so) = 0.1514 ± 0.0005 nm, of the X ˜ 2 A ″ state of S 2O −, are deduced by employing an iterative Franck–Condon analysis procedure in the spectral simulation. Furthermore, the well-resolved theoretical spectrum is derived by improving the resolution of photoelectron spectra in simulations.

Ab initio calculations and spectral simulation of the [formula omitted] photodetachment process

Geometry optimization and harmonic vibrational frequency calculations were performed on the X ˜ 2 A ′ state of P 2 H and X ˜ 1 A ′ state of P 2 H − at the MP2, B3LYP, QCISD and CCSD levels. Franck–Condon factors were computed to simulate the recently observed, photoelectron spectrum of the P 2 H − anions [K. M. Ervin and W. C. Lineberger, J. Chem. Phys. 122 (2005) 194303]. The theoretical spectra obtained by employing density-functional theory (DFT) with 6-311+G(2p,d) values are in excellent agreement with the observed one. In addition, Franck–Condon analyses and spectral simulations confirm the assignment of the observed spectrum to the X ˜ 2 A ′ − X ˜ 1 A ′ photodetachment process of P 2 H −. Using iterative Franck–Condon analysis procedure in the spectral simulations, it is concluded that the reliable variation of geometry is 0.0026 nm for bond length R(PP), 0.0068 nm for bond length R(PH), and 8.3 ° for bond angle ∠( PPH) from X ˜ 2 A ′ state of P 2 H to X ˜ 1 A ′ state of P 2 H −.

Franck–Condon simulation of the photoelectron spectrum of [formula omitted] including Duschinsky effects

A theoretical method to calculate multidimensional Franck–Condon factors including Duschinsky effects is described and used to simulate the photoelectron spectrum of the anion SO 2 - . Geometry optimizations and harmonic vibrational frequency calculations have been performed on the X ˜ 1 A 1 state of SO 2 and X ˜ 2 B 1 state of SO 2 - . Franck–Condon analyses and spectral simulation were carried out on the first photoelectron band of SO 2 - . The theoretical spectra obtained by employing CCSD(T)/6-311+G(2d,p) values are in excellent agreement with the experiment. In addition, the equilibrium geometric parameters, r e(OS) = 0.1508 ± 0.0005 nm and θ e(O–S–O) = 113.5 ± 0.5°, of the X ˜ 2 B 1 state of SO 2 - , are derived by employing an iterative Franck–Condon analysis procedure in the spectral simulation.

Franck-Condon simulation of the single-vibronic-level emission spectra of HPCl/DPCl and the chemiluminescence spectrum of HPCl, including anharmonicity.

Restricted-spin coupled-cluster single-double plus perturbative triple excitation [RCCSD(T)] potential energy functions (PEFs) were calculated for the X (2)A" and A (2)A' states of HPCl employing the augmented correlation-consistent polarized-valence-quadruple-zeta (aug-cc-pVQZ) basis set. Further geometry optimization calculations were carried out on both electronic states of HPCl at the RCCSD(T) level with all electron and quasirelativistic effective core potential basis sets of better than the aug-cc-pVQZ quality, and also including some core electrons, in order to obtain more reliable geometrical parameters and relative electronic energy of the two states. Anharmonic vibrational wave functions of the two states of HPCl and DPCl, and Franck-Condon (FC) factors of the A (2)A'-X (2)A" transition were computed employing the RCCSD(T)/aug-cc-pVQZ PEFs. Calculated FC factors with allowance for Duschinsky rotation and anharmonicity were used to simulate the single-vibronic-level (SVL) emission spectra of HPCl and DPCl reported by Brandon et al. [J. Chem. Phys. 119, 2037 (2003)] and the chemiluminescence spectrum reported by Bramwell et al. [Chem. Phys. Lett. 331, 483 (2000)]. Comparison between simulated and observed SVL emission spectra gives the experimentally derived equilibrium geometry of the A (2)A' state of HPCl of r(e)(PCl) = 2.0035 +/- 0.0015 A, theta(e) = 116.08 +/- 0.60 degrees, and r(e)(HP) = 1.4063+/-0.0015 A via the iterative Franck-Condon analysis procedure. Comparison between simulated and observed chemiluminescence spectra confirms that the vibrational population distribution of the A (2)A' state of HPCl is non-Boltzmann, as proposed by Baraille et al. [Chem. Phys. 289, 263 (2003)].

The geometry of the NO 2− anion: ab initio calculations and Franck–Condon analysis

Geometry optimization and harmonic vibrational frequency calculations have been performed on the X ̃ 2 A 1 state of NO 2 and X ̃ 1 A 1 state of NO 2 −. Franck–Condon analyses and spectral simulations were carried out on the NO 2( X ̃ 2 A 1 )–NO 2 −( X ̃ 1 A 1 ) photo-detachment process. In addition, the equilibrium geometry parameters, r(NO)=1.248 ± 0.005 Å and ∠(ONO)=116.8 ± 0.5°, of the X ̃ 1 A 1 state of NO 2 −, are derived by employing an iterative Franck–Condon analysis procedure in the spectral simulation. Our conclusions regarding the anion geometry suggest a reinterpretation of the results of Woo et al. [Phys. Rev. A 24 (1981) 1380].

Geometries of the Halocarbene anions HCF − and CF 2−: ab initio calculation and Franck–Condon analysis

A theoretical method to calculate multidimensional Franck–Condon factors including Duschinsky effects is described and used to simulate the photoelectron spectra of HCF − and CF 2 − radicals. Geometry optimization and harmonic vibrational frequency calculations have been performed on the X̃ 1A′ state of HCF and X̃ 2A″ state of HCF −, and X̃ 1A 1 state of CF 2 and X̃ 2B 1 state of CF 2 −. Franck–Condon analyses and spectral simulation were carried out on the first photoelectron band of HCF − and CF 2 −, respectively. The theoretical spectra obtained by employing B3LYP/6-311+G(2d,p) values are in excellent agreement with the observed ones. In addition, the equilibrium geometry parameters, R(CF)=0.1475±0.0005 nm, of the X̃ 2A″ state of HCF −, and r(FC)=0.1425±0.0005 nm and ∠(FCF)=100.5±0.5°, of the X̃ 2B i state of CF 2 −, are derived by employing an iterative Franck–Condon analysis procedure in the spectral simulation.

Franck–Condon simulation of photoelectron spectroscopy of HNO− and DNO−: including Duschinsky effects

A theoretical method to calculate multidimensional Franck–Condon factors including Duschinsky effects is described and used to simulate the photoelectron spectra of HNO − and DNO − radicals. Geometry optimization and harmonic vibrational frequency calculations have been performed on the X ̃ 1 A ′ and a ̃ 3 A ″ states of HNO/DNO and X ̃ 2 A ″ state of HNO − /DNO − . The theoretical spectra obtained by employing B3LYP/6-311+G(2d,p) values are in excellent agreement with the observed ones. In addition, the equilibrium geometry parameters, R (NO)=0.1335±0.0005 nm, ∠(HNO)=106.3°±0.5°, of the X ̃ 2 A ″ state of HNO − /DNO − are derived by employing an iterative Franck–Condon analysis procedure in the spectral simulation.

Potential energy functions of the X̃ 2B1, Ã 2B2, B̃ 2A1, and C̃ 2A2 states of Cl2O+ and the X̃ 1A1 state of Cl2O: Franck–Condon simulations of photoelectron bands of Cl2O which include anharmonicity

Restricted-spin coupled-cluster single and double plus perturbative triple excitations [RCCSD (T)] and/or complete-active-space self-consistent-field multireference internally-contracted configuration interaction (CASSCF/MRCI) potential energy functions of the X̃ 2B1, à 2B2, B̃ 2A1, and C̃ 2A2 states of Cl2O+ and the X̃ 1A1 state of Cl2O, with basis sets of up to the augmented-correlation-consistent-polarized-valence-quadruple-zeta quality, have been reported. For each of these states, vibrational wave functions of the symmetric stretching and bending modes have been computed, employing the potential energy function obtained at the highest level of calculation, with Watson’s Hamiltonian and anharmonic vibrational wave functions expressed as linear combinations of harmonic basis functions. The helium I photoelectron spectrum of Cl2O has been simulated with Franck–Condon factors calculated using computed anharmonic vibrational wave functions and allowing for Duschinsky rotation. The adiabatic ionization energies (AIEs) to the four lowest cationic states of Cl2O+ have been evaluated at the RCCSD(T) level with basis sets of up to polarized-valence-quintuple-zeta quality and by various extrapolation techniques to the basis set limit. Revised equilibrium geometrical parameters of the X̃ 2B1 and C̃ 2A2 states of Cl2O+ were obtained from the iterative Franck–Condon analysis procedure, and revised AIEs for the à 2B2 and B̃ 2A1 states of Cl2O+ were estimated based on comparison between the simulated and observed photoelectron spectra. It was found that inclusion of anharmonicity in the Franck–Condon factor calculations for each electronic state improves the quality of the simulated spectrum. The computed T1 diagnostics from the RCCSD calculations suggest that the B̃ 2A1 state of Cl2O+, with the ⋯(9a1)1(3b2)2(10a1)0 electronic configuration, possesses multiconfigurational character in the region of r(ClO)=1.87 Å and θ(ClOCl)=125°. CASSCF/MRCI/aug-cc-pVQZ(no g) calculations show an avoided crossing between the B̃ 2A1 state and (2)2A1 state [with the ⋯(9a1)2(3b1)0(10a1)1 electronic configuration], in the region of 1.96&amp;gt;r&amp;gt;1.80 Å and 137.0&amp;gt;θ&amp;gt;132.0°.

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.

Franck—Condon analyses on the first photoelectron bands of formaldehyde and its deuterated species

Abstract An iterative Franck—Condon analysis procedure was applied to a three-mode system to deduce ionic geometries of formaldehyde and its deuterated species upon X 2 B 2 ←X 1 A 1 ionization processes. The ionic geometries thus derived agree with one another and are parallel in signs with those predicted from changes in overlap populations and the nodal repulsive forces model. Furthermore, the calculated ionic geometries were found to be insensitive to the Duschinsky matrix relating to the two combining states.

An iterative franck-condon analysis procedure for the determination of ionic geometries for H 2O and H 2S

The pattern search optimization procedure is applied to a Franck-Condon analysis method with the use of the formulation proposed by Kupka and Cribb for evaluating vibrational overlap integrals [H. Kupka and P.H. Cribb, J. Chem. Phys., 85 (1986) 1303; H. Kupka and G. Olbrich, J. Chem. Phys., 80 (1985) 3163]. The method was applied to three different ionization processes accompanied by small to large changes in molecular geometry. It was found that ionic geometries thus deduced for transitions with small structural changes agree well with observation. However, the agreement is poor for transitions with large geometric changes although the method can still account for the observed vibrational intensity distributions to a certain extent. In addition, properties of the pattern search method as well as Kupka and Cribb's formulation for Franck-Condon analyses are discussed.