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.     

Ultrafast Relaxation Dynamics of the Excited States of 1‐Amino‐ and 1‐(N,N‐Dimethylamino)‐fluoren‐9‐ones

The dynamics of the excited states of 1‐aminofluoren‐9‐one (1AF) and 1‐(N,N‐dimethylamino)‐fluoren‐9‐one (1DMAF) are investigated by using steady‐state absorption and fluorescence as well as subpicosecond time‐resolved absorption spectroscopic techniques. Following photoexcitation of 1AF, which exists in the intramolecular hydrogen‐bonded form in aprotic solvents, the excited‐state intramolecular proton‐transfer reaction is the only relaxation process observed in the excited singlet (S₁) state. However, in protic solvents, the intramolecular hydrogen bond is disrupted in the excited state and an intermolecular hydrogen bond is formed with the solvent leading to reorganization of the hydrogen‐bond network structure of the solvent. The latter takes place in the timescale of the process of solvation dynamics. In the case of 1DMAF, the main relaxation pathway for the locally excited singlet, S₁(LE), or S₁(ICT) state is the configurational relaxation, via nearly barrierless twisting of the dimethylamino group to form the twisted intramolecular charge‐transfer, S₁(TICT), state. A crossing between the excited‐state and ground‐state potential energy curves is responsible for the fast, radiationless deactivation and nonemissive character of the S₁(TICT) state in polar solvents, both aprotic and protic. However, in viscous but strong hydrogen‐bond‐donating solvents, such as ethylene glycol and glycerol, crossing between the potential energy surfaces for the ground electronic state and the hydrogen‐bonded complex formed between the S₁(TICT) state and the solvent is possibly avoided and the hydrogen‐bonded complex is weakly emissive.     

On the use of multipole expansion of the Coulomb potential in quantum chemistry

Some features of the multipole expansion of the Coulomb potential V for a system of point charges are studied. It is shown that multipole expansion is convergent both locally in L₂(R³) and weakly on some classes of functions. One-particle Hamiltonians H_n = H₀ + V_n, where H₀ is the kinetic energy operator and V_n is the n-th partial sum of the multipole expansion of V, are discussed, and the convergence of their eigenvalues to those of H = H₀ + V with increasing n is proved. It is also shown that the discrete spectrum eigenfunctions of H_n converge to those of H both in L₂(R³) (together with their first and second derivatives) and uniformly on R³. © 1996 John Wiley & Sons, Inc.     

Reinvestigation of intramolecular hydrogen bond in malonaldehyde derivatives: An ab initio,AIM and NBO study

The RAHB systems in malonaldehyde and its derivatives at MP2/ 6‐311++G(d,p) level of theory were studied and their intramolecular hydrogen bond energies by using the related rotamers method was obtained. The topological properties of electron density distribution in O? H···O intramolecular hydrogen bond have been analyzed in term of quantum theory of atoms in molecules (QTAIM). Correlations between the H‐bond strength and topological parameters are probed. The results of QTAIM clearly showed that the linear correlation between the electron density distribution at HB critical point and RAHB ring critical point with the corresponding hydrogen bond energies was obtained. Moreover, it was found a linear correlation between the electronic potential energy density, V(r_cp), and hydrogen bond energy which can be used as a simple equation for evaluation of HB energy in complex RAHB systems. Finally, the similar linear treatment between the geometrical parameters, such as O···O or O? H distance, and Lp(O)→σ*_OH charge transfer energy with the intramolecular hydrogen bond energy is observed. © 2010 Wiley Periodicals, Inc., Int J Quantum Chem, 2011     

Behavior of the electron density near an impurity

The behavior of the electron density n(r) and potential energy V(r) near the origin, where an impurity of charge Z is located, is studied using the Lindhard dielectric theory of the free-electron gas. The leading odd terms in the power-series expansion of n(r) and V(r) are obtained. It is shown that the derivative n′(0) = ?2Zn₀/a₀, where n₀ is the free-electron gas density and a₀ is the Bohr radius.     

Perturbed structures generated using coordinates derived from compliance constants in internal vibrations for QTAIM dual functional analysis: Intrinsic dynamic nature of interactions

Perturbed structures for QTAIM dual functional analysis (QTAIM‐DFA) are proposed to generate using the coordinates corresponding to the compliance force constants in internal vibrations (CIV). In QTAIM‐DFA, total electron energy densities H_b( r _c) are plotted versus H_b( r _c) – V_b( r _c)/2 at bond critical points (BCPs) of interactions in question, where V_b( r _c) are potential energy densities at BCPs. Each plot of an interaction based on the data from both perturbed structures and fully optimized one takes the form (θ_p, κ_p), where θ_p corresponds to the tangent line of the plot and κ_p is the curvature. The θ_p values evaluated with CIV are equal to those obtained by partial optimizations with the interaction distance in question being fixed suitably, within the calculation errors. Very high applicability of CIV is demonstrated to generate the perturbed structures for QTAIM‐DFA. Dynamic nature of interactions based on (θ_p, κ_p) with CIV is called “intrinsic dynamic nature of interactions.”     

An empirical potential for the O-H·O Hydrogen bond: Part I. Comparison with ab initio molecular orbital results

An empirical potential energy function has been devised for the O-H·O hydrogen bond, for use with the MMI force field. The energy of the hydrogen bond is described as the sum of van der Waals, electrostatic and Morse components. The function has been used to calculate the potential energy hypersurface of the water dimer, and the results are compared with published ab initio molecular orbital studies. Satisfactory agreement is obtained except for orientations involving very short H·H contacts. The geometry and hydrogen bond energy of the equilibrium linear form of (H₂O)₂ are calculated to be r(O·O) = 2.84 Å, θ = 36°, ΔE = ?5.35 kcal mol^?1, which are close to the values obtained by experiment, and from molecular orbital calculations. The relative importance of the electrostatic component of the empirical hydrogen bond energy is consistent with molecular orbital energy decomposition studies. The empirical function has also been used to calculate the energy of the water trimer in orientations which serve as models for the crystallographic bifurcated hydrogen bond. The results indicate that, in these orientations, the trimer is typically 0–3 kcal mol^?1 more stable than the dimer, a result which is consistent with ab initio calculations.     

OH自由基的高精度量子化学研究

采用内收缩MRCI方法(Internally Contracted Multiconfiguration-Reference Configuration Interaction)研究了OH自由基, 计算得到其基态稳定构型的键长是0.09708 nm, 对应的实验值是0.096966 nm, 第一激发态的键长是0.10137 nm,实验值是0.10121 nm. 同时得到势能曲线PECs (Potential Energy Curve), 再分别由Murrell-Sorbie势能函数拟合计算和POLFIT程序计算得到OH自由基在基态X²Π和第一激发态A²Σ⁺时的光谱数据：平衡振动频率ω_e, 非谐性常数ω_eχ_e以及高阶修正ω_eY_e, 平衡转动常数B_e, 振转耦合系数α_e, 解离能D₀和垂直跃迁能量ν₀₀. 这些理论计算结果与最新的实验值非常吻合, 精确度比前人也有很大提高. 其中我们计算得到基态OH(X²Π)的解离能D₀＝35568.86 cm^－1, 第一激发态OH (A²Σ⁺)的解离能D₀＝18953.93 cm^－1, 从第一激发态A²Σ⁺ (ν＝0)到基态X²Π (v＝0)的垂直跃迁能ν₀₀＝32496.42 cm^－1.     

Statistical–mechanical models with separable many-body interactions: especially partition functions and thermodynamic consequences

We start from a classical statistical–mechanical theory for the internal energy in terms of three- and four-body correlation functions g ₃ and g ₄ for homogeneous atomic liquids like argon, with assumed central pair interactions f(r_ij){\phi(r_{ij})} . The importance of constructing the partition function (pf) as spatial integrals over g ₃, g ₄ and f{\phi} is stressed, together with some basic thermodynamic consequences of such a pf. A second classical example taken for two-body interactions is the so-called one-component plasma in two dimensions, for a particular coupling strength treated by Alastuey and Jancovici (J Phys (France) 42:1, 1981) and by Fantoni and Tellez (J Stat Phys 133:449, 2008). Again thermodynamic consequences provide a particular focus. Then quantum–mechanical assemblies are treated, again with separable many-body interactions. The example chosen is that of an N-body inhomogeneous extended system generated by a one-body potential energy V(r). The focus here is on the diagonal element of the canonical density matrix: the so-called Slater sum S(r, β), related to the pf by pf(b) = òS(r, b)d[(r)\vec]{{\rm pf}(\beta) = \int {S({\bf r}, \beta)}d\vec {r}}, β = (k _B T)⁻¹. The Slater sum S(r, β) can be related exactly, via a partial differential equation, to the one-body potential V(r), for specific choices of V which are cited. The work of Green (J Chem Phys 18:1123, 1950), is referred to for a generalization, but now perturbative, to two-body forces. Finally, to avoid perturbation series, the work concludes with some proposals to allow the treatment of extended assemblies in which regions of long-range ordered magnetism exist in the phase diagram. One of us (Z.D.Z.) has recently proposed a putative pf for a three-dimensional (3D) Ising model, based on two, as yet unproved, conjectures and has pointed out some important thermodynamic consequences of this pf. It would obviously be of considerable interest if such a pf, together with conjectures, could be rigorously proved.     

On the relationship between the electron-pair distribution function and the correlation energy of an atom

The pair distribution function h(r₁₂;r₁, γ) and the virial theorem are used to derive a general expression for the local contributions to the total correlation energy of an atom. A direct link between correlation effects and the correlation energy is obtained by use of G(r₁, γ) and Γ(r₁, y). The former is the probability associated with a given choice of r₁ and γ, while the latter describes the local contribution to the correlation energy. Explicit calculations for the ground state of helium indicate that the angular dependence of the local contribution to the correlation energy is essentially independent of r₁, whereas the local correlation energy shows a strong r₁ dependence. The maximum contribution to the correlation energy occurs at intermediate values of γ where there is close agreement between the Hartree–Fock and exact densities.     

Electron density and its derivatives at the nucleus for spherically confined hydrogen atom

It is shown that the energy of a hydrogen‐like atom confined inside a spherical cavity of radius, R, and potential barrier, V₀, is quantitatively defined by the ratio . Here, the conventional spherical density (r) is scaled as η_l(r) = and the ratio of the second derivative η(r) to η_l(r) is evaluated at the nucleus. Numerical results of the ratios are presented for 1s, 2s, 2p, and 3d states at several values of V₀. For such states, the characteristic radii of confinement leading to the well‐defined values of energy are identified. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

A multi-configuration LCAO–MO study for complex unsaturated molecules. I. General theory and its application to the benzene anion

A multi-configuration LCAO –MO approach using a π-bond order–bond length linear relation is introduced to predict the geometrical structures for the electronic ground and excited states of unsaturated hydrocarbons. The procedure is designed to include configuration interaction in each iterative computation where the π-electron approximation is employed under the Pariser–Parr type semi-empirical treatment. The π-bond order–bond length relation is determined as r_pq = 1.523 – 0.193P_pq, when the bond lengths of ethylene, benzene and naphthalene are used and the groundstate functions including the singly and doubly excited configurations are taken into account to obtain the bond orders P_pq. The iterative calculation is applied to the ground state and the two lowest excited states of the benzene anion in both D_6h and D_2h molecular geometries. The geometrical structures and the π-electron energies are computed for the ground and excited states of the anion; for the latter, two types of configuration species are used. It is found that the first lowest excited state is not subjected to the Jahn–Teller effect and the calculated excited state energies do not agree with the observed values (c. 1.0 ～ 2.5 eV higher than the observed values). The latter point is discussed in detail. It is also found that the resultant ground state energy depression due to configuration mixing is not very large and the two types of configuration species used give different CI effects on the energy levels of the two lowest excited states of the anion. Finally, the stabilization energy due to the Jahn–Teller distortion is estimated for the ground state of the anion.     

Towards an exact orbital-free single-particle kinetic energy density for the inhomogeneous electron liquid in the Be atom

Holas and March [Phys. Rev. A51, 2040 (1995)] wrote the gradient of the one-body potential V(r) in terms of low-order derivatives of the idempotent Dirac density matrix built from a single Slater determinant of Kohn–Sham orbitals. Here, this is first combined with the study of Dawson and March [J. Chem. Phys. 81, 5850 (1984)] to express the single-particle kinetic energy density of the Be atom ground-state in terms of both the electron density n(r) and potential V(r). While this is the more compact formulation, we then, by removing V(r), demonstrate that the ratio t(r)/n(r) depends, though non-locally, only on the single variable n′(r)/n(r), no high-order gradients entering for the spherical Be atom.     

Intramolecular O-H··N hydrogen bonds in 2-hydroxybenzaldimine compounds: Spectroscopic and quantum chemical investigations

Summary Spectroscopic and quantum chemical data that characterize the intramolecular hydrogen bonds in a series of 2-hydroxybenzaldimine compounds (Ph(OH)(CH=N-R);R=-NH(Ph), -OH,-OCH₃, -NH₂, -Ph, -CHO, -H, -CH₃, -(CO)(CH₃) are reported. Optimized geometries and vibrational spectra were calculated at HF/6-31G(d,p) and B3LYP/6-31G(d,p) levels, NMR spectra at GIAO-BLYP/6-311++G(d,p)//B3LYP/6-31G(d,p) level of theory. The sequences of calculated frequencies (v(OH) andv(CN)) and chemical shifts ((OH)) agree reasonably well with the corresponding experimental data. The consistency of the calculated data is demonstrated by exploiting several correlations between bond distances, vibrational frequencies, chemical shifts, and hydrogen bond energies. In particular, there exists an almost perfect relation between the hydrogen bond strengths, as measured byr(O-H) distances, and the hydrogen bond distancesr(H··N) andr(O··N) It is shown that electrostic potentials and several kinds of partial charges (Mulliken, CHELPG, MK, and NPA) of the nitrogen atoms, to a first approximation, may serve as a means for characterizing the proton acceptor capabilities of the different imino groups.

---

Intramolekulare O-H··N-Wasserstoffbrückenbindungen in 2-Hydroxybenzaldiminverbindungen: Spektroskopische und quantenchemische Untersuchungen

Zusammenfassung Intramolekulare Wasserstoffbrückenbindungen in einer Reihe von 2-Hydroxybenzaldiminverbindungen (Ph(OH)(CH=N-R);R=-NH(Ph), -OH, -OCH₃, -NH₂, -Ph, -CHO,-H, -CH₃, -(CO)CH₃) werden mit Hilfe von spektroskopischen und quantenchemischen Daten charakterisiert und untersucht. Optimierte Geometrien und Schwingungsspektren wurden mittelsab initio (HF/6-31G(d,p)) und DFT Methoden (B3LYP/6-31G(d,p)) berechnet, NMR-Spektren mittels GIAO-BLYP/6-311++G(d,p)//B3LYP/6-31G(d,p). Die berechneten Frequenzen (v(OH) undv(CN)) sowie die chemischen Verschiebungen ((OH)) stimmen mit den entsprechenden experimentellen Daten gut überein. Die innere Konsistenz der Rechnungen wird durch verschiedene Korrelationen zwischen Bindungsabständen, Schwingungsfrequenzen, chemischen Verschiebungen und Wasserstoffbrückenbindungsenergien demonstriert. Insbesondere findet man eine nahezu perfekte Relation zwischen den Bindungsabständenr(O-H), die als Maß für die Stärken der Wasserstoffbrückenbindungen dienen, und den Wasserstoffbrückenbindungsabständenr(H··N) undr(O··N). Es wird gezeigt, daß die Protonenakzeptorfähigkeiten der unterschiedlichen Iminogruppen in erster Näherung durch elektrostatische Potentiale und verschiedene Partialladungen (Mulliken, CHELPG, MK und NPA) der Stickstoffatome charakterisiert werden können.

     

Linear relationships in α,β-unsaturated carbonyl compounds between the half-wave reduction potentials, the frontier orbital energies and the Hammett σp values

A linear relationship between the half-wave reduction potentials of α,β-unsaturated carbonyl compounds R–CHCH–COX and the Hammett σ_p values of R and X is proposed: E_1/2=−1.341σ_p(X)σ_p(R)+1.123σ_p(X)+1.746σ_p(R)−1.694. A linear relationship is also observed for the LUMO's energy values, the absolute chemical hardness η, the chemical potential μ, the electrophilicity power ω, or the polarisation of the ethylenic double bond with the Hammett σ_p values of R and X.     

Hydrothermal Syntheses, Structures, and Properties of Two New Potassium Vanadium Phosphates: K3(VO) (V2O3) (PO4)2(HPO4) and K3 (VO) (HV2O3) (PO4)2(HPO4)

Two new potassium vanadium phosphates have been prepared and their structures have been determined from analysis of single crystal X-ray data. The two compounds, K₃(VO)(V₂O₃) (PO₄)₂(HPO₄) and K₃(VO)(HV₂O₃)(PO₄)₂(HPO₄), are isostructural, except for the incorporation of an extra hydrogen atom into the nearly identical frameworks. The structures consist of a three-dimensional network of [VO]_n chains connected through phosphate groups to a [V₂O₃] moiety. Magnetic susceptibility experiments indicate that in the case of the di-hydrogen compound, there are no significant magnetic interactions between the three independent vanadium (IV) centers. Crystal data: for K₃(VO)(V₂O₃)(PO₄)₂ (HPO₄), M_r = 620.02, orthorhombic space group Pnma (No. 62), a = 7.023(4) Å, b = 13.309(7) Å, c = 14.294(7) Å, V = 1336(2) ų, Z = 4, R = 5.02%, and R_w = 5.24% for 1238 observed reflections [I > 3σ(I)]; for K₃(VO)(HV₂O₃)(PO₄)₂(HPO₄), M_r = 621.04, orthorhombic space group Pnma (No. 62), a = 6.975(3) Å, b = 13.559(7) Å, c = 14.130(7) Å, V = 1336(1) ų, Z = 4, R = 6.02%, and R_w = 6.34% for 1465 observed reflections [I > 3σ(I)].     

The exchange-correlation potential of DFT obtained from a semiempirically fine-tuned Hartree–Fock density for inhomogeneous electron liquids

The present authors have given an exact theory of the exchange-correlation potential V _xc (r) in terms of (i) the exact ground-state electron density n(r) and (ii) the idempotent Dirac density matrix γ(r,?r′) generated by the DFT one-body potential V(r), having n(r) as its diagonal element. Here, we display two approximate consequences: (a) a form of V _xc (r) generated by the semiempirically fine-tuned HF density of Cordero et al. (N.A. Cordero, N.H. March, and J.A. Alonso, Phys. Rev. A 75, 052502 (2007)) and (b) the exchange-only potential V _x (r) determined solely by the HF ground state density for the Be atom.