Determination of vibrational level spacings of van der waals molecules from the Lennard-Jones potential

An approximate expression for the eigenvalues for van der Waals molecules by use of the Lennard-Jones (12-6) potential in the WKB approximation is presented. The expression is applied to the rare gas molecules. Ar₂, Kr₂, and Xe₂ by fitting the potential function to the observed potential parameters. Calculated results of vibrational energy spacings for these molecules agree well with the experiment and other calculations which are based on numerical integration of the Schrödinger equation. For Xe₂, the energy spacing expression is used to determine the thermodynamic functions of the van der Waals bond.     

The lifetimes of gas phase CO2˙− and N2O˙− calculated from the transition probability of the autodetachment process A− → A + e−

A procedure to calculate the quantum mechanical transition probability of a unimolecular primary chemical process, A^? → A + e^? is investigated for the circumstance where A^? and A have different numbers of vibrational and rotational degrees of freedom (one is linear, the other not). A procedure is introduced to deal with the coupling between the vibrational and rotational motions. The proposed method was applied to calculating the lifetimes of CO₂^˙? and N₂O^˙? in the gas phase. The geometry optimizations and frequency calculations for CO₂, CO₂^˙?, N₂O, and N₂O^˙? are performed at HF, MP2, and QCISD(T) levels with 6-31G* or 6–31+G* basis sets, in order to obtain reliable geometric and spectroscopic information on these systems. Lifetimes are calculated for several of the lower vibrational–rotational states of the anions, as well as for the Boltzmann distribution of states at 298 K. The lifetime of the lowest vibrational–rotational state of CO₂^˙?, is 1.03 × 10^?4 s, and of the lowest vibrational state with rotational levels weighted by Boltzmann distribution at 298 K, 1.50 × 10^?4 s. These values are in good agreement with the experimental number, 9.0 ± 2.0 × 10^?5 s, and support the experimental evidence that CO₂^˙? was formed in its ground vibrational level by the techniques used. The lifetime of CO₂^˙? calculated with Boltzmann distribution over its vibrational and rotational levels at 298 K, is 1.51 × 10^?5 s. There are no direct measurements of the lifetime of N₂O^˙?, but it was estimated to be greater than 10^?4 s from experimental evidence. The predicted lifetimes of N₂O^˙?, at its lowest vibrational–rotational state (0 K) and lowest vibrational state with rotational levels weighted by the Boltzmann distribution at 298 K, are 238 and 19.1 s, respectively. The lifetime of N₂O^˙? at thermal equilibrium at 298 K is 6.66 × 10^?2 s, indicating that electron loss from the excited vibrational states of N₂O^˙? is significant. This study represents the first theoretical investigation of CO₂^˙? and N₂O^˙? lifetimes. © 1994 John Wiley & Sons, Inc.     

Spectroscopic constants of SiH2, GeH2, SnH2, and their cations and anions from density functional computations

Local (LSD ) and nonlocal (NLSD ) spin density calculations using different exchangecorrelation functionals have been performed to determine equilibrium geometries, harmonic vibrational frequencies (ωe), ionization potentials (IP ), electron affinities (EA ), dipole moments (μ), and singlet-triplet energy gaps (Δ E_ST) of SiH₂, GeH₂, and SnH₂. Geometrical structures as well as vibrational frequencies are in agreement with the available experimental data and compare favorably with the most sophisticated postHartree-Fock computations performed until now. Both computed IPS (9.15 and 9.25 eV for SiH₂ and GeH₂, respectively) and EA of SiH₂ (1.17 eV) compare favorably with experimental data (9.17, 9.21, and 1.2 eV). Accurate values are obtained also for singlet-triplet energy gaps. We report for the first time the electron affinities of all neutral systems and the spectroscopic constants of the cations and anions. © 1995 John Wiley & Sons, Inc.     

Chemical bond breaking/forming as a Franck–Condon electronic process

We describe chemical bond changes as Franck–Condon electronic processes within a new theoretical ansatz that we call ‘rigged’ Born–Oppenheimer (R-BO) approach. The notion of the separability of nuclear and electron states implied in the standard Born–Oppenheimer (BO) scheme is retained. However, in the present scheme the electronic wave functions do not depend upon the nuclear coordinate (R-space). The new functions are obtained from an auxiliary Hamiltonian corresponding to the electronic system (r-coordinates) submitted to a Coulomb potential generated by external sources of charges in real space (α-coordinates) instead of massive nuclear objects. A stationary arrangement characterized by the coordinates α_0A, is determined by a particular electronic wave function, ψ(r;α_0A); it is only at this stationary point, where an electronic Schrödinger equation: H_e(r,α_0A)|Ψ(r;α_0A)=E(α_0A)|Ψ(r;α_0A) must hold. This equation permits us to use modern electronic methods based upon analytic first and second derivatives to construct model electronic wave functions and stationary geometry for external sources. If the set of wave functions {Ψ(r;α_0A)} is made orthogonal, the energy functional in α-space, E(α;α_0A)=Ψ(r;α_0A)|H_e(r,α_0A)|Ψ(r;α_0A) is isomorphic to a potential energy function in R-space: E(R;α_0A)=Ψ(r;α_0A)|H_e(r,R)|Ψ(r;α_0A). This functional defines, by hypothesis, a trapping convex potential in R-space and the nuclear quantum states are determined by a particular Schrödinger equation. The total wave function for the chemical species A reads as a product of our electronic wave function with the nuclear wave function (Ξ_ik(R;α_0A)): Φ_ik(r,R)=Ψ_i(r,α_0A)Ξ_ik(R;α_0A). This approach facilitates the introduction of molecular frame without restrictions in the R-space. Two molecules (characterized with different electronic spectra) that are decomposable into the same number of particles (isomers) have the same Coulomb Hamiltonian and they are then characterized by different electronic wave functions for which no R-coordinate ‘deformation’ can possibly change its electronic structure. A bond breaking/forming process must be formally described as a spectroscopic-like electronic process. The theory provides an alternative to the adiabatic as well as the diabatic scheme for understanding molecular processes. As an illustration of the present ideas, the reaction of H₂+CO leading to formaldehyde is examined in some detail.     

The formula of anchoring energy for a nematic liquid crystal

The interface energy for a nematic liquid crystal (NLC) is considered as the sum of potential energy between LC molecules and molecules of the substrate surface, and a formula for anchoring energy is derived by elementary principles. The anchoring energy for a NLC should have two terms, the first term is the same as the Rapini–Papoular expression, the second is related to the normal of interface and resultes from the biaxial property of a NLC induced by interface. Hence there are two anchoring coefficients, W ₁ and W ₂. We demonstrate that W ₁ is equal to the tilt angle strength A_θ , and W ₂ corresponds to the difference between A_θ and the azimuthal strength A_? . Thus A_θ –A_? is due to the biaxial property of the NLC near the interface. Applying this formula to the twisted NLC cell, we discuss the threshold and saturation field, as well as the maximal tilt angel θ _m with respect to A_θ /A_? . Previously proposed formulae are discussed from our point view.     

Ab initio study of low-lying electronic states of the PF2 radical

The equilibrium geometries, excitation energies, force constants, and vibrational frequencies of the low-lying electronic states X²B₁, ²A₁, ²B₂, and ²A₂ of the PF₂ radical have been calculated at the MRSDCI level with a double zeta plus polarization basis set. Our calculated geometry, force constants, and vibrational frequencies for the X²B₁ state are in good agreement with experimental data. The electronic transition moments, oscillator strengths for the ²A₁ → X²B₁ and ²A₂ → X²B₁ transitions, and radiative lifetimes for the ²A₁ and ²A₂ states are calculated based on the MRSDCI wave functions. © 1994 by John Wiley & Sons, Inc.     

Vibrational–rotational energies of all H2 isotopomers using Monte Carlo methods

Using variational Monte Carlo techniques, we have computed several of the lowest rotational–vibrational energies of all the hydrogen molecule isotopomers (H₂, HD, HT, D₂, DT, and T₂). These calculations do not require the excited states to be explicitly orthogonalized. We have examined both the usual Gaussian wave function form as well as a rapidly convergent Padé form. The high‐quality potential energy surfaces used in these calculations are taken from our earlier work and include the Born–Oppenheimer energy, the diagonal correction to the Born–Oppenheimer approximation, and the lowest‐order relativistic corrections at 24 internuclear points. Our energies are in good agreement with those determined by other methods. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Size consistency corrections for configurational interaction calculations

A size consistency error formula, correcting for the erratic dependence on the number of particles in a doubly substituted (or otherwise restricted) configurational interaction (CI ) treatment, is derived. The formula is expressed in terms of the particle pair eneriges ?_k and the normalization integrals Δ_k of the corrections to the unperturbed normalized wave function. The theory as well as the results for the ¹A₁ ground state of H₂O and the ²B₁ state of H²O₊ show close agreement with the coupled electron pair approach (CEPA ).     

AB initio Calculations of the structure and spectra of chromium, molybdenum, and tungsten tetrafluorides. nontetrahedral molecular structure of WF4

The properties of MF₄ molecules (M = Cr, Mo, W) are investigated by the restricted Hartree-Fock method using Möller-Plesset second-order perturbation theory and by the second-order configuration interaction method using the multiconfigumtion wave function derived in a complete active space approximation, in wide bases complemented with polarization d and f functions. Relativistic effective potentials are used to describe the core electrons. For CrF₄ and MoF₄, the tetrahedral configuration of nuclei in the electronic state of³k₂ symmetry is energetically most favorable. In the WF4 molecule, the least-energy structure is a D_2h structure in the singlet state ¹A₁. The D_4h,(¹ A_1g) and T_d ( ³A₂) configurations in the WF₄ molecule are higher on the energy scale than the ground state by 4 and 6305 cm’1 and are saddle points. For all of the analyzed configurations of MF₄ molecules, the geometrical parameters, the vibrational spectra, and the energies of vertical electronic transitions are found. The chemical bonding is analyzed and a simple model is proposed to explain the variation of the relative energies of states in the series CrF₄→MoF₄→WF₄.     

Molecular electrostatic potentials and partial atomic charges from correlated wave functions: Applications to the electronic ground and excited states of 3-methylindole

Procedures have been developed to generate molecular electrostatic potentials based on correlated wave function from ab initio or semiempirical electronic structure programs. A new algorithm for point-wise sampling of the potential is described and used to obtain partial atomic charges via a linear, least squares fit between classical and quantum mechanical electrostatic potentials. The proposed sampling algorithm is efficient and promises to introduce less rotational variance in the potential derived partial charges than algorithms applied previously. Electrostatic potentials and fitted atomic charges from ab initio (HF/6–31G* and MP2/6-31G*) and semiempirical (INDO/S; HF, SECI, and SDCI) wave functions are presented for the electronic ground (S₀) and excited (¹L_b, ¹L_a) states of 3-methylindole. © 1992 by John Wiley & Sons, Inc.     

Die Photoelektron-Spektren der Monohalogenacetylene

The photoelectron spectra of the four monohaloacetylenes X? C?C? H with X ? F, Cl, Br, I have been recorded. The first four bands of these spectra are assigned (in order of increasing ionization potentials) to the following states: band 1: ²Π_3/2(1), ²Π_1/2(1); band 2: ²Π_3/2(2), ²Π_1/2(2); band 3: ²Σ⁺(3); band 4: ²Σ⁺(4). A correlation diagram based on a simple ZDO-MO model shows that the observed band positions and the size of the splits due to spin-orbit coupling can be satisfactorily explained in terms of such a model. It is found that the orbital energies A_X of the postulated halogen n p λ(X)-basis orbitals are a linear function of the ionization potentials I(X) of the free atoms X. The validity of the ZDO-MO-model is confirmed by the excellent qualitative agreement between the observed and predicted spacings of the vibrational fine structure of the π-bands.     

Is cluster evaporation statistical? A comparison of simulation results for Ar13 with exact classical phase space theory

We have simulated the dissociation reactionA _n →A _n?1+A for small clusters (L-J argon) with well defined internal energy and total angular momentum. Reaction rates and kinetic energy release distributions are compared to the predictions of several statistical theories, including RRK, the “Engelking” model, and phase space theory (PST). We have applied classical phase space theory in an essentially exact formulation using accurate anharmonic vibrational densities of states (and no adjustable parameters). We present a critical evaluation of the different theories and sensitivity of the results to the underlying assumptions.     

Hylleraas method for many‐electron atoms. I. The Hamiltonian

A general expression for the nonrelativistic Hamiltonian for n‐electron atoms with the fixed nucleus approximation is derived in a straightforward manner using the chain rule. The kinetic energy part is transformed into the mutually independent distance coordinates r_i, r_ij, and the polar angles θ_i, and φ_i. This form of the Hamiltonian is very appropriate for calculating integrals using Slater orbitals, not only of states of S symmetry, but also of states with higher angular momentum, as P states. As a first step in a study of the Hylleraas method for five‐electron systems, variational calculations on the ²P ground state of boron atom are performed without any interelectronic distance. The orbital exponents are optimized. The single‐term reference wave function leads to an energy of ?24.498369 atomic units (a.u.) with a virial factor of η = 2.0000000009, which coincides with the Hartree–Fock energy ?24.498369 a.u. A 150‐term wave function expansion leads to an energy of ?24.541246 a.u., with a factor of η = 1.9999999912, which represents 28% of the correlation energy. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Difluorocarbene Studied with Threshold Photoelectron Spectroscopy (TPES): Measurement of the First Adiabatic Ionization Energy (AIE) of CF2

The first photoelectron band of difluorocarbene CF₂, has been studied by threshold photoelectron (TPE) spectroscopy. CF₂ was prepared by microwave discharge of a flowing mixture of hexafluoropropene, C₃F₆, and argon. A vibrationally resolved band was observed in which at least twenty‐two components were observed. In the first PE band of CF₂, the adiabatic ionization energy differs significantly from the vertical ionization energy because, for the ionization CF₂⁺ (X?²A₁)+e^? ← CF₂ (X?¹A₁), there is an increase in the FCF bond angle (by ≈20°) and a decrease in the C? F bond length (by ≈0.7 Å). The adiabatic component was not observed in the experimental TPE spectrum. However, on comparing this spectrum with an ab initio/Franck–Condon simulation of this band, using results from high‐level ab initio calculations, the structure associated with the vibrational components could be assigned. This led to alignment of the experimental TPE spectrum and the computed Franck–Condon envelope, and a determination of the first adiabatic ionization energy of CF₂ as (11.362±0.005) eV. From the assignment of the vibrational structure, values were obtained for the harmonic and fundamental frequencies of the symmetric stretching mode (ν₁′) and symmetric bending mode (ν₂′) in CF₂⁺ (X?²A₁).     

Improved interaction potential for alkali halide molecules

An improved interaction potential has been devised for diatomic alkali halide molecules. This potential, in addition to similar attraction terms as in the Rittner potential, includes a new exponential for the short-range repulsion. The constant m in the exponential is seen to be well expressible in terms of the parameters of the Rittner potential. The new potential is also correlated with different properties, as for example, effective charges, effective radii, effective principal quantum numbers, etc., of the combining ions. Various spectroscopic constants, viz., the ionic dissociation energy D_i, the vibrational–rotational coupling constant α_e, the vibrational anharmonicity constant ω_ex_e, as well as two second-order spectroscopic constants γ_e and β_e have been calculated for this and for the Rittner potential. From comparisons between these two potentials, the new one has been observed better than the other.     

New approach to the rapid semiempirical calculation of molecular electrostatic potentials based on the am1 wave function: Comparison with ab initio hf/6-31g* results

A new approach to the computation of molecular electrostatic potentials based on the AM1 wave function is described. In contrast to the prevailing philosophy, but consistent with the underlying NDDO approximation, no deorthogonalization of the wave function is carried out. The integrals required for the computation of the electronic contributions to the molecular electrostatic potential are evaluated in a manner similar to that of the AM1 core-electron attraction integrals, while the nuclear contributions are computed using a new semiempirical function—Z_A(S^AS^A, S^pS^p)[1 + exp[ – ω_A(R_Ai – δ_A)]]—where the atomic parameters ω_A and δ_A are obtained by calibration against the results of ab initio HF/6-31G* calculations. Isopotential contour maps for guanine and cytosine obtained with the new method are qualitatively almost indistinguishable from their HF/6-31G* counterparts, while quantitative comparisons for the minima for a wide range of molecules are reproduced with an rms error of 5.2 kcal mol^?1. The locations of the “lone-pair” minima for a wide range of heterosubstituted organic molecules generally fall within 0.02 Å of the corresponding HF/6-31G* minima while those in the π-regions of unsaturated molecules are generally within 0.2 Å. Because of the rapid integral evaluation, the fully semiempirical method described here is extremely economical. For example, for the guanine–cytosine base pair it is >500 times faster than calculations in which the complete integral matrix is computed analytically from the deorthogonalized AM1 wave function. © John Wiley & Sons, Inc.     

Optische Absorptions-Spektroskopie an ionischen Ozoniden

Optical Absorption Spectroscopy on Ionic Ozonides Electronic transitions of ionic ozonides of alkali-metal and tetraorganylonium cations have been investigated in the range from 200 to 2000 nm. Optical absorption measurements of diffuse reflexion on powder and transmission on single crystals displayed a broad absorption at ∼480 nm and 490 nm, respectively. Optical spectra of the O₃^–-radical in solution gave a well resolved vibronic finestructure with spacings of ∼790 cm^–1, symmetrically to the maximum of the absorption band at 460 nm, and uneffected of the counter ion and solvent used. The finestructure allows a correlation to the ν_s-vibration of the ozonide ion in the excited state. The observed electronic transition is assigned to ²B₁ → ²A₂, based on MO-observations and ab initio calculations. Earlier reports of a strong absorption band at low energy for O₃^– cannot be reconfirmed.     

Thermochemical properties of free radicals from G3MP2B3 calculations

For a set of 32 selected free radicals, energy minimum structures, harmonic vibrational wave numbers ω_e, principal moments of inertia I_A, I_B, and I_C, heat capacities C°_p(T), entropies S°(T), thermal energy contents H°(T) ? H°(0), and standard enthalpies of formation Δ_fH°(T) were calculated at the G3MP2B3 level of theory in the temperature range 200–3000 K. In this article, thermodynamic functions at T = 298.15 K are presented and compared with recent experimental values. The mean absolute deviation between calculated and experimental Δ_fH°(298.15) values resulted in 3.91 kJ mol^?1, which is close to the average experimental uncertainty of ± 3.55 kJ mol^?1. The influence of hindered rotation on thermodynamic functions is studied for isopropyl and tert‐butyl radicals. © 2002 Wiley Periodicals, Inc. Int J Chem Kinet 34: 550–560, 2002     

Correlation energy,correlated electron density,and exchange‐correlation potential in some spherically confined atoms

We report correlation energies, electron densities, and exchange‐correlation potentials obtained from configuration interaction and density functional calculations on spherically confined He, Be, Be²⁺, and Ne atoms. The variation of the correlation energy with the confinement radius R_c is relatively small for the He, Be²⁺, and Ne systems. Curiously, the Lee–Yang–Parr (LYP) functional works well for weak confinements but fails completely for small R_c. However, in the neutral beryllium atom the CI correlation energy increases markedly with decreasing R_c. This effect is less pronounced at the density‐functional theory level. The LYP functional performs very well for the unconfined Be atom, but fails badly for small R_c. The standard exchange‐correlation potentials exhibit significant deviation from the “exact” potential obtained by inversion of Kohn–Sham equation. The LYP correlation potential behaves erratically at strong confinements. © 2016 Wiley Periodicals, Inc.