Hellmann–Feynman theorem near the threshold

The features of the spectrum structure are considered for situations where some parameter λ of the N‐electron Hamiltonian reaches the threshold value η under which the discrete energy level falls into the continuous spectrum. The electron density properties are also studied. It is proved that for a sequence of the wave functions converging in energy to the lower bound of the continuous spectrum as λ approaches η the corresponding sequence of the electron densities converges to the density of the (N ? 1)‐electron ground state. The results generalize the Hellmann–Feynman theorem for the cases where only the one‐side energy derivatives exist or there is no limiting wave function. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Wellenmechanische Absolutrechnungen an Molekülen und Atomsystemen mit der SCF?MO?LC(LCGO) Methode. III. Das Cyclopropan (C3H6)

The C₃H₆ has been investigated ab initio, taking all 24 electrons into account, using the Allgemeines Programmsystem/SCF ? MO ? LC (LCGO ) Methode. Variation of the C? C distance gives a total energy of ?116.02 a.u. at a C? C distance of 2.91 a.u. The ionization energy was found to be 10.33 eV.     

ACS symposium

The SCF potential surface of the ground state for NO₂ was calculated by using program JAMOL 3. The McLean–Loew–Berkowitz CGTO 's were used as basis functions. One of the two N? O distance R is fixed to 2.25 a.u. and the other one r and the ONO angle θ are varied from 2.25 to 5.0 a.u. and from 0° to 180°, respectively. The potential surface has the minimum around r = 2.50 a.u. and θ = 120°, where the energy is found to be ?203.954 a.u.     

New insight into the electronic structure of iron(IV)‐oxo porphyrin compound I. A quantum chemical topological analysis

The electronic structure of iron‐oxo porphyrin π‐cation radical complex Por^·+Fe^IV?O (S? H) has been studied for doublet and quartet electronic states by means of two methods of the quantum chemical topology analysis: electron localization function (ELF) η(r) and electron density ρ(r). The formation of this complex leads to essential perturbation of the topological structure of the carbon–carbon bonds in porphyrin moiety. The double C?C bonds in the pyrrole anion subunits, represented by pair of bonding disynaptic basins V_i=1,2(C,C) in isolated porphyrin, are replaced by single attractor V(C,C)_i=1–20 after complexation with the Fe cation. The iron–nitrogen bonds are covalent dative bonds, N→Fe, described by the disynaptic bonding basins V(Fe,N)_i=1–4, where electron density is almost formed by the lone pairs of the N atoms. The nature of the iron–oxygen bond predicted by the ELF topological analysis, shows a main contribution of the electrostatic interaction, Fe^δ+···O^δ?, as long as no attractors between the C(Fe) and C(O) core basins were found, although there are common surfaces between the iron and oxygen basines and coupling between iron and oxygen lone pairs, that could be interpreted as a charge‐shift bond. The Fe? S bond, characterized by the disynaptic bonding basin V(Fe,S), is partially a dative bond with the lone pair donated from sulfur atom. The change of electronic state from the doublet (M = 2) to quartet (M = 4) leads to reorganization of spin polarization, which is observed only for the porphyrin skeleton (?0.43e to 0.50e) and S? H bond (?0.55e to 0.52e). © 2012 Wiley Periodicals, Inc.     

Quantum mechanics and quasiclassical study of the H/D+FO → OH/OD+F,HF/DF+O reactions: Chemical stereodynamics

The time‐dependent quantum wave packet and the quasi‐classical trajectory (QCT) calculations for the title reactions are carried out using three recent‐developed accurate potential energy surfaces of the 1¹A′, 1³A′, and 1³A″ states. The two commonly used polarization‐dependent differential cross sections, dσ₀₀/dω_t, dσ₂₀/dω_t, with ω_t being the polar coordinates of the product velocity ω′, and the three angular distributions, P(θ_r), P(Φ_r), and P(θ_r,Φ_r), with θ_r, Φ_r being the polar angles of the product angular momentum, are generated in the center‐of‐mass frame using the QCT method to gain insight into the alignment and the orientation of the product molecules. Influences of the potential energy surface, the collision energy, and the isotope mass on the stereodynamics are shown and discussed. Validity of the QCT calculation has been examined and proved in the comparison with the quantum wave packet calculation. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

Initial Radical Separation after Photolysis of 2,2′‐Azobis(isobutyronitrile) (AIBN) in Solution: Modeling the Primary Cage Effect for Polar Radicals

There is a contradiction as to the initial spatial separation r_i of the two transient 2‐cyanoprop‐2‐yl radicals (Me₂ ? CN) formed by flash photolysis of 2,2′‐azobis(isobutyronitrile) (AIBN) in solvents of various viscosities. The cage effect, expressed in terms of the in‐cage termination probability of the resulting radicals, is predicted correctly by classical Langevin models assuming a decrease of r_i with increasing viscosity. However, the electron‐spin polarization of the radicals escaping the primary cage clearly indicates that the initial separation distance r_i is independent of the solution viscosity. This obvious discrepancy can be reconciled by accounting for the strong electric dipole moments of these radicals and the resulting inter‐radical dipole? dipole interaction potential. We propose a primary‐caging model for polar radicals in solution based on an attractive inter‐radical mean‐force potential. The model is applied to the flash photolysis of AIBN and shown to describe properly the viscosity dependence of both the in‐cage termination probability (cage effect) and the electron‐spin polarization of the escaping 2‐cyanoprop‐2‐yl radicals.     

On the formulation of spin-free quantum chemistry

A set of spin-free functions ?_i(r),i = 1 …? f, is obtained which form the basis of spin-free quantum chemistry. The ?_i(r) show a one-to-one correspondence to antisymmetric space-spin functions Ψ_i(r, σ) with spin functions constructed according to Löwdin's projector operator method.     

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 AMO method at small internuclear distances

The AMO function of the hydrogen molecule ψ = ψ_c + η ψ_i, where ψ_c is the covalent part and ψ_i the ionic part, is investigated for small internuclear distances R. We found η → ?1 as R →,?1 as R → 0, contrary to the intuitively expected limit η → 1. However, near R = 0 an analytical expression of ψ is derived, showing that ψ reduces to the helium ground state as R → 0. We have proved that the empirical concept ?covalent and ionic character”? should be replaced by the symmetry argument in the case of small R.     

Two-Dimensional fully numerical solutions of molecular Schrödinger equations. II. Solution of the Poisson equation and results for singlet states of H2 and HeH+

Two-dimensional fully numerical solutions of the Hartree–Fock problem are reported for the singlet ground states of H^?, He, H₂, and HeH⁺. The H₂ energy at R = 1.4 a.u. is ?1.13362957 a.u.     

Floating spherical gaussian orbital open-shell calculations on the four-electron H4 system

Floating spherical Gaussian orbital (FSGO ) open-shell calculations have been made to determine the potential energy surface of planar square and rectangular arrangements of the four-electron system H₄. This surface is discussed in relation to the bimolecular isotope exchange reaction H2+D2-→ 2HD. The changes in energy and geometry accompanying the coplanar approach of two hydrogen molecules interacting chemically have also been investigated. Calculations on the electronic energies of planar T-shaped and kite arrangements of H4 of various sizes show that it is unlikely that these configurations can serve as transition states for the exchange reaction. However, the energy curve for linear configurations of H4 (H? H? H … H), calculated as a function of the H₃ … H distance with the symmetric linear H₃ (H-H-H) unit fixed at the internuclear distance of 1.9080 a.u., is found to have a deep minimum (?1.9176 a.u.) at an r(H₃ … H) distance of 1.5846 a.u. The overall results suggest that the following mechanism for the exchange reaction, H₂+H₂→H₂+H+H→H₃+H→H+H₂+ H→H₂+H₂ could be advantageous as it requires a barrier height of 0.1604 a.u. which is significantly lower than that calculated from the saddle point energy (0.1950 a.u.). However, the problem of reconciling this with the experimental activation energy of 0.0685 a.u. still remains.     

Theoretical treatment of predissociation of the CO (3sσ) B and (3pσ) C1Σ+ Rydberg states based on a rigorous adiabatic representation

Ab initio multireference configuration interaction calculations for adiabatic potential curves, nonadiabatic couplings 〈φ _i (R,r)|d/dR|φ _j (R,r)〉 and 〈φ _i (R,r)|d²/dR ²|φ _j (R,r)〉, and nuclear kinetic energy corrections 〈dφ _i (R,r)/dR|dφ _i (R,r)/dR〉 for the (3sσ) B and (3pσ) C¹Σ⁺ Rydberg states of the CO molecule have been carried out. The energy positions and predissociation linewidths for the observed vibrational levels of these two states have been determined in a rigorous adiabatic representation by the complex scaling method employing a basis of complex scaled harmonic vibrational functions in conjunction with the Gauss-Hermite quadrature method to evaluate the complex Hamiltonian matrix elements. The present treatment correctly reproduces the observed trends in energies and line broadening for vibrational levels of the B¹Σ⁺ state and represents an improvement over the previous treatment in literature. The errors in the determined spacings of the v = 0–4 vibrational levels of the C¹Σ⁺ state are less than 2% compared with measured data. The predissociation linewidths for the v=3,4 levels of the C¹Σ⁺ state are found to be 4.9 and 8.9 cm⁻¹, respectively, in good agreement with the observed values. Received: 23 March 1998 / Accepted: 27 July 1998 / Published online: 9 October 1998     

Asymptotic behaviour of molecular bound state wavefunctions

Upper bounds are derived for |r^μ_i|, μ = 1,2, ·, where ? denotes an exact electronic bound state wavefunction of a molecular system in the Born-Oppenheimer approximation, and r_i is the distance of the ith electron from an appropriately chosen point, e.g., the molecular center. It is further shown that ? decays exponentially if r_i → ∞.     

On the use of Gaussian shell type basis orbitals for single-center expansions. I. Evaluation of integrals

Formulas are derived for all Hamiltonian integrals required for molecular computations using a novel basis for single-center expansions. The basis orbitals depend exponentially upon α(r ? ρ)² where r and ρ are the distance from center to electron and to a variationally scaled spherical shell, respectively. Comparisons are made between these so-called Gaussian shell orbitals (GSO ) and the conventional GTO and STO bases for single-center calculations. A preliminary comparison on H using a single GSO , a non-integer STO , and a GTO gives the optimized energies: ?0.51089 a.u., ?0.50504 a.u., and ?0.50422 a.u., respectively.     

Microwave Spectra of o-Fluorotoluene and Its 13C Isotopic Species: Methyl Internal Rotation and Molecular Structure

The rotational spectra of o-fluorotoluene and its seven ¹³C isotopic species were recorded in the frequency range from 4 to 20 GHz with employment of pulsed molecular beam Fourier-transform microwave (MB-FTMW) spectrometers. The analysis of the spectra in the two lowest states of methyl internal rotation (torsional ground state, A and E species) was based on a asymmetric frame-rigid symmetric top Hamiltonian with inclusion of centrifugal distortion terms, yielding structural rotational constants, as well as the threefold barrier V ₃ to internal rotation and the angle(a,i) between the principal moment of inertia a axis and the internal rotor axis i. The rotational constants of all eight isotopomeres were used to derive the seven ¹³C r _s coordinates of the molecule.     

Correlation between thermal conductivity and bond length alternation in carbon nanotubes: A combined reverse nonequilibrium molecular dynamics—Crystal orbital analysis

The thermal conductivity (λ) of carbon nanotubes (CNTs) with chirality indices (5,0), (10,0), (5,5), and (10,10) has been studied by reverse nonequilibrium molecular dynamics (RNEMD) simulations as a function of different bond length alternation patterns (Δr_i). The Δr_i dependence of the bond force constant (k_rx) in the molecular dynamics force field has been modeled with the help of an electronic band structure approach. These calculations show that the Δr_i dependence of k_rx in tubes with not too small a diameter can be mapped by a simple linear bond length–bond order correlation. A bond length alternation with an overall reduction in the length of the nanotube causes an enhancement of λ, whereas an alternation scheme leading to an elongation of the tube is coupled to a decrease of the thermal conductivity. This effect is more pronounced in carbon nanotubes with larger diameters. The formation of a polyene‐like structure in the direction of the longitudinal axis has a negligible influence on λ. A comparative analysis of the RNEMD and crystal orbital results indicates that Δr_i‐dependent modifications of λ and the electrical conductivity are uncorrelated. This behavior is in‐line with a heat transfer that is not carried by electrons. Modifications of λ as a function of the bond alternation in the (10,10) nanotube are explained with the help of power spectra, which provide access to the density of vibrational states. We have suggested longitudinal low‐energy modes in the spectra that might be responsible for the Δr_i dependence of λ. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Generalized quantum mechanical two-centre problems

For the one-electron Schrödinger equation among the solutions of which the Slater-Zener-type functions can be found, it is shown, that it can be generalized to the two-centre case only in one way, if one demands separability in prolate spheroidal coordinates, and if in addition to the Coulomb term of the potential energy there shall be an additional function of the product r ₁ · r ₂ only. The generalized problem with a potential energy of the form V(r) = ? Z₁/r₁ ? Z₂/r₂ ? Q(R)/r₁r₂ is studied for the case of two equal centres Z ₁=Z₂=Z≧0 with regard to the existence and number of bound states. The results are extended as far as possible also to the case with unequal centres. For some examples with equal centres wave functions and correlation diagrams have been computed exactly for the lowest electronic states.     

Theoretical electronic studies of aluminum monobromide and aluminum monoiodide

By using CASSCF/MRCI methods, theoretical molecular calculations have been performed for 12 electronic states for AlBr molecule and 12 electronic states for AlI molecule in the representation ^2s+1Λ (neglecting spin‐orbit effects). Calculated potential energy curves are displayed. Spectroscopic constants including the harmonic vibrational wave number ω_e, the electronic energy T_e referred to the ground state and the equilibrium internuclear distance R_e are predicted for these singlet and triplet electronic states for both AlBr and AlI molecules. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Wellenmechanische Absolutrechnungen an Molekülen und Atomsystemen mit der SCF?MO?LC(LCGO) Methode. VI. Das Methan (CH4)

The CH₄ molecule has been investigated ab initio, for four different distances R_{C? H} and one distortion of a triangle HCH about the equilibrium condition, taking all 10 electrons into account, using the Allgemeines Programmsystem/SCF ? MO ? LC (LCGO ) Methode. The equilibrium distance R_{C? H} was estimated to 2.053 a.u., the minimum of the total energy to ?40.06 a.u., the heat of formation to 17.0 eV and the ionization energy to 14.8 eV. For the breathing force constant and the force constant of the twisting vibration a ω₁ of 3139 cm^?1 and a ω₂ of 1865 cm^?1 were found.     

Poly(silyens)s: Charge Carrier Photogeneration and Transport: S. Nešpůrek,A. Eckhardt Polym. Adv. Technol. 12, 427–440 (2001)

The original article to which this Erratum refers was published in Polym. Adv. Technol. 12 , 427–440 (2001) In the following paragraph, ‘µ’ was inserted instead of ‘γ’. The paragraph should have read: Xerographic Discharge The electric field (F) dependences of photogeneration efficiency as measured by the xerographic discharge method are shown in Figure 4 for PMPSi and PBMSi. The electric field and temperature (T) dependences of the photogeneration efficiency can be described using Onsager theory of geminate recombination (full lines). This theory can be used to describe one step in the photogeneration process, the thermal dissociation of ion‐pairs formed by light into free charge carriers in the external electric field. Under the assumption that the distribution of bound pairs is spherically symmetrical, the overall photogeneration efficiency ν is expressed as where ?₀ is the voltage and temperature independent primary quantum yield (fraction of the ion‐pair generated per photon), g(r) is the spatial distribution of the pairs, and r is the pair separation distance. Function f(r,F,T) represents the Onsager dissociation probability [9] of the ion‐pair. The best theoretical fits of the experimental data given in Figure 4 were found using the Onsager theory with the Gaussian distribution of radii [17] ­ exp (?r²/γ²) using parameters: γ = 1.3 nm, η₀ = 0.45 and γ = 1.6 nm, η₀ = 0.85 for PMPSi and PBMSi, respectively. According to Eq. 2 the photogeneration quantum efficiency η is also temperature dependent. The activation energy was determined as E_η = 0.05 eV for PMPSi (inset in Figure 4) and 0.04 eV for PBMSi. The temperature dependence of the photogeneration efficiency is also in good agreement with the Onsager dissociation theory (full line in inset of Figure 4).